This is ../../info/calc, produced by makeinfo version 4.11 from calc.texi. This file documents Calc, the GNU Emacs calculator. Copyright (C) 1990, 1991, 2001, 2002, 2003, 2004, 2005, 2006, 2007, 2008, 2009, 2010 Free Software Foundation, Inc. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.3 or any later version published by the Free Software Foundation; with the Invariant Sections being just "GNU GENERAL PUBLIC LICENSE", with the Front-Cover texts being "A GNU Manual," and with the Back-Cover Texts as in (a) below. A copy of the license is included in the section entitled "GNU Free Documentation License." (a) The FSF's Back-Cover Text is: "You have the freedom to copy and modify this GNU manual. Buying copies from the FSF supports it in developing GNU and promoting software freedom." INFO-DIR-SECTION Emacs START-INFO-DIR-ENTRY * Calc: (calc). Advanced desk calculator and mathematical tool. END-INFO-DIR-ENTRY  File: calc, Node: Introduction, Next: Data Types, Prev: Tutorial, Up: Top 5 Introduction ************** This chapter is the beginning of the Calc reference manual. It covers basic concepts such as the stack, algebraic and numeric entry, undo, numeric prefix arguments, etc. * Menu: * Basic Commands:: * Help Commands:: * Stack Basics:: * Numeric Entry:: * Algebraic Entry:: * Quick Calculator:: * Prefix Arguments:: * Undo:: * Error Messages:: * Multiple Calculators:: * Troubleshooting Commands::  File: calc, Node: Basic Commands, Next: Help Commands, Prev: Introduction, Up: Introduction 5.1 Basic Commands ================== To start the Calculator in its standard interface, type `M-x calc'. By default this creates a pair of small windows, `*Calculator*' and `*Calc Trail*'. The former displays the contents of the Calculator stack and is manipulated exclusively through Calc commands. It is possible (though not usually necessary) to create several Calc mode buffers each of which has an independent stack, undo list, and mode settings. There is exactly one Calc Trail buffer; it records a list of the results of all calculations that have been done. The Calc Trail buffer uses a variant of Calc mode, so Calculator commands still work when the trail buffer's window is selected. It is possible to turn the trail window off, but the `*Calc Trail*' buffer itself still exists and is updated silently. *Note Trail Commands::. In most installations, the `C-x * c' key sequence is a more convenient way to start the Calculator. Also, `C-x * *' is a synonym for `C-x * c' unless you last used Calc in its Keypad mode. Most Calc commands use one or two keystrokes. Lower- and upper-case letters are distinct. Commands may also be entered in full `M-x' form; for some commands this is the only form. As a convenience, the `x' key (`calc-execute-extended-command') is like `M-x' except that it enters the initial string `calc-' for you. For example, the following key sequences are equivalent: `S', `M-x calc-sin ', `x sin '. Although Calc is designed to be used from the keyboard, some of Calc's more common commands are available from a menu. In the menu, the arguments to the functions are given by referring to their stack level numbers. The Calculator exists in many parts. When you type `C-x * c', the Emacs "auto-load" mechanism will bring in only the first part, which contains the basic arithmetic functions. The other parts will be auto-loaded the first time you use the more advanced commands like trig functions or matrix operations. This is done to improve the response time of the Calculator in the common case when all you need to do is a little arithmetic. If for some reason the Calculator fails to load an extension module automatically, you can force it to load all the extensions by using the `C-x * L' (`calc-load-everything') command. *Note Mode Settings::. If you type `M-x calc' or `C-x * c' with any numeric prefix argument, the Calculator is loaded if necessary, but it is not actually started. If the argument is positive, the `calc-ext' extensions are also loaded if necessary. User-written Lisp code that wishes to make use of Calc's arithmetic routines can use `(calc 0)' or `(calc 1)' to auto-load the Calculator. If you type `C-x * b', then next time you use `C-x * c' you will get a Calculator that uses the full height of the Emacs screen. When full-screen mode is on, `C-x * c' runs the `full-calc' command instead of `calc'. From the Unix shell you can type `emacs -f full-calc' to start a new Emacs specifically for use as a calculator. When Calc is started from the Emacs command line like this, Calc's normal "quit" commands actually quit Emacs itself. The `C-x * o' command is like `C-x * c' except that the Calc window is not actually selected. If you are already in the Calc window, `C-x * o' switches you out of it. (The regular Emacs `C-x o' command would also work for this, but it has a tendency to drop you into the Calc Trail window instead, which `C-x * o' takes care not to do.) For one quick calculation, you can type `C-x * q' (`quick-calc') which prompts you for a formula (like `2+3/4'). The result is displayed at the bottom of the Emacs screen without ever creating any special Calculator windows. *Note Quick Calculator::. Finally, if you are using the X window system you may want to try `C-x * k' (`calc-keypad') which runs Calc with a "calculator keypad" picture as well as a stack display. Click on the keys with the mouse to operate the calculator. *Note Keypad Mode::. The `q' key (`calc-quit') exits Calc mode and closes the Calculator's window(s). It does not delete the Calculator buffers. If you type `M-x calc' again, the Calculator will reappear with the contents of the stack intact. Typing `C-x * c' or `C-x * *' again from inside the Calculator buffer is equivalent to executing `calc-quit'; you can think of `C-x * *' as toggling the Calculator on and off. The `C-x * x' command also turns the Calculator off, no matter which user interface (standard, Keypad, or Embedded) is currently active. It also cancels `calc-edit' mode if used from there. The `d ' key sequence (`calc-refresh') redraws the contents of the Calculator buffer from memory. Use this if the contents of the buffer have been damaged somehow. The `o' key (`calc-realign') moves the cursor back to its "home" position at the bottom of the Calculator buffer. The `<' and `>' keys are bound to `calc-scroll-left' and `calc-scroll-right'. These are just like the normal horizontal scrolling commands except that they scroll one half-screen at a time by default. (Calc formats its output to fit within the bounds of the window whenever it can.) The `{' and `}' keys are bound to `calc-scroll-down' and `calc-scroll-up'. They scroll up or down by one-half the height of the Calc window. The `C-x * 0' command (`calc-reset'; that's `C-x *' followed by a zero) resets the Calculator to its initial state. This clears the stack, resets all the modes to their initial values (the values that were saved with `m m' (`calc-save-modes')), clears the caches (*note Caches::), and so on. (It does _not_ erase the values of any variables.) With an argument of 0, Calc will be reset to its default state; namely, the modes will be given their default values. With a positive prefix argument, `C-x * 0' preserves the contents of the stack but resets everything else to its initial state; with a negative prefix argument, `C-x * 0' preserves the contents of the stack but resets everything else to its default state.  File: calc, Node: Help Commands, Next: Stack Basics, Prev: Basic Commands, Up: Introduction 5.2 Help Commands ================= The `?' key (`calc-help') displays a series of brief help messages. Some keys (such as `b' and `d') are prefix keys, like Emacs' and `C-x' prefixes. You can type `?' after a prefix to see a list of commands beginning with that prefix. (If the message includes `[MORE]', press `?' again to see additional commands for that prefix.) The `h h' (`calc-full-help') command displays all the `?' responses at once. When printed, this makes a nice, compact (three pages) summary of Calc keystrokes. In general, the `h' key prefix introduces various commands that provide help within Calc. Many of the `h' key functions are Calc-specific analogues to the `C-h' functions for Emacs help. The `h i' (`calc-info') command runs the Emacs Info system to read this manual on-line. This is basically the same as typing `C-h i' (the regular way to run the Info system), then, if Info is not already in the Calc manual, selecting the beginning of the manual. The `C-x * i' command is another way to read the Calc manual; it is different from `h i' in that it works any time, not just inside Calc. The plain `i' key is also equivalent to `h i', though this key is obsolete and may be replaced with a different command in a future version of Calc. The `h t' (`calc-tutorial') command runs the Info system on the Tutorial section of the Calc manual. It is like `h i', except that it selects the starting node of the tutorial rather than the beginning of the whole manual. (It actually selects the node "Interactive Tutorial" which tells a few things about using the Info system before going on to the actual tutorial.) The `C-x * t' key is equivalent to `h t' (but it works at all times). The `h s' (`calc-info-summary') command runs the Info system on the Summary node of the Calc manual. *Note Summary::. The `C-x * s' key is equivalent to `h s'. The `h k' (`calc-describe-key') command looks up a key sequence in the Calc manual. For example, `h k H a S' looks up the documentation on the `H a S' (`calc-solve-for') command. This works by looking up the textual description of the key(s) in the Key Index of the manual, then jumping to the node indicated by the index. Most Calc commands do not have traditional Emacs documentation strings, since the `h k' command is both more convenient and more instructive. This means the regular Emacs `C-h k' (`describe-key') command will not be useful for Calc keystrokes. The `h c' (`calc-describe-key-briefly') command reads a key sequence and displays a brief one-line description of it at the bottom of the screen. It looks for the key sequence in the Summary node of the Calc manual; if it doesn't find the sequence there, it acts just like its regular Emacs counterpart `C-h c' (`describe-key-briefly'). For example, `h c H a S' gives the description: H a S runs calc-solve-for: a `H a S' v => fsolve(a,v) (?=notes) which means the command `H a S' or `H M-x calc-solve-for' takes a value `a' from the stack, prompts for a value `v', then applies the algebraic function `fsolve' to these values. The `?=notes' message means you can now type `?' to see additional notes from the summary that apply to this command. The `h f' (`calc-describe-function') command looks up an algebraic function or a command name in the Calc manual. Enter an algebraic function name to look up that function in the Function Index or enter a command name beginning with `calc-' to look it up in the Command Index. This command will also look up operator symbols that can appear in algebraic formulas, like `%' and `=>'. The `h v' (`calc-describe-variable') command looks up a variable in the Calc manual. Enter a variable name like `pi' or `PlotRejects'. The `h b' (`calc-describe-bindings') command is just like `C-h b', except that only local (Calc-related) key bindings are listed. The `h n' or `h C-n' (`calc-view-news') command displays the "news" or change history of Calc. This is kept in the file `README', which Calc looks for in the same directory as the Calc source files. The `h C-c', `h C-d', and `h C-w' keys display copying, distribution, and warranty information about Calc. These work by pulling up the appropriate parts of the "Copying" or "Reporting Bugs" sections of the manual.  File: calc, Node: Stack Basics, Next: Numeric Entry, Prev: Help Commands, Up: Introduction 5.3 Stack Basics ================ Calc uses RPN notation. If you are not familiar with RPN, *note RPN Tutorial::. To add the numbers 1 and 2 in Calc you would type the keys: `1 2 +'. ( corresponds to the key on most calculators.) The first three keystrokes "push" the numbers 1 and 2 onto the stack. The `+' key always "pops" the top two numbers from the stack, adds them, and pushes the result (3) back onto the stack. This number is ready for further calculations: `5 -' pushes 5 onto the stack, then pops the 3 and 5, subtracts them, and pushes the result (-2). Note that the "top" of the stack actually appears at the _bottom_ of the buffer. A line containing a single `.' character signifies the end of the buffer; Calculator commands operate on the number(s) directly above this line. The `d t' (`calc-truncate-stack') command allows you to move the `.' marker up and down in the stack; *note Truncating the Stack::. Stack elements are numbered consecutively, with number 1 being the top of the stack. These line numbers are ordinarily displayed on the lefthand side of the window. The `d l' (`calc-line-numbering') command controls whether these numbers appear. (Line numbers may be turned off since they slow the Calculator down a bit and also clutter the display.) The unshifted letter `o' (`calc-realign') command repositions the cursor to its top-of-stack "home" position. It also undoes any horizontal scrolling in the window. If you give it a numeric prefix argument, it instead moves the cursor to the specified stack element. The (or equivalent ) key is only required to separate two consecutive numbers. (After all, if you typed `1 2' by themselves the Calculator would enter the number 12.) If you press or _not_ right after typing a number, the key duplicates the number on the top of the stack. ` *' is thus a handy way to square a number. The key pops and throws away the top number on the stack. The key swaps the top two objects on the stack. *Note Stack and Trail::, for descriptions of these and other stack-related commands.  File: calc, Node: Numeric Entry, Next: Algebraic Entry, Prev: Stack Basics, Up: Introduction 5.4 Numeric Entry ================= Pressing a digit or other numeric key begins numeric entry using the minibuffer. The number is pushed on the stack when you press the or keys. If you press any other non-numeric key, the number is pushed onto the stack and the appropriate operation is performed. If you press a numeric key which is not valid, the key is ignored. There are three different concepts corresponding to the word "minus," typified by `a-b' (subtraction), `-x' (change-sign), and `-5' (negative number). Calc uses three different keys for these operations, respectively: `-', `n', and `_' (the underscore). The `-' key subtracts the two numbers on the top of the stack. The `n' key changes the sign of the number on the top of the stack or the number currently being entered. The `_' key begins entry of a negative number or changes the sign of the number currently being entered. The following sequences all enter the number -5 onto the stack: `0 5 -', `5 n ', `5 n', `_ 5 ', `5 _ '. Some other keys are active during numeric entry, such as `#' for non-decimal numbers, `:' for fractions, and `@' for HMS forms. These notations are described later in this manual with the corresponding data types. *Note Data Types::. During numeric entry, the only editing key available is .  File: calc, Node: Algebraic Entry, Next: Quick Calculator, Prev: Numeric Entry, Up: Introduction 5.5 Algebraic Entry =================== The `'' (`calc-algebraic-entry') command can be used to enter calculations in algebraic form. This is accomplished by typing the apostrophe key, ', followed by the expression in standard format: ' 2+3*4 . This will compute `2+(3*4) = 14' and push it on the stack. If you wish you can ignore the RPN aspect of Calc altogether and simply enter algebraic expressions in this way. You may want to use every so often to clear previous results off the stack. You can press the apostrophe key during normal numeric entry to switch the half-entered number into Algebraic entry mode. One reason to do this would be to fix a typo, as the full Emacs cursor motion and editing keys are available during algebraic entry but not during numeric entry. In the same vein, during either numeric or algebraic entry you can press ``' (backquote) to switch to `calc-edit' mode, where you complete your half-finished entry in a separate buffer. *Note Editing Stack Entries::. If you prefer algebraic entry, you can use the command `m a' (`calc-algebraic-mode') to set Algebraic mode. In this mode, digits and other keys that would normally start numeric entry instead start full algebraic entry; as long as your formula begins with a digit you can omit the apostrophe. Open parentheses and square brackets also begin algebraic entry. You can still do RPN calculations in this mode, but you will have to press to terminate every number: `2 3 * 4 +' would accomplish the same thing as `2*3+4 '. If you give a numeric prefix argument like `C-u' to the `m a' command, it enables Incomplete Algebraic mode; this is like regular Algebraic mode except that it applies to the `(' and `[' keys only. Numeric keys still begin a numeric entry in this mode. The `m t' (`calc-total-algebraic-mode') gives you an even stronger algebraic-entry mode, in which _all_ regular letter and punctuation keys begin algebraic entry. Use this if you prefer typing `sqrt( )' instead of `Q', `factor( )' instead of `a f', and so on. To type regular Calc commands when you are in Total Algebraic mode, hold down the key. Thus `M-q' is the command to quit Calc, `M-p' sets the precision, and `M-m t' (or `M-m M-t', if you prefer) turns Total Algebraic mode back off again. Meta keys also terminate algebraic entry, so that `2+3 M-S' is equivalent to `2+3 M-S'. The symbol `Alg*' will appear in the mode line whenever you are in this mode. Pressing `'' (the apostrophe) a second time re-enters the previous algebraic formula. You can then use the normal Emacs editing keys to modify this formula to your liking before pressing . Within a formula entered from the keyboard, the symbol `$' represents the number on the top of the stack. If an entered formula contains any `$' characters, the Calculator replaces the top of stack with that formula rather than simply pushing the formula onto the stack. Thus, `' 1+2 ' pushes 3 on the stack, and `$*2 ' replaces it with 6. Note that the `$' key always initiates algebraic entry; the `'' is unnecessary if `$' is the first character in the new formula. Higher stack elements can be accessed from an entered formula with the symbols `$$', `$$$', and so on. The number of stack elements removed (to be replaced by the entered values) equals the number of dollar signs in the longest such symbol in the formula. For example, `$$+$$$' adds the second and third stack elements, replacing the top three elements with the answer. (All information about the top stack element is thus lost since no single `$' appears in this formula.) A slightly different way to refer to stack elements is with a dollar sign followed by a number: `$1', `$2', and so on are much like `$', `$$', etc., except that stack entries referred to numerically are not replaced by the algebraic entry. That is, while `$+1' replaces 5 on the stack with 6, `$1+1' leaves the 5 on the stack and pushes an additional 6. If a sequence of formulas are entered separated by commas, each formula is pushed onto the stack in turn. For example, `1,2,3' pushes those three numbers onto the stack (leaving the 3 at the top), and `$+1,$-1' replaces a 5 on the stack with 4 followed by 6. Also, `$,$$' exchanges the top two elements of the stack, just like the key. You can finish an algebraic entry with `M-=' or `M-' instead of . This uses `=' to evaluate the variables in each formula that goes onto the stack. (Thus `' pi ' pushes the variable `pi', but `' pi M-' pushes 3.1415.) If you finish your algebraic entry by pressing (or `C-j') instead of , Calc disables the default simplifications (as if by `m O'; *note Simplification Modes::) while the entry is being pushed on the stack. Thus `' 1+2 ' pushes 3 on the stack, but `' 1+2 ' pushes the formula `1+2'; you might then press `=' when it is time to evaluate this formula.  File: calc, Node: Quick Calculator, Next: Prefix Arguments, Prev: Algebraic Entry, Up: Introduction 5.6 "Quick Calculator" Mode =========================== There is another way to invoke the Calculator if all you need to do is make one or two quick calculations. Type `C-x * q' (or `M-x quick-calc'), then type any formula as an algebraic entry. The Calculator will compute the result and display it in the echo area, without ever actually putting up a Calc window. You can use the `$' character in a Quick Calculator formula to refer to the previous Quick Calculator result. Older results are not retained; the Quick Calculator has no effect on the full Calculator's stack or trail. If you compute a result and then forget what it was, just run `C-x * q' again and enter `$' as the formula. If this is the first time you have used the Calculator in this Emacs session, the `C-x * q' command will create the `*Calculator*' buffer and perform all the usual initializations; it simply will refrain from putting that buffer up in a new window. The Quick Calculator refers to the `*Calculator*' buffer for all mode settings. Thus, for example, to set the precision that the Quick Calculator uses, simply run the full Calculator momentarily and use the regular `p' command. If you use `C-x * q' from inside the Calculator buffer, the effect is the same as pressing the apostrophe key (algebraic entry). The result of a Quick calculation is placed in the Emacs "kill ring" as well as being displayed. A subsequent `C-y' command will yank the result into the editing buffer. You can also use this to yank the result into the next `C-x * q' input line as a more explicit alternative to `$' notation, or to yank the result into the Calculator stack after typing `C-x * c'. If you finish your formula by typing (or `C-j') instead of , the result is inserted immediately into the current buffer rather than going into the kill ring. Quick Calculator results are actually evaluated as if by the `=' key (which replaces variable names by their stored values, if any). If the formula you enter is an assignment to a variable using the `:=' operator, say, `foo := 2 + 3' or `foo := foo + 1', then the result of the evaluation is stored in that Calc variable. *Note Store and Recall::. If the result is an integer and the current display radix is decimal, the number will also be displayed in hex, octal and binary formats. If the integer is in the range from 1 to 126, it will also be displayed as an ASCII character. For example, the quoted character `"x"' produces the vector result `[120]' (because 120 is the ASCII code of the lower-case `x'; *note Strings::). Since this is a vector, not an integer, it is displayed only according to the current mode settings. But running Quick Calc again and entering `120' will produce the result `120 (16#78, 8#170, x)' which shows the number in its decimal, hexadecimal, octal, and ASCII forms. Please note that the Quick Calculator is not any faster at loading or computing the answer than the full Calculator; the name "quick" merely refers to the fact that it's much less hassle to use for small calculations.  File: calc, Node: Prefix Arguments, Next: Undo, Prev: Quick Calculator, Up: Introduction 5.7 Numeric Prefix Arguments ============================ Many Calculator commands use numeric prefix arguments. Some, such as `d s' (`calc-sci-notation'), set a parameter to the value of the prefix argument or use a default if you don't use a prefix. Others (like `d f' (`calc-fix-notation')) require an argument and prompt for a number if you don't give one as a prefix. As a rule, stack-manipulation commands accept a numeric prefix argument which is interpreted as an index into the stack. A positive argument operates on the top N stack entries; a negative argument operates on the Nth stack entry in isolation; and a zero argument operates on the entire stack. Most commands that perform computations (such as the arithmetic and scientific functions) accept a numeric prefix argument that allows the operation to be applied across many stack elements. For unary operations (that is, functions of one argument like absolute value or complex conjugate), a positive prefix argument applies that function to the top N stack entries simultaneously, and a negative argument applies it to the Nth stack entry only. For binary operations (functions of two arguments like addition, GCD, and vector concatenation), a positive prefix argument "reduces" the function across the top N stack elements (for example, `C-u 5 +' sums the top 5 stack entries; *note Reducing and Mapping::), and a negative argument maps the next-to-top N stack elements with the top stack element as a second argument (for example, `7 c-u -5 +' adds 7 to the top 5 stack elements). This feature is not available for operations which use the numeric prefix argument for some other purpose. Numeric prefixes are specified the same way as always in Emacs: Press a sequence of -digits, or press followed by digits, or press `C-u' followed by digits. Some commands treat plain `C-u' (without any actual digits) specially. You can type `~' (`calc-num-prefix') to pop an integer from the top of the stack and enter it as the numeric prefix for the next command. For example, `C-u 16 p' sets the precision to 16 digits; an alternate (silly) way to do this would be `2 4 ^ ~ p', i.e., compute 2 to the fourth power and set the precision to that value. Conversely, if you have typed a numeric prefix argument the `~' key pushes it onto the stack in the form of an integer.  File: calc, Node: Undo, Next: Error Messages, Prev: Prefix Arguments, Up: Introduction 5.8 Undoing Mistakes ==================== The shift-`U' key (`calc-undo') undoes the most recent operation. If that operation added or dropped objects from the stack, those objects are removed or restored. If it was a "store" operation, you are queried whether or not to restore the variable to its original value. The `U' key may be pressed any number of times to undo successively farther back in time; with a numeric prefix argument it undoes a specified number of operations. When the Calculator is quit, as with the `q' (`calc-quit') command, the undo history will be truncated to the length of the customizable variable `calc-undo-length' (*note Customizing Calc::), which by default is `100'. (Recall that `C-x * c' is synonymous with `calc-quit' while inside the Calculator; this also truncates the undo history.) Currently the mode-setting commands (like `calc-precision') are not undoable. You can undo past a point where you changed a mode, but you will need to reset the mode yourself. The shift-`D' key (`calc-redo') redoes an operation that was mistakenly undone. Pressing `U' with a negative prefix argument is equivalent to executing `calc-redo'. You can redo any number of times, up to the number of recent consecutive undo commands. Redo information is cleared whenever you give any command that adds new undo information, i.e., if you undo, then enter a number on the stack or make any other change, then it will be too late to redo. The `M-' key (`calc-last-args') is like undo in that it restores the arguments of the most recent command onto the stack; however, it does not remove the result of that command. Given a numeric prefix argument, this command applies to the `n'th most recent command which removed items from the stack; it pushes those items back onto the stack. The `K' (`calc-keep-args') command provides a related function to `M-'. *Note Stack and Trail::. It is also possible to recall previous results or inputs using the trail. *Note Trail Commands::. The standard Emacs `C-_' undo key is recognized as a synonym for `U'.  File: calc, Node: Error Messages, Next: Multiple Calculators, Prev: Undo, Up: Introduction 5.9 Error Messages ================== Many situations that would produce an error message in other calculators simply create unsimplified formulas in the Emacs Calculator. For example, `1 0 /' pushes the formula `1 / 0'; `0 L' pushes the formula `ln(0)'. Floating-point overflow and underflow are also reasons for this to happen. When a function call must be left in symbolic form, Calc usually produces a message explaining why. Messages that are probably surprising or indicative of user errors are displayed automatically. Other messages are simply kept in Calc's memory and are displayed only if you type `w' (`calc-why'). You can also press `w' if the same computation results in several messages. (The first message will end with `[w=more]' in this case.) The `d w' (`calc-auto-why') command controls when error messages are displayed automatically. (Calc effectively presses `w' for you after your computation finishes.) By default, this occurs only for "important" messages. The other possible modes are to report _all_ messages automatically, or to report none automatically (so that you must always press `w' yourself to see the messages).  File: calc, Node: Multiple Calculators, Next: Troubleshooting Commands, Prev: Error Messages, Up: Introduction 5.10 Multiple Calculators ========================= It is possible to have any number of Calc mode buffers at once. Usually this is done by executing `M-x another-calc', which is similar to `C-x * c' except that if a `*Calculator*' buffer already exists, a new, independent one with a name of the form `*Calculator*' is created. You can also use the command `calc-mode' to put any buffer into Calculator mode, but this would ordinarily never be done. The `q' (`calc-quit') command does not destroy a Calculator buffer; it only closes its window. Use `M-x kill-buffer' to destroy a Calculator buffer. Each Calculator buffer keeps its own stack, undo list, and mode settings such as precision, angular mode, and display formats. In Emacs terms, variables such as `calc-stack' are buffer-local variables. The global default values of these variables are used only when a new Calculator buffer is created. The `calc-quit' command saves the stack and mode settings of the buffer being quit as the new defaults. There is only one trail buffer, `*Calc Trail*', used by all Calculator buffers.  File: calc, Node: Troubleshooting Commands, Prev: Multiple Calculators, Up: Introduction 5.11 Troubleshooting Commands ============================= This section describes commands you can use in case a computation incorrectly fails or gives the wrong answer. *Note Reporting Bugs::, if you find a problem that appears to be due to a bug or deficiency in Calc. * Menu: * Autoloading Problems:: * Recursion Depth:: * Caches:: * Debugging Calc::  File: calc, Node: Autoloading Problems, Next: Recursion Depth, Prev: Troubleshooting Commands, Up: Troubleshooting Commands 5.11.1 Autoloading Problems --------------------------- The Calc program is split into many component files; components are loaded automatically as you use various commands that require them. Occasionally Calc may lose track of when a certain component is necessary; typically this means you will type a command and it won't work because some function you've never heard of was undefined. If this happens, the easiest workaround is to type `C-x * L' (`calc-load-everything') to force all the parts of Calc to be loaded right away. This will cause Emacs to take up a lot more memory than it would otherwise, but it's guaranteed to fix the problem.  File: calc, Node: Recursion Depth, Next: Caches, Prev: Autoloading Problems, Up: Troubleshooting Commands 5.11.2 Recursion Depth ---------------------- Calc uses recursion in many of its calculations. Emacs Lisp keeps a variable `max-lisp-eval-depth' which limits the amount of recursion possible in an attempt to recover from program bugs. If a calculation ever halts incorrectly with the message "Computation got stuck or ran too long," use the `M' command (`calc-more-recursion-depth') to increase this limit. (Of course, this will not help if the calculation really did get stuck due to some problem inside Calc.) The limit is always increased (multiplied) by a factor of two. There is also an `I M' (`calc-less-recursion-depth') command which decreases this limit by a factor of two, down to a minimum value of 200. The default value is 1000. These commands also double or halve `max-specpdl-size', another internal Lisp recursion limit. The minimum value for this limit is 600.  File: calc, Node: Caches, Next: Debugging Calc, Prev: Recursion Depth, Up: Troubleshooting Commands 5.11.3 Caches ------------- Calc saves certain values after they have been computed once. For example, the `P' (`calc-pi') command initially "knows" the constant `pi' to about 20 decimal places; if the current precision is greater than this, it will recompute `pi' using a series approximation. This value will not need to be recomputed ever again unless you raise the precision still further. Many operations such as logarithms and sines make use of similarly cached values such as `pi/4' and `ln(2)'. The visible effect of caching is that high-precision computations may seem to do extra work the first time. Other things cached include powers of two (for the binary arithmetic functions), matrix inverses and determinants, symbolic integrals, and data points computed by the graphing commands. If you suspect a Calculator cache has become corrupt, you can use the `calc-flush-caches' command to reset all caches to the empty state. (This should only be necessary in the event of bugs in the Calculator.) The `C-x * 0' (with the zero key) command also resets caches along with all other aspects of the Calculator's state.  File: calc, Node: Debugging Calc, Prev: Caches, Up: Troubleshooting Commands 5.11.4 Debugging Calc --------------------- A few commands exist to help in the debugging of Calc commands. *Note Programming::, to see the various ways that you can write your own Calc commands. The `Z T' (`calc-timing') command turns on and off a mode in which the timing of slow commands is reported in the Trail. Any Calc command that takes two seconds or longer writes a line to the Trail showing how many seconds it took. This value is accurate only to within one second. All steps of executing a command are included; in particular, time taken to format the result for display in the stack and trail is counted. Some prompts also count time taken waiting for them to be answered, while others do not; this depends on the exact implementation of the command. For best results, if you are timing a sequence that includes prompts or multiple commands, define a keyboard macro to run the whole sequence at once. Calc's `X' command (*note Keyboard Macros::) will then report the time taken to execute the whole macro. Another advantage of the `X' command is that while it is executing, the stack and trail are not updated from step to step. So if you expect the output of your test sequence to leave a result that may take a long time to format and you don't wish to count this formatting time, end your sequence with a keystroke to clear the result from the stack. When you run the sequence with `X', Calc will never bother to format the large result. Another thing `Z T' does is to increase the Emacs variable `gc-cons-threshold' to a much higher value (two million; the usual default in Calc is 250,000) for the duration of each command. This generally prevents garbage collection during the timing of the command, though it may cause your Emacs process to grow abnormally large. (Garbage collection time is a major unpredictable factor in the timing of Emacs operations.) Another command that is useful when debugging your own Lisp extensions to Calc is `M-x calc-pass-errors', which disables the error handler that changes the "`max-lisp-eval-depth' exceeded" message to the much more friendly "Computation got stuck or ran too long." This handler interferes with the Emacs Lisp debugger's `debug-on-error' mode. Errors are reported in the handler itself rather than at the true location of the error. After you have executed `calc-pass-errors', Lisp errors will be reported correctly but the user-friendly message will be lost.  