# Author: Travis Oliphant 2001 import _quadpack import sys import numpy from numpy import inf, Inf __all__ = ['quad', 'dblquad', 'tplquad', 'quad_explain'] error = _quadpack.error def quad_explain(output=sys.stdout): """ Print extra information about integrate.quad() parameters and returns. Parameters ---------- output : instance with "write" method Information about `quad` is passed to ``output.write()``. Default is ``sys.stdout``. Returns ------- None """ output.write(""" Extra information for quad() inputs and outputs: If full_output is non-zero, then the third output argument (infodict) is a dictionary with entries as tabulated below. For infinite limits, the range is transformed to (0,1) and the optional outputs are given with respect to this transformed range. Let M be the input argument limit and let K be infodict['last']. The entries are: 'neval' : The number of function evaluations. 'last' : The number, K, of subintervals produced in the subdivision process. 'alist' : A rank-1 array of length M, the first K elements of which are the left end points of the subintervals in the partition of the integration range. 'blist' : A rank-1 array of length M, the first K elements of which are the right end points of the subintervals. 'rlist' : A rank-1 array of length M, the first K elements of which are the integral approximations on the subintervals. 'elist' : A rank-1 array of length M, the first K elements of which are the moduli of the absolute error estimates on the subintervals. 'iord' : A rank-1 integer array of length M, the first L elements of which are pointers to the error estimates over the subintervals with L=K if K<=M/2+2 or L=M+1-K otherwise. Let I be the sequence infodict['iord'] and let E be the sequence infodict['elist']. Then E[I[1]], ..., E[I[L]] forms a decreasing sequence. If the input argument points is provided (i.e. it is not None), the following additional outputs are placed in the output dictionary. Assume the points sequence is of length P. 'pts' : A rank-1 array of length P+2 containing the integration limits and the break points of the intervals in ascending order. This is an array giving the subintervals over which integration will occur. 'level' : A rank-1 integer array of length M (=limit), containing the subdivision levels of the subintervals, i.e., if (aa,bb) is a subinterval of (pts[1], pts[2]) where pts[0] and pts[2] are adjacent elements of infodict['pts'], then (aa,bb) has level l if |bb-aa|=|pts[2]-pts[1]| * 2**(-l). 'ndin' : A rank-1 integer array of length P+2. After the first integration over the intervals (pts[1], pts[2]), the error estimates over some of the intervals may have been increased artificially in order to put their subdivision forward. This array has ones in slots corresponding to the subintervals for which this happens. Weighting the integrand: The input variables, weight and wvar, are used to weight the integrand by a select list of functions. Different integration methods are used to compute the integral with these weighting functions. The possible values of weight and the corresponding weighting functions are. 'cos' : cos(w*x) : wvar = w 'sin' : sin(w*x) : wvar = w 'alg' : g(x) = ((x-a)**alpha)*((b-x)**beta) : wvar = (alpha, beta) 'alg-loga': g(x)*log(x-a) : wvar = (alpha, beta) 'alg-logb': g(x)*log(b-x) : wvar = (alpha, beta) 'alg-log' : g(x)*log(x-a)*log(b-x) : wvar = (alpha, beta) 'cauchy' : 1/(x-c) : wvar = c wvar holds the parameter w, (alpha, beta), or c depending on the weight selected. In these expressions, a and b are the integration limits. For the 'cos' and 'sin' weighting, additional inputs and outputs are available. For finite integration limits, the integration is performed using a Clenshaw-Curtis method which uses Chebyshev moments. For repeated calculations, these moments are saved in the output dictionary: 'momcom' : The maximum level of Chebyshev moments that have been computed, i.e., if M_c is infodict['momcom'] then the moments have been computed for intervals of length |b-a|* 2**(-l), l=0,1,...,M_c. 'nnlog' : A rank-1 integer array of length M(=limit), containing the subdivision levels of the subintervals, i.e., an element of this array is equal to l if the corresponding subinterval is |b-a|* 2**(-l). 'chebmo' : A rank-2 array of shape (25, maxp1) containing the computed Chebyshev moments. These can be passed on to an integration over the same interval by passing this array as the second element of the sequence wopts and passing infodict['momcom'] as the first element. If one of the integration limits is infinite, then a Fourier integral is computed (assuming w neq 0). If full_output is 1 and a numerical error is encountered, besides the error message attached to the output tuple, a dictionary is also appended to the output tuple which translates the error codes in the array info['ierlst'] to English messages. The output information dictionary contains the following entries instead