File: calc, Node: Data Types, Next: Stack and Trail, Prev: Introduction, Up: Top 6 Data Types ************ This chapter discusses the various types of objects that can be placed on the Calculator stack, how they are displayed, and how they are entered. (*Note Data Type Formats::, for information on how these data types are represented as underlying Lisp objects.) Integers, fractions, and floats are various ways of describing real numbers. HMS forms also for many purposes act as real numbers. These types can be combined to form complex numbers, modulo forms, error forms, or interval forms. (But these last four types cannot be combined arbitrarily: error forms may not contain modulo forms, for example.) Finally, all these types of numbers may be combined into vectors, matrices, or algebraic formulas. * Menu: * Integers:: The most basic data type. * Fractions:: This and above are called "rationals". * Floats:: This and above are called "reals". * Complex Numbers:: This and above are called "numbers". * Infinities:: * Vectors and Matrices:: * Strings:: * HMS Forms:: * Date Forms:: * Modulo Forms:: * Error Forms:: * Interval Forms:: * Incomplete Objects:: * Variables:: * Formulas::  File: calc, Node: Integers, Next: Fractions, Prev: Data Types, Up: Data Types 6.1 Integers ============ The Calculator stores integers to arbitrary precision. Addition, subtraction, and multiplication of integers always yields an exact integer result. (If the result of a division or exponentiation of integers is not an integer, it is expressed in fractional or floating-point form according to the current Fraction mode. *Note Fraction Mode::.) A decimal integer is represented as an optional sign followed by a sequence of digits. Grouping (*note Grouping Digits::) can be used to insert a comma at every third digit for display purposes, but you must not type commas during the entry of numbers. A non-decimal integer is represented as an optional sign, a radix between 2 and 36, a `#' symbol, and one or more digits. For radix 11 and above, the letters A through Z (upper- or lower-case) count as digits and do not terminate numeric entry mode. *Note Radix Modes::, for how to set the default radix for display of integers. Numbers of any radix may be entered at any time. If you press `#' at the beginning of a number, the current display radix is used.  File: calc, Node: Fractions, Next: Floats, Prev: Integers, Up: Data Types 6.2 Fractions ============= A "fraction" is a ratio of two integers. Fractions are traditionally written "2/3" but Calc uses the notation `2:3'. (The `/' key performs RPN division; the following two sequences push the number `2:3' on the stack: `2 : 3 ', or `2 3 /' assuming Fraction mode has been enabled.) When the Calculator produces a fractional result it always reduces it to simplest form, which may in fact be an integer. Fractions may also be entered in a three-part form, where `2:3:4' represents two-and-three-quarters. *Note Fraction Formats::, for fraction display formats. Non-decimal fractions are entered and displayed as `RADIX#NUM:DENOM' (or in the analogous three-part form). The numerator and denominator always use the same radix.  File: calc, Node: Floats, Next: Complex Numbers, Prev: Fractions, Up: Data Types 6.3 Floats ========== A floating-point number or "float" is a number stored in scientific notation. The number of significant digits in the fractional part is governed by the current floating precision (*note Precision::). The range of acceptable values is from `10^-3999999' (inclusive) to `10^4000000' (exclusive), plus the corresponding negative values and zero. Calculations that would exceed the allowable range of values (such as `exp(exp(20))') are left in symbolic form by Calc. The messages "floating-point overflow" or "floating-point underflow" indicate that during the calculation a number would have been produced that was too large or too close to zero, respectively, to be represented by Calc. This does not necessarily mean the final result would have overflowed, just that an overflow occurred while computing the result. (In fact, it could report an underflow even though the final result would have overflowed!) If a rational number and a float are mixed in a calculation, the result will in general be expressed as a float. Commands that require an integer value (such as `k g' [`gcd']) will also accept integer-valued floats, i.e., floating-point numbers with nothing after the decimal point. Floats are identified by the presence of a decimal point and/or an exponent. In general a float consists of an optional sign, digits including an optional decimal point, and an optional exponent consisting of an `e', an optional sign, and up to seven exponent digits. For example, `23.5e-2' is 23.5 times ten to the minus-second power, or 0.235. Floating-point numbers are normally displayed in decimal notation with all significant figures shown. Exceedingly large or small numbers are displayed in scientific notation. Various other display options are available. *Note Float Formats::. Floating-point numbers are stored in decimal, not binary. The result of each operation is rounded to the nearest value representable in the number of significant digits specified by the current precision, rounding away from zero in the case of a tie. Thus (in the default display mode) what you see is exactly what you get. Some operations such as square roots and transcendental functions are performed with several digits of extra precision and then rounded down, in an effort to make the final result accurate to the full requested precision. However, accuracy is not rigorously guaranteed. If you suspect the validity of a result, try doing the same calculation in a higher precision. The Calculator's arithmetic is not intended to be IEEE-conformant in any way. While floats are always _stored_ in decimal, they can be entered and displayed in any radix just like integers and fractions. Since a float that is entered in a radix other that 10 will be converted to decimal, the number that Calc stores may not be exactly the number that was entered, it will be the closest decimal approximation given the current precison. The notation `RADIX#DDD.DDD' is a floating-point number whose digits are in the specified radix. Note that the `.' is more aptly referred to as a "radix point" than as a decimal point in this case. The number `8#123.4567' is defined as `8#1234567 * 8^-4'. If the radix is 14 or less, you can use `e' notation to write a non-decimal number in scientific notation. The exponent is written in decimal, and is considered to be a power of the radix: `8#1234567e-4'. If the radix is 15 or above, the letter `e' is a digit, so scientific notation must be written out, e.g., `16#123.4567*16^2'. The first two exercises of the Modes Tutorial explore some of the properties of non-decimal floats.  File: calc, Node: Complex Numbers, Next: Infinities, Prev: Floats, Up: Data Types 6.4 Complex Numbers =================== There are two supported formats for complex numbers: rectangular and polar. The default format is rectangular, displayed in the form `(REAL,IMAG)' where REAL is the real part and IMAG is the imaginary part, each of which may be any real number. Rectangular complex numbers can also be displayed in `A+Bi' notation; *note Complex Formats::. Polar complex numbers are displayed in the form `(R;THETA)' where R is the nonnegative magnitude and THETA is the argument or phase angle. The range of THETA depends on the current angular mode (*note Angular Modes::); it is generally between -180 and +180 degrees or the equivalent range in radians. Complex numbers are entered in stages using incomplete objects. *Note Incomplete Objects::. Operations on rectangular complex numbers yield rectangular complex results, and similarly for polar complex numbers. Where the two types are mixed, or where new complex numbers arise (as for the square root of a negative real), the current "Polar mode" is used to determine the type. *Note Polar Mode::. A complex result in which the imaginary part is zero (or the phase angle is 0 or 180 degrees or `pi' radians) is automatically converted to a real number.  File: calc, Node: Infinities, Next: Vectors and Matrices, Prev: Complex Numbers, Up: Data Types 6.5 Infinities ============== The word `inf' represents the mathematical concept of "infinity". Calc actually has three slightly different infinity-like values: `inf', `uinf', and `nan'. These are just regular variable names (*note Variables::); you should avoid using these names for your own variables because Calc gives them special treatment. Infinities, like all variable names, are normally entered using algebraic entry. Mathematically speaking, it is not rigorously correct to treat "infinity" as if it were a number, but mathematicians often do so informally. When they say that `1 / inf = 0', what they really mean is that `1 / x', as `x' becomes larger and larger, becomes arbitrarily close to zero. So you can imagine that if `x' got "all the way to infinity," then `1 / x' would go all the way to zero. Similarly, when they say that `exp(inf) = inf', they mean that `exp(x)' grows without bound as `x' grows. The symbol `-inf' likewise stands for an infinitely negative real value; for example, we say that `exp(-inf) = 0'. You can have an infinity pointing in any direction on the complex plane: `sqrt(-inf) = i inf'. The same concept of limits can be used to define `1 / 0'. We really want the value that `1 / x' approaches as `x' approaches zero. But if all we have is `1 / 0', we can't tell which direction `x' was coming from. If `x' was positive and decreasing toward zero, then we should say that `1 / 0 = inf'. But if `x' was negative and increasing toward zero, the answer is `1 / 0 = -inf'. In fact, `x' could be an imaginary number, giving the answer `i inf' or `-i inf'. Calc uses the special symbol `uinf' to mean "undirected infinity", i.e., a value which is infinitely large but with an unknown sign (or direction on the complex plane). Calc actually has three modes that say how infinities are handled. Normally, infinities never arise from calculations that didn't already have them. Thus, `1 / 0' is treated simply as an error and left unevaluated. The `m i' (`calc-infinite-mode') command (*note Infinite Mode::) enables a mode in which `1 / 0' evaluates to `uinf' instead. There is also an alternative type of infinite mode which says to treat zeros as if they were positive, so that `1 / 0 = inf'. While this is less mathematically correct, it may be the answer you want in some cases. Since all infinities are "as large" as all others, Calc simplifies, e.g., `5 inf' to `inf'. Another example is `5 - inf = -inf', where the `-inf' is so large that adding a finite number like five to it does not affect it. Note that `a - inf' also results in `-inf'; Calc assumes that variables like `a' always stand for finite quantities. Just to show that infinities really are all the same size, note that `sqrt(inf) = inf^2 = exp(inf) = inf' in Calc's notation. It's not so easy to define certain formulas like `0 * inf' and `inf / inf'. Depending on where these zeros and infinities came from, the answer could be literally anything. The latter formula could be the limit of `x / x' (giving a result of one), or `2 x / x' (giving two), or `x^2 / x' (giving `inf'), or `x / x^2' (giving zero). Calc uses the symbol `nan' to represent such an "indeterminate" value. (The name "nan" comes from analogy with the "NAN" concept of IEEE standard arithmetic; it stands for "Not A Number." This is somewhat of a misnomer, since `nan' _does_ stand for some number or infinity, it's just that _which_ number it stands for cannot be determined.) In Calc's notation, `0 * inf = nan' and `inf / inf = nan'. A few other common indeterminate expressions are `inf - inf' and `inf ^ 0'. Also, `0 / 0 = nan' if you have turned on Infinite mode (as described above). Infinities are especially useful as parts of "intervals". *Note Interval Forms::.  File: calc, Node: Vectors and Matrices, Next: Strings, Prev: Infinities, Up: Data Types 6.6 Vectors and Matrices ======================== The "vector" data type is flexible and general. A vector is simply a list of zero or more data objects. When these objects are numbers, the whole is a vector in the mathematical sense. When these objects are themselves vectors of equal (nonzero) length, the whole is a "matrix". A vector which is not a matrix is referred to here as a "plain vector". A vector is displayed as a list of values separated by commas and enclosed in square brackets: `[1, 2, 3]'. Thus the following is a 2 row by 3 column matrix: `[[1, 2, 3], [4, 5, 6]]'. Vectors, like complex numbers, are entered as incomplete objects. *Note Incomplete Objects::. During algebraic entry, vectors are entered all at once in the usual brackets-and-commas form. Matrices may be entered algebraically as nested vectors, or using the shortcut notation `[1, 2, 3; 4, 5, 6]', with rows separated by semicolons. The commas may usually be omitted when entering vectors: `[1 2 3]'. Curly braces may be used in place of brackets: `{1, 2, 3}', but the commas are required in this case. Traditional vector and matrix arithmetic is also supported; *note Basic Arithmetic:: and *note Matrix Functions::. Many other operations are applied to vectors element-wise. For example, the complex conjugate of a vector is a vector of the complex conjugates of its elements. Algebraic functions for building vectors include `vec(a, b, c)' to build `[a, b, c]', `cvec(a, n, m)' to build an NxM matrix of `a's, and `index(n)' to build a vector of integers from 1 to `n'.  File: calc, Node: Strings, Next: HMS Forms, Prev: Vectors and Matrices, Up: Data Types 6.7 Strings =========== Character strings are not a special data type in the Calculator. Rather, a string is represented simply as a vector all of whose elements are integers in the range 0 to 255 (ASCII codes). You can enter a string at any time by pressing the `"' key. Quotation marks and backslashes are written `\"' and `\\', respectively, inside strings. Other notations introduced by backslashes are: \a 7 \^@ 0 \b 8 \^a-z 1-26 \e 27 \^[ 27 \f 12 \^\\ 28 \n 10 \^] 29 \r 13 \^^ 30 \t 9 \^_ 31 \^? 127 Finally, a backslash followed by three octal digits produces any character from its ASCII code. Strings are normally displayed in vector-of-integers form. The `d "' (`calc-display-strings') command toggles a mode in which any vectors of small integers are displayed as quoted strings instead. The backslash notations shown above are also used for displaying strings. Characters 128 and above are not translated by Calc; unless you have an Emacs modified for 8-bit fonts, these will show up in backslash-octal-digits notation. For characters below 32, and for character 127, Calc uses the backslash-letter combination if there is one, or otherwise uses a `\^' sequence. The only Calc feature that uses strings is "compositions"; *note Compositions::. Strings also provide a convenient way to do conversions between ASCII characters and integers. There is a `string' function which provides a different display format for strings. Basically, `string(S)', where S is a vector of integers in the proper range, is displayed as the corresponding string of characters with no surrounding quotation marks or other modifications. Thus `string("ABC")' (or `string([65 66 67])') will look like `ABC' on the stack. This happens regardless of whether `d "' has been used. The only way to turn it off is to use `d U' (unformatted language mode) which will display `string("ABC")' instead. Control characters are displayed somewhat differently by `string'. Characters below 32, and character 127, are shown using `^' notation (same as shown above, but without the backslash). The quote and backslash characters are left alone, as are characters 128 and above. The `bstring' function is just like `string' except that the resulting string is breakable across multiple lines if it doesn't fit all on one line. Potential break points occur at every space character in the string.  File: calc, Node: HMS Forms, Next: Date Forms, Prev: Strings, Up: Data Types 6.8 HMS Forms ============= "HMS" stands for Hours-Minutes-Seconds; when used as an angular argument, the interpretation is Degrees-Minutes-Seconds. All functions that operate on angles accept HMS forms. These are interpreted as degrees regardless of the current angular mode. It is also possible to use HMS as the angular mode so that calculated angles are expressed in degrees, minutes, and seconds. The default format for HMS values is `HOURS@ MINS' SECS"'. During entry, the letters `h' (for "hours") or `o' (approximating the "degrees" symbol) are accepted as well as `@', `m' is accepted in place of `'', and `s' is accepted in place of `"'. The HOURS value is an integer (or integer-valued float). The MINS value is an integer or integer-valued float between 0 and 59. The SECS value is a real number between 0 (inclusive) and 60 (exclusive). A positive HMS form is interpreted as HOURS + MINS/60 + SECS/3600. A negative HMS form is interpreted as - HOURS - MINS/60 - SECS/3600. Display format for HMS forms is quite flexible. *Note HMS Formats::. HMS forms can be added and subtracted. When they are added to numbers, the numbers are interpreted according to the current angular mode. HMS forms can also be multiplied and divided by real numbers. Dividing two HMS forms produces a real-valued ratio of the two angles. Just for kicks, `M-x calc-time' pushes the current time of day on the stack as an HMS form.  File: calc, Node: Date Forms, Next: Modulo Forms, Prev: HMS Forms, Up: Data Types 6.9 Date Forms ============== A "date form" represents a date and possibly an associated time. Simple date arithmetic is supported: Adding a number to a date produces a new date shifted by that many days; adding an HMS form to a date shifts it by that many hours. Subtracting two date forms computes the number of days between them (represented as a simple number). Many other operations, such as multiplying two date forms, are nonsensical and are not allowed by Calc. Date forms are entered and displayed enclosed in `< >' brackets. The default format is, e.g., `' for dates, or `<3:32:20pm Wed Jan 9, 1991>' for dates with times. Input is flexible; date forms can be entered in any of the usual notations for dates and times. *Note Date Formats::. Date forms are stored internally as numbers, specifically the number of days since midnight on the morning of January 1 of the year 1 AD. If the internal number is an integer, the form represents a date only; if the internal number is a fraction or float, the form represents a date and time. For example, `<6:00am Wed Jan 9, 1991>' is represented by the number 726842.25. The standard precision of 12 decimal digits is enough to ensure that a (reasonable) date and time can be stored without roundoff error. If the current precision is greater than 12, date forms will keep additional digits in the seconds position. For example, if the precision is 15, the seconds will keep three digits after the decimal point. Decreasing the precision below 12 may cause the time part of a date form to become inaccurate. This can also happen if astronomically high years are used, though this will not be an issue in everyday (or even everymillennium) use. Note that date forms without times are stored as exact integers, so roundoff is never an issue for them. You can use the `v p' (`calc-pack') and `v u' (`calc-unpack') commands to get at the numerical representation of a date form. *Note Packing and Unpacking::. Date forms can go arbitrarily far into the future or past. Negative year numbers represent years BC. Calc uses a combination of the Gregorian and Julian calendars, following the history of Great Britain and the British colonies. This is the same calendar that is used by the `cal' program in most Unix implementations. Some historical background: The Julian calendar was created by Julius Caesar in the year 46 BC as an attempt to fix the gradual drift caused by the lack of leap years in the calendar used until that time. The Julian calendar introduced an extra day in all years divisible by four. After some initial confusion, the calendar was adopted around the year we call 8 AD. Some centuries later it became apparent that the Julian year of 365.25 days was itself not quite right. In 1582 Pope Gregory XIII introduced the Gregorian calendar, which added the new rule that years divisible by 100, but not by 400, were not to be considered leap years despite being divisible by four. Many countries delayed adoption of the Gregorian calendar because of religious differences; in Britain it was put off until the year 1752, by which time the Julian calendar had fallen eleven days behind the true seasons. So the switch to the Gregorian calendar in early September 1752 introduced a discontinuity: The day after Sep 2, 1752 is Sep 14, 1752. Calc follows this convention. To take another example, Russia waited until 1918 before adopting the new calendar, and thus needed to remove thirteen days (between Feb 1, 1918 and Feb 14, 1918). This means that Calc's reckoning will be inconsistent with Russian history between 1752 and 1918, and similarly for various other countries. Today's timekeepers introduce an occasional "leap second" as well, but Calc does not take these minor effects into account. (If it did, it would have to report a non-integer number of days between, say, `<12:00am Mon Jan 1, 1900>' and `<12:00am Sat Jan 1, 2000>'.) Calc uses the Julian calendar for all dates before the year 1752, including dates BC when the Julian calendar technically had not yet been invented. Thus the claim that day number -10000 is called "August 16, 28 BC" should be taken with a grain of salt. Please note that there is no "year 0"; the day before `' is `'. These are days 0 and -1 respectively in Calc's internal numbering scheme. Another day counting system in common use is, confusingly, also called "Julian." The Julian day number is the numbers of days since 12:00 noon (GMT) on Jan 1, 4713 BC, which in Calc's scheme (in GMT) is -1721423.5 (recall that Calc starts at midnight instead of noon). Thus to convert a Calc date code obtained by unpacking a date form into a Julian day number, simply add 1721423.5 after compensating for the time zone difference. The built-in `t J' command performs this conversion for you. The Julian day number is based on the Julian cycle, which was invented in 1583 by Joseph Justus Scaliger. Scaliger named it the Julian cycle since it involves the Julian calendar, but some have suggested that Scaliger named it in honor of his father, Julius Caesar Scaliger. The Julian cycle is based on three other cycles: the indiction cycle, the Metonic cycle, and the solar cycle. The indiction cycle is a 15 year cycle originally used by the Romans for tax purposes but later used to date medieval documents. The Metonic cycle is a 19 year cycle; 19 years is close to being a common multiple of a solar year and a lunar month, and so every 19 years the phases of the moon will occur on the same days of the year. The solar cycle is a 28 year cycle; the Julian calendar repeats itself every 28 years. The smallest time period which contains multiples of all three cycles is the least common multiple of 15 years, 19 years and 28 years, which (since they're pairwise relatively prime) is 15*19*28 = 7980 years. This is the length of a Julian cycle. Working backwards, the previous year in which all three cycles began was 4713 BC, and so Scalinger chose that year as the beginning of a Julian cycle. Since at the time there were no historical records from before 4713 BC, using this year as a starting point had the advantage of avoiding negative year numbers. In 1849, the astronomer John Herschel (son of William Herschel) suggested using the number of days since the beginning of the Julian cycle as an astronomical dating system; this idea was taken up by other astronomers. (At the time, noon was the start of the astronomical day. Herschel originally suggested counting the days since Jan 1, 4713 BC at noon Alexandria time; this was later amended to noon GMT.) Julian day numbering is largely used in astronomy. The Unix operating system measures time as an integer number of seconds since midnight, Jan 1, 1970. To convert a Calc date value into a Unix time stamp, first subtract 719164 (the code for `'), then multiply by 86400 (the number of seconds in a day) and press `R' to round to the nearest integer. If you have a date form, you can simply subtract the day `' instead of unpacking and subtracting 719164. Likewise, divide by 86400 and add `' to convert from Unix time to a Calc date form. (Note that Unix normally maintains the time in the GMT time zone; you may need to subtract five hours to get New York time, or eight hours for California time. The same is usually true of Julian day counts.) The built-in `t U' command performs these conversions.  File: calc, Node: Modulo Forms, Next: Error Forms, Prev: Date Forms, Up: Data Types 6.10 Modulo Forms ================= A "modulo form" is a real number which is taken modulo (i.e., within an integer multiple of) some value M. Arithmetic modulo M often arises in number theory. Modulo forms are written `A mod M', where A and M are real numbers or HMS forms, and `0 <= a < M'. In many applications `a' and `M' will be integers but this is not required. To create a modulo form during numeric entry, press the shift-`M' key to enter the word `mod'. As a special convenience, pressing shift-`M' a second time automatically enters the value of `M' that was most recently used before. During algebraic entry, either type `mod' by hand or press `M-m' (that's `-m'). Once again, pressing this a second time enters the current modulo. Modulo forms are not to be confused with the modulo operator `%'. The expression `27 % 10' means to compute 27 modulo 10 to produce the result 7. Further computations treat this 7 as just a regular integer. The expression `27 mod 10' produces the result `7 mod 10'; further computations with this value are again reduced modulo 10 so that the result always lies in the desired range. When two modulo forms with identical `M''s are added or multiplied, the Calculator simply adds or multiplies the values, then reduces modulo `M'. If one argument is a modulo form and the other a plain number, the plain number is treated like a compatible modulo form. It is also possible to raise modulo forms to powers; the result is the value raised to the power, then reduced modulo `M'. (When all values involved are integers, this calculation is done much more efficiently than actually computing the power and then reducing.) Two modulo forms `A mod M' and `B mod M' can be divided if `a', `b', and `M' are all integers. The result is the modulo form which, when multiplied by `B mod M', produces `A mod M'. If there is no solution to this equation (which can happen only when `M' is non-prime), or if any of the arguments are non-integers, the division is left in symbolic form. Other operations, such as square roots, are not yet supported for modulo forms. (Note that, although `(A mod M)^.5' will compute a "modulo square root" in the sense of reducing `sqrt(a)' modulo `M', this is not a useful definition from the number-theoretical point of view.) It is possible to mix HMS forms and modulo forms. For example, an HMS form modulo 24 could be used to manipulate clock times; an HMS form modulo 360 would be suitable for angles. Making the modulo `M' also be an HMS form eliminates troubles that would arise if the angular mode were inadvertently set to Radians, in which case `2@ 0' 0" mod 24' would be interpreted as two degrees modulo 24 radians! Modulo forms cannot have variables or formulas for components. If you enter the formula `(x + 2) mod 5', Calc propagates the modulus to each of the coefficients: `(1 mod 5) x + (2 mod 5)'. You can use `v p' and `%' to modify modulo forms. *Note Packing and Unpacking::. *Note Basic Arithmetic::. The algebraic function `makemod(a, m)' builds the modulo form `a mod m'.  File: calc, Node: Error Forms, Next: Interval Forms, Prev: Modulo Forms, Up: Data Types 6.11 Error Forms ================ An "error form" is a number with an associated standard deviation, as in `2.3 +/- 0.12'. The notation `X +/- sigma' stands for an uncertain value which follows a normal or Gaussian distribution of mean `x' and standard deviation or "error" `sigma'. Both the mean and the error can be either numbers or formulas. Generally these are real numbers but the mean may also be complex. If the error is negative or complex, it is changed to its absolute value. An error form with zero error is converted to a regular number by the Calculator. All arithmetic and transcendental functions accept error forms as input. Operations on the mean-value part work just like operations on regular numbers. The error part for any function `f(x)' (such as `sin(x)') is defined by the error of `x' times the derivative of `f' evaluated at the mean value of `x'. For a two-argument function `f(x,y)' (such as addition) the error is the square root of the sum of the squares of the errors due to `x' and `y'. Note that this definition assumes the errors in `x' and `y' are uncorrelated. A side effect of this definition is that `(2 +/- 1) * (2 +/- 1)' is not the same as `(2 +/- 1)^2'; the former represents the product of two independent values which happen to have the same probability distributions, and the latter is the product of one random value with itself. The former will produce an answer with less error, since on the average the two independent errors can be expected to cancel out. Consult a good text on error analysis for a discussion of the proper use of standard deviations. Actual errors often are neither Gaussian-distributed nor uncorrelated, and the above formulas are valid only when errors are small. As an example, the error arising from `sin(X +/- SIGMA)' is `SIGMA abs(cos(X))'. When `x' is close to zero, `cos(x)' is close to one so the error in the sine is close to `sigma'; this makes sense, since `sin(x)' is approximately `x' near zero, so a given error in `x' will produce about the same error in the sine. Likewise, near 90 degrees `cos(x)' is nearly zero and so the computed error is small: The sine curve is nearly flat in that region, so an error in `x' has relatively little effect on the value of `sin(x)'. However, consider `sin(90 +/- 1000)'. The cosine of 90 is zero, so Calc will report zero error! We get an obviously wrong result because we have violated the small-error approximation underlying the error analysis. If the error in `x' had been small, the error in `sin(x)' would indeed have been negligible. To enter an error form during regular numeric entry, use the `p' ("plus-or-minus") key to type the `+/-' symbol. (If you try actually typing `+/-' the `+' key will be interpreted as the Calculator's `+' command!) Within an algebraic formula, you can press `M-+' to type the `+/-' symbol, or type it out by hand. Error forms and complex numbers can be mixed; the formulas shown above are used for complex numbers, too; note that if the error part evaluates to a complex number its absolute value (or the square root of the sum of the squares of the absolute values of the two error contributions) is used. Mathematically, this corresponds to a radially symmetric Gaussian distribution of numbers on the complex plane. However, note that Calc considers an error form with real components to represent a real number, not a complex distribution around a real mean. Error forms may also be composed of HMS forms. For best results, both the mean and the error should be HMS forms if either one is. The algebraic function `sdev(a, b)' builds the error form `a +/- b'.  File: calc, Node: Interval Forms, Next: Incomplete Objects, Prev: Error Forms, Up: Data Types 6.12 Interval Forms =================== An "interval" is a subset of consecutive real numbers. For example, the interval `[2 .. 4]' represents all the numbers from 2 to 4, inclusive. If you multiply it by the interval `[0.5 .. 2]' you obtain `[1 .. 8]'. This calculation represents the fact that if you multiply some number in the range `[2 .. 4]' by some other number in the range `[0.5 .. 2]', your result will lie in the range from 1 to 8. Interval arithmetic is used to get a worst-case estimate of the possible range of values a computation will produce, given the set of possible values of the input. Calc supports several varieties of intervals, including "closed" intervals of the type shown above, "open" intervals such as `(2 .. 4)', which represents the range of numbers from 2 to 4 _exclusive_, and "semi-open" intervals in which one end uses a round parenthesis and the other a square bracket. In mathematical terms, `[2 .. 4]' means `2 <= x <= 4', whereas `[2 .. 4)' represents `2 <= x < 4', `(2 .. 4]' represents `2 < x <= 4', and `(2 .. 4)' represents `2 < x < 4'. The lower and upper limits of an interval must be either real numbers (or HMS or date forms), or symbolic expressions which are assumed to be real-valued, or `-inf' and `inf'. In general the lower limit must be less than the upper limit. A closed interval containing only one value, `[3 .. 3]', is converted to a plain number (3) automatically. An interval containing no values at all (such as `[3 .. 2]' or `[2 .. 2)') can be represented but is not guaranteed to behave well when used in arithmetic. Note that the interval `[3 .. inf)' represents all real numbers greater than or equal to 3, and `(-inf .. inf)' represents all real numbers. In fact, `[-inf .. inf]' represents all real numbers including the real infinities. Intervals are entered in the notation shown here, either as algebraic formulas, or using incomplete forms. (*Note Incomplete Objects::.) In algebraic formulas, multiple periods in a row are collected from left to right, so that `1...1e2' is interpreted as `1.0 .. 1e2' rather than `1 .. 0.1e2'. Add spaces or zeros if you want to get the other interpretation. If you omit the lower or upper limit, a default of `-inf' or `inf' (respectively) is furnished. Infinite mode also affects operations on intervals (*note Infinities::). Calc will always introduce an open infinity, as in `1 / (0 .. 2] = [0.5 .. inf)'. But closed infinities, `1 / [0 .. 2] = [0.5 .. inf]', arise only in Infinite mode; otherwise they are left unevaluated. Note that the "direction" of a zero is not an issue in this case since the zero is always assumed to be continuous with the rest of the interval. For intervals that contain zero inside them Calc is forced to give the result, `1 / (-2 .. 2) = [-inf .. inf]'. While it may seem that intervals and error forms are similar, they are based on entirely different concepts of inexact quantities. An error form `X +/- SIGMA' means a variable is random, and its value could be anything but is "probably" within one SIGMA of the mean value `x'. An interval `[A .. B]' means a variable's value is unknown, but guaranteed to lie in the specified range. Error forms are statistical or "average case" approximations; interval arithmetic tends to produce "worst case" bounds on an answer. Intervals may not contain complex numbers, but they may contain HMS forms or date forms. *Note Set Operations::, for commands that interpret interval forms as subsets of the set of real numbers. The algebraic function `intv(n, a, b)' builds an interval form from `a' to `b'; `n' is an integer code which must be 0 for `(..)', 1 for `(..]', 2 for `[..)', or 3 for `[..]'. Please note that in fully rigorous interval arithmetic, care would be taken to make sure that the computation of the lower bound rounds toward minus infinity, while upper bound computations round toward plus infinity. Calc's arithmetic always uses a round-to-nearest mode, which means that roundoff errors could creep into an interval calculation to produce intervals slightly smaller than they ought to be. For example, entering `[1..2]' and pressing `Q 2 ^' should yield the interval `[1..2]' again, but in fact it yields the (slightly too small) interval `[1..1.9999999]' due to roundoff error.  File: calc, Node: Incomplete Objects, Next: Variables, Prev: Interval Forms, Up: Data Types 6.13 Incomplete Objects ======================= When `(' or `[' is typed to begin entering a complex number or vector, respectively, the effect is to push an "incomplete" complex number or vector onto the stack. The `,' key adds the value(s) at the top of the stack onto the current incomplete object. The `)' and `]' keys "close" the incomplete object after adding any values on the top of the stack in front of the incomplete object. As a result, the sequence of keystrokes `[ 2 , 3 2 * , 9 ]' pushes the vector `[2, 6, 9]' onto the stack. Likewise, `( 1 , 2 Q )' pushes the complex number `(1, 1.414)' (approximately). If several values lie on the stack in front of the incomplete object, all are collected and appended to the object. Thus the `,' key is redundant: `[ 2 3 2 * 9 ]'. Some people prefer the equivalent key to . As a special case, typing `,' immediately after `(', `[', or `,' adds a zero or duplicates the preceding value in the list being formed. Typing during incomplete entry removes the last item from the list. The `;' key is used in the same way as `,' to create polar complex numbers: `( 1 ; 2 )'. When entering a vector, `;' is useful for creating a matrix. In particular, `[ [ 1 , 2 ; 3 , 4 ; 5 , 6 ] ]' is equivalent to `[ [ 1 , 2 ] , [ 3 , 4 ] , [ 5 , 6 ] ]'. Incomplete entry is also used to enter intervals. For example, `[ 2 .. 4 )' enters a semi-open interval. Note that when you type the first period, it will be interpreted as a decimal point, but when you type a second period immediately afterward, it is re-interpreted as part of the interval symbol. Typing `..' corresponds to executing the `calc-dots' command. If you find incomplete entry distracting, you may wish to enter vectors and complex numbers as algebraic formulas by pressing the apostrophe key.  File: calc, Node: Variables, Next: Formulas, Prev: Incomplete Objects, Up: Data Types 6.14 Variables ============== A "variable" is somewhere between a storage register on a conventional calculator, and a variable in a programming language. (In fact, a Calc variable is really just an Emacs Lisp variable that contains a Calc number or formula.) A variable's name is normally composed of letters and digits. Calc also allows apostrophes and `#' signs in variable names. (The Calc variable `foo' corresponds to the Emacs Lisp variable `var-foo', but unless you access the variable from within Emacs Lisp, you don't need to worry about it. Variable names in algebraic formulas implicitly have `var-' prefixed to their names. The `#' character in variable names used in algebraic formulas corresponds to a dash `-' in the Lisp variable name. If the name contains any dashes, the prefix `var-' is _not_ automatically added. Thus the two formulas `foo + 1' and `var#foo + 1' both refer to the same variable.) In a command that takes a variable name, you can either type the full name of a variable, or type a single digit to use one of the special convenience variables `q0' through `q9'. For example, `3 s s 2' stores the number 3 in variable `q2', and `3 s s foo ' stores that number in variable `foo'. To push a variable itself (as opposed to the variable's value) on the stack, enter its name as an algebraic expression using the apostrophe (<'>) key. The `=' (`calc-evaluate') key "evaluates" a formula by replacing all variables in the formula which have been given values by a `calc-store' or `calc-let' command by their stored values. Other variables are left alone. Thus a variable that has not been stored acts like an abstract variable in algebra; a variable that has been stored acts more like a register in a traditional calculator. With a positive numeric prefix argument, `=' evaluates the top N stack entries; with a negative argument, `=' evaluates the Nth stack entry. A few variables are called "special constants". Their names are `e', `pi', `i', `phi', and `gamma'. (*Note Scientific Functions::.) When they are evaluated with `=', their values are calculated if necessary according to the current precision or complex polar mode. If you wish to use these symbols for other purposes, simply undefine or redefine them using `calc-store'. The variables `inf', `uinf', and `nan' stand for infinite or indeterminate values. It's best not to use them as regular variables, since Calc uses special algebraic rules when it manipulates them. Calc displays a warning message if you store a value into any of these special variables. *Note Store and Recall::, for a discussion of commands dealing with variables.  File: calc, Node: Formulas, Prev: Variables, Up: Data Types 6.15 Formulas ============= When you press the apostrophe key you may enter any expression or formula in algebraic form. (Calc uses the terms "expression" and "formula" interchangeably.) An expression is built up of numbers, variable names, and function calls, combined with various arithmetic operators. Parentheses may be used to indicate grouping. Spaces are ignored within formulas, except that spaces are not permitted within variable names or numbers. Arithmetic operators, in order from highest to lowest precedence, and with their equivalent function names, are: `_' [`subscr'] (subscripts); postfix `%' [`percent'] (as in `25% = 0.25'); prefix `!' [`lnot'] (logical "not," as in `!x'); `+/-' [`sdev'] (the standard deviation symbol) and `mod' [`makemod'] (the symbol for modulo forms); postfix `!' [`fact'] (factorial, as in `n!') and postfix `!!' [`dfact'] (double factorial); `^' [`pow'] (raised-to-the-power-of); prefix `+' and `-' [`neg'] (as in `-x'); `*' [`mul']; `/' [`div'], `%' [`mod'] (modulo), and `\' [`idiv'] (integer division); infix `+' [`add'] and `-' [`sub'] (as in `x-y'); `|' [`vconcat'] (vector concatenation); relations `=' [`eq'], `!=' [`neq'], `<' [`lt'], `>' [`gt'], `<=' [`leq'], and `>=' [`geq']; `&&' [`land'] (logical "and"); `||' [`lor'] (logical "or"); the C-style "if" operator `a?b:c' [`if']; `!!!' [`pnot'] (rewrite pattern "not"); `&&&' [`pand'] (rewrite pattern "and"); `|||' [`por'] (rewrite pattern "or"); `:=' [`assign'] (for assignments and rewrite rules); `::' [`condition'] (rewrite pattern condition); `=>' [`evalto']. Note that, unlike in usual computer notation, multiplication binds more strongly than division: `a*b/c*d' is equivalent to `(a*b)/(c*d)'. The multiplication sign `*' may be omitted in many cases. In particular, if the righthand side is a number, variable name, or parenthesized expression, the `*' may be omitted. Implicit multiplication has the same precedence as the explicit `*' operator. The one exception to the rule is that a variable name followed by a parenthesized expression, as in `f(x)', is interpreted as a function call, not an implicit `*'. In many cases you must use a space if you omit the `*': `2a' is the same as `2*a', and `a b' is the same as `a*b', but `ab' is a variable called `ab', _not_ the product of `a' and `b'! Also note that `f (x)' is still a function call. The rules are slightly different for vectors written with square brackets. In vectors, the space character is interpreted (like the comma) as a separator of elements of the vector. Thus `[ 2a b+c d ]' is equivalent to `[2*a, b+c, d]', whereas `2a b+c d' is equivalent to `2*a*b + c*d'. Note that spaces around the brackets, and around explicit commas, are ignored. To force spaces to be interpreted as multiplication you can enclose a formula in parentheses as in `[(a b) 2(c d)]', which is interpreted as `[a*b, 2*c*d]'. An implicit comma is also inserted between `][', as in the matrix `[[1 2][3 4]]'. Vectors that contain commas (not embedded within nested parentheses or brackets) do not treat spaces specially: `[a b, 2 c d]' is a vector of two elements. Also, if it would be an error to treat spaces as separators, but not otherwise, then Calc will ignore spaces: `[a - b]' is a vector of one element, but `[a -b]' is a vector of two elements. Finally, vectors entered with curly braces instead of square brackets do not give spaces any special treatment. When Calc displays a vector that does not contain any commas, it will insert parentheses if necessary to make the meaning clear: `[(a b)]'. The expression `5%-2' is ambiguous; is this five-percent minus two, or five modulo minus-two? Calc always interprets the leftmost symbol as an infix operator preferentially (modulo, in this case), so you would need to write `(5%)-2' to get the former interpretation. A function call is, e.g., `sin(1+x)'. (The Calc algebraic function `foo' corresponds to the Emacs Lisp function `calcFunc-foo', but unless you access the function from within Emacs Lisp, you don't need to worry about it.) Most mathematical Calculator commands like `calc-sin' have function equivalents like `sin'. If no Lisp function is defined for a function called by a formula, the call is left as it is during algebraic manipulation: `f(x+y)' is left alone. Beware that many innocent-looking short names like `in' and `re' have predefined meanings which could surprise you; however, single letters or single letters followed by digits are always safe to use for your own function names. *Note Function Index::. In the documentation for particular commands, the notation `H S' (`calc-sinh') [`sinh'] means that the key sequence `H S', the command `M-x calc-sinh', and the algebraic function `sinh(x)' all represent the same operation. Commands that interpret ("parse") text as algebraic formulas include algebraic entry (`''), editing commands like ``' which parse the contents of the editing buffer when you finish, the `C-x * g' and `C-x * r' commands, the `C-y' command, the X window system "paste" mouse operation, and Embedded mode. All of these operations use the same rules for parsing formulas; in particular, language modes (*note Language Modes::) affect them all in the same way. When you read a large amount of text into the Calculator (say a vector which represents a big set of rewrite rules; *note Rewrite Rules::), you may wish to include comments in the text. Calc's formula parser ignores the symbol `%%' and anything following it on a line: [ a + b, %% the sum of "a" and "b" c + d, %% last line is coming up: e + f ] This is parsed exactly the same as `[ a + b, c + d, e + f ]'. *Note Syntax Tables::, for a way to create your own operators and other input notations. *Note Compositions::, for a way to create new display formats. *Note Algebra::, for commands for manipulating formulas symbolically.  File: calc, Node: Stack and Trail, Next: Mode Settings, Prev: Data Types, Up: Top 7 Stack and Trail Commands ************************** This chapter describes the Calc commands for manipulating objects on the stack and in the trail buffer. (These commands operate on objects of any type, such as numbers, vectors, formulas, and incomplete objects.) * Menu: * Stack Manipulation:: * Editing Stack Entries:: * Trail Commands:: * Keep Arguments::  File: calc, Node: Stack Manipulation, Next: Editing Stack Entries, Prev: Stack and Trail, Up: Stack and Trail 7.1 Stack Manipulation Commands =============================== To duplicate the top object on the stack, press or (two equivalent keys for the `calc-enter' command). Given a positive numeric prefix argument, these commands duplicate several elements at the top of the stack. Given a negative argument, these commands duplicate the specified element of the stack. Given an argument of zero, they duplicate the entire stack. For example, with `10 20 30' on the stack, creates `10 20 30 30', `C-u 2 ' creates `10 20 30 20 30', `C-u - 2 ' creates `10 20 30 20', and `C-u 0 ' creates `10 20 30 10 20 30'. The (`calc-over') command (on a key marked Line-Feed if you have it, else on `C-j') is like `calc-enter' except that the sign of the numeric prefix argument is interpreted oppositely. Also, with no prefix argument the default argument is 2. Thus with `10 20 30' on the stack, and `C-u 2 ' are both equivalent to `C-u - 2 ', producing `10 20 30 20'. To remove the top element from the stack, press (`calc-pop'). The `C-d' key is a synonym for . (If the top element is an incomplete object with at least one element, the last element is removed from it.) Given a positive numeric prefix argument, several elements are removed. Given a negative argument, the specified element of the stack is deleted. Given an argument of zero, the entire stack is emptied. For example, with `10 20 30' on the stack, leaves `10 20', `C-u 2 ' leaves `10', `C-u - 2 ' leaves `10 30', and `C-u 0 ' leaves an empty stack. The `M-' (`calc-pop-above') command is to what is to : It interprets the sign of the numeric prefix argument in the opposite way, and the default argument is 2. Thus `M-' by itself removes the second-from-top stack element, leaving the first, third, fourth, and so on; `M-3 M-' deletes the third stack element. To exchange the top two elements of the stack, press (`calc-roll-down'). Given a positive numeric prefix argument, the specified number of elements at the top of the stack are rotated downward. Given a negative argument, the entire stack is rotated downward the specified number of times. Given an argument of zero, the entire stack is reversed top-for-bottom. For example, with `10 20 30 40 50' on the stack, creates `10 20 30 50 40', `C-u 3 ' creates `10 20 50 30 40', `C-u - 2 ' creates `40 50 10 20 30', and `C-u 0 ' creates `50 40 30 20 10'. The command `M-' (`calc-roll-up') is analogous to except that it rotates upward instead of downward. Also, the default with no prefix argument is to rotate the top 3 elements. For example, with `10 20 30 40 50' on the stack, `M-' creates `10 20 40 50 30', `C-u 4 M-' creates `10 30 40 50 20', `C-u - 2 M-' creates `30 40 50 10 20', and `C-u 0 M-' creates `50 40 30 20 10'. A good way to view the operation of and `M-' is in terms of moving a particular element to a new position in the stack. With a positive argument N, moves the top stack element down to level N, making room for it by pulling all the intervening stack elements toward the top. `M-' moves the element at level N up to the top. (Compare with , which copies instead of moving the element in level N.) With a negative argument -N, rotates the stack to move the object in level N to the deepest place in the stack, and the object in level N+1 to the top. `M-' rotates the deepest stack element to be in level N, also putting the top stack element in level N+1. *Note Selecting Subformulas::, for a way to apply these commands to any portion of a vector or formula on the stack. The command `C-x C-t' (`calc-transpose-lines') will transpose the stack object determined by the point with the stack object at the next higher level. For example, with `10 20 30 40 50' on the stack and the point on the line containing `30', `C-x C-t' creates `10 20 40 30 50'. More generally, `C-x C-t' acts on the stack objects determined by the current point (and mark) similar to how the text-mode command `transpose-lines' acts on lines. With argument N, `C-x C-t' will move the stack object at the level above the current point and move it past N other objects; for example, with `10 20 30 40 50' on the stack and the point on the line containing `30', `C-u 2 C-x C-t' creates `10 40 20 30 50'. With an argument of 0, `C-x C-t' will switch the stack objects at the levels determined by the point and the mark.  File: calc, Node: Editing Stack Entries, Next: Trail Commands, Prev: Stack Manipulation, Up: Stack and Trail 7.2 Editing Stack Entries ========================= The ``' (`calc-edit') command creates a temporary buffer (`*Calc Edit*') for editing the top-of-stack value using regular Emacs commands. Note that ``' is a backquote, not a quote. With a numeric prefix argument, it edits the specified number of stack entries at once. (An argument of zero edits the entire stack; a negative argument edits one specific stack entry.) When you are done editing, press `C-c C-c' to finish and return to Calc. The and keys also work to finish most sorts of editing, though in some cases Calc leaves with its usual meaning ("insert a newline") if it's a situation where you might want to insert new lines into the editing buffer. When you finish editing, the Calculator parses the lines of text in the `*Calc Edit*' buffer as numbers or formulas, replaces the original stack elements in the original buffer with these new values, then kills the `*Calc Edit*' buffer. The original Calculator buffer continues to exist during editing, but for best results you should be careful not to change it until you have finished the edit. You can also cancel the edit by killing the buffer with `C-x k'. The formula is normally reevaluated as it is put onto the stack. For example, editing `a + 2' to `3 + 2' and pressing `C-c C-c' will push 5 on the stack. If you use to finish, Calc will put the result on the stack without evaluating it. If you give a prefix argument to `C-c C-c', Calc will not kill the `*Calc Edit*' buffer. You can switch back to that buffer and continue editing if you wish. However, you should understand that if you initiated the edit with ``', the `C-c C-c' operation will be programmed to replace the top of the stack with the new edited value, and it will do this even if you have rearranged the stack in the meanwhile. This is not so much of a problem with other editing commands, though, such as `s e' (`calc-edit-variable'; *note Operations on Variables::). If the `calc-edit' command involves more than one stack entry, each line of the `*Calc Edit*' buffer is interpreted as a separate formula. Otherwise, the entire buffer is interpreted as one formula, with line breaks ignored. (You can use `C-o' or `C-q C-j' to insert a newline in the buffer without pressing .) The ``' key also works during numeric or algebraic entry. The text entered so far is moved to the `*Calc Edit*' buffer for more extensive editing than is convenient in the minibuffer.  File: calc, Node: Trail Commands, Next: Keep Arguments, Prev: Editing Stack Entries, Up: Stack and Trail 7.3 Trail Commands ================== The commands for manipulating the Calc Trail buffer are two-key sequences beginning with the `t' prefix. The `t d' (`calc-trail-display') command turns display of the trail on and off. Normally the trail display is toggled on if it was off, off if it was on. With a numeric prefix of zero, this command always turns the trail off; with a prefix of one, it always turns the trail on. The other trail-manipulation commands described here automatically turn the trail on. Note that when the trail is off values are still recorded there; they are simply not displayed. To set Emacs to turn the trail off by default, type `t d' and then save the mode settings with `m m' (`calc-save-modes'). The `t i' (`calc-trail-in') and `t o' (`calc-trail-out') commands switch the cursor into and out of the Calc Trail window. In practice they are rarely used, since the commands shown below are a more convenient way to move around in the trail, and they work "by remote control" when the cursor is still in the Calculator window. There is a "trail pointer" which selects some entry of the trail at any given time. The trail pointer looks like a `>' symbol right before the selected number. The following commands operate on the trail pointer in various ways. The `t y' (`calc-trail-yank') command reads the selected value in the trail and pushes it onto the Calculator stack. It allows you to re-use any previously computed value without retyping. With a numeric prefix argument N, it yanks the value N lines above the current trail pointer. The `t <' (`calc-trail-scroll-left') and `t >' (`calc-trail-scroll-right') commands horizontally scroll the trail window left or right by one half of its width. The `t n' (`calc-trail-next') and `t p' (`calc-trail-previous)' commands move the trail pointer down or up one line. The `t f' (`calc-trail-forward') and `t b' (`calc-trail-backward') commands move the trail pointer down or up one screenful at a time. All of these commands accept numeric prefix arguments to move several lines or screenfuls at a time. The `t [' (`calc-trail-first') and `t ]' (`calc-trail-last') commands move the trail pointer to the first or last line of the trail. The `t h' (`calc-trail-here') command moves the trail pointer to the cursor position; unlike the other trail commands, `t h' works only when Calc Trail is the selected window. The `t s' (`calc-trail-isearch-forward') and `t r' (`calc-trail-isearch-backward') commands perform an incremental search forward or backward through the trail. You can press to terminate the search; the trail pointer moves to the current line. If you cancel the search with `C-g', the trail pointer stays where it was when the search began. The `t m' (`calc-trail-marker') command allows you to enter a line of text of your own choosing into the trail. The text is inserted after the line containing the trail pointer; this usually means it is added to the end of the trail. Trail markers are useful mainly as the targets for later incremental searches in the trail. The `t k' (`calc-trail-kill') command removes the selected line from the trail. The line is saved in the Emacs kill ring suitable for yanking into another buffer, but it is not easy to yank the text back into the trail buffer. With a numeric prefix argument, this command kills the N lines below or above the selected one. The `t .' (`calc-full-trail-vectors') command is described elsewhere; *note Vector and Matrix Formats::.  File: calc, Node: Keep Arguments, Prev: Trail Commands, Up: Stack and Trail 7.4 Keep Arguments ================== The `K' (`calc-keep-args') command acts like a prefix for the following command. It prevents that command from removing its arguments from the stack. For example, after `2 3 +', the stack contains the sole number 5, but after `2 3 K +', the stack contains the arguments and the result: `2 3 5'. With the exception of keyboard macros, this works for all commands that take arguments off the stack. (To avoid potentially unpleasant behavior, a `K' prefix before a keyboard macro will be ignored. A `K' prefix called _within_ the keyboard macro will still take effect.) As another example, `K a s' simplifies a formula, pushing the simplified version of the formula onto the stack after the original formula (rather than replacing the original formula). Note that you could get the same effect by typing ` a s', copying the formula and then simplifying the copy. One difference is that for a very large formula the time taken to format the intermediate copy in ` a s' could be noticeable; `K a s' would avoid this extra work. Even stack manipulation commands are affected. works by popping two values and pushing them back in the opposite order, so `2 3 K ' produces `2 3 3 2'. A few Calc commands provide other ways of doing the same thing. For example, `' sin($)' replaces the number on the stack with its sine using algebraic entry; to push the sine and keep the original argument you could use either `' sin($1)' or `K ' sin($)'. *Note Algebraic Entry::. Also, the `s s' command is effectively the same as `K s t'. *Note Storing Variables::. If you execute a command and then decide you really wanted to keep the argument, you can press `M-' (`calc-last-args'). This command pushes the last arguments that were popped by any command onto the stack. Note that the order of things on the stack will be different than with `K': `2 3 + M-' leaves `5 2 3' on the stack instead of `2 3 5'. *Note Undo::.  File: calc, Node: Mode Settings, Next: Arithmetic, Prev: Stack and Trail, Up: Top 8 Mode Settings *************** This chapter describes commands that set modes in the Calculator. They do not affect the contents of the stack, although they may change the _appearance_ or _interpretation_ of the stack's contents. * Menu: * General Mode Commands:: * Precision:: * Inverse and Hyperbolic:: * Calculation Modes:: * Simplification Modes:: * Declarations:: * Display Modes:: * Language Modes:: * Modes Variable:: * Calc Mode Line::  File: calc, Node: General Mode Commands, Next: Precision, Prev: Mode Settings, Up: Mode Settings 8.1 General Mode Commands ========================= You can save all of the current mode settings in your Calc init file (the file given by the variable `calc-settings-file', typically `~/.emacs.d/calc.el') with the `m m' (`calc-save-modes') command. This will cause Emacs to reestablish these modes each time it starts up. The modes saved in the file include everything controlled by the `m' and `d' prefix keys, the current precision and binary word size, whether or not the trail is displayed, the current height of the Calc window, and more. The current interface (used when you type `C-x * *') is also saved. If there were already saved mode settings in the file, they are replaced. Otherwise, the new mode information is appended to the end of the file. The `m R' (`calc-mode-record-mode') command tells Calc to record all the mode settings (as if by pressing `m m') every time a mode setting changes. If the modes are saved this way, then this "automatic mode recording" mode is also saved. Type `m R' again to disable this method of recording the mode settings. To turn it off permanently, the `m m' command will also be necessary. (If Embedded mode is enabled, other options for recording the modes are available; *note Mode Settings in Embedded Mode::.) The `m F' (`calc-settings-file-name') command allows you to choose a different file than the current value of `calc-settings-file' for `m m', `Z P', and similar commands to save permanent information. You are prompted for a file name. All Calc modes are then reset to their default values, then settings from the file you named are loaded if this file exists, and this file becomes the one that Calc will use in the future for commands like `m m'. The default settings file name is `~/.calc.el'. You can see the current file name by giving a blank response to the `m F' prompt. See also the discussion of the `calc-settings-file' variable; *note Customizing Calc::. If the file name you give is your user init file (typically `~/.emacs'), `m F' will not automatically load the new file. This is because your user init file may contain other things you don't want to reread. You can give a numeric prefix argument of 1 to `m F' to force it to read the file no matter what. Conversely, an argument of -1 tells `m F' _not_ to read the new file. An argument of 2 or -2 tells `m F' not to reset the modes to their defaults beforehand, which is useful if you intend your new file to have a variant of the modes present in the file you were using before. The `m x' (`calc-always-load-extensions') command enables a mode in which the first use of Calc loads the entire program, including all extensions modules. Otherwise, the extensions modules will not be loaded until the various advanced Calc features are used. Since this mode only has effect when Calc is first loaded, `m x' is usually followed by `m m' to make the mode-setting permanent. To load all of Calc just once, rather than always in the future, you can press `C-x * L'. The `m S' (`calc-shift-prefix') command enables a mode in which all of Calc's letter prefix keys may be typed shifted as well as unshifted. If you are typing, say, `a S' (`calc-solve-for') quite often you might find it easier to turn this mode on so that you can type `A S' instead. When this mode is enabled, the commands that used to be on those single shifted letters (e.g., `A' (`calc-abs')) can now be invoked by pressing the shifted letter twice: `A A'. Note that the `v' prefix key always works both shifted and unshifted, and the `z' and `Z' prefix keys are always distinct. Also, the `h' prefix is not affected by this mode. Press `m S' again to disable shifted-prefix mode.  File: calc, Node: Precision, Next: Inverse and Hyperbolic, Prev: General Mode Commands, Up: Mode Settings 8.2 Precision ============= The `p' (`calc-precision') command controls the precision to which floating-point calculations are carried. The precision must be at least 3 digits and may be arbitrarily high, within the limits of memory and time. This affects only floats: Integer and rational calculations are always carried out with as many digits as necessary. The `p' key prompts for the current precision. If you wish you can instead give the precision as a numeric prefix argument. Many internal calculations are carried to one or two digits higher precision than normal. Results are rounded down afterward to the current precision. Unless a special display mode has been selected, floats are always displayed with their full stored precision, i.e., what you see is what you get. Reducing the current precision does not round values already on the stack, but those values will be rounded down before being used in any calculation. The `c 0' through `c 9' commands (*note Conversions::) can be used to round an existing value to a new precision. It is important to distinguish the concepts of "precision" and "accuracy". In the normal usage of these words, the number 123.4567 has a precision of 7 digits but an accuracy of 4 digits. The precision is the total number of digits not counting leading or trailing zeros (regardless of the position of the decimal point). The accuracy is simply the number of digits after the decimal point (again not counting trailing zeros). In Calc you control the precision, not the accuracy of computations. If you were to set the accuracy instead, then calculations like `exp(100)' would generate many more digits than you would typically need, while `exp(-100)' would probably round to zero! In Calc, both these computations give you exactly 12 (or the requested number of) significant digits. The only Calc features that deal with accuracy instead of precision are fixed-point display mode for floats (`d f'; *note Float Formats::), and the rounding functions like `floor' and `round' (*note Integer Truncation::). Also, `c 0' through `c 9' deal with both precision and accuracy depending on the magnitudes of the numbers involved. If you need to work with a particular fixed accuracy (say, dollars and cents with two digits after the decimal point), one solution is to work with integers and an "implied" decimal point. For example, $8.99 divided by 6 would be entered `899 6 /', yielding 149.833 (actually $1.49833 with our implied decimal point); pressing `R' would round this to 150 cents, i.e., $1.50. *Note Floats::, for still more on floating-point precision and related issues.  File: calc, Node: Inverse and Hyperbolic, Next: Calculation Modes, Prev: Precision, Up: Mode Settings 8.3 Inverse and Hyperbolic Flags ================================ There is no single-key equivalent to the `calc-arcsin' function. Instead, you must first press `I' (`calc-inverse') to set the "Inverse Flag", then press `S' (`calc-sin'). The `I' key actually toggles the Inverse Flag. When this flag is set, the word `Inv' appears in the mode line. Likewise, the `H' key (`calc-hyperbolic') sets or clears the Hyperbolic Flag, which transforms `calc-sin' into `calc-sinh'. If both of these flags are set at once, the effect will be `calc-arcsinh'. (The Hyperbolic flag is also used by some non-trigonometric commands; for example `H L' computes a base-10, instead of base-e, logarithm.) Command names like `calc-arcsin' are provided for completeness, and may be executed with `x' or `M-x'. Their effect is simply to toggle the Inverse and/or Hyperbolic flags and then execute the corresponding base command (`calc-sin' in this case). The Inverse and Hyperbolic flags apply only to the next Calculator command, after which they are automatically cleared. (They are also cleared if the next keystroke is not a Calc command.) Digits you type after `I' or `H' (or `K') are treated as prefix arguments for the next command, not as numeric entries. The same is true of `C-u', but not of the minus sign (`K -' means to subtract and keep arguments). The third Calc prefix flag, `K' (keep-arguments), is discussed elsewhere. *Note Keep Arguments::.  File: calc, Node: Calculation Modes, Next: Simplification Modes, Prev: Inverse and Hyperbolic, Up: Mode Settings 8.4 Calculation Modes ===================== The commands in this section are two-key sequences beginning with the `m' prefix. (That's the letter `m', not the key.) The `m a' (`calc-algebraic-mode') command is described elsewhere (*note Algebraic Entry::). * Menu: * Angular Modes:: * Polar Mode:: * Fraction Mode:: * Infinite Mode:: * Symbolic Mode:: * Matrix Mode:: * Automatic Recomputation:: * Working Message::  File: calc, Node: Angular Modes, Next: Polar Mode, Prev: Calculation Modes, Up: Calculation Modes 8.4.1 Angular Modes ------------------- The Calculator supports three notations for angles: radians, degrees, and degrees-minutes-seconds. When a number is presented to a function like `sin' that requires an angle, the current angular mode is used to interpret the number as either radians or degrees. If an HMS form is presented to `sin', it is always interpreted as degrees-minutes-seconds. Functions that compute angles produce a number in radians, a number in degrees, or an HMS form depending on the current angular mode. If the result is a complex number and the current mode is HMS, the number is instead expressed in degrees. (Complex-number calculations would normally be done in Radians mode, though. Complex numbers are converted to degrees by calculating the complex result in radians and then multiplying by 180 over `pi'.) The `m r' (`calc-radians-mode'), `m d' (`calc-degrees-mode'), and `m h' (`calc-hms-mode') commands control the angular mode. The current angular mode is displayed on the Emacs mode line. The default angular mode is Degrees.  File: calc, Node: Polar Mode, Next: Fraction Mode, Prev: Angular Modes, Up: Calculation Modes 8.4.2 Polar Mode ---------------- The Calculator normally "prefers" rectangular complex numbers in the sense that rectangular form is used when the proper form can not be decided from the input. This might happen by multiplying a rectangular number by a polar one, by taking the square root of a negative real number, or by entering `( 2 3 )'. The `m p' (`calc-polar-mode') command toggles complex-number preference between rectangular and polar forms. In Polar mode, all of the above example situations would produce polar complex numbers.  File: calc, Node: Fraction Mode, Next: Infinite Mode, Prev: Polar Mode, Up: Calculation Modes 8.4.3 Fraction Mode ------------------- Division of two integers normally yields a floating-point number if the result cannot be expressed as an integer. In some cases you would rather get an exact fractional answer. One way to accomplish this is to use the `:' (`calc-fdiv') [`fdiv'] command, which divides the two integers on the top of the stack to produce a fraction: `6 4 :' produces `3:2' even though `6 4 /' produces `1.5'. To set the Calculator to produce fractional results for normal integer divisions, use the `m f' (`calc-frac-mode') command. For example, `8/4' produces `2' in either mode, but `6/4' produces `3:2' in Fraction mode, `1.5' in Float mode. At any time you can use `c f' (`calc-float') to convert a fraction to a float, or `c F' (`calc-fraction') to convert a float to a fraction. *Note Conversions::.  File: calc, Node: Infinite Mode, Next: Symbolic Mode, Prev: Fraction Mode, Up: Calculation Modes 8.4.4 Infinite Mode ------------------- The Calculator normally treats results like `1 / 0' as errors; formulas like this are left in unsimplified form. But Calc can be put into a mode where such calculations instead produce "infinite" results. The `m i' (`calc-infinite-mode') command turns this mode on and off. When the mode is off, infinities do not arise except in calculations that already had infinities as inputs. (One exception is that infinite open intervals like `[0 .. inf)' can be generated; however, intervals closed at infinity (`[0 .. inf]') will not be generated when Infinite mode is off.) With Infinite mode turned on, `1 / 0' will generate `uinf', an undirected infinity. *Note Infinities::, for a discussion of the difference between `inf' and `uinf'. Also, `0 / 0' evaluates to `nan', the "indeterminate" symbol. Various other functions can also return infinities in this mode; for example, `ln(0) = -inf', and `gamma(-7) = uinf'. Once again, note that `exp(inf) = inf' regardless of Infinite mode because this calculation has infinity as an input. The `m i' command with a numeric prefix argument of zero, i.e., `C-u 0 m i', turns on a Positive Infinite mode in which zero is treated as positive instead of being directionless. Thus, `1 / 0 = inf' and `-1 / 0 = -inf' in this mode. Note that zero never actually has a sign in Calc; there are no separate representations for +0 and -0. Positive Infinite mode merely changes the interpretation given to the single symbol, `0'. One consequence of this is that, while you might expect `1 / -0 = -inf', actually `1 / -0' is equivalent to `1 / 0', which is equal to positive `inf'.  File: calc, Node: Symbolic Mode, Next: Matrix Mode, Prev: Infinite Mode, Up: Calculation Modes 8.4.5 Symbolic Mode ------------------- Calculations are normally performed numerically wherever possible. For example, the `calc-sqrt' command, or `sqrt' function in an algebraic expression, produces a numeric answer if the argument is a number or a symbolic expression if the argument is an expression: `2 Q' pushes 1.4142 but `<'> x+1 Q' pushes `sqrt(x+1)'. In "Symbolic mode", controlled by the `m s' (`calc-symbolic-mode') command, functions which would produce inexact, irrational results are left in symbolic form. Thus `16 Q' pushes 4, but `2 Q' pushes `sqrt(2)'. The shift-`N' (`calc-eval-num') command evaluates numerically the expression at the top of the stack, by temporarily disabling `calc-symbolic-mode' and executing `=' (`calc-evaluate'). Given a numeric prefix argument, it also sets the floating-point precision to the specified value for the duration of the command. To evaluate a formula numerically without expanding the variables it contains, you can use the key sequence `m s a v m s' (this uses `calc-alg-evaluate', which resimplifies but doesn't evaluate variables.)  File: calc, Node: Matrix Mode, Next: Automatic Recomputation, Prev: Symbolic Mode, Up: Calculation Modes 8.4.6 Matrix and Scalar Modes ----------------------------- Calc sometimes makes assumptions during algebraic manipulation that are awkward or incorrect when vectors and matrices are involved. Calc has two modes, "Matrix mode" and "Scalar mode", which modify its behavior around vectors in useful ways. Press `m v' (`calc-matrix-mode') once to enter Matrix mode. In this mode, all objects are assumed to be matrices unless provably otherwise. One major effect is that Calc will no longer consider multiplication to be commutative. (Recall that in matrix arithmetic, `A*B' is not the same as `B*A'.) This assumption affects rewrite rules and algebraic simplification. Another effect of this mode is that calculations that would normally produce constants like 0 and 1 (e.g., `a - a' and `a / a', respectively) will now produce function calls that represent "generic" zero or identity matrices: `idn(0)', `idn(1)'. The `idn' function `idn(A,N)' returns A times an NxN identity matrix; if N is omitted, it doesn't know what dimension to use and so the `idn' call remains in symbolic form. However, if this generic identity matrix is later combined with a matrix whose size is known, it will be converted into a true identity matrix of the appropriate size. On the other hand, if it is combined with a scalar (as in `idn(1) + 2'), Calc will assume it really was a scalar after all and produce, e.g., 3. Press `m v' a second time to get Scalar mode. Here, objects are assumed _not_ to be vectors or matrices unless provably so. For example, normally adding a variable to a vector, as in `[x, y, z] + a', will leave the sum in symbolic form because as far as Calc knows, `a' could represent either a number or another 3-vector. In Scalar mode, `a' is assumed to be a non-vector, and the addition is evaluated to `[x+a, y+a, z+a]'. Press `m v' a third time to return to the normal mode of operation. If you press `m v' with a numeric prefix argument N, you get a special "dimensioned" Matrix mode in which matrices of unknown size are assumed to be NxN square matrices. Then, the function call `idn(1)' will expand into an actual matrix rather than representing a "generic" matrix. Simply typing `C-u m v' will get you a square Matrix mode, in which matrices of unknown size are assumed to be square matrices of unspecified size. Of course these modes are approximations to the true state of affairs, which is probably that some quantities will be matrices and others will be scalars. One solution is to "declare" certain variables or functions to be scalar-valued. *Note Declarations::, to see how to make declarations in Calc. There is nothing stopping you from declaring a variable to be scalar and then storing a matrix in it; however, if you do, the results you get from Calc may not be valid. Suppose you let Calc get the result `[x+a, y+a, z+a]' shown above, and then stored `[1, 2, 3]' in `a'. The result would not be the same as for `[x, y, z] + [1, 2, 3]', but that's because you have broken your earlier promise to Calc that `a' would be scalar. Another way to mix scalars and matrices is to use selections (*note Selecting Subformulas::). Use Matrix mode when operating on your formula normally; then, to apply Scalar mode to a certain part of the formula without affecting the rest just select that part, change into Scalar mode and press `=' to resimplify the part under this mode, then change back to Matrix mode before deselecting.  File: calc, Node: Automatic Recomputation, Next: Working Message, Prev: Matrix Mode, Up: Calculation Modes 8.4.7 Automatic Recomputation ----------------------------- The "evaluates-to" operator, `=>', has the special property that any `=>' formulas on the stack are recomputed whenever variable values or mode settings that might affect them are changed. *Note Evaluates-To Operator::. The `m C' (`calc-auto-recompute') command turns this automatic recomputation on and off. If you turn it off, Calc will not update `=>' operators on the stack (nor those in the attached Embedded mode buffer, if there is one). They will not be updated unless you explicitly do so by pressing `=' or until you press `m C' to turn recomputation back on. (While automatic recomputation is off, you can think of `m C m C' as a command to update all `=>' operators while leaving recomputation off.) To update `=>' operators in an Embedded buffer while automatic recomputation is off, use `C-x * u'. *Note Embedded Mode::.  File: calc, Node: Working Message, Prev: Automatic Recomputation, Up: Calculation Modes 8.4.8 Working Messages ---------------------- Since the Calculator is written entirely in Emacs Lisp, which is not designed for heavy numerical work, many operations are quite slow. The Calculator normally displays the message `Working...' in the echo area during any command that may be slow. In addition, iterative operations such as square roots and trigonometric functions display the intermediate result at each step. Both of these types of messages can be disabled if you find them distracting. Type `m w' (`calc-working') with a numeric prefix of 0 to disable all "working" messages. Use a numeric prefix of 1 to enable only the plain `Working...' message. Use a numeric prefix of 2 to see intermediate results as well. With no numeric prefix this displays the current mode. While it may seem that the "working" messages will slow Calc down considerably, experiments have shown that their impact is actually quite small. But if your terminal is slow you may find that it helps to turn the messages off.  File: calc, Node: Simplification Modes, Next: Declarations, Prev: Calculation Modes, Up: Mode Settings 8.5 Simplification Modes ======================== The current "simplification mode" controls how numbers and formulas are "normalized" when being taken from or pushed onto the stack. Some normalizations are unavoidable, such as rounding floating-point results to the current precision, and reducing fractions to simplest form. Others, such as simplifying a formula like `a+a' (or `2+3'), are done by default but can be turned off when necessary. When you press a key like `+' when `2' and `3' are on the stack, Calc pops these numbers, normalizes them, creates the formula `2+3', normalizes it, and pushes the result. Of course the standard rules for normalizing `2+3' will produce the result `5'. Simplification mode commands consist of the lower-case `m' prefix key followed by a shifted letter. The `m O' (`calc-no-simplify-mode') command turns off all optional simplifications. These would leave a formula like `2+3' alone. In fact, nothing except simple numbers are ever affected by normalization in this mode. The `m N' (`calc-num-simplify-mode') command turns off simplification of any formulas except those for which all arguments are constants. For example, `1+2' is simplified to `3', and `a+(2-2)' is simplified to `a+0' but no further, since one argument of the sum is not a constant. Unfortunately, `(a+2)-2' is _not_ simplified because the top-level `-' operator's arguments are not both constant numbers (one of them is the formula `a+2'). A constant is a number or other numeric object (such as a constant error form or modulo form), or a vector all of whose elements are constant. The `m D' (`calc-default-simplify-mode') command restores the default simplifications for all formulas. This includes many easy and fast algebraic simplifications such as `a+0' to `a', and `a + 2 a' to `3 a', as well as evaluating functions like `deriv(x^2, x)' to `2 x'. The `m B' (`calc-bin-simplify-mode') mode applies the default simplifications to a result and then, if the result is an integer, uses the `b c' (`calc-clip') command to clip the integer according to the current binary word size. *Note Binary Functions::. Real numbers are rounded to the nearest integer and then clipped; other kinds of results (after the default simplifications) are left alone. The `m A' (`calc-alg-simplify-mode') mode does algebraic simplification; it applies all the default simplifications, and also the more powerful (and slower) simplifications made by `a s' (`calc-simplify'). *Note Algebraic Simplifications::. The `m E' (`calc-ext-simplify-mode') mode does "extended" algebraic simplification, as by the `a e' (`calc-simplify-extended') command. *Note Unsafe Simplifications::. The `m U' (`calc-units-simplify-mode') mode does units simplification; it applies the command `u s' (`calc-simplify-units'), which in turn is a superset of `a s'. In this mode, variable names which are identifiable as unit names (like `mm' for "millimeters") are simplified with their unit definitions in mind. A common technique is to set the simplification mode down to the lowest amount of simplification you will allow to be applied automatically, then use manual commands like `a s' and `c c' (`calc-clean') to perform higher types of simplifications on demand. *Note Algebraic Definitions::, for another sample use of No-Simplification mode.  File: calc, Node: Declarations, Next: Display Modes, Prev: Simplification Modes, Up: Mode Settings 8.6 Declarations ================ A "declaration" is a statement you make that promises you will use a certain variable or function in a restricted way. This may give Calc the freedom to do things that it couldn't do if it had to take the fully general situation into account. * Menu: * Declaration Basics:: * Kinds of Declarations:: * Functions for Declarations::  File: calc, Node: Declaration Basics, Next: Kinds of Declarations, Prev: Declarations, Up: Declarations 8.6.1 Declaration Basics ------------------------ The `s d' (`calc-declare-variable') command is the easiest way to make a declaration for a variable. This command prompts for the variable name, then prompts for the declaration. The default at the declaration prompt is the previous declaration, if any. You can edit this declaration, or press `C-k' to erase it and type a new declaration. (Or, erase it and press to clear the declaration, effectively "undeclaring" the variable.) A declaration is in general a vector of "type symbols" and "range" values. If there is only one type symbol or range value, you can write it directly rather than enclosing it in a vector. For example, `s d foo real ' declares `foo' to be a real number, and `s d bar [int, const, [1..6]] ' declares `bar' to be a constant integer between 1 and 6. (Actually, you can omit the outermost brackets and Calc will provide them for you: `s d bar int, const, [1..6] '.) Declarations in Calc are kept in a special variable called `Decls'. This variable encodes the set of all outstanding declarations in the form of a matrix. Each row has two elements: A variable or vector of variables declared by that row, and the declaration specifier as described above. You can use the `s D' command to edit this variable if you wish to see all the declarations at once. *Note Operations on Variables::, for a description of this command and the `s p' command that allows you to save your declarations permanently if you wish. Items being declared can also be function calls. The arguments in the call are ignored; the effect is to say that this function returns values of the declared type for any valid arguments. The `s d' command declares only variables, so if you wish to make a function declaration you will have to edit the `Decls' matrix yourself. For example, the declaration matrix [ [ foo, real ] [ [j, k, n], int ] [ f(1,2,3), [0 .. inf) ] ] declares that `foo' represents a real number, `j', `k' and `n' represent integers, and the function `f' always returns a real number in the interval shown. If there is a declaration for the variable `All', then that declaration applies to all variables that are not otherwise declared. It does not apply to function names. For example, using the row `[All, real]' says that all your variables are real unless they are explicitly declared without `real' in some other row. The `s d' command declares `All' if you give a blank response to the variable-name prompt.  File: calc, Node: Kinds of Declarations, Next: Functions for Declarations, Prev: Declaration Basics, Up: Declarations 8.6.2 Kinds of Declarations --------------------------- The type-specifier part of a declaration (that is, the second prompt in the `s d' command) can be a type symbol, an interval, or a vector consisting of zero or more type symbols followed by zero or more intervals or numbers that represent the set of possible values for the variable. [ [ a, [1, 2, 3, 4, 5] ] [ b, [1 .. 5] ] [ c, [int, 1 .. 5] ] ] Here `a' is declared to contain one of the five integers shown; `b' is any number in the interval from 1 to 5 (any real number since we haven't specified), and `c' is any integer in that interval. Thus the declarations for `a' and `c' are nearly equivalent (see below). The type-specifier can be the empty vector `[]' to say that nothing is known about a given variable's value. This is the same as not declaring the variable at all except that it overrides any `All' declaration which would otherwise apply. The initial value of `Decls' is the empty vector `[]'. If `Decls' has no stored value or if the value stored in it is not valid, it is ignored and there are no declarations as far as Calc is concerned. (The `s d' command will replace such a malformed value with a fresh empty matrix, `[]', before recording the new declaration.) Unrecognized type symbols are ignored. The following type symbols describe what sorts of numbers will be stored in a variable: `int' Integers. `numint' Numerical integers. (Integers or integer-valued floats.) `frac' Fractions. (Rational numbers which are not integers.) `rat' Rational numbers. (Either integers or fractions.) `float' Floating-point numbers. `real' Real numbers. (Integers, fractions, or floats. Actually, intervals and error forms with real components also count as reals here.) `pos' Positive real numbers. (Strictly greater than zero.) `nonneg' Nonnegative real numbers. (Greater than or equal to zero.) `number' Numbers. (Real or complex.) Calc uses this information to determine when certain simplifications of formulas are safe. For example, `(x^y)^z' cannot be simplified to `x^(y z)' in general; for example, `((-3)^2)^1:2' is 3, but `(-3)^(2*1:2) = (-3)^1' is -3. However, this simplification _is_ safe if `z' is known to be an integer, or if `x' is known to be a nonnegative real number. If you have given declarations that allow Calc to deduce either of these facts, Calc will perform this simplification of the formula. Calc can apply a certain amount of logic when using declarations. For example, `(x^y)^(2n+1)' will be simplified if `n' has been declared `int'; Calc knows that an integer times an integer, plus an integer, must always be an integer. (In fact, Calc would simplify `(-x)^(2n+1)' to `-(x^(2n+1))' since it is able to determine that `2n+1' must be an odd integer.) Similarly, `(abs(x)^y)^z' will be simplified to `abs(x)^(y z)' because Calc knows that the `abs' function always returns a nonnegative real. If you had a `myabs' function that also had this property, you could get Calc to recognize it by adding the row `[myabs(), nonneg]' to the `Decls' matrix. One instance of this simplification is `sqrt(x^2)' (since the `sqrt' function is effectively a one-half power). Normally Calc leaves this formula alone. After the command `s d x real ', however, it can simplify the formula to `abs(x)'. And after `s d x nonneg ', Calc can simplify this formula all the way to `x'. If there are any intervals or real numbers in the type specifier, they comprise the set of possible values that the variable or function being declared can have. In particular, the type symbol `real' is effectively the same as the range `[-inf .. inf]' (note that infinity is included in the range of possible values); `pos' is the same as `(0 .. inf]', and `nonneg' is the same as `[0 .. inf]'. Saying `[real, [-5 .. 5]]' is redundant because the fact that the variable is real can be deduced just from the interval, but `[int, [-5 .. 5]]' and `[rat, [-5 .. 5]]' are useful combinations. Note that the vector of intervals or numbers is in the same format used by Calc's set-manipulation commands. *Note Set Operations::. The type specifier `[1, 2, 3]' is equivalent to `[numint, 1, 2, 3]', _not_ to `[int, 1, 2, 3]'. In other words, the range of possible values means only that the variable's value must be numerically equal to a number in that range, but not that it must be equal in type as well. Calc's set operations act the same way; `in(2, [1., 2., 3.])' and `in(1.5, [1:2, 3:2, 5:2])' both report "true." If you use a conflicting combination of type specifiers, the results are unpredictable. An example is `[pos, [0 .. 5]]', where the interval does not lie in the range described by the type symbol. "Real" declarations mostly affect simplifications involving powers like the one described above. Another case where they are used is in the `a P' command which returns a list of all roots of a polynomial; if the variable has been declared real, only the real roots (if any) will be included in the list. "Integer" declarations are used for simplifications which are valid only when certain values are integers (such as `(x^y)^z' shown above). Another command that makes use of declarations is `a s', when simplifying equations and inequalities. It will cancel `x' from both sides of `a x = b x' only if it is sure `x' is non-zero, say, because it has a `pos' declaration. To declare specifically that `x' is real and non-zero, use `[[-inf .. 0), (0 .. inf]]'. (There is no way in the current notation to say that `x' is nonzero but not necessarily real.) The `a e' command does "unsafe" simplifications, including cancelling `x' from the equation when `x' is not known to be nonzero. Another set of type symbols distinguish between scalars and vectors. `scalar' The value is not a vector. `vector' The value is a vector. `matrix' The value is a matrix (a rectangular vector of vectors). `sqmatrix' The value is a square matrix. These type symbols can be combined with the other type symbols described above; `[int, matrix]' describes an object which is a matrix of integers. Scalar/vector declarations are used to determine whether certain algebraic operations are safe. For example, `[a, b, c] + x' is normally not simplified to `[a + x, b + x, c + x]', but it will be if `x' has been declared `scalar'. On the other hand, multiplication is usually assumed to be commutative, but the terms in `x y' will never be exchanged if both `x' and `y' are known to be vectors or matrices. (Calc currently never distinguishes between `vector' and `matrix' declarations.) *Note Matrix Mode::, for a discussion of Matrix mode and Scalar mode, which are similar to declaring `[All, matrix]' or `[All, scalar]' but much more convenient. One more type symbol that is recognized is used with the `H a d' command for taking total derivatives of a formula. *Note Calculus::. `const' The value is a constant with respect to other variables. Calc does not check the declarations for a variable when you store a value in it. However, storing -3.5 in a variable that has been declared `pos', `int', or `matrix' may have unexpected effects; Calc may evaluate `sqrt(x^2)' to `3.5' if it substitutes the value first, or to `-3.5' if `x' was declared `pos' and the formula `sqrt(x^2)' is simplified to `x' before the value is substituted. Before using a variable for a new purpose, it is best to use `s d' or `s D' to check to make sure you don't still have an old declaration for the variable that will conflict with its new meaning.  File: calc, Node: Functions for Declarations, Prev: Kinds of Declarations, Up: Declarations 8.6.3 Functions for Declarations -------------------------------- Calc has a set of functions for accessing the current declarations in a convenient manner. These functions return 1 if the argument can be shown to have the specified property, or 0 if the argument can be shown _not_ to have that property; otherwise they are left unevaluated. These functions are suitable for use with rewrite rules (*note Conditional Rewrite Rules::) or programming constructs (*note Conditionals in Macros::). They can be entered only using algebraic notation. *Note Logical Operations::, for functions that perform other tests not related to declarations. For example, `dint(17)' returns 1 because 17 is an integer, as do `dint(n)' and `dint(2 n - 3)' if `n' has been declared `int', but `dint(2.5)' and `dint(n + 0.5)' return 0. Calc consults knowledge of its own built-in functions as well as your own declarations: `dint(floor(x))' returns 1. The `dint' function checks if its argument is an integer. The `dnatnum' function checks if its argument is a natural number, i.e., a nonnegative integer. The `dnumint' function checks if its argument is numerically an integer, i.e., either an integer or an integer-valued float. Note that these and the other data type functions also accept vectors or matrices composed of suitable elements, and that real infinities `inf' and `-inf' are considered to be integers for the purposes of these functions. The `drat' function checks if its argument is rational, i.e., an integer or fraction. Infinities count as rational, but intervals and error forms do not. The `dreal' function checks if its argument is real. This includes integers, fractions, floats, real error forms, and intervals. The `dimag' function checks if its argument is imaginary, i.e., is mathematically equal to a real number times `i'. The `dpos' function checks for positive (but nonzero) reals. The `dneg' function checks for negative reals. The `dnonneg' function checks for nonnegative reals, i.e., reals greater than or equal to zero. Note that the `a s' command can simplify an expression like `x > 0' to 1 or 0 using `dpos', and that `a s' is effectively applied to all conditions in rewrite rules, so the actual functions `dpos', `dneg', and `dnonneg' are rarely necessary. The `dnonzero' function checks that its argument is nonzero. This includes all nonzero real or complex numbers, all intervals that do not include zero, all nonzero modulo forms, vectors all of whose elements are nonzero, and variables or formulas whose values can be deduced to be nonzero. It does not include error forms, since they represent values which could be anything including zero. (This is also the set of objects considered "true" in conditional contexts.) The `deven' function returns 1 if its argument is known to be an even integer (or integer-valued float); it returns 0 if its argument is known not to be even (because it is known to be odd or a non-integer). The `a s' command uses this to simplify a test of the form `x % 2 = 0'. There is also an analogous `dodd' function. The `drange' function returns a set (an interval or a vector of intervals and/or numbers; *note Set Operations::) that describes the set of possible values of its argument. If the argument is a variable or a function with a declaration, the range is copied from the declaration. Otherwise, the possible signs of the expression are determined using a method similar to `dpos', etc., and a suitable set like `[0 .. inf]' is returned. If the expression is not provably real, the `drange' function remains unevaluated. The `dscalar' function returns 1 if its argument is provably scalar, or 0 if its argument is provably non-scalar. It is left unevaluated if this cannot be determined. (If Matrix mode or Scalar mode is in effect, this function returns 1 or 0, respectively, if it has no other information.) When Calc interprets a condition (say, in a rewrite rule) it considers an unevaluated formula to be "false." Thus, `dscalar(a)' is "true" only if `a' is provably scalar, and `!dscalar(a)' is "true" only if `a' is provably non-scalar; both are "false" if there is insufficient information to tell.  File: calc, Node: Display Modes, Next: Language Modes, Prev: Declarations, Up: Mode Settings 8.7 Display Modes ================= The commands in this section are two-key sequences beginning with the `d' prefix. The `d l' (`calc-line-numbering') and `d b' (`calc-line-breaking') commands are described elsewhere; *note Stack Basics:: and *note Normal Language Modes::, respectively. Display formats for vectors and matrices are also covered elsewhere; *note Vector and Matrix Formats::. One thing all display modes have in common is their treatment of the `H' prefix. This prefix causes any mode command that would normally refresh the stack to leave the stack display alone. The word "Dirty" will appear in the mode line when Calc thinks the stack display may not reflect the latest mode settings. The `d ' (`calc-refresh-top') command reformats the top stack entry according to all the current modes. Positive prefix arguments reformat the top N entries; negative prefix arguments reformat the specified entry, and a prefix of zero is equivalent to `d ' (`calc-refresh'), which reformats the entire stack. For example, `H d s M-2 d ' changes to scientific notation but reformats only the top two stack entries in the new mode. The `I' prefix has another effect on the display modes. The mode is set only temporarily; the top stack entry is reformatted according to that mode, then the original mode setting is restored. In other words, `I d s' is equivalent to `H d s d H d (OLD MODE)'. * Menu: * Radix Modes:: * Grouping Digits:: * Float Formats:: * Complex Formats:: * Fraction Formats:: * HMS Formats:: * Date Formats:: * Truncating the Stack:: * Justification:: * Labels::  File: calc, Node: Radix Modes, Next: Grouping Digits, Prev: Display Modes, Up: Display Modes 8.7.1 Radix Modes ----------------- Calc normally displays numbers in decimal ("base-10" or "radix-10") notation. Calc can actually display in any radix from two (binary) to 36. When the radix is above 10, the letters `A' to `Z' are used as digits. When entering such a number, letter keys are interpreted as potential digits rather than terminating numeric entry mode. The key sequences `d 2', `d 8', `d 6', and `d 0' select binary, octal, hexadecimal, and decimal as the current display radix, respectively. Numbers can always be entered in any radix, though the current radix is used as a default if you press `#' without any initial digits. A number entered without a `#' is _always_ interpreted as decimal. To set the radix generally, use `d r' (`calc-radix') and enter an integer from 2 to 36. You can specify the radix as a numeric prefix argument; otherwise you will be prompted for it. Integers normally are displayed with however many digits are necessary to represent the integer and no more. The `d z' (`calc-leading-zeros') command causes integers to be padded out with leading zeros according to the current binary word size. (*Note Binary Functions::, for a discussion of word size.) If the absolute value of the word size is `w', all integers are displayed with at least enough digits to represent `(2^w)-1' in the current radix. (Larger integers will still be displayed in their entirety.) With the binary, octal and hexadecimal display modes, Calc can display `w'-bit integers using two's complement notation. This option is selected with the key sequences `C-u d 2', `C-u d 8' and `C-u d 6', respectively, and a negative word size might be appropriate (*note Binary Functions::). In two's complement notation, the integers in the (nearly) symmetric interval from `-2^(w-1)' to `2^(w-1)-1' are represented by the integers from `0' to `2^w-1': the integers from `0' to `2^(w-1)-1' are represented by themselves and the integers from `-2^(w-1)' to `-1' are represented by the integers from `2^(w-1)' to `2^w-1' (the integer `k' is represented by `k+2^w'). Calc will display a two's complement integer by the radix (either `2', `8' or `16'), two `#' symbols, and then its representation (including any leading zeros necessary to include all `w' bits). In a two's complement display mode, numbers that are not displayed in two's complement notation (i.e., that aren't integers from `-2^(w-1)' to `2^(w-1)-1') will be represented using Calc's usual notation (in the appropriate radix).  File: calc, Node: Grouping Digits, Next: Float Formats, Prev: Radix Modes, Up: Display Modes 8.7.2 Grouping Digits --------------------- Long numbers can be hard to read if they have too many digits. For example, the factorial of 30 is 33 digits long! Press `d g' (`calc-group-digits') to enable "Grouping" mode, in which digits are displayed in clumps of 3 or 4 (depending on the current radix) separated by commas. The `d g' command toggles grouping on and off. With a numeric prefix of 0, this command displays the current state of the grouping flag; with an argument of minus one it disables grouping; with a positive argument `N' it enables grouping on every `N' digits. For floating-point numbers, grouping normally occurs only before the decimal point. A negative prefix argument `-N' enables grouping every `N' digits both before and after the decimal point. The `d ,' (`calc-group-char') command allows you to choose any character as the grouping separator. The default is the comma character. If you find it difficult to read vectors of large integers grouped with commas, you may wish to use spaces or some other character instead. This command takes the next character you type, whatever it is, and uses it as the digit separator. As a special case, `d , \' selects `\,' (TeX's thin-space symbol) as the digit separator. Please note that grouped numbers will not generally be parsed correctly if re-read in textual form, say by the use of `C-x * y' and `C-x * g'. (*Note Kill and Yank::, for details on these commands.) One exception is the `\,' separator, which doesn't interfere with parsing because it is ignored by TeX language mode.  File: calc, Node: Float Formats, Next: Complex Formats, Prev: Grouping Digits, Up: Display Modes 8.7.3 Float Formats ------------------- Floating-point quantities are normally displayed in standard decimal form, with scientific notation used if the exponent is especially high or low. All significant digits are normally displayed. The commands in this section allow you to choose among several alternative display formats for floats. The `d n' (`calc-normal-notation') command selects the normal display format. All significant figures in a number are displayed. With a positive numeric prefix, numbers are rounded if necessary to that number of significant digits. With a negative numerix prefix, the specified number of significant digits less than the current precision is used. (Thus `C-u -2 d n' displays 10 digits if the current precision is 12.) The `d f' (`calc-fix-notation') command selects fixed-point notation. The numeric argument is the number of digits after the decimal point, zero or more. This format will relax into scientific notation if a nonzero number would otherwise have been rounded all the way to zero. Specifying a negative number of digits is the same as for a positive number, except that small nonzero numbers will be rounded to zero rather than switching to scientific notation. The `d s' (`calc-sci-notation') command selects scientific notation. A positive argument sets the number of significant figures displayed, of which one will be before and the rest after the decimal point. A negative argument works the same as for `d n' format. The default is to display all significant digits. The `d e' (`calc-eng-notation') command selects engineering notation. This is similar to scientific notation except that the exponent is rounded down to a multiple of three, with from one to three digits before the decimal point. An optional numeric prefix sets the number of significant digits to display, as for `d s'. It is important to distinguish between the current _precision_ and the current _display format_. After the commands `C-u 10 p' and `C-u 6 d n' the Calculator computes all results to ten significant figures but displays only six. (In fact, intermediate calculations are often carried to one or two more significant figures, but values placed on the stack will be rounded down to ten figures.) Numbers are never actually rounded to the display precision for storage, except by commands like `C-k' and `C-x * y' which operate on the actual displayed text in the Calculator buffer. The `d .' (`calc-point-char') command selects the character used as a decimal point. Normally this is a period; users in some countries may wish to change this to a comma. Note that this is only a display style; on entry, periods must always be used to denote floating-point numbers, and commas to separate elements in a list.  File: calc, Node: Complex Formats, Next: Fraction Formats, Prev: Float Formats, Up: Display Modes 8.7.4 Complex Formats --------------------- There are three supported notations for complex numbers in rectangular form. The default is as a pair of real numbers enclosed in parentheses and separated by a comma: `(a,b)'. The `d c' (`calc-complex-notation') command selects this style. The other notations are `d i' (`calc-i-notation'), in which numbers are displayed in `a+bi' form, and `d j' (`calc-j-notation') which displays the form `a+bj' preferred in some disciplines. Complex numbers are normally entered in `(a,b)' format. If you enter `2+3i' as an algebraic formula, it will be stored as the formula `2 + 3 * i'. However, if you use `=' to evaluate this formula and you have not changed the variable `i', the `i' will be interpreted as `(0,1)' and the formula will be simplified to `(2,3)'. Other commands (like `calc-sin') will _not_ interpret the formula `2 + 3 * i' as a complex number. *Note Variables::, under "special constants."  File: calc, Node: Fraction Formats, Next: HMS Formats, Prev: Complex Formats, Up: Display Modes 8.7.5 Fraction Formats ---------------------- Display of fractional numbers is controlled by the `d o' (`calc-over-notation') command. By default, a number like eight thirds is displayed in the form `8:3'. The `d o' command prompts for a one- or two-character format. If you give one character, that character is used as the fraction separator. Common separators are `:' and `/'. (During input of numbers, the `:' key must be used regardless of the display format; in particular, the `/' is used for RPN-style division, _not_ for entering fractions.) If you give two characters, fractions use "integer-plus-fractional-part" notation. For example, the format `+/' would display eight thirds as `2+2/3'. If two colons are present in a number being entered, the number is interpreted in this form (so that the entries `2:2:3' and `8:3' are equivalent). It is also possible to follow the one- or two-character format with a number. For example: `:10' or `+/3'. In this case, Calc adjusts all fractions that are displayed to have the specified denominator, if possible. Otherwise it adjusts the denominator to be a multiple of the specified value. For example, in `:6' mode the fraction `1:6' will be unaffected, but `2:3' will be displayed as `4:6', `1:2' will be displayed as `3:6', and `1:8' will be displayed as `3:24'. Integers are also affected by this mode: 3 is displayed as `18:6'. Note that the format `:1' writes fractions the same as `:', but it writes integers as `n:1'. The fraction format does not affect the way fractions or integers are stored, only the way they appear on the screen. The fraction format never affects floats.  File: calc, Node: HMS Formats, Next: Date Formats, Prev: Fraction Formats, Up: Display Modes 8.7.6 HMS Formats ----------------- The `d h' (`calc-hms-notation') command controls the display of HMS (hours-minutes-seconds) forms. It prompts for a string which consists basically of an "hours" marker, optional punctuation, a "minutes" marker, more optional punctuation, and a "seconds" marker. Punctuation is zero or more spaces, commas, or semicolons. The hours marker is one or more non-punctuation characters. The minutes and seconds markers must be single non-punctuation characters. The default HMS format is `@ ' "', producing HMS values of the form `23@ 30' 15.75"'. The format `deg, ms' would display this same value as `23deg, 30m15.75s'. During numeric entry, the `h' or `o' keys are recognized as synonyms for `@' regardless of display format. The `m' and `s' keys are recognized as synonyms for `'' and `"', respectively, but only if an `@' (or `h' or `o') has already been typed; otherwise, they have their usual meanings (`m-' prefix and `s-' prefix). Thus, `5 "', `0 @ 5 "', and `0 h 5 s' are some of the ways to enter the quantity "five seconds." The `'' key is recognized as "minutes" only if `@' (or `h' or `o') has already been pressed; otherwise it means to switch to algebraic entry.  File: calc, Node: Date Formats, Next: Truncating the Stack, Prev: HMS Formats, Up: Display Modes 8.7.7 Date Formats ------------------ The `d d' (`calc-date-notation') command controls the display of date forms (*note Date Forms::). It prompts for a string which contains letters that represent the various parts of a date and time. To show which parts should be omitted when the form represents a pure date with no time, parts of the string can be enclosed in `< >' marks. If you don't include `< >' markers in the format, Calc guesses at which parts, if any, should be omitted when formatting pure dates. The default format is: `Www Mmm D, YYYY'. An example string in this format is `3:32pm Wed Jan 9, 1991'. If you enter a blank format string, this default format is reestablished. Calc uses `< >' notation for nameless functions as well as for dates. *Note Specifying Operators::. To avoid confusion with nameless functions, your date formats should avoid using the `#' character. * Menu: * Date Formatting Codes:: * Free-Form Dates:: * Standard Date Formats::  File: calc, Node: Date Formatting Codes, Next: Free-Form Dates, Prev: Date Formats, Up: Date Formats 8.7.7.1 Date Formatting Codes ............................. When displaying a date, the current date format is used. All characters except for letters and `<' and `>' are copied literally when dates are formatted. The portion between `< >' markers is omitted for pure dates, or included for date/time forms. Letters are interpreted according to the table below. When dates are read in during algebraic entry, Calc first tries to match the input string to the current format either with or without the time part. The punctuation characters (including spaces) must match exactly; letter fields must correspond to suitable text in the input. If this doesn't work, Calc checks if the input is a simple number; if so, the number is interpreted as a number of days since Jan 1, 1 AD. Otherwise, Calc tries a much more relaxed and flexible algorithm which is described in the next section. Weekday names are ignored during reading. Two-digit year numbers are interpreted as lying in the range from 1941 to 2039. Years outside that range are always entered and displayed in full. Year numbers with a leading `+' sign are always interpreted exactly, allowing the entry and display of the years 1 through 99 AD. Here is a complete list of the formatting codes for dates: Y Year: "91" for 1991, "7" for 2007, "+23" for 23 AD. YY Year: "91" for 1991, "07" for 2007, "+23" for 23 AD. BY Year: "91" for 1991, " 7" for 2007, "+23" for 23 AD. YYY Year: "1991" for 1991, "23" for 23 AD. YYYY Year: "1991" for 1991, "+23" for 23 AD. aa Year: "ad" or blank. AA Year: "AD" or blank. aaa Year: "ad " or blank. (Note trailing space.) AAA Year: "AD " or blank. aaaa Year: "a.d." or blank. AAAA Year: "A.D." or blank. bb Year: "bc" or blank. BB Year: "BC" or blank. bbb Year: " bc" or blank. (Note leading space.) BBB Year: " BC" or blank. bbbb Year: "b.c." or blank. BBBB Year: "B.C." or blank. M Month: "8" for August. MM Month: "08" for August. BM Month: " 8" for August. MMM Month: "AUG" for August. Mmm Month: "Aug" for August. mmm Month: "aug" for August. MMMM Month: "AUGUST" for August. Mmmm Month: "August" for August. D Day: "7" for 7th day of month. DD Day: "07" for 7th day of month. BD Day: " 7" for 7th day of month. W Weekday: "0" for Sunday, "6" for Saturday. WWW Weekday: "SUN" for Sunday. Www Weekday: "Sun" for Sunday. www Weekday: "sun" for Sunday. WWWW Weekday: "SUNDAY" for Sunday. Wwww Weekday: "Sunday" for Sunday. d Day of year: "34" for Feb. 3. ddd Day of year: "034" for Feb. 3. bdd Day of year: " 34" for Feb. 3. h Hour: "5" for 5 AM; "17" for 5 PM. hh Hour: "05" for 5 AM; "17" for 5 PM. bh Hour: " 5" for 5 AM; "17" for 5 PM. H Hour: "5" for 5 AM and 5 PM. HH Hour: "05" for 5 AM and 5 PM. BH Hour: " 5" for 5 AM and 5 PM. p AM/PM: "a" or "p". P AM/PM: "A" or "P". pp AM/PM: "am" or "pm". PP AM/PM: "AM" or "PM". pppp AM/PM: "a.m." or "p.m.". PPPP AM/PM: "A.M." or "P.M.". m Minutes: "7" for 7. mm Minutes: "07" for 7. bm Minutes: " 7" for 7. s Seconds: "7" for 7; "7.23" for 7.23. ss Seconds: "07" for 7; "07.23" for 7.23. bs Seconds: " 7" for 7; " 7.23" for 7.23. SS Optional seconds: "07" for 7; blank for 0. BS Optional seconds: " 7" for 7; blank for 0. N Numeric date/time: "726842.25" for 6:00am Wed Jan 9, 1991. n Numeric date: "726842" for any time on Wed Jan 9, 1991. J Julian date/time: "2448265.75" for 6:00am Wed Jan 9, 1991. j Julian date: "2448266" for any time on Wed Jan 9, 1991. U Unix time: "663400800" for 6:00am Wed Jan 9, 1991. X Brackets suppression. An "X" at the front of the format causes the surrounding `< >' delimiters to be omitted when formatting dates. Note that the brackets are still required for algebraic entry. If "SS" or "BS" (optional seconds) is preceded by a colon, the colon is also omitted if the seconds part is zero. If "bb," "bbb" or "bbbb" or their upper-case equivalents appear in the format, then negative year numbers are displayed without a minus sign. Note that "aa" and "bb" are mutually exclusive. Some typical usages would be `YYYY AABB'; `AAAYYYYBBB'; `YYYYBBB'. The formats "YY," "YYYY," "MM," "DD," "ddd," "hh," "HH," "mm," "ss," and "SS" actually match any number of digits during reading unless several of these codes are strung together with no punctuation in between, in which case the input must have exactly as many digits as there are letters in the format. The "j," "J," and "U" formats do not make any time zone adjustment. They effectively use `julian(x,0)' and `unixtime(x,0)' to make the conversion; *note Date Arithmetic::.  File: calc, Node: Free-Form Dates, Next: Standard Date Formats, Prev: Date Formatting Codes, Up: Date Formats 8.7.7.2 Free-Form Dates ....................... When reading a date form during algebraic entry, Calc falls back on the algorithm described here if the input does not exactly match the current date format. This algorithm generally "does the right thing" and you don't have to worry about it, but it is described here in full detail for the curious. Calc does not distinguish between upper- and lower-case letters while interpreting dates. First, the time portion, if present, is located somewhere in the text and then removed. The remaining text is then interpreted as the date. A time is of the form `hh:mm:ss', possibly with the seconds part omitted and possibly with an AM/PM indicator added to indicate 12-hour time. If the AM/PM is present, the minutes may also be omitted. The AM/PM part may be any of the words `am', `pm', `noon', or `midnight'; each of these may be abbreviated to one letter, and the alternate forms `a.m.', `p.m.', and `mid' are also understood. Obviously `noon' and `midnight' are allowed only on 12:00:00. The words `noon', `mid', and `midnight' are also recognized with no number attached. If there is no AM/PM indicator, the time is interpreted in 24-hour format. To read the date portion, all words and numbers are isolated from the string; other characters are ignored. All words must be either month names or day-of-week names (the latter of which are ignored). Names can be written in full or as three-letter abbreviations. Large numbers, or numbers with `+' or `-' signs, are interpreted as years. If one of the other numbers is greater than 12, then that must be the day and the remaining number in the input is therefore the month. Otherwise, Calc assumes the month, day and year are in the same order that they appear in the current date format. If the year is omitted, the current year is taken from the system clock. If there are too many or too few numbers, or any unrecognizable words, then the input is rejected. If there are any large numbers (of five digits or more) other than the year, they are ignored on the assumption that they are something like Julian dates that were included along with the traditional date components when the date was formatted. One of the words `ad', `a.d.', `bc', or `b.c.' may optionally be used; the latter two are equivalent to a minus sign on the year value. If you always enter a four-digit year, and use a name instead of a number for the month, there is no danger of ambiguity.  File: calc, Node: Standard Date Formats, Prev: Free-Form Dates, Up: Date Formats 8.7.7.3 Standard Date Formats ............................. There are actually ten standard date formats, numbered 0 through 9. Entering a blank line at the `d d' command's prompt gives you format number 1, Calc's usual format. You can enter any digit to select the other formats. To create your own standard date formats, give a numeric prefix argument from 0 to 9 to the `d d' command. The format you enter will be recorded as the new standard format of that number, as well as becoming the new current date format. You can save your formats permanently with the `m m' command (*note Mode Settings::). 0 `N' (Numerical format) 1 `Www Mmm D, YYYY' (American format) 2 `D Mmm YYYY<, h:mm:SS>' (European format) 3 `Www Mmm BD< hh:mm:ss> YYYY' (Unix written date format) 4 `M/D/Y< H:mm:SSpp>' (American slashed format) 5 `D.M.Y< h:mm:SS>' (European dotted format) 6 `M-D-Y< H:mm:SSpp>' (American dashed format) 7 `D-M-Y< h:mm:SS>' (European dashed format) 8 `j<, h:mm:ss>' (Julian day plus time) 9 `YYddd< hh:mm:ss>' (Year-day format)  File: calc, Node: Truncating the Stack, Next: Justification, Prev: Date Formats, Up: Display Modes 8.7.8 Truncating the Stack -------------------------- The `d t' (`calc-truncate-stack') command moves the `.' line that marks the top-of-stack up or down in the Calculator buffer. The number right above that line is considered to the be at the top of the stack. Any numbers below that line are "hidden" from all stack operations (although still visible to the user). This is similar to the Emacs "narrowing" feature, except that the values below the `.' are _visible_, just temporarily frozen. This feature allows you to keep several independent calculations running at once in different parts of the stack, or to apply a certain command to an element buried deep in the stack. Pressing `d t' by itself moves the `.' to the line the cursor is on. Thus, this line and all those below it become hidden. To un-hide these lines, move down to the end of the buffer and press `d t'. With a positive numeric prefix argument `n', `d t' hides the bottom `n' values in the buffer. With a negative argument, it hides all but the top `n' values. With an argument of zero, it hides zero values, i.e., moves the `.' all the way down to the bottom. The `d [' (`calc-truncate-up') and `d ]' (`calc-truncate-down') commands move the `.' up or down one line at a time (or several lines with a prefix argument).  File: calc, Node: Justification, Next: Labels, Prev: Truncating the Stack, Up: Display Modes 8.7.9 Justification ------------------- Values on the stack are normally left-justified in the window. You can control this arrangement by typing `d <' (`calc-left-justify'), `d >' (`calc-right-justify'), or `d =' (`calc-center-justify'). For example, in Right-Justification mode, stack entries are displayed flush-right against the right edge of the window. If you change the width of the Calculator window you may have to type `d ' (`calc-refresh') to re-align right-justified or centered text. Right-justification is especially useful together with fixed-point notation (see `d f'; `calc-fix-notation'). With these modes together, the decimal points on numbers will always line up. With a numeric prefix argument, the justification commands give you a little extra control over the display. The argument specifies the horizontal "origin" of a display line. It is also possible to specify a maximum line width using the `d b' command (*note Normal Language Modes::). For reference, the precise rules for formatting and breaking lines are given below. Notice that the interaction between origin and line width is slightly different in each justification mode. In Left-Justified mode, the line is indented by a number of spaces given by the origin (default zero). If the result is longer than the maximum line width, if given, or too wide to fit in the Calc window otherwise, then it is broken into lines which will fit; each broken line is indented to the origin. In Right-Justified mode, lines are shifted right so that the rightmost character is just before the origin, or just before the current window width if no origin was specified. If the line is too long for this, then it is broken; the current line width is used, if specified, or else the origin is used as a width if that is specified, or else the line is broken to fit in the window. In Centering mode, the origin is the column number of the center of each stack entry. If a line width is specified, lines will not be allowed to go past that width; Calc will either indent less or break the lines if necessary. If no origin is specified, half the line width or Calc window width is used. Note that, in each case, if line numbering is enabled the display is indented an additional four spaces to make room for the line number. The width of the line number is taken into account when positioning according to the current Calc window width, but not when positioning by explicit origins and widths. In the latter case, the display is formatted as specified, and then uniformly shifted over four spaces to fit the line numbers.  File: calc, Node: Labels, Prev: Justification, Up: Display Modes 8.7.10 Labels ------------- The `d {' (`calc-left-label') command prompts for a string, then displays that string to the left of every stack entry. If the entries are left-justified (*note Justification::), then they will appear immediately after the label (unless you specified an origin greater than the length of the label). If the entries are centered or right-justified, the label appears on the far left and does not affect the horizontal position of the stack entry. Give a blank string (with `d { ') to turn the label off. The `d }' (`calc-right-label') command similarly adds a label on the righthand side. It does not affect positioning of the stack entries unless they are right-justified. Also, if both a line width and an origin are given in Right-Justified mode, the stack entry is justified to the origin and the righthand label is justified to the line width. One application of labels would be to add equation numbers to formulas you are manipulating in Calc and then copying into a document (possibly using Embedded mode). The equations would typically be centered, and the equation numbers would be on the left or right as you prefer.  File: calc, Node: Language Modes, Next: Modes Variable, Prev: Display Modes, Up: Mode Settings 8.8 Language Modes ================== The commands in this section change Calc to use a different notation for entry and display of formulas, corresponding to the conventions of some other common language such as Pascal or LaTeX. Objects displayed on the stack or yanked from the Calculator to an editing buffer will be formatted in the current language; objects entered in algebraic entry or yanked from another buffer will be interpreted according to the current language. The current language has no effect on things written to or read from the trail buffer, nor does it affect numeric entry. Only algebraic entry is affected. You can make even algebraic entry ignore the current language and use the standard notation by giving a numeric prefix, e.g., `C-u ''. For example, suppose the formula `2*a[1] + atan(a[2])' occurs in a C program; elsewhere in the program you need the derivatives of this formula with respect to `a[1]' and `a[2]'. First, type `d C' to switch to C notation. Now use `C-u C-x * g' to grab the formula into the Calculator, `a d a[1] ' to differentiate with respect to the first variable, and `C-x * y' to yank the formula for the derivative back into your C program. Press `U' to undo the differentiation and repeat with `a d a[2] ' for the other derivative. Without being switched into C mode first, Calc would have misinterpreted the brackets in `a[1]' and `a[2]', would not have known that `atan' was equivalent to Calc's built-in `arctan' function, and would have written the formula back with notations (like implicit multiplication) which would not have been valid for a C program. As another example, suppose you are maintaining a C program and a LaTeX document, each of which needs a copy of the same formula. You can grab the formula from the program in C mode, switch to LaTeX mode, and yank the formula into the document in LaTeX math-mode format. Language modes are selected by typing the letter `d' followed by a shifted letter key. * Menu: * Normal Language Modes:: * C FORTRAN Pascal:: * TeX and LaTeX Language Modes:: * Eqn Language Mode:: * Yacas Language Mode:: * Maxima Language Mode:: * Giac Language Mode:: * Mathematica Language Mode:: * Maple Language Mode:: * Compositions:: * Syntax Tables::  File: calc, Node: Normal Language Modes, Next: C FORTRAN Pascal, Prev: Language Modes, Up: Language Modes 8.8.1 Normal Language Modes --------------------------- The `d N' (`calc-normal-language') command selects the usual notation for Calc formulas, as described in the rest of this manual. Matrices are displayed in a multi-line tabular format, but all other objects are written in linear form, as they would be typed from the keyboard. The `d O' (`calc-flat-language') command selects a language identical with the normal one, except that matrices are written in one-line form along with everything else. In some applications this form may be more suitable for yanking data into other buffers. Even in one-line mode, long formulas or vectors will still be split across multiple lines if they exceed the width of the Calculator window. The `d b' (`calc-line-breaking') command turns this line-breaking feature on and off. (It works independently of the current language.) If you give a numeric prefix argument of five or greater to the `d b' command, that argument will specify the line width used when breaking long lines. The `d B' (`calc-big-language') command selects a language which uses textual approximations to various mathematical notations, such as powers, quotients, and square roots: ____________ | a + 1 2 | ----- + c \| b in place of `sqrt((a+1)/b + c^2)'. Subscripts like `a_i' are displayed as actual subscripts in Big mode. Double subscripts, `a_i_j' (`subscr(subscr(a, i), j)') are displayed as `a' with subscripts separated by commas: `i, j'. They must still be entered in the usual underscore notation. One slight ambiguity of Big notation is that 3 - - 4 can represent either the negative rational number `-3:4', or the actual expression `-(3/4)'; but the latter formula would normally never be displayed because it would immediately be evaluated to `-3:4' or `-0.75', so this ambiguity is not a problem in typical use. Non-decimal numbers are displayed with subscripts. Thus there is no way to tell the difference between `16#C2' and `C2_16', though generally you will know which interpretation is correct. Logarithms `log(x,b)' and `log10(x)' also use subscripts in Big mode. In Big mode, stack entries often take up several lines. To aid readability, stack entries are separated by a blank line in this mode. You may find it useful to expand the Calc window's height using `C-x ^' (`enlarge-window') or to make the Calc window the only one on the screen with `C-x 1' (`delete-other-windows'). Long lines are currently not rearranged to fit the window width in Big mode, so you may need to use the `<' and `>' keys to scroll across a wide formula. For really big formulas, you may even need to use `{' and `}' to scroll up and down. The `d U' (`calc-unformatted-language') command altogether disables the use of operator notation in formulas. In this mode, the formula shown above would be displayed: sqrt(add(div(add(a, 1), b), pow(c, 2))) These four modes differ only in display format, not in the format expected for algebraic entry. The standard Calc operators work in all four modes, and unformatted notation works in any language mode (except that Mathematica mode expects square brackets instead of parentheses).  File: calc, Node: C FORTRAN Pascal, Next: TeX and LaTeX Language Modes, Prev: Normal Language Modes, Up: Language Modes 8.8.2 C, FORTRAN, and Pascal Modes ---------------------------------- The `d C' (`calc-c-language') command selects the conventions of the C language for display and entry of formulas. This differs from the normal language mode in a variety of (mostly minor) ways. In particular, C language operators and operator precedences are used in place of Calc's usual ones. For example, `a^b' means `xor(a,b)' in C mode; a value raised to a power is written as a function call, `pow(a,b)'. In C mode, vectors and matrices use curly braces instead of brackets. Octal and hexadecimal values are written with leading `0' or `0x' rather than using the `#' symbol. Array subscripting is translated into `subscr' calls, so that `a[i]' in C mode is the same as `a_i' in Normal mode. Assignments turn into the `assign' function, which Calc normally displays using the `:=' symbol. The variables `pi' and `e' would be displayed `pi' and `e' in Normal mode, but in C mode they are displayed as `M_PI' and `M_E', corresponding to the names of constants typically provided in the `' header. Functions whose names are different in C are translated automatically for entry and display purposes. For example, entering `asin(x)' will push the formula `arcsin(x)' onto the stack; this formula will be displayed as `asin(x)' as long as C mode is in effect. The `d P' (`calc-pascal-language') command selects Pascal conventions. Like C mode, Pascal mode interprets array brackets and uses a different table of operators. Hexadecimal numbers are entered and displayed with a preceding dollar sign. (Thus the regular meaning of `$2' during algebraic entry does not work in Pascal mode, though `$' (and `$$', etc.) not followed by digits works the same as always.) No special provisions are made for other non-decimal numbers, vectors, and so on, since there is no universally accepted standard way of handling these in Pascal. The `d F' (`calc-fortran-language') command selects FORTRAN conventions. Various function names are transformed into FORTRAN equivalents. Vectors are written as `/1, 2, 3/', and may be entered this way or using square brackets. Since FORTRAN uses round parentheses for both function calls and array subscripts, Calc displays both in the same way; `a(i)' is interpreted as a function call upon reading, and subscripts must be entered as `subscr(a, i)'. If the variable `a' has been declared to have type `vector' or `matrix', however, then `a(i)' will be parsed as a subscript. (*Note Declarations::.) Usually it doesn't matter, though; if you enter the subscript expression `a(i)' and Calc interprets it as a function call, you'll never know the difference unless you switch to another language mode or replace `a' with an actual vector (or unless `a' happens to be the name of a built-in function!). Underscores are allowed in variable and function names in all of these language modes. The underscore here is equivalent to the `#' in Normal mode, or to hyphens in the underlying Emacs Lisp variable names. FORTRAN and Pascal modes normally do not adjust the case of letters in formulas. Most built-in Calc names use lower-case letters. If you use a positive numeric prefix argument with `d P' or `d F', these modes will use upper-case letters exclusively for display, and will convert to lower-case on input. With a negative prefix, these modes convert to lower-case for display and input.  File: calc, Node: TeX and LaTeX Language Modes, Next: Eqn Language Mode, Prev: C FORTRAN Pascal, Up: Language Modes 8.8.3 TeX and LaTeX Language Modes ---------------------------------- The `d T' (`calc-tex-language') command selects the conventions of "math mode" in Donald Knuth's TeX typesetting language, and the `d L' (`calc-latex-language') command selects the conventions of "math mode" in LaTeX, a typesetting language that uses TeX as its formatting engine. Calc's LaTeX language mode can read any formula that the TeX language mode can, although LaTeX mode may display it differently. Formulas are entered and displayed in the appropriate notation; `sin(a/b)' will appear as `\sin\left( {a \over b} \right)' in TeX mode and `\sin\left(\frac{a}{b}\right)' in LaTeX mode. Math formulas are often enclosed by `$ $' signs in TeX and LaTeX; these should be omitted when interfacing with Calc. To Calc, the `$' sign has the same meaning it always does in algebraic formulas (a reference to an existing entry on the stack). Complex numbers are displayed as in `3 + 4i'. Fractions and quotients are written using `\over' in TeX mode (as in `{a \over b}') and `\frac' in LaTeX mode (as in `\frac{a}{b}'); binomial coefficients are written with `\choose' in TeX mode (as in `{a \choose b}') and `\binom' in LaTeX mode (as in `\binom{a}{b}'). Interval forms are written with `\ldots', and error forms are written with `\pm'. Absolute values are written as in `|x + 1|', and the floor and ceiling functions are written with `\lfloor', `\rfloor', etc. The words `\left' and `\right' are ignored when reading formulas in TeX and LaTeX modes. Both `inf' and `uinf' are written as `\infty'; when read, `\infty' always translates to `inf'. Function calls are written the usual way, with the function name followed by the arguments in parentheses. However, functions for which TeX and LaTeX have special names (like `\sin') will use curly braces instead of parentheses for very simple arguments. During input, curly braces and parentheses work equally well for grouping, but when the document is formatted the curly braces will be invisible. Thus the printed result is `sin 2x' but `sin(2 + x)'. Function and variable names not treated specially by TeX and LaTeX are simply written out as-is, which will cause them to come out in italic letters in the printed document. If you invoke `d T' or `d L' with a positive numeric prefix argument, names of more than one character will instead be enclosed in a protective commands that will prevent them from being typeset in the math italics; they will be written `\hbox{NAME}' in TeX mode and `\text{NAME}' in LaTeX mode. The `\hbox{ }' and `\text{ }' notations are ignored during reading. If you use a negative prefix argument, such function names are written `\NAME', and function names that begin with `\' during reading have the `\' removed. (Note that in this mode, long variable names are still written with `\hbox' or `\text'. However, you can always make an actual variable name like `\bar' in any TeX mode.) During reading, text of the form `\matrix{ ... }' is replaced by `[ ... ]'. The same also applies to `\pmatrix' and `\bmatrix'. In LaTeX mode this also applies to `\begin{matrix} ... \end{matrix}', `\begin{bmatrix} ... \end{bmatrix}', `\begin{pmatrix} ... \end{pmatrix}', as well as `\begin{smallmatrix} ... \end{smallmatrix}'. The symbol `&' is interpreted as a comma, and the symbols `\cr' and `\\' are interpreted as semicolons. During output, matrices are displayed in `\matrix{ a & b \\ c & d}' format in TeX mode and in `\begin{pmatrix} a & b \\ c & d \end{pmatrix}' format in LaTeX mode; you may need to edit this afterwards to change to your preferred matrix form. If you invoke `d T' or `d L' with an argument of 2 or -2, then matrices will be displayed in two-dimensional form, such as \begin{pmatrix} a & b \\ c & d \end{pmatrix} This may be convenient for isolated matrices, but could lead to expressions being displayed like \begin{pmatrix} \times x a & b \\ c & d \end{pmatrix} While this wouldn't bother Calc, it is incorrect LaTeX. (Similarly for TeX.) Accents like `\tilde' and `\bar' translate into function calls internally (`tilde(x)', `bar(x)'). The `\underline' sequence is treated as an accent. The `\vec' accent corresponds to the function name `Vec', because `vec' is the name of a built-in Calc function. The following table shows the accents in Calc, TeX, LaTeX and "eqn" (described in the next section): Calc TeX LaTeX eqn ---- --- ----- --- acute \acute \acute Acute \Acute bar \bar \bar bar Bar \Bar breve \breve \breve Breve \Breve check \check \check Check \Check dddot \dddot ddddot \ddddot dot \dot \dot dot Dot \Dot dotdot \ddot \ddot dotdot DotDot \Ddot dyad dyad grave \grave \grave Grave \Grave hat \hat \hat hat Hat \Hat Prime prime tilde \tilde \tilde tilde Tilde \Tilde under \underline \underline under Vec \vec \vec vec VEC \Vec The `=>' (evaluates-to) operator appears as a `\to' symbol: `{A \to B}'. TeX defines `\to' as an alias for `\rightarrow'. However, if the `=>' is the top-level expression being formatted, a slightly different notation is used: `\evalto A \to B'. The `\evalto' word is ignored by Calc's input routines, and is undefined in TeX. You will typically want to include one of the following definitions at the top of a TeX file that uses `\evalto': \def\evalto{} \def\evalto#1\to{} The first definition formats evaluates-to operators in the usual way. The second causes only the B part to appear in the printed document; the A part and the arrow are hidden. Another definition you may wish to use is `\let\to=\Rightarrow' which causes `\to' to appear more like Calc's `=>' symbol. *Note Evaluates-To Operator::, for a discussion of `evalto'. The complete set of TeX control sequences that are ignored during reading is: \hbox \mbox \text \left \right \, \> \: \; \! \quad \qquad \hfil \hfill \displaystyle \textstyle \dsize \tsize \scriptstyle \scriptscriptstyle \ssize \ssize \rm \bf \it \sl \roman \bold \italic \slanted \cal \mit \Cal \Bbb \frak \goth \evalto Note that, because these symbols are ignored, reading a TeX or LaTeX formula into Calc and writing it back out may lose spacing and font information. Also, the "discretionary multiplication sign" `\*' is read the same as `*'. The TeX version of this manual includes some printed examples at the end of this section.  File: calc, Node: Eqn Language Mode, Next: Yacas Language Mode, Prev: TeX and LaTeX Language Modes, Up: Language Modes 8.8.4 Eqn Language Mode ----------------------- "Eqn" is another popular formatter for math formulas. It is designed for use with the TROFF text formatter, and comes standard with many versions of Unix. The `d E' (`calc-eqn-language') command selects "eqn" notation. The "eqn" language's main idiosyncrasy is that whitespace plays a significant part in the parsing of the language. For example, `sqrt x+1 + y' treats `x+1' as the argument of the `sqrt' operator. "Eqn" also understands more conventional grouping using curly braces: `sqrt{x+1} + y'. Braces are required only when the argument contains spaces. In Calc's "eqn" mode, however, curly braces are required to delimit arguments of operators like `sqrt'. The first of the above examples would treat only the `x' as the argument of `sqrt', and in fact `sin x+1' would be interpreted as `sin * x + 1', because `sin' is not a special operator in the "eqn" language. If you always surround the argument with curly braces, Calc will never misunderstand. Calc also understands parentheses as grouping characters. Another peculiarity of "eqn"'s syntax makes it advisable to separate words with spaces from any surrounding characters that aren't curly braces, so Calc writes `sin ( x + y )' in "eqn" mode. (The spaces around `sin' are important to make "eqn" recognize that `sin' should be typeset in a roman font, and the spaces around `x' and `y' are a good idea just in case the "eqn" document has defined special meanings for these names, too.) Powers and subscripts are written with the `sub' and `sup' operators, respectively. Note that the caret symbol `^' is treated the same as a space in "eqn" mode, as is the `~' symbol (these are used to introduce spaces of various widths into the typeset output of "eqn"). As in LaTeX mode, Calc's formatter omits parentheses around the arguments of functions like `ln' and `sin' if they are "simple-looking"; in this case Calc surrounds the argument with braces, separated by a `~' from the function name: `sin~{x}'. Font change codes (like `roman X') and positioning codes (like `~' and `down N X') are ignored by the "eqn" reader. Also ignored are the words `left', `right', `mark', and `lineup'. Quotation marks in "eqn" mode input are treated the same as curly braces: `sqrt "1+x"' is equivalent to `sqrt {1+x}'; this is only an approximation to the true meaning of quotes in "eqn", but it is good enough for most uses. Accent codes (`X dot') are handled by treating them as function calls (`dot(X)') internally. *Note TeX and LaTeX Language Modes::, for a table of these accent functions. The `prime' accent is treated specially if it occurs on a variable or function name: `f prime prime ( x prime )' is stored internally as `f''(x')'. For example, taking the derivative of `f(2 x)' with `a d x' will produce `2 f'(2 x)', which "eqn" mode will display as `2 f prime ( 2 x )'. Assignments are written with the `<-' (left-arrow) symbol, and `evalto' operators are written with `->' or `evalto ... ->' (*note TeX and LaTeX Language Modes::, for a discussion of this). The regular Calc symbols `:=' and `=>' are also recognized for these operators during reading. Vectors in "eqn" mode use regular Calc square brackets, but matrices are formatted as `matrix { ccol { a above b } ... }'. The words `lcol' and `rcol' are recognized as synonyms for `ccol' during input, and are generated instead of `ccol' if the matrix justification mode so specifies.  File: calc, Node: Yacas Language Mode, Next: Maxima Language Mode, Prev: Eqn Language Mode, Up: Language Modes 8.8.5 Yacas Language Mode ------------------------- The `d Y' (`calc-yacas-language') command selects the conventions of Yacas, a free computer algebra system. While the operators and functions in Yacas are similar to those of Calc, the names of built-in functions in Yacas are capitalized. The Calc formula `sin(2 x)', for example, is entered and displayed `Sin(2 x)' in Yacas mode, and ``arcsin(x^2)' is `ArcSin(x^2)' in Yacas mode. Complex numbers are written are written `3 + 4 I'. The standard special constants are written `Pi', `E', `I', `GoldenRatio' and `Gamma'. `Infinity' represents both `inf' and `uinf', and `Undefined' represents `nan'. Certain operators on functions, such as `D' for differentiation and `Integrate' for integration, take a prefix form in Yacas. For example, the derivative of `e^x sin(x)' can be computed with `D(x) Exp(x)*Sin(x)'. Other notable differences between Yacas and standard Calc expressions are that vectors and matrices use curly braces in Yacas, and subscripts use square brackets. If, for example, `A' represents the list `{a,2,c,4}', then `A[3]' would equal `c'.  File: calc, Node: Maxima Language Mode, Next: Giac Language Mode, Prev: Yacas Language Mode, Up: Language Modes 8.8.6 Maxima Language Mode -------------------------- The `d X' (`calc-maxima-language') command selects the conventions of Maxima, another free computer algebra system. The function names in Maxima are similar, but not always identical, to Calc. For example, instead of `arcsin(x)', Maxima will use `asin(x)'. Complex numbers are written `3 + 4 %i'. The standard special constants are written `%pi', `%e', `%i', `%phi' and `%gamma'. In Maxima, `inf' means the same as in Calc, but `infinity' represents Calc's `uinf'. Underscores as well as percent signs are allowed in function and variable names in Maxima mode. The underscore again is equivalent to the `#' in Normal mode, and the percent sign is equivalent to `o'o'. Maxima uses square brackets for lists and vectors, and matrices are written as calls to the function `matrix', given the row vectors of the matrix as arguments. Square brackets are also used as subscripts.  File: calc, Node: Giac Language Mode, Next: Mathematica Language Mode, Prev: Maxima Language Mode, Up: Language Modes 8.8.7 Giac Language Mode ------------------------ The `d A' (`calc-giac-language') command selects the conventions of Giac, another free computer algebra system. The function names in Giac are similar to Maxima. Complex numbers are written `3 + 4 i'. The standard special constants in Giac are the same as in Calc, except that `infinity' represents both Calc's `inf' and `uinf'. Underscores are allowed in function and variable names in Giac mode. Brackets are used for subscripts. In Giac, indexing of lists begins at 0, instead of 1 as in Calc. So if `A' represents the list `[a,2,c,4]', then `A[2]' would equal `c'. In general, `A[n]' in Giac mode corresponds to `A_(n+1)' in Normal mode. The Giac interval notation `2 .. 3' has no surrounding brackets; Calc reads `2 .. 3' as the closed interval `[2 .. 3]' and writes any kind of interval as `2 .. 3'. This means you cannot see the difference between an open and a closed interval while in Giac mode.  File: calc, Node: Mathematica Language Mode, Next: Maple Language Mode, Prev: Giac Language Mode, Up: Language Modes 8.8.8 Mathematica Language Mode ------------------------------- The `d M' (`calc-mathematica-language') command selects the conventions of Mathematica. Notable differences in Mathematica mode are that the names of built-in functions are capitalized, and function calls use square brackets instead of parentheses. Thus the Calc formula `sin(2 x)' is entered and displayed `Sin[2 x]' in Mathematica mode. Vectors and matrices use curly braces in Mathematica. Complex numbers are written `3 + 4 I'. The standard special constants in Calc are written `Pi', `E', `I', `GoldenRatio', `EulerGamma', `Infinity', `ComplexInfinity', and `Indeterminate' in Mathematica mode. Non-decimal numbers are written, e.g., `16^^7fff'. Floating-point numbers in scientific notation are written `1.23*10.^3'. Subscripts use double square brackets: `a[[i]]'.  File: calc, Node: Maple Language Mode, Next: Compositions, Prev: Mathematica Language Mode, Up: Language Modes 8.8.9 Maple Language Mode ------------------------- The `d W' (`calc-maple-language') command selects the conventions of Maple. Maple's language is much like C. Underscores are allowed in symbol names; square brackets are used for subscripts; explicit `*'s for multiplications are required. Use either `^' or `**' to denote powers. Maple uses square brackets for lists and curly braces for sets. Calc interprets both notations as vectors, and displays vectors with square brackets. This means Maple sets will be converted to lists when they pass through Calc. As a special case, matrices are written as calls to the function `matrix', given a list of lists as the argument, and can be read in this form or with all-capitals `MATRIX'. The Maple interval notation `2 .. 3' is like Giac's interval notation, and is handled the same by Calc. Maple writes complex numbers as `3 + 4*I'. Its special constants are `Pi', `E', `I', and `infinity' (all three of `inf', `uinf', and `nan' display as `infinity'). Floating-point numbers are written `1.23*10.^3'. Among things not currently handled by Calc's Maple mode are the various quote symbols, procedures and functional operators, and inert (`&') operators.  File: calc, Node: Compositions, Next: Syntax Tables, Prev: Maple Language Mode, Up: Language Modes 8.8.10 Compositions ------------------- There are several "composition functions" which allow you to get displays in a variety of formats similar to those in Big language mode. Most of these functions do not evaluate to anything; they are placeholders which are left in symbolic form by Calc's evaluator but are recognized by Calc's display formatting routines. Two of these, `string' and `bstring', are described elsewhere. *Note Strings::. For example, `string("ABC")' is displayed as `ABC'. When viewed on the stack it will be indistinguishable from the variable `ABC', but internally it will be stored as `string([65, 66, 67])' and can still be manipulated this way; for example, the selection and vector commands `j 1 v v j u' would select the vector portion of this object and reverse the elements, then deselect to reveal a string whose characters had been reversed. The composition functions do the same thing in all language modes (although their components will of course be formatted in the current language mode). The one exception is Unformatted mode (`d U'), which does not give the composition functions any special treatment. The functions are discussed here because of their relationship to the language modes. * Menu: * Composition Basics:: * Horizontal Compositions:: * Vertical Compositions:: * Other Compositions:: * Information about Compositions:: * User-Defined Compositions::  File: calc, Node: Composition Basics, Next: Horizontal Compositions, Prev: Compositions, Up: Compositions 8.8.10.1 Composition Basics ........................... Compositions are generally formed by stacking formulas together horizontally or vertically in various ways. Those formulas are themselves compositions. TeX users will find this analogous to TeX's "boxes." Each multi-line composition has a "baseline"; horizontal compositions use the baselines to decide how formulas should be positioned relative to one another. For example, in the Big mode formula 2 a + b 17 + ------ c the second term of the sum is four lines tall and has line three as its baseline. Thus when the term is combined with 17, line three is placed on the same level as the baseline of 17. Another important composition concept is "precedence". This is an integer that represents the binding strength of various operators. For example, `*' has higher precedence (195) than `+' (180), which means that `(a * b) + c' will be formatted without the parentheses, but `a * (b + c)' will keep the parentheses. The operator table used by normal and Big language modes has the following precedences: _ 1200 (subscripts) % 1100 (as in n%) ! 1000 (as in !n) mod 400 +/- 300 !! 210 (as in n!!) ! 210 (as in n!) ^ 200 - 197 (as in -n) * 195 (or implicit multiplication) / % \ 190 + - 180 (as in a+b) | 170 < = 160 (and other relations) && 110 || 100 ? : 90 !!! 85 &&& 80 ||| 75 := 50 :: 45 => 40 The general rule is that if an operator with precedence `n' occurs as an argument to an operator with precedence `m', then the argument is enclosed in parentheses if `n < m'. Top-level expressions and expressions which are function arguments, vector components, etc., are formatted with precedence zero (so that they normally never get additional parentheses). For binary left-associative operators like `+', the righthand argument is actually formatted with one-higher precedence than shown in the table. This makes sure `(a + b) + c' omits the parentheses, but the unnatural form `a + (b + c)' keeps its parentheses. Right-associative operators like `^' format the lefthand argument with one-higher precedence. The `cprec' function formats an expression with an arbitrary precedence. For example, `cprec(abc, 185)' will combine into sums and products as follows: `7 + abc', `7 (abc)' (because this `cprec' form has higher precedence than addition, but lower precedence than multiplication). A final composition issue is "line breaking". Calc uses two different strategies for "flat" and "non-flat" compositions. A non-flat composition is anything that appears on multiple lines (not counting line breaking). Examples would be matrices and Big mode powers and quotients. Non-flat compositions are displayed exactly as specified. If they come out wider than the current window, you must use horizontal scrolling (`<' and `>') to view them. Flat compositions, on the other hand, will be broken across several lines if they are too wide to fit the window. Certain points in a composition are noted internally as "break points". Calc's general strategy is to fill each line as much as possible, then to move down to the next line starting at the first break point that didn't fit. However, the line breaker understands the hierarchical structure of formulas. It will not break an "inner" formula if it can use an earlier break point from an "outer" formula instead. For example, a vector of sums might be formatted as: [ a + b + c, d + e + f, g + h + i, j + k + l, m ] If the `m' can fit, then so, it seems, could the `g'. But Calc prefers to break at the comma since the comma is part of a "more outer" formula. Calc would break at a plus sign only if it had to, say, if the very first sum in the vector had itself been too large to fit. Of the composition functions described below, only `choriz' generates break points. The `bstring' function (*note Strings::) also generates breakable items: A break point is added after every space (or group of spaces) except for spaces at the very beginning or end of the string. Composition functions themselves count as levels in the formula hierarchy, so a `choriz' that is a component of a larger `choriz' will be less likely to be broken. As a special case, if a `bstring' occurs as a component of a `choriz' or `choriz'-like object (such as a vector or a list of arguments in a function call), then the break points in that `bstring' will be on the same level as the break points of the surrounding object.  File: calc, Node: Horizontal Compositions, Next: Vertical Compositions, Prev: Composition Basics, Up: Compositions 8.8.10.2 Horizontal Compositions ................................ The `choriz' function takes a vector of objects and composes them horizontally. For example, `choriz([17, a b/c, d])' formats as `17a b / cd' in Normal language mode, or as a b 17---d c in Big language mode. This is actually one case of the general function `choriz(VEC, SEP, PREC)', where either or both of SEP and PREC may be omitted. PREC gives the "precedence" to use when formatting each of the components of VEC. The default precedence is the precedence from the surrounding environment. SEP is a string (i.e., a vector of character codes as might be entered with `" "' notation) which should separate components of the composition. Also, if SEP is given, the line breaker will allow lines to be broken after each occurrence of SEP. If SEP is omitted, the composition will not be breakable (unless any of its component compositions are breakable). For example, `2 choriz([a, b c, d = e], " + ", 180)' is formatted as `2 a + b c + (d = e)'. To get the `choriz' to have precedence 180 "outwards" as well as "inwards," enclose it in a `cprec' form: `2 cprec(choriz(...), 180)' formats as `2 (a + b c + (d = e))'. The baseline of a horizontal composition is the same as the baselines of the component compositions, which are all aligned.  File: calc, Node: Vertical Compositions, Next: Other Compositions, Prev: Horizontal Compositions, Up: Compositions 8.8.10.3 Vertical Compositions .............................. The `cvert' function makes a vertical composition. Each component of the vector is centered in a column. The baseline of the result is by default the top line of the resulting composition. For example, `f(cvert([a, bb, ccc]), cvert([a^2 + 1, b^2]))' formats in Big mode as f( a , 2 ) bb a + 1 ccc 2 b There are several special composition functions that work only as components of a vertical composition. The `cbase' function controls the baseline of the vertical composition; the baseline will be the same as the baseline of whatever component is enclosed in `cbase'. Thus `f(cvert([a, cbase(bb), ccc]), cvert([a^2 + 1, cbase(b^2)]))' displays as 2 a + 1 a 2 f(bb , b ) ccc There are also `ctbase' and `cbbase' functions which make the baseline of the vertical composition equal to the top or bottom line (rather than the baseline) of that component. Thus `cvert([cbase(a / b)]) + cvert([ctbase(a / b)]) + cvert([cbbase(a / b)])' gives a a - - + a + b b - b There should be only one `cbase', `ctbase', or `cbbase' function in a given vertical composition. These functions can also be written with no arguments: `ctbase()' is a zero-height object which means the baseline is the top line of the following item, and `cbbase()' means the baseline is the bottom line of the preceding item. The `crule' function builds a "rule," or horizontal line, across a vertical composition. By itself `crule()' uses `-' characters to build the rule. You can specify any other character, e.g., `crule("=")'. The argument must be a character code or vector of exactly one character code. It is repeated to match the width of the widest item in the stack. For example, a quotient with a thick line is `cvert([a + 1, cbase(crule("=")), b^2])': a + 1 ===== 2 b Finally, the functions `clvert' and `crvert' act exactly like `cvert' except that the items are left- or right-justified in the stack. Thus `clvert([a, bb, ccc]) + crvert([a, bb, ccc])' gives: a + a bb bb ccc ccc Like `choriz', the vertical compositions accept a second argument which gives the precedence to use when formatting the components. Vertical compositions do not support separator strings.  File: calc, Node: Other Compositions, Next: Information about Compositions, Prev: Vertical Compositions, Up: Compositions 8.8.10.4 Other Compositions ........................... The `csup' function builds a superscripted expression. For example, `csup(a, b)' looks the same as `a^b' does in Big language mode. This is essentially a horizontal composition of `a' and `b', where `b' is shifted up so that its bottom line is one above the baseline. Likewise, the `csub' function builds a subscripted expression. This shifts `b' down so that its top line is one below the bottom line of `a' (note that this is not quite analogous to `csup'). Other arrangements can be obtained by using `choriz' and `cvert' directly. The `cflat' function formats its argument in "flat" mode, as obtained by `d O', if the current language mode is normal or Big. It has no effect in other language modes. For example, `a^(b/c)' is formatted by Big mode like `csup(a, cflat(b/c))' to improve its readability. The `cspace' function creates horizontal space. For example, `cspace(4)' is effectively the same as `string(" ")'. A second string (i.e., vector of characters) argument is repeated instead of the space character. For example, `cspace(4, "ab")' looks like `abababab'. If the second argument is not a string, it is formatted in the normal way and then several copies of that are composed together: `cspace(4, a^2)' yields 2 2 2 2 a a a a If the number argument is zero, this is a zero-width object. The `cvspace' function creates vertical space, or a vertical stack of copies of a certain string or formatted object. The baseline is the center line of the resulting stack. A numerical argument of zero will produce an object which contributes zero height if used in a vertical composition. There are also `ctspace' and `cbspace' functions which create vertical space with the baseline the same as the baseline of the top or bottom copy, respectively, of the second argument. Thus `cvspace(2, a/b) + ctspace(2, a/b) + cbspace(2, a/b)' displays as: a - a b - a a b + - + - a b b - a b - b  File: calc, Node: Information about Compositions, Next: User-Defined Compositions, Prev: Other Compositions, Up: Compositions 8.8.10.5 Information about Compositions ....................................... The functions in this section are actual functions; they compose their arguments according to the current language and other display modes, then return a certain measurement of the composition as an integer. The `cwidth' function measures the width, in characters, of a composition. For example, `cwidth(a + b)' is 5, and `cwidth(a / b)' is 5 in Normal mode, 1 in Big mode, and 11 in TeX mode (for `{a \over b}'). The argument may involve the composition functions described in this section. The `cheight' function measures the height of a composition. This is the total number of lines in the argument's printed form. The functions `cascent' and `cdescent' measure the amount of the height that is above (and including) the baseline, or below the baseline, respectively. Thus `cascent(X) + cdescent(X)' always equals `cheight(X)'. For a one-line formula like `a + b', `cascent' returns 1 and `cdescent' returns 0. For `a / b' in Big mode, `cascent' returns 2 and `cdescent' returns 1. The only formula for which `cascent' will return zero is `cvspace(0)' or equivalents.  File: calc, Node: User-Defined Compositions, Prev: Information about Compositions, Up: Compositions 8.8.10.6 User-Defined Compositions .................................. The `Z C' (`calc-user-define-composition') command lets you define the display format for any algebraic function. You provide a formula containing a certain number of argument variables on the stack. Any time Calc formats a call to the specified function in the current language mode and with that number of arguments, Calc effectively replaces the function call with that formula with the arguments replaced. Calc builds the default argument list by sorting all the variable names that appear in the formula into alphabetical order. You can edit this argument list before pressing if you wish. Any variables in the formula that do not appear in the argument list will be displayed literally; any arguments that do not appear in the formula will not affect the display at all. You can define formats for built-in functions, for functions you have defined with `Z F' (*note Algebraic Definitions::), or for functions which have no definitions but are being used as purely syntactic objects. You can define different formats for each language mode, and for each number of arguments, using a succession of `Z C' commands. When Calc formats a function call, it first searches for a format defined for the current language mode (and number of arguments); if there is none, it uses the format defined for the Normal language mode. If neither format exists, Calc uses its built-in standard format for that function (usually just `FUNC(ARGS)'). If you execute `Z C' with the number 0 on the stack instead of a formula, any defined formats for the function in the current language mode will be removed. The function will revert to its standard format. For example, the default format for the binomial coefficient function `choose(n, m)' in the Big language mode is n ( ) m You might prefer the notation, C n m To define this notation, first make sure you are in Big mode, then put the formula choriz([cvert([cvspace(1), n]), C, cvert([cvspace(1), m])]) on the stack and type `Z C'. Answer the first prompt with `choose'. The second prompt will be the default argument list of `(C m n)'. Edit this list to be `(n m)' and press . Now, try it out: For example, turn simplification off with `m O' and enter `choose(a,b) + choose(7,3)' as an algebraic entry. C + C a b 7 3 As another example, let's define the usual notation for Stirling numbers of the first kind, `stir1(n, m)'. This is just like the regular format for binomial coefficients but with square brackets instead of parentheses. choriz([string("["), cvert([n, cbase(cvspace(1)), m]), string("]")]) Now type `Z C stir1 ', edit the argument list to `(n m)', and type . The formula provided to `Z C' usually will involve composition functions, but it doesn't have to. Putting the formula `a + b + c' onto the stack and typing `Z C foo ' would define the function `foo(x,y,z)' to display like `x + y + z'. This "sum" will act exactly like a real sum for all formatting purposes (it will be parenthesized the same, and so on). However it will be computationally unrelated to a sum. For example, the formula `2 * foo(1, 2, 3)' will display as `2 (1 + 2 + 3)'. Operator precedences have caused the "sum" to be written in parentheses, but the arguments have not actually been summed. (Generally a display format like this would be undesirable, since it can easily be confused with a real sum.) The special function `eval' can be used inside a `Z C' composition formula to cause all or part of the formula to be evaluated at display time. For example, if the formula is `a + eval(b + c)', then `foo(1, 2, 3)' will be displayed as `1 + 5'. Evaluation will use the default simplifications, regardless of the current simplification mode. There are also `evalsimp' and `evalextsimp' which simplify as if by `a s' and `a e' (respectively). Note that these "functions" operate only in the context of composition formulas (and also in rewrite rules, where they serve a similar purpose; *note Rewrite Rules::). On the stack, a call to `eval' will be left in symbolic form. It is not a good idea to use `eval' except as a last resort. It can cause the display of formulas to be extremely slow. For example, while `eval(a + b)' might seem quite fast and simple, there are several situations where it could be slow. For example, `a' and/or `b' could be polar complex numbers, in which case doing the sum requires trigonometry. Or, `a' could be the factorial `fact(100)' which is unevaluated because you have typed `m O'; `eval' will evaluate it anyway to produce a large, unwieldy integer. You can save your display formats permanently using the `Z P' command (*note Creating User Keys::).  File: calc, Node: Syntax Tables, Prev: Compositions, Up: Language Modes 8.8.11 Syntax Tables -------------------- Syntax tables do for input what compositions do for output: They allow you to teach custom notations to Calc's formula parser. Calc keeps a separate syntax table for each language mode. (Note that the Calc "syntax tables" discussed here are completely unrelated to the syntax tables described in the Emacs manual.) The `Z S' (`calc-edit-user-syntax') command edits the syntax table for the current language mode. If you want your syntax to work in any language, define it in the Normal language mode. Type `C-c C-c' to finish editing the syntax table, or `C-x k' to cancel the edit. The `m m' command saves all the syntax tables along with the other mode settings; *note General Mode Commands::. * Menu: * Syntax Table Basics:: * Precedence in Syntax Tables:: * Advanced Syntax Patterns:: * Conditional Syntax Rules::  File: calc, Node: Syntax Table Basics, Next: Precedence in Syntax Tables, Prev: Syntax Tables, Up: Syntax Tables 8.8.11.1 Syntax Table Basics ............................ "Parsing" is the process of converting a raw string of characters, such as you would type in during algebraic entry, into a Calc formula. Calc's parser works in two stages. First, the input is broken down into "tokens", such as words, numbers, and punctuation symbols like `+', `:=', and `+/-'. Space between tokens is ignored (except when it serves to separate adjacent words). Next, the parser matches this string of tokens against various built-in syntactic patterns, such as "an expression followed by `+' followed by another expression" or "a name followed by `(', zero or more expressions separated by commas, and `)'." A "syntax table" is a list of user-defined "syntax rules", which allow you to specify new patterns to define your own favorite input notations. Calc's parser always checks the syntax table for the current language mode, then the table for the Normal language mode, before it uses its built-in rules to parse an algebraic formula you have entered. Each syntax rule should go on its own line; it consists of a "pattern", a `:=' symbol, and a Calc formula with an optional "condition". (Syntax rules resemble algebraic rewrite rules, but the notation for patterns is completely different.) A syntax pattern is a list of tokens, separated by spaces. Except for a few special symbols, tokens in syntax patterns are matched literally, from left to right. For example, the rule, foo ( ) := 2+3 would cause Calc to parse the formula `4+foo()*5' as if it were `4+(2+3)*5'. Notice that the parentheses were written as two separate tokens in the rule. As a result, the rule works for both `foo()' and `foo ( )'. If we had written the rule as `foo () := 2+3', then Calc would treat `()' as a single, indivisible token, so that `foo( )' would not be recognized by the rule. (It would be parsed as a regular zero-argument function call instead.) In fact, this rule would also make trouble for the rest of Calc's parser: An unrelated formula like `bar()' would now be tokenized into `bar ()' instead of `bar ( )', so that the standard parser for function calls would no longer recognize it! While it is possible to make a token with a mixture of letters and punctuation symbols, this is not recommended. It is better to break it into several tokens, as we did with `foo()' above. The symbol `#' in a syntax pattern matches any Calc expression. On the righthand side, the things that matched the `#'s can be referred to as `#1', `#2', and so on (where `#1' matches the leftmost `#' in the pattern). For example, these rules match a user-defined function, prefix operator, infix operator, and postfix operator, respectively: foo ( # ) := myfunc(#1) foo # := myprefix(#1) # foo # := myinfix(#1,#2) # foo := mypostfix(#1) Thus `foo(3)' will parse as `myfunc(3)', and `2+3 foo' will parse as `mypostfix(2+3)'. It is important to write the first two rules in the order shown, because Calc tries rules in order from first to last. If the pattern `foo #' came first, it would match anything that could match the `foo ( # )' rule, since an expression in parentheses is itself a valid expression. Thus the `foo ( # )' rule would never get to match anything. Likewise, the last two rules must be written in the order shown or else `3 foo 4' will be parsed as `mypostfix(3) * 4'. (Of course, the best way to avoid these ambiguities is not to use the same symbol in more than one way at the same time! In case you're not convinced, try the following exercise: How will the above rules parse the input `foo(3,4)', if at all? Work it out for yourself, then try it in Calc and see.) Calc is quite flexible about what sorts of patterns are allowed. The only rule is that every pattern must begin with a literal token (like `foo' in the first two patterns above), or with a `#' followed by a literal token (as in the last two patterns). After that, any mixture is allowed, although putting two `#'s in a row will not be very useful since two expressions with nothing between them will be parsed as one expression that uses implicit multiplication. As a more practical example, Maple uses the notation `sum(a(i), i=1..10)' for sums, which Calc's Maple mode doesn't recognize at present. To handle this syntax, we simply add the rule, sum ( # , # = # .. # ) := sum(#1,#2,#3,#4) to the Maple mode syntax table. As another example, C mode can't read assignment operators like `++' and `*='. We can define these operators quite easily: # *= # := muleq(#1,#2) # ++ := postinc(#1) ++ # := preinc(#1) To complete the job, we would use corresponding composition functions and `Z C' to cause these functions to display in their respective Maple and C notations. (Note that the C example ignores issues of operator precedence, which are discussed in the next section.) You can enclose any token in quotes to prevent its usual interpretation in syntax patterns: # ":=" # := becomes(#1,#2) Quotes also allow you to include spaces in a token, although once again it is generally better to use two tokens than one token with an embedded space. To include an actual quotation mark in a quoted token, precede it with a backslash. (This also works to include backslashes in tokens.) # "bad token" # "/\"\\" # := silly(#1,#2,#3) This will parse `3 bad token 4 /"\ 5' to `silly(3,4,5)'. The token `#' has a predefined meaning in Calc's formula parser; it is not valid to use `"#"' in a syntax rule. However, longer tokens that include the `#' character are allowed. Also, while `"$"' and `"\""' are allowed as tokens, their presence in the syntax table will prevent those characters from working in their usual ways (referring to stack entries and quoting strings, respectively). Finally, the notation `%%' anywhere in a syntax table causes the rest of the line to be ignored as a comment.  File: calc, Node: Precedence in Syntax Tables, Next: Advanced Syntax Patterns, Prev: Syntax Table Basics, Up: Syntax Tables 8.8.11.2 Precedence ................... Different operators are generally assigned different "precedences". By default, an operator defined by a rule like # foo # := foo(#1,#2) will have an extremely low precedence, so that `2*3+4 foo 5 == 6' will be parsed as `(2*3+4) foo (5 == 6)'. To change the precedence of an operator, use the notation `#/P' in place of `#', where P is an integer precedence level. For example, 185 lies between the precedences for `+' and `*', so if we change this rule to #/185 foo #/186 := foo(#1,#2) then `2+3 foo 4*5' will be parsed as `2+(3 foo (4*5))'. Also, because we've given the righthand expression slightly higher precedence, our new operator will be left-associative: `1 foo 2 foo 3' will be parsed as `(1 foo 2) foo 3'. By raising the precedence of the lefthand expression instead, we can create a right-associative operator. *Note Composition Basics::, for a table of precedences of the standard Calc operators. For the precedences of operators in other language modes, look in the Calc source file `calc-lang.el'.  File: calc, Node: Advanced Syntax Patterns, Next: Conditional Syntax Rules, Prev: Precedence in Syntax Tables, Up: Syntax Tables 8.8.11.3 Advanced Syntax Patterns ................................. To match a function with a variable number of arguments, you could write foo ( # ) := myfunc(#1) foo ( # , # ) := myfunc(#1,#2) foo ( # , # , # ) := myfunc(#1,#2,#3) but this isn't very elegant. To match variable numbers of items, Calc uses some notations inspired regular expressions and the "extended BNF" style used by some language designers. foo ( { # }*, ) := apply(myfunc,#1) The token `{' introduces a repeated or optional portion. One of the three tokens `}*', `}+', or `}?' ends the portion. These will match zero or more, one or more, or zero or one copies of the enclosed pattern, respectively. In addition, `}*' and `}+' can be followed by a separator token (with no space in between, as shown above). Thus `{ # }*,' matches nothing, or one expression, or several expressions separated by commas. A complete `{ ... }' item matches as a vector of the items that matched inside it. For example, the above rule will match `foo(1,2,3)' to get `apply(myfunc,[1,2,3])'. The Calc `apply' function takes a function name and a vector of arguments and builds a call to the function with those arguments, so the net result is the formula `myfunc(1,2,3)'. If the body of a `{ ... }' contains several `#'s (or nested `{ ... }' constructs), then the items will be strung together into the resulting vector. If the body does not contain anything but literal tokens, the result will always be an empty vector. foo ( { # , # }+, ) := bar(#1) foo ( { { # }*, }*; ) := matrix(#1) will parse `foo(1, 2, 3, 4)' as `bar([1, 2, 3, 4])', and `foo(1, 2; 3, 4)' as `matrix([[1, 2], [3, 4]])'. Also, after some thought it's easy to see how this pair of rules will parse `foo(1, 2, 3)' as `matrix([[1, 2, 3]])', since the first rule will only match an even number of arguments. The rule foo ( # { , # , # }? ) := bar(#1,#2) will parse `foo(2,3,4)' as `bar(2,[3,4])', and `foo(2)' as `bar(2,[])'. The notation `{ ... }?.' (note the trailing period) works just the same as regular `{ ... }?', except that it does not count as an argument; the following two rules are equivalent: foo ( # , { also }? # ) := bar(#1,#3) foo ( # , { also }?. # ) := bar(#1,#2) Note that in the first case the optional text counts as `#2', which will always be an empty vector, but in the second case no empty vector is produced. Another variant is `{ ... }?$', which means the body is optional only at the end of the input formula. All built-in syntax rules in Calc use this for closing delimiters, so that during algebraic entry you can type `[sqrt(2), sqrt(3 ', omitting the closing parenthesis and bracket. Calc does this automatically for trailing `)', `]', and `>' tokens in syntax rules, but you can use `{ ... }?$' explicitly to get this effect with any token (such as `"}"' or `end'). Like `{ ... }?.', this notation does not count as an argument. Conversely, you can use quotes, as in `")"', to prevent a closing-delimiter token from being automatically treated as optional. Calc's parser does not have full backtracking, which means some patterns will not work as you might expect: foo ( { # , }? # , # ) := bar(#1,#2,#3) Here we are trying to make the first argument optional, so that `foo(2,3)' parses as `bar([],2,3)'. Unfortunately, Calc first tries to match `2,' against the optional part of the pattern, finds a match, and so goes ahead to match the rest of the pattern. Later on it will fail to match the second comma, but it doesn't know how to go back and try the other alternative at that point. One way to get around this would be to use two rules: foo ( # , # , # ) := bar([#1],#2,#3) foo ( # , # ) := bar([],#1,#2) More precisely, when Calc wants to match an optional or repeated part of a pattern, it scans forward attempting to match that part. If it reaches the end of the optional part without failing, it "finalizes" its choice and proceeds. If it fails, though, it backs up and tries the other alternative. Thus Calc has "partial" backtracking. A fully backtracking parser would go on to make sure the rest of the pattern matched before finalizing the choice.  File: calc, Node: Conditional Syntax Rules, Prev: Advanced Syntax Patterns, Up: Syntax Tables 8.8.11.4 Conditional Syntax Rules ................................. It is possible to attach a "condition" to a syntax rule. For example, the rules foo ( # ) := ifoo(#1) :: integer(#1) foo ( # ) := gfoo(#1) will parse `foo(3)' as `ifoo(3)', but will parse `foo(3.5)' and `foo(x)' as calls to `gfoo'. Any number of conditions may be attached; all must be true for the rule to succeed. A condition is "true" if it evaluates to a nonzero number. *Note Logical Operations::, for a list of Calc functions like `integer' that perform logical tests. The exact sequence of events is as follows: When Calc tries a rule, it first matches the pattern as usual. It then substitutes `#1', `#2', etc., in the conditions, if any. Next, the conditions are simplified and evaluated in order from left to right, as if by the `a s' algebra command (*note Simplifying Formulas::). Each result is true if it is a nonzero number, or an expression that can be proven to be nonzero (*note Declarations::). If the results of all conditions are true, the expression (such as `ifoo(#1)') has its `#'s substituted, and that is the result of the parse. If the result of any condition is false, Calc goes on to try the next rule in the syntax table. Syntax rules also support `let' conditions, which operate in exactly the same way as they do in algebraic rewrite rules. *Note Other Features of Rewrite Rules::, for details. A `let' condition is always true, but as a side effect it defines a variable which can be used in later conditions, and also in the expression after the `:=' sign: foo ( # ) := hifoo(x) :: let(x := #1 + 0.5) :: dnumint(x) The `dnumint' function tests if a value is numerically an integer, i.e., either a true integer or an integer-valued float. This rule will parse `foo' with a half-integer argument, like `foo(3.5)', to a call like `hifoo(4.)'. The lefthand side of a syntax rule `let' must be a simple variable, not the arbitrary pattern that is allowed in rewrite rules. The `matches' function is also treated specially in syntax rule conditions (again, in the same way as in rewrite rules). *Note Matching Commands::. If the matching pattern contains meta-variables, then those meta-variables may be used in later conditions and in the result expression. The arguments to `matches' are not evaluated in this situation. sum ( # , # ) := sum(#1,a,b,c) :: matches(#2, a=[b..c]) This is another way to implement the Maple mode `sum' notation. In this approach, we allow `#2' to equal the whole expression `i=1..10'. Then, we use `matches' to break it apart into its components. If the expression turns out not to match the pattern, the syntax rule will fail. Note that `Z S' always uses Calc's Normal language mode for editing expressions in syntax rules, so we must use regular Calc notation for the interval `[b..c]' that will correspond to the Maple mode interval `1..10'.  File: calc, Node: Modes Variable, Next: Calc Mode Line, Prev: Language Modes, Up: Mode Settings 8.9 The `Modes' Variable ======================== The `m g' (`calc-get-modes') command pushes onto the stack a vector of numbers that describes the various mode settings that are in effect. With a numeric prefix argument, it pushes only the Nth mode, i.e., the Nth element of this vector. Keyboard macros can use the `m g' command to modify their behavior based on the current mode settings. The modes vector is also available in the special variable `Modes'. In other words, `m g' is like `s r Modes '. It will not work to store into this variable; in fact, if you do, `Modes' will cease to track the current modes. (The `m g' command will continue to work, however.) In general, each number in this vector is suitable as a numeric prefix argument to the associated mode-setting command. (Recall that the `~' key takes a number from the stack and gives it as a numeric prefix to the next command.) The elements of the modes vector are as follows: 1. Current precision. Default is 12; associated command is `p'. 2. Binary word size. Default is 32; associated command is `b w'. 3. Stack size (not counting the value about to be pushed by `m g'). This is zero if `m g' is executed with an empty stack. 4. Number radix. Default is 10; command is `d r'. 5. Floating-point format. This is the number of digits, plus the constant 0 for normal notation, 10000 for scientific notation, 20000 for engineering notation, or 30000 for fixed-point notation. These codes are acceptable as prefix arguments to the `d n' command, but note that this may lose information: For example, `d s' and `C-u 12 d s' have similar (but not quite identical) effects if the current precision is 12, but they both produce a code of 10012, which will be treated by `d n' as `C-u 12 d s'. If the precision then changes, the float format will still be frozen at 12 significant figures. 6. Angular mode. Default is 1 (degrees). Other values are 2 (radians) and 3 (HMS). The `m d' command accepts these prefixes. 7. Symbolic mode. Value is 0 or 1; default is 0. Command is `m s'. 8. Fraction mode. Value is 0 or 1; default is 0. Command is `m f'. 9. Polar mode. Value is 0 (rectangular) or 1 (polar); default is 0. Command is `m p'. 10. Matrix/Scalar mode. Default value is -1. Value is 0 for Scalar mode, -2 for Matrix mode, -3 for square Matrix mode, or N for NxN Matrix mode. Command is `m v'. 11. Simplification mode. Default is 1. Value is -1 for off (`m O'), 0 for `m N', 2 for `m B', 3 for `m A', 4 for `m E', or 5 for `m U'. The `m D' command accepts these prefixes. 12. Infinite mode. Default is -1 (off). Value is 1 if the mode is on, or 0 if the mode is on with positive zeros. Command is `m i'. For example, the sequence `M-1 m g 2 + ~ p' increases the precision by two, leaving a copy of the old precision on the stack. Later, `~ p' will restore the original precision using that stack value. (This sequence might be especially useful inside a keyboard macro.) As another example, `M-3 m g 1 - ~ ' deletes all but the oldest (bottommost) stack entry. Yet another example: The HP-48 "round" command rounds a number to the current displayed precision. You could roughly emulate this in Calc with the sequence `M-5 m g 10000 % ~ c c'. (This would not work for fixed-point mode, but it wouldn't be hard to do a full emulation with the help of the `Z [' and `Z ]' programming commands. *Note Conditionals in Macros::.)  File: calc, Node: Calc Mode Line, Prev: Modes Variable, Up: Mode Settings 8.10 The Calc Mode Line ======================= This section is a summary of all symbols that can appear on the Calc mode line, the highlighted bar that appears under the Calc stack window (or under an editing window in Embedded mode). The basic mode line format is: --%*-Calc: 12 Deg OTHER MODES (Calculator) The `%*' indicates that the buffer is "read-only"; it shows that regular Emacs commands are not allowed to edit the stack buffer as if it were text. The word `Calc:' changes to `CalcEmbed:' if Embedded mode is enabled. The words after this describe the various Calc modes that are in effect. The first mode is always the current precision, an integer. The second mode is always the angular mode, either `Deg', `Rad', or `Hms'. Here is a complete list of the remaining symbols that can appear on the mode line: `Alg' Algebraic mode (`m a'; *note Algebraic Entry::). `Alg[(' Incomplete algebraic mode (`C-u m a'). `Alg*' Total algebraic mode (`m t'). `Symb' Symbolic mode (`m s'; *note Symbolic Mode::). `Matrix' Matrix mode (`m v'; *note Matrix Mode::). `MatrixN' Dimensioned Matrix mode (`C-u N m v'; *note Matrix Mode::). `SqMatrix' Square Matrix mode (`C-u m v'; *note Matrix Mode::). `Scalar' Scalar mode (`m v'; *note Matrix Mode::). `Polar' Polar complex mode (`m p'; *note Polar Mode::). `Frac' Fraction mode (`m f'; *note Fraction Mode::). `Inf' Infinite mode (`m i'; *note Infinite Mode::). `+Inf' Positive Infinite mode (`C-u 0 m i'). `NoSimp' Default simplifications off (`m O'; *note Simplification Modes::). `NumSimp' Default simplifications for numeric arguments only (`m N'). `BinSimpW' Binary-integer simplification mode; word size W (`m B', `b w'). `AlgSimp' Algebraic simplification mode (`m A'). `ExtSimp' Extended algebraic simplification mode (`m E'). `UnitSimp' Units simplification mode (`m U'). `Bin' Current radix is 2 (`d 2'; *note Radix Modes::). `Oct' Current radix is 8 (`d 8'). `Hex' Current radix is 16 (`d 6'). `RadixN' Current radix is N (`d r'). `Zero' Leading zeros (`d z'; *note Radix Modes::). `Big' Big language mode (`d B'; *note Normal Language Modes::). `Flat' One-line normal language mode (`d O'). `Unform' Unformatted language mode (`d U'). `C' C language mode (`d C'; *note C FORTRAN Pascal::). `Pascal' Pascal language mode (`d P'). `Fortran' FORTRAN language mode (`d F'). `TeX' TeX language mode (`d T'; *note TeX and LaTeX Language Modes::). `LaTeX' LaTeX language mode (`d L'; *note TeX and LaTeX Language Modes::). `Eqn' "Eqn" language mode (`d E'; *note Eqn Language Mode::). `Math' Mathematica language mode (`d M'; *note Mathematica Language Mode::). `Maple' Maple language mode (`d W'; *note Maple Language Mode::). `NormN' Normal float mode with N digits (`d n'; *note Float Formats::). `FixN' Fixed point mode with N digits after the point (`d f'). `Sci' Scientific notation mode (`d s'). `SciN' Scientific notation with N digits (`d s'). `Eng' Engineering notation mode (`d e'). `EngN' Engineering notation with N digits (`d e'). `LeftN' Left-justified display indented by N (`d <'; *note Justification::). `Right' Right-justified display (`d >'). `RightN' Right-justified display with width N (`d >'). `Center' Centered display (`d ='). `CenterN' Centered display with center column N (`d ='). `WidN' Line breaking with width N (`d b'; *note Normal Language Modes::). `Wide' No line breaking (`d b'). `Break' Selections show deep structure (`j b'; *note Making Selections::). `Save' Record modes in `~/.calc.el' (`m R'; *note General Mode Commands::). `Local' Record modes in Embedded buffer (`m R'). `LocEdit' Record modes as editing-only in Embedded buffer (`m R'). `LocPerm' Record modes as permanent-only in Embedded buffer (`m R'). `Global' Record modes as global in Embedded buffer (`m R'). `Manual' Automatic recomputation turned off (`m C'; *note Automatic Recomputation::). `Graph' GNUPLOT process is alive in background (*note Graphics::). `Sel' Top-of-stack has a selection (Embedded only; *note Making Selections::). `Dirty' The stack display may not be up-to-date (*note Display Modes::). `Inv' "Inverse" prefix was pressed (`I'; *note Inverse and Hyperbolic::). `Hyp' "Hyperbolic" prefix was pressed (`H'). `Keep' "Keep-arguments" prefix was pressed (`K'). `Narrow' Stack is truncated (`d t'; *note Truncating the Stack::). In addition, the symbols `Active' and `~Active' can appear as minor modes on an Embedded buffer's mode line. *Note Embedded Mode::.  File: calc, Node: Arithmetic, Next: Scientific Functions, Prev: Mode Settings, Up: Top 9 Arithmetic Functions ********************** This chapter describes the Calc commands for doing simple calculations on numbers, such as addition, absolute value, and square roots. These commands work by removing the top one or two values from the stack, performing the desired operation, and pushing the result back onto the stack. If the operation cannot be performed, the result pushed is a formula instead of a number, such as `2/0' (because division by zero is invalid) or `sqrt(x)' (because the argument `x' is a formula). Most of the commands described here can be invoked by a single keystroke. Some of the more obscure ones are two-letter sequences beginning with the `f' ("functions") prefix key. *Note Prefix Arguments::, for a discussion of the effect of numeric prefix arguments on commands in this chapter which do not otherwise interpret a prefix argument. * Menu: * Basic Arithmetic:: * Integer Truncation:: * Complex Number Functions:: * Conversions:: * Date Arithmetic:: * Financial Functions:: * Binary Functions::  File: calc, Node: Basic Arithmetic, Next: Integer Truncation, Prev: Arithmetic, Up: Arithmetic 9.1 Basic Arithmetic ==================== The `+' (`calc-plus') command adds two numbers. The numbers may be any of the standard Calc data types. The resulting sum is pushed back onto the stack. If both arguments of `+' are vectors or matrices (of matching dimensions), the result is a vector or matrix sum. If one argument is a vector and the other a scalar (i.e., a non-vector), the scalar is added to each of the elements of the vector to form a new vector. If the scalar is not a number, the operation is left in symbolic form: Suppose you added `x' to the vector `[1,2]'. You may want the result `[1+x,2+x]', or you may plan to substitute a 2-vector for `x' in the future. Since the Calculator can't tell which interpretation you want, it makes the safest assumption. *Note Reducing and Mapping::, for a way to add `x' to every element of a vector. If either argument of `+' is a complex number, the result will in general be complex. If one argument is in rectangular form and the other polar, the current Polar mode determines the form of the result. If Symbolic mode is enabled, the sum may be left as a formula if the necessary conversions for polar addition are non-trivial. If both arguments of `+' are HMS forms, the forms are added according to the usual conventions of hours-minutes-seconds notation. If one argument is an HMS form and the other is a number, that number is converted from degrees or radians (depending on the current Angular mode) to HMS format and then the two HMS forms are added. If one argument of `+' is a date form, the other can be either a real number, which advances the date by a certain number of days, or an HMS form, which advances the date by a certain amount of time. Subtracting two date forms yields the number of days between them. Adding two date forms is meaningless, but Calc interprets it as the subtraction of one date form and the negative of the other. (The negative of a date form can be understood by remembering that dates are stored as the number of days before or after Jan 1, 1 AD.) If both arguments of `+' are error forms, the result is an error form with an appropriately computed standard deviation. If one argument is an error form and the other is a number, the number is taken to have zero error. Error forms may have symbolic formulas as their mean and/or error parts; adding these will produce a symbolic error form result. However, adding an error form to a plain symbolic formula (as in `(a +/- b) + c') will not work, for the same reasons just mentioned for vectors. Instead you must write `(a +/- b) + (c +/- 0)'. If both arguments of `+' are modulo forms with equal values of `M', or if one argument is a modulo form and the other a plain number, the result is a modulo form which represents the sum, modulo `M', of the two values. If both arguments of `+' are intervals, the result is an interval which describes all possible sums of the possible input values. If one argument is a plain number, it is treated as the interval `[x .. x]'. If one argument of `+' is an infinity and the other is not, the result is that same infinity. If both arguments are infinite and in the same direction, the result is the same infinity, but if they are infinite in different directions the result is `nan'. The `-' (`calc-minus') command subtracts two values. The top number on the stack is subtracted from the one behind it, so that the computation `5 2 -' produces 3, not -3. All options available for `+' are available for `-' as well. The `*' (`calc-times') command multiplies two numbers. If one argument is a vector and the other a scalar, the scalar is multiplied by the elements of the vector to produce a new vector. If both arguments are vectors, the interpretation depends on the dimensions of the vectors: If both arguments are matrices, a matrix multiplication is done. If one argument is a matrix and the other a plain vector, the vector is interpreted as a row vector or column vector, whichever is dimensionally correct. If both arguments are plain vectors, the result is a single scalar number which is the dot product of the two vectors. If one argument of `*' is an HMS form and the other a number, the HMS form is multiplied by that amount. It is an error to multiply two HMS forms together, or to attempt any multiplication involving date forms. Error forms, modulo forms, and intervals can be multiplied; see the comments for addition of those forms. When two error forms or intervals are multiplied they are considered to be statistically independent; thus, `[-2 .. 3] * [-2 .. 3]' is `[-6 .. 9]', whereas `[-2 .. 3] ^ 2' is `[0 .. 9]'. The `/' (`calc-divide') command divides two numbers. When combining multiplication and division in an algebraic formula, it is good style to use parentheses to distinguish between possible interpretations; the expression `a/b*c' should be written `(a/b)*c' or `a/(b*c)', as appropriate. Without the parentheses, Calc will interpret `a/b*c' as `a/(b*c)', since in algebraic entry Calc gives division a lower precedence than multiplication. (This is not standard across all computer languages, and Calc may change the precedence depending on the language mode being used. *Note Language Modes::.) This default ordering can be changed by setting the customizable variable `calc-multiplication-has-precedence' to `nil' (*note Customizing Calc::); this will give multiplication and division equal precedences. Note that Calc's default choice of precedence allows `a b / c d' to be used as a shortcut for a b ---. c d When dividing a scalar `B' by a square matrix `A', the computation performed is `B' times the inverse of `A'. This also occurs if `B' is itself a vector or matrix, in which case the effect is to solve the set of linear equations represented by `B'. If `B' is a matrix with the same number of rows as `A', or a plain vector (which is interpreted here as a column vector), then the equation `A X = B' is solved for the vector or matrix `X'. Otherwise, if `B' is a non-square matrix with the same number of _columns_ as `A', the equation `X A = B' is solved. If you wish a vector `B' to be interpreted as a row vector to be solved as `X A = B', make it into a one-row matrix with `C-u 1 v p' first. To force a left-handed solution with a square matrix `B', transpose `A' and `B' before dividing, then transpose the result. HMS forms can be divided by real numbers or by other HMS forms. Error forms can be divided in any combination of ways. Modulo forms where both values and the modulo are integers can be divided to get an integer modulo form result. Intervals can be divided; dividing by an interval that encompasses zero or has zero as a limit will result in an infinite interval. The `^' (`calc-power') command raises a number to a power. If the power is an integer, an exact result is computed using repeated multiplications. For non-integer powers, Calc uses Newton's method or logarithms and exponentials. Square matrices can be raised to integer powers. If either argument is an error (or interval or modulo) form, the result is also an error (or interval or modulo) form. If you press the `I' (inverse) key first, the `I ^' command computes an Nth root: `125 3 I ^' computes the number 5. (This is entirely equivalent to `125 1:3 ^'.) The `\' (`calc-idiv') command divides two numbers on the stack to produce an integer result. It is equivalent to dividing with , then rounding down with `F' (`calc-floor'), only a bit more convenient and efficient. Also, since it is an all-integer operation when the arguments are integers, it avoids problems that `/ F' would have with floating-point roundoff. The `%' (`calc-mod') command performs a "modulo" (or "remainder") operation. Mathematically, `a%b = a - (a\b)*b', and is defined for all real numbers `a' and `b' (except `b=0'). For positive `b', the result will always be between 0 (inclusive) and `b' (exclusive). Modulo does not work for HMS forms and error forms. If `a' is a modulo form, its modulo is changed to `b', which must be positive real number. The `:' (`calc-fdiv') [`fdiv'] command divides the two integers on the top of the stack to produce a fractional result. This is a convenient shorthand for enabling Fraction mode (with `m f') temporarily and using `/'. Note that during numeric entry the `:' key is interpreted as a fraction separator, so to divide 8 by 6 you would have to type `8 6 :'. (Of course, in this case, it would be much easier simply to enter the fraction directly as `8:6 '!) The `n' (`calc-change-sign') command negates the number on the top of the stack. It works on numbers, vectors and matrices, HMS forms, date forms, error forms, intervals, and modulo forms. The `A' (`calc-abs') [`abs'] command computes the absolute value of a number. The result of `abs' is always a nonnegative real number: With a complex argument, it computes the complex magnitude. With a vector or matrix argument, it computes the Frobenius norm, i.e., the square root of the sum of the squares of the absolute values of the elements. The absolute value of an error form is defined by replacing the mean part with its absolute value and leaving the error part the same. The absolute value of a modulo form is undefined. The absolute value of an interval is defined in the obvious way. The `f A' (`calc-abssqr') [`abssqr'] command computes the absolute value squared of a number, vector or matrix, or error form. The `f s' (`calc-sign') [`sign'] command returns 1 if its argument is positive, -1 if its argument is negative, or 0 if its argument is zero. In algebraic form, you can also write `sign(a,x)' which evaluates to `x * sign(a)', i.e., either `x', `-x', or zero depending on the sign of `a'. The `&' (`calc-inv') [`inv'] command computes the reciprocal of a number, i.e., `1 / x'. Operating on a square matrix, it computes the inverse of that matrix. The `Q' (`calc-sqrt') [`sqrt'] command computes the square root of a number. For a negative real argument, the result will be a complex number whose form is determined by the current Polar mode. The `f h' (`calc-hypot') [`hypot'] command computes the square root of the sum of the squares of two numbers. That is, `hypot(a,b)' is the length of the hypotenuse of a right triangle with sides `a' and `b'. If the arguments are complex numbers, their squared magnitudes are used. The `f Q' (`calc-isqrt') [`isqrt'] command computes the integer square root of an integer. This is the true square root of the number, rounded down to an integer. For example, `isqrt(10)' produces 3. Note that, like `\' [`idiv'], this uses exact integer arithmetic throughout to avoid roundoff problems. If the input is a floating-point number or other non-integer value, this is exactly the same as `floor(sqrt(x))'. The `f n' (`calc-min') [`min'] and `f x' (`calc-max') [`max'] commands take the minimum or maximum of two real numbers, respectively. These commands also work on HMS forms, date forms, intervals, and infinities. (In algebraic expressions, these functions take any number of arguments and return the maximum or minimum among all the arguments.) The `f M' (`calc-mant-part') [`mant'] function extracts the "mantissa" part `m' of its floating-point argument; `f X' (`calc-xpon-part') [`xpon'] extracts the "exponent" part `e'. The original number is equal to `m * 10^e', where `m' is in the interval `[1.0 .. 10.0)' except that `m=e=0' if the original number is zero. For integers and fractions, `mant' returns the number unchanged and `xpon' returns zero. The `v u' (`calc-unpack') command can also be used to "unpack" a floating-point number; this produces an integer mantissa and exponent, with the constraint that the mantissa is not a multiple of ten (again except for the `m=e=0' case). The `f S' (`calc-scale-float') [`scf'] function scales a number by a given power of ten. Thus, `scf(mant(x), xpon(x)) = x' for any real `x'. The second argument must be an integer, but the first may actually be any numeric value. For example, `scf(5,-2) = 0.05' or `1:20' depending on the current Fraction mode. The `f [' (`calc-decrement') [`decr'] and `f ]' (`calc-increment') [`incr'] functions decrease or increase a number by one unit. For integers, the effect is obvious. For floating-point numbers, the change is by one unit in the last place. For example, incrementing `12.3456' when the current precision is 6 digits yields `12.3457'. If the current precision had been 8 digits, the result would have been `12.345601'. Incrementing `0.0' produces `10^-p', where `p' is the current precision. These operations are defined only on integers and floats. With numeric prefix arguments, they change the number by `n' units. Note that incrementing followed by decrementing, or vice-versa, will almost but not quite always cancel out. Suppose the precision is 6 digits and the number `9.99999' is on the stack. Incrementing will produce `10.0000'; decrementing will produce `9.9999'. One digit has been dropped. This is an unavoidable consequence of the way floating-point numbers work. Incrementing a date/time form adjusts it by a certain number of seconds. Incrementing a pure date form adjusts it by a certain number of days.  File: calc, Node: Integer Truncation, Next: Complex Number Functions, Prev: Basic Arithmetic, Up: Arithmetic 9.2 Integer Truncation ====================== There are four commands for truncating a real number to an integer, differing mainly in their treatment of negative numbers. All of these commands have the property that if the argument is an integer, the result is the same integer. An integer-valued floating-point argument is converted to integer form. If you press `H' (`calc-hyperbolic') first, the result will be expressed as an integer-valued floating-point number. The `F' (`calc-floor') [`floor' or `ffloor'] command truncates a real number to the next lower integer, i.e., toward minus infinity. Thus `3.6 F' produces 3, but `_3.6 F' produces -4. The `I F' (`calc-ceiling') [`ceil' or `fceil'] command truncates toward positive infinity. Thus `3.6 I F' produces 4, and `_3.6 I F' produces -3. The `R' (`calc-round') [`round' or `fround'] command rounds to the nearest integer. When the fractional part is .5 exactly, this command rounds away from zero. (All other rounding in the Calculator uses this convention as well.) Thus `3.5 R' produces 4 but `3.4 R' produces 3; `_3.5 R' produces -4. The `I R' (`calc-trunc') [`trunc' or `ftrunc'] command truncates toward zero. In other words, it "chops off" everything after the decimal point. Thus `3.6 I R' produces 3 and `_3.6 I R' produces -3. These functions may not be applied meaningfully to error forms, but they do work for intervals. As a convenience, applying `floor' to a modulo form floors the value part of the form. Applied to a vector, these functions operate on all elements of the vector one by one. Applied to a date form, they operate on the internal numerical representation of dates, converting a date/time form into a pure date. There are two more rounding functions which can only be entered in algebraic notation. The `roundu' function is like `round' except that it rounds up, toward plus infinity, when the fractional part is .5. This distinction matters only for negative arguments. Also, `rounde' rounds to an even number in the case of a tie, rounding up or down as necessary. For example, `rounde(3.5)' and `rounde(4.5)' both return 4, but `rounde(5.5)' returns 6. The advantage of round-to-even is that the net error due to rounding after a long calculation tends to cancel out to zero. An important subtle point here is that the number being fed to `rounde' will already have been rounded to the current precision before `rounde' begins. For example, `rounde(2.500001)' with a current precision of 6 will incorrectly, or at least surprisingly, yield 2 because the argument will first have been rounded down to `2.5' (which `rounde' sees as an exact tie between 2 and 3). Each of these functions, when written in algebraic formulas, allows a second argument which specifies the number of digits after the decimal point to keep. For example, `round(123.4567, 2)' will produce the answer 123.46, and `round(123.4567, -1)' will produce 120 (i.e., the cutoff is one digit to the _left_ of the decimal point). A second argument of zero is equivalent to no second argument at all. To compute the fractional part of a number (i.e., the amount which, when added to `floor(N)', will produce N) just take N modulo 1 using the `%' command. Note also the `\' (integer quotient), `f I' (integer logarithm), and `f Q' (integer square root) commands, which are analogous to `/', `B', and `Q', respectively, except that they take integer arguments and return the result rounded down to an integer.  File: calc, Node: Complex Number Functions, Next: Conversions, Prev: Integer Truncation, Up: Arithmetic 9.3 Complex Number Functions ============================ The `J' (`calc-conj') [`conj'] command computes the complex conjugate of a number. For complex number `a+bi', the complex conjugate is `a-bi'. If the argument is a real number, this command leaves it the same. If the argument is a vector or matrix, this command replaces each element by its complex conjugate. The `G' (`calc-argument') [`arg'] command computes the "argument" or polar angle of a complex number. For a number in polar notation, this is simply the second component of the pair `(R;THETA)'. The result is expressed according to the current angular mode and will be in the range -180 degrees (exclusive) to +180 degrees (inclusive), or the equivalent range in radians. The `calc-imaginary' command multiplies the number on the top of the stack by the imaginary number `i = (0,1)'. This command is not normally bound to a key in Calc, but it is available on the button in Keypad mode. The `f r' (`calc-re') [`re'] command replaces a complex number by its real part. This command has no effect on real numbers. (As an added convenience, `re' applied to a modulo form extracts the value part.) The `f i' (`calc-im') [`im'] command replaces a complex number by its imaginary part; real numbers are converted to zero. With a vector or matrix argument, these functions operate element-wise. The `v p' (`calc-pack') command can pack the top two numbers on the stack into a composite object such as a complex number. With a prefix argument of -1, it produces a rectangular complex number; with an argument of -2, it produces a polar complex number. (Also, *note Building Vectors::.) The `v u' (`calc-unpack') command takes the complex number (or other composite object) on the top of the stack and unpacks it into its separate components.  File: calc, Node: Conversions, Next: Date Arithmetic, Prev: Complex Number Functions, Up: Arithmetic 9.4 Conversions =============== The commands described in this section convert numbers from one form to another; they are two-key sequences beginning with the letter `c'. The `c f' (`calc-float') [`pfloat'] command converts the number on the top of the stack to floating-point form. For example, `23' is converted to `23.0', `3:2' is converted to `1.5', and `2.3' is left the same. If the value is a composite object such as a complex number or vector, each of the components is converted to floating-point. If the value is a formula, all numbers in the formula are converted to floating-point. Note that depending on the current floating-point precision, conversion to floating-point format may lose information. As a special exception, integers which appear as powers or subscripts are not floated by `c f'. If you really want to float a power, you can use a `j s' command to select the power followed by `c f'. Because `c f' cannot examine the formula outside of the selection, it does not notice that the thing being floated is a power. *Note Selecting Subformulas::. The normal `c f' command is "pervasive" in the sense that it applies to all numbers throughout the formula. The `pfloat' algebraic function never stays around in a formula; `pfloat(a + 1)' changes to `a + 1.0' as soon as it is evaluated. With the Hyperbolic flag, `H c f' [`float'] operates only on the number or vector of numbers at the top level of its argument. Thus, `float(1)' is 1.0, but `float(a + 1)' is left unevaluated because its argument is not a number. You should use `H c f' if you wish to guarantee that the final value, once all the variables have been assigned, is a float; you would use `c f' if you wish to do the conversion on the numbers that appear right now. The `c F' (`calc-fraction') [`pfrac'] command converts a floating-point number into a fractional approximation. By default, it produces a fraction whose decimal representation is the same as the input number, to within the current precision. You can also give a numeric prefix argument to specify a tolerance, either directly, or, if the prefix argument is zero, by using the number on top of the stack as the tolerance. If the tolerance is a positive integer, the fraction is correct to within that many significant figures. If the tolerance is a non-positive integer, it specifies how many digits fewer than the current precision to use. If the tolerance is a floating-point number, the fraction is correct to within that absolute amount. The `pfrac' function is pervasive, like `pfloat'. There is also a non-pervasive version, `H c F' [`frac'], which is analogous to `H c f' discussed above. The `c d' (`calc-to-degrees') [`deg'] command converts a number into degrees form. The value on the top of the stack may be an HMS form (interpreted as degrees-minutes-seconds), or a real number which will be interpreted in radians regardless of the current angular mode. The `c r' (`calc-to-radians') [`rad'] command converts an HMS form or angle in degrees into an angle in radians. The `c h' (`calc-to-hms') [`hms'] command converts a real number, interpreted according to the current angular mode, to an HMS form describing the same angle. In algebraic notation, the `hms' function also accepts three arguments: `hms(H, M, S)'. (The three-argument version is independent of the current angular mode.) The `calc-from-hms' command converts the HMS form on the top of the stack into a real number according to the current angular mode. The `c p' (`calc-polar') command converts the complex number on the top of the stack from polar to rectangular form, or from rectangular to polar form, whichever is appropriate. Real numbers are left the same. This command is equivalent to the `rect' or `polar' functions in algebraic formulas, depending on the direction of conversion. (It uses `polar', except that if the argument is already a polar complex number, it uses `rect' instead. The `I c p' command always uses `rect'.) The `c c' (`calc-clean') [`pclean'] command "cleans" the number on the top of the stack. Floating point numbers are re-rounded according to the current precision. Polar numbers whose angular components have strayed from the -180 to +180 degree range are normalized. (Note that results will be undesirable if the current angular mode is different from the one under which the number was produced!) Integers and fractions are generally unaffected by this operation. Vectors and formulas are cleaned by cleaning each component number (i.e., pervasively). If the simplification mode is set below the default level, it is raised to the default level for the purposes of this command. Thus, `c c' applies the default simplifications even if their automatic application is disabled. *Note Simplification Modes::. A numeric prefix argument to `c c' sets the floating-point precision to that value for the duration of the command. A positive prefix (of at least 3) sets the precision to the specified value; a negative or zero prefix decreases the precision by the specified amount. The keystroke sequences `c 0' through `c 9' are equivalent to `c c' with the corresponding negative prefix argument. If roundoff errors have changed 2.0 into 1.999999, typing `c 1' to clip off one decimal place often conveniently does the trick. The `c c' command with a numeric prefix argument, and the `c 0' through `c 9' commands, also "clip" very small floating-point numbers to zero. If the exponent is less than or equal to the negative of the specified precision, the number is changed to 0.0. For example, if the current precision is 12, then `c 2' changes the vector `[1e-8, 1e-9, 1e-10, 1e-11]' to `[1e-8, 1e-9, 0, 0]'. Numbers this small generally arise from roundoff noise. If the numbers you are using really are legitimately this small, you should avoid using the `c 0' through `c 9' commands. (The plain `c c' command rounds to the current precision but does not clip small numbers.) One more property of `c 0' through `c 9', and of `c c' with a prefix argument, is that integer-valued floats are converted to plain integers, so that `c 1' on `[1., 1.5, 2., 2.5, 3.]' produces `[1, 1.5, 2, 2.5, 3]'. This is not done for huge numbers (`1e100' is technically an integer-valued float, but you wouldn't want it automatically converted to a 100-digit integer). With the Hyperbolic flag, `H c c' and `H c 0' through `H c 9' operate non-pervasively [`clean'].  File: calc, Node: Date Arithmetic, Next: Financial Functions, Prev: Conversions, Up: Arithmetic 9.5 Date Arithmetic =================== The commands described in this section perform various conversions and calculations involving date forms (*note Date Forms::). They use the `t' (for time/date) prefix key followed by shifted letters. The simplest date arithmetic is done using the regular `+' and `-' commands. In particular, adding a number to a date form advances the date form by a certain number of days; adding an HMS form to a date form advances the date by a certain amount of time; and subtracting two date forms produces a difference measured in days. The commands described here provide additional, more specialized operations on dates. Many of these commands accept a numeric prefix argument; if you give plain `C-u' as the prefix, these commands will instead take the additional argument from the top of the stack. * Menu: * Date Conversions:: * Date Functions:: * Time Zones:: * Business Days::  File: calc, Node: Date Conversions, Next: Date Functions, Prev: Date Arithmetic, Up: Date Arithmetic 9.5.1 Date Conversions ---------------------- The `t D' (`calc-date') [`date'] command converts a date form into a number, measured in days since Jan 1, 1 AD. The result will be an integer if DATE is a pure date form, or a fraction or float if DATE is a date/time form. Or, if its argument is a number, it converts this number into a date form. With a numeric prefix argument, `t D' takes that many objects (up to six) from the top of the stack and interprets them in one of the following ways: The `date(YEAR, MONTH, DAY)' function builds a pure date form out of the specified year, month, and day, which must all be integers. YEAR is a year number, such as 1991 (_not_ the same as 91!). MONTH must be an integer in the range 1 to 12; DAY must be in the range 1 to 31. If the specified month has fewer than 31 days and DAY is too large, the equivalent day in the following month will be used. The `date(MONTH, DAY)' function builds a pure date form using the current year, as determined by the real-time clock. The `date(YEAR, MONTH, DAY, HMS)' function builds a date/time form using an HMS form. The `date(YEAR, MONTH, DAY, HOUR, MINUTE, SECOND)' function builds a date/time form. HOUR should be an integer in the range 0 to 23; MINUTE should be an integer in the range 0 to 59; SECOND should be any real number in the range `[0 .. 60)'. The last two arguments default to zero if omitted. The `t J' (`calc-julian') [`julian'] command converts a date form into a Julian day count, which is the number of days since noon (GMT) on Jan 1, 4713 BC. A pure date is converted to an integer Julian count representing noon of that day. A date/time form is converted to an exact floating-point Julian count, adjusted to interpret the date form in the current time zone but the Julian day count in Greenwich Mean Time. A numeric prefix argument allows you to specify the time zone; *note Time Zones::. Use a prefix of zero to suppress the time zone adjustment. Note that pure date forms are never time-zone adjusted. This command can also do the opposite conversion, from a Julian day count (either an integer day, or a floating-point day and time in the GMT zone), into a pure date form or a date/time form in the current or specified time zone. The `t U' (`calc-unix-time') [`unixtime'] command converts a date form into a Unix time value, which is the number of seconds since midnight on Jan 1, 1970, or vice-versa. The numeric result will be an integer if the current precision is 12 or less; for higher precisions, the result may be a float with (PRECISION-12) digits after the decimal. Just as for `t J', the numeric time is interpreted in the GMT time zone and the date form is interpreted in the current or specified zone. Some systems use Unix-like numbering but with the local time zone; give a prefix of zero to suppress the adjustment if so. The `t C' (`calc-convert-time-zones') [`tzconv'] command converts a date form from one time zone to another. You are prompted for each time zone name in turn; you can answer with any suitable Calc time zone expression (*note Time Zones::). If you answer either prompt with a blank line, the local time zone is used for that prompt. You can also answer the first prompt with `$' to take the two time zone names from the stack (and the date to be converted from the third stack level).  File: calc, Node: Date Functions, Next: Business Days, Prev: Date Conversions, Up: Date Arithmetic 9.5.2 Date Functions -------------------- The `t N' (`calc-now') [`now'] command pushes the current date and time on the stack as a date form. The time is reported in terms of the specified time zone; with no numeric prefix argument, `t N' reports for the current time zone. The `t P' (`calc-date-part') command extracts one part of a date form. The prefix argument specifies the part; with no argument, this command prompts for a part code from 1 to 9. The various part codes are described in the following paragraphs. The `M-1 t P' [`year'] function extracts the year number from a date form as an integer, e.g., 1991. This and the following functions will also accept a real number for an argument, which is interpreted as a standard Calc day number. Note that this function will never return zero, since the year 1 BC immediately precedes the year 1 AD. The `M-2 t P' [`month'] function extracts the month number from a date form as an integer in the range 1 to 12. The `M-3 t P' [`day'] function extracts the day number from a date form as an integer in the range 1 to 31. The `M-4 t P' [`hour'] function extracts the hour from a date form as an integer in the range 0 (midnight) to 23. Note that 24-hour time is always used. This returns zero for a pure date form. This function (and the following two) also accept HMS forms as input. The `M-5 t P' [`minute'] function extracts the minute from a date form as an integer in the range 0 to 59. The `M-6 t P' [`second'] function extracts the second from a date form. If the current precision is 12 or less, the result is an integer in the range 0 to 59. For higher precisions, the result may instead be a floating-point number. The `M-7 t P' [`weekday'] function extracts the weekday number from a date form as an integer in the range 0 (Sunday) to 6 (Saturday). The `M-8 t P' [`yearday'] function extracts the day-of-year number from a date form as an integer in the range 1 (January 1) to 366 (December 31 of a leap year). The `M-9 t P' [`time'] function extracts the time portion of a date form as an HMS form. This returns `0@ 0' 0"' for a pure date form. The `t M' (`calc-new-month') [`newmonth'] command computes a new date form that represents the first day of the month specified by the input date. The result is always a pure date form; only the year and month numbers of the input are retained. With a numeric prefix argument N in the range from 1 to 31, `t M' computes the Nth day of the month. (If N is greater than the actual number of days in the month, or if N is zero, the last day of the month is used.) The `t Y' (`calc-new-year') [`newyear'] command computes a new pure date form that represents the first day of the year specified by the input. The month, day, and time of the input date form are lost. With a numeric prefix argument N in the range from 1 to 366, `t Y' computes the Nth day of the year (366 is treated as 365 in non-leap years). A prefix argument of 0 computes the last day of the year (December 31). A negative prefix argument from -1 to -12 computes the first day of the Nth month of the year. The `t W' (`calc-new-week') [`newweek'] command computes a new pure date form that represents the Sunday on or before the input date. With a numeric prefix argument, it can be made to use any day of the week as the starting day; the argument must be in the range from 0 (Sunday) to 6 (Saturday). This function always subtracts between 0 and 6 days from the input date. Here's an example use of `newweek': Find the date of the next Wednesday after a given date. Using `M-3 t W' or `newweek(d, 3)' will give you the _preceding_ Wednesday, so `newweek(d+7, 3)' will give you the following Wednesday. A further look at the definition of `newweek' shows that if the input date is itself a Wednesday, this formula will return the Wednesday one week in the future. An exercise for the reader is to modify this formula to yield the same day if the input is already a Wednesday. Another interesting exercise is to preserve the time-of-day portion of the input (`newweek' resets the time to midnight; hint: how can `newweek' be defined in terms of the `weekday' function?). The `pwday(DATE)' function (not on any key) computes the day-of-month number of the Sunday on or before DATE. With two arguments, `pwday(DATE, DAY)' computes the day number of the Sunday on or before day number DAY of the month specified by DATE. The DAY must be in the range from 7 to 31; if the day number is greater than the actual number of days in the month, the true number of days is used instead. Thus `pwday(DATE, 7)' finds the first Sunday of the month, and `pwday(DATE, 31)' finds the last Sunday of the month. With a third WEEKDAY argument, `pwday' can be made to look for any day of the week instead of Sunday. The `t I' (`calc-inc-month') [`incmonth'] command increases a date form by one month, or by an arbitrary number of months specified by a numeric prefix argument. The time portion, if any, of the date form stays the same. The day also stays the same, except that if the new month has fewer days the day number may be reduced to lie in the valid range. For example, `incmonth()' produces `'. Because of this, `t I t I' and `M-2 t I' do not always give the same results (`' versus `' in this case). The `incyear(DATE, STEP)' function increases a date form by the specified number of years, which may be any positive or negative integer. Note that `incyear(d, n)' is equivalent to `incmonth(d, 12*n)', but these do not have simple equivalents in terms of day arithmetic because months and years have varying lengths. If the STEP argument is omitted, 1 year is assumed. There is no keyboard command for this function; use `C-u 12 t I' instead. There is no `newday' function at all because `F' [`floor'] serves this purpose. Similarly, instead of `incday' and `incweek' simply use `d + n' or `d + 7 n'. *Note Basic Arithmetic::, for the `f ]' [`incr'] command which can adjust a date/time form by a certain number of seconds.  File: calc, Node: Business Days, Next: Time Zones, Prev: Date Functions, Up: Date Arithmetic 9.5.3 Business Days ------------------- Often time is measured in "business days" or "working days," where weekends and holidays are skipped. Calc's normal date arithmetic functions use calendar days, so that subtracting two consecutive Mondays will yield a difference of 7 days. By contrast, subtracting two consecutive Mondays would yield 5 business days (assuming two-day weekends and the absence of holidays). The `t +' (`calc-business-days-plus') [`badd'] and `t -' (`calc-business-days-minus') [`bsub'] commands perform arithmetic using business days. For `t +', one argument must be a date form and the other must be a real number (positive or negative). If the number is not an integer, then a certain amount of time is added as well as a number of days; for example, adding 0.5 business days to a time in Friday evening will produce a time in Monday morning. It is also possible to add an HMS form; adding `12@ 0' 0"' also adds half a business day. For `t -', the arguments are either a date form and a number or HMS form, or two date forms, in which case the result is the number of business days between the two dates. By default, Calc considers any day that is not a Saturday or Sunday to be a business day. You can define any number of additional holidays by editing the variable `Holidays'. (There is an `s H' convenience command for editing this variable.) Initially, `Holidays' contains the vector `[sat, sun]'. Entries in the `Holidays' vector may be any of the following kinds of objects: * Date forms (pure dates, not date/time forms). These specify particular days which are to be treated as holidays. * Intervals of date forms. These specify a range of days, all of which are holidays (e.g., Christmas week). *Note Interval Forms::. * Nested vectors of date forms. Each date form in the vector is considered to be a holiday. * Any Calc formula which evaluates to one of the above three things. If the formula involves the variable `y', it stands for a yearly repeating holiday; `y' will take on various year numbers like 1992. For example, `date(y, 12, 25)' specifies Christmas day, and `newweek(date(y, 11, 7), 4) + 21' specifies Thanksgiving (which is held on the fourth Thursday of November). If the formula involves the variable `m', that variable takes on month numbers from 1 to 12: `date(y, m, 15)' is a holiday that takes place on the 15th of every month. * A weekday name, such as `sat' or `sun'. This is really a variable whose name is a three-letter, lower-case day name. * An interval of year numbers (integers). This specifies the span of years over which this holiday list is to be considered valid. Any business-day arithmetic that goes outside this range will result in an error message. Use this if you are including an explicit list of holidays, rather than a formula to generate them, and you want to make sure you don't accidentally go beyond the last point where the holidays you entered are complete. If there is no limiting interval in the `Holidays' vector, the default `[1 .. 2737]' is used. (This is the absolute range of years for which Calc's business-day algorithms will operate.) * An interval of HMS forms. This specifies the span of hours that are to be considered one business day. For example, if this range is `[9@ 0' 0" .. 17@ 0' 0"]' (i.e., 9am to 5pm), then the business day is only eight hours long, so that `1.5 t +' on `<4:00pm Fri Dec 13, 1991>' will add one business day and four business hours to produce `<12:00pm Tue Dec 17, 1991>'. Likewise, `t -' will now express differences in time as fractions of an eight-hour day. Times before 9am will be treated as 9am by business date arithmetic, and times at or after 5pm will be treated as 4:59:59pm. If there is no HMS interval in `Holidays', the full 24-hour day `[0 0' 0" .. 24 0' 0"]' is assumed. (Regardless of the type of bounds you specify, the interval is treated as inclusive on the low end and exclusive on the high end, so that the work day goes from 9am up to, but not including, 5pm.) If the `Holidays' vector is empty, then `t +' and `t -' will act just like `+' and `-' because there will then be no difference between business days and calendar days. Calc expands the intervals and formulas you give into a complete list of holidays for internal use. This is done mainly to make sure it can detect multiple holidays. (For example, `' is both New Year's Day and a Sunday, but Calc's algorithms take care to count it only once when figuring the number of holidays between two dates.) Since the complete list of holidays for all the years from 1 to 2737 would be huge, Calc actually computes only the part of the list between the smallest and largest years that have been involved in business-day calculations so far. Normally, you won't have to worry about this. Keep in mind, however, that if you do one calculation for 1992, and another for 1792, even if both involve only a small range of years, Calc will still work out all the holidays that fall in that 200-year span. If you add a (positive) number of days to a date form that falls on a weekend or holiday, the date form is treated as if it were the most recent business day. (Thus adding one business day to a Friday, Saturday, or Sunday will all yield the following Monday.) If you subtract a number of days from a weekend or holiday, the date is effectively on the following business day. (So subtracting one business day from Saturday, Sunday, or Monday yields the preceding Friday.) The difference between two dates one or both of which fall on holidays equals the number of actual business days between them. These conventions are consistent in the sense that, if you add N business days to any date, the difference between the result and the original date will come out to N business days. (It can't be completely consistent though; a subtraction followed by an addition might come out a bit differently, since `t +' is incapable of producing a date that falls on a weekend or holiday.) There is a `holiday' function, not on any keys, that takes any date form and returns 1 if that date falls on a weekend or holiday, as defined in `Holidays', or 0 if the date is a business day.  File: calc, Node: Time Zones, Prev: Business Days, Up: Date Arithmetic 9.5.4 Time Zones ---------------- Time zones and daylight saving time are a complicated business. The conversions to and from Julian and Unix-style dates automatically compute the correct time zone and daylight saving adjustment to use, provided they can figure out this information. This section describes Calc's time zone adjustment algorithm in detail, in case you want to do conversions in different time zones or in case Calc's algorithms can't determine the right correction to use. Adjustments for time zones and daylight saving time are done by `t U', `t J', `t N', and `t C', but not by any other commands. In particular, ` - ' evaluates to exactly 30 days even though there is a daylight-saving transition in between. This is also true for Julian pure dates: `julian() - julian()'. But Julian and Unix date/times will adjust for daylight saving time: using Calc's default daylight saving time rule (see the explanation below), `julian(<12am may 1 1991>) - julian(<12am apr 1 1991>)' evaluates to `29.95833' (that's 29 days and 23 hours) because one hour was lost when daylight saving commenced on April 7, 1991. In brief, the idiom `julian(DATE1) - julian(DATE2)' computes the actual number of 24-hour periods between two dates, whereas `DATE1 - DATE2' computes the number of calendar days between two dates without taking daylight saving into account. The `calc-time-zone' [`tzone'] command converts the time zone specified by its numeric prefix argument into a number of seconds difference from Greenwich mean time (GMT). If the argument is a number, the result is simply that value multiplied by 3600. Typical arguments for North America are 5 (Eastern) or 8 (Pacific). If Daylight Saving time is in effect, one hour should be subtracted from the normal difference. If you give a prefix of plain `C-u', `calc-time-zone' (like other date arithmetic commands that include a time zone argument) takes the zone argument from the top of the stack. (In the case of `t J' and `t U', the normal argument is then taken from the second-to-top stack position.) This allows you to give a non-integer time zone adjustment. The time-zone argument can also be an HMS form, or it can be a variable which is a time zone name in upper- or lower-case. For example `tzone(PST) = tzone(8)' and `tzone(pdt) = tzone(7)' (for Pacific standard and daylight saving times, respectively). North American and European time zone names are defined as follows; note that for each time zone there is one name for standard time, another for daylight saving time, and a third for "generalized" time in which the daylight saving adjustment is computed from context. YST PST MST CST EST AST NST GMT WET MET MEZ 9 8 7 6 5 4 3.5 0 -1 -2 -2 YDT PDT MDT CDT EDT ADT NDT BST WETDST METDST MESZ 8 7 6 5 4 3 2.5 -1 -2 -3 -3 YGT PGT MGT CGT EGT AGT NGT BGT WEGT MEGT MEGZ 9/8 8/7 7/6 6/5 5/4 4/3 3.5/2.5 0/-1 -1/-2 -2/-3 -2/-3 To define time zone names that do not appear in the above table, you must modify the Lisp variable `math-tzone-names'. This is a list of lists describing the different time zone names; its structure is best explained by an example. The three entries for Pacific Time look like this: ( ( "PST" 8 0 ) ; Name as an upper-case string, then standard ( "PDT" 8 -1 ) ; adjustment, then daylight saving adjustment. ( "PGT" 8 "PST" "PDT" ) ) ; Generalized time zone. With no arguments, `calc-time-zone' or `tzone()' will by default get the time zone and daylight saving information from the calendar (*note Calendar/Diary: (emacs)Daylight Saving.). To use a different time zone, or if the calendar does not give the desired result, you can set the Calc variable `TimeZone' (which is by default `nil') to an appropriate time zone name. (The easiest way to do this is to edit the `TimeZone' variable using Calc's `s T' command, then use the `s p' (`calc-permanent-variable') command to save the value of `TimeZone' permanently.) If the time zone given by `TimeZone' is a generalized time zone, e.g., `EGT', Calc examines the date being converted to tell whether to use standard or daylight saving time. But if the current time zone is explicit, e.g., `EST' or `EDT', then that adjustment is used exactly and Calc's daylight saving algorithm is not consulted. The special time zone name `local' is equivalent to no argument; i.e., it uses the information obtained from the calendar. The `t J' and `t U' commands with no numeric prefix arguments do the same thing as `tzone()'; namely, use the information from the calendar if `TimeZone' is `nil', otherwise use the time zone given by `TimeZone'. When Calc computes the daylight saving information itself (i.e., when the `TimeZone' variable is set), it will by default consider daylight saving time to begin at 2 a.m. on the second Sunday of March (for years from 2007 on) or on the last Sunday in April (for years before 2007), and to end at 2 a.m. on the first Sunday of November. (for years from 2007 on) or the last Sunday in October (for years before 2007). These are the rules that have been in effect in much of North America since 1966 and take into account the rule change that began in 2007. If you are in a country that uses different rules for computing daylight saving time, you have two choices: Write your own daylight saving hook, or control time zones explicitly by setting the `TimeZone' variable and/or always giving a time-zone argument for the conversion functions. The Lisp variable `math-daylight-savings-hook' holds the name of a function that is used to compute the daylight saving adjustment for a given date. The default is `math-std-daylight-savings', which computes an adjustment (either 0 or -1) using the North American rules given above. The daylight saving hook function is called with four arguments: The date, as a floating-point number in standard Calc format; a six-element list of the date decomposed into year, month, day, hour, minute, and second, respectively; a string which contains the generalized time zone name in upper-case, e.g., `"WEGT"'; and a special adjustment to be applied to the hour value when converting into a generalized time zone (see below). The Lisp function `math-prev-weekday-in-month' is useful for daylight saving computations. This is an internal version of the user-level `pwday' function described in the previous section. It takes four arguments: The floating-point date value, the corresponding six-element date list, the day-of-month number, and the weekday number (0-6). The default daylight saving hook ignores the time zone name, but a more sophisticated hook could use different algorithms for different time zones. It would also be possible to use different algorithms depending on the year number, but the default hook always uses the algorithm for 1987 and later. Here is a listing of the default daylight saving hook: (defun math-std-daylight-savings (date dt zone bump) (cond ((< (nth 1 dt) 4) 0) ((= (nth 1 dt) 4) (let ((sunday (math-prev-weekday-in-month date dt 7 0))) (cond ((< (nth 2 dt) sunday) 0) ((= (nth 2 dt) sunday) (if (>= (nth 3 dt) (+ 3 bump)) -1 0)) (t -1)))) ((< (nth 1 dt) 10) -1) ((= (nth 1 dt) 10) (let ((sunday (math-prev-weekday-in-month date dt 31 0))) (cond ((< (nth 2 dt) sunday) -1) ((= (nth 2 dt) sunday) (if (>= (nth 3 dt) (+ 2 bump)) 0 -1)) (t 0)))) (t 0)) ) The `bump' parameter is equal to zero when Calc is converting from a date form in a generalized time zone into a GMT date value. It is -1 when Calc is converting in the other direction. The adjustments shown above ensure that the conversion behaves correctly and reasonably around the 2 a.m. transition in each direction. There is a "missing" hour between 2 a.m. and 3 a.m. at the beginning of daylight saving time; converting a date/time form that falls in this hour results in a time value for the following hour, from 3 a.m. to 4 a.m. At the end of daylight saving time, the hour from 1 a.m. to 2 a.m. repeats itself; converting a date/time form that falls in this hour results in a time value for the first manifestation of that time (_not_ the one that occurs one hour later). If `math-daylight-savings-hook' is `nil', then the daylight saving adjustment is always taken to be zero. In algebraic formulas, `tzone(ZONE, DATE)' computes the time zone adjustment for a given zone name at a given date. The DATE is ignored unless ZONE is a generalized time zone. If DATE is a date form, the daylight saving computation is applied to it as it appears. If DATE is a numeric date value, it is adjusted for the daylight-saving version of ZONE before being given to the daylight saving hook. This odd-sounding rule ensures that the daylight-saving computation is always done in local time, not in the GMT time that a numeric DATE is typically represented in. The `dsadj(DATE, ZONE)' function computes the daylight saving adjustment that is appropriate for DATE in time zone ZONE. If ZONE is explicitly in or not in daylight saving time (e.g., `PDT' or `PST') the DATE is ignored. If ZONE is a generalized time zone, the algorithms described above are used. If ZONE is omitted, the computation is done for the current time zone.  File: calc, Node: Financial Functions, Next: Binary Functions, Prev: Date Arithmetic, Up: Arithmetic 9.6 Financial Functions ======================= Calc's financial or business functions use the `b' prefix key followed by a shifted letter. (The `b' prefix followed by a lower-case letter is used for operations on binary numbers.) Note that the rate and the number of intervals given to these functions must be on the same time scale, e.g., both months or both years. Mixing an annual interest rate with a time expressed in months will give you very wrong answers! It is wise to compute these functions to a higher precision than you really need, just to make sure your answer is correct to the last penny; also, you may wish to check the definitions at the end of this section to make sure the functions have the meaning you expect. * Menu: * Percentages:: * Future Value:: * Present Value:: * Related Financial Functions:: * Depreciation Functions:: * Definitions of Financial Functions::  File: calc, Node: Percentages, Next: Future Value, Prev: Financial Functions, Up: Financial Functions 9.6.1 Percentages ----------------- The `M-%' (`calc-percent') command takes a percentage value, say 5.4, and converts it to an equivalent actual number. For example, `5.4 M-%' enters 0.054 on the stack. (That's the or key combined with `%'.) Actually, `M-%' creates a formula of the form `5.4%'. You can enter `5.4%' yourself during algebraic entry. The `%' operator simply means, "the preceding value divided by 100." The `%' operator has very high precedence, so that `1+8%' is interpreted as `1+(8%)', not as `(1+8)%'. (The `%' operator is just a postfix notation for the `percent' function, just like `20!' is the notation for `fact(20)', or twenty-factorial.) The formula `5.4%' would normally evaluate immediately to 0.054, but the `M-%' command suppresses evaluation as it puts the formula onto the stack. However, the next Calc command that uses the formula `5.4%' will evaluate it as its first step. The net effect is that you get to look at `5.4%' on the stack, but Calc commands see it as `0.054', which is what they expect. In particular, `5.4%' and `0.054' are suitable values for the RATE arguments of the various financial functions, but the number `5.4' is probably _not_ suitable--it represents a rate of 540 percent! The key sequence `M-% *' effectively means "percent-of." For example, `68 25 M-% *' computes 17, which is 25% of 68 (and also 68% of 25, which comes out to the same thing). The `c %' (`calc-convert-percent') command converts the value on the top of the stack from numeric to percentage form. For example, if 0.08 is on the stack, `c %' converts it to `8%'. The quantity is the same, it's just represented differently. (Contrast this with `M-%', which would convert this number to `0.08%'.) The `=' key is a convenient way to convert a formula like `8%' back to numeric form, 0.08. To compute what percentage one quantity is of another quantity, use `/ c %'. For example, `17 68 / c %' displays `25%'. The `b %' (`calc-percent-change') [`relch'] command calculates the percentage change from one number to another. For example, `40 50 b %' produces the answer `25%', since 50 is 25% larger than 40. A negative result represents a decrease: `50 40 b %' produces `-20%', since 40 is 20% smaller than 50. (The answers are different in magnitude because, in the first case, we're increasing by 25% of 40, but in the second case, we're decreasing by 20% of 50.) The effect of `40 50 b %' is to compute `(50-40)/40', converting the answer to percentage form as if by `c %'.  File: calc, Node: Future Value, Next: Present Value, Prev: Percentages, Up: Financial Functions 9.6.2 Future Value ------------------ The `b F' (`calc-fin-fv') [`fv'] command computes the future value of an investment. It takes three arguments from the stack: `fv(RATE, N, PAYMENT)'. If you give payments of PAYMENT every year for N years, and the money you have paid earns interest at RATE per year, then this function tells you what your investment would be worth at the end of the period. (The actual interval doesn't have to be years, as long as N and RATE are expressed in terms of the same intervals.) This function assumes payments occur at the _end_ of each interval. The `I b F' [`fvb'] command does the same computation, but assuming your payments are at the beginning of each interval. Suppose you plan to deposit $1000 per year in a savings account earning 5.4% interest, starting right now. How much will be in the account after five years? `fvb(5.4%, 5, 1000) = 5870.73'. Thus you will have earned $870 worth of interest over the years. Using the stack, this calculation would have been `5.4 M-% 5 1000 I b F'. Note that the rate is expressed as a number between 0 and 1, _not_ as a percentage. The `H b F' [`fvl'] command computes the future value of an initial lump sum investment. Suppose you could deposit those five thousand dollars in the bank right now; how much would they be worth in five years? `fvl(5.4%, 5, 5000) = 6503.89'. The algebraic functions `fv' and `fvb' accept an optional fourth argument, which is used as an initial lump sum in the sense of `fvl'. In other words, `fv(RATE, N, PAYMENT, INITIAL) = fv(RATE, N, PAYMENT) + fvl(RATE, N, INITIAL)'. To illustrate the relationships between these functions, we could do the `fvb' calculation "by hand" using `fvl'. The final balance will be the sum of the contributions of our five deposits at various times. The first deposit earns interest for five years: `fvl(5.4%, 5, 1000) = 1300.78'. The second deposit only earns interest for four years: `fvl(5.4%, 4, 1000) = 1234.13'. And so on down to the last deposit, which earns one year's interest: `fvl(5.4%, 1, 1000) = 1054.00'. The sum of these five values is, sure enough, $5870.73, just as was computed by `fvb' directly. What does `fv(5.4%, 5, 1000) = 5569.96' mean? The payments are now at the ends of the periods. The end of one year is the same as the beginning of the next, so what this really means is that we've lost the payment at year zero (which contributed $1300.78), but we're now counting the payment at year five (which, since it didn't have a chance to earn interest, counts as $1000). Indeed, `5569.96 = 5870.73 - 1300.78 + 1000' (give or take a bit of roundoff error).  File: calc, Node: Present Value, Next: Related Financial Functions, Prev: Future Value, Up: Financial Functions 9.6.3 Present Value ------------------- The `b P' (`calc-fin-pv') [`pv'] command computes the present value of an investment. Like `fv', it takes three arguments: `pv(RATE, N, PAYMENT)'. It computes the present value of a series of regular payments. Suppose you have the chance to make an investment that will pay $2000 per year over the next four years; as you receive these payments you can put them in the bank at 9% interest. You want to know whether it is better to make the investment, or to keep the money in the bank where it earns 9% interest right from the start. The calculation `pv(9%, 4, 2000)' gives the result 6479.44. If your initial investment must be less than this, say, $6000, then the investment is worthwhile. But if you had to put up $7000, then it would be better just to leave it in the bank. Here is the interpretation of the result of `pv': You are trying to compare the return from the investment you are considering, which is `fv(9%, 4, 2000) = 9146.26', with the return from leaving the money in the bank, which is `fvl(9%, 4, X)' where X is the amount of money you would have to put up in advance. The `pv' function finds the break-even point, `x = 6479.44', at which `fvl(9%, 4, 6479.44)' is also equal to 9146.26. This is the largest amount you should be willing to invest. The `I b P' [`pvb'] command solves the same problem, but with payments occurring at the beginning of each interval. It has the same relationship to `fvb' as `pv' has to `fv'. For example `pvb(9%, 4, 2000) = 7062.59', a larger number than `pv' produced because we get to start earning interest on the return from our investment sooner. The `H b P' [`pvl'] command computes the present value of an investment that will pay off in one lump sum at the end of the period. For example, if we get our $8000 all at the end of the four years, `pvl(9%, 4, 8000) = 5667.40'. This is much less than `pv' reported, because we don't earn any interest on the return from this investment. Note that `pvl' and `fvl' are simple inverses: `fvl(9%, 4, 5667.40) = 8000'. You can give an optional fourth lump-sum argument to `pv' and `pvb'; this is handled in exactly the same way as the fourth argument for `fv' and `fvb'. The `b N' (`calc-fin-npv') [`npv'] command computes the net present value of a series of irregular investments. The first argument is the interest rate. The second argument is a vector which represents the expected return from the investment at the end of each interval. For example, if the rate represents a yearly interest rate, then the vector elements are the return from the first year, second year, and so on. Thus, `npv(9%, [2000,2000,2000,2000]) = pv(9%, 4, 2000) = 6479.44'. Obviously this function is more interesting when the payments are not all the same! The `npv' function can actually have two or more arguments. Multiple arguments are interpreted in the same way as for the vector statistical functions like `vsum'. *Note Single-Variable Statistics::. Basically, if there are several payment arguments, each either a vector or a plain number, all these values are collected left-to-right into the complete list of payments. A numeric prefix argument on the `b N' command says how many payment values or vectors to take from the stack. The `I b N' [`npvb'] command computes the net present value where payments occur at the beginning of each interval rather than at the end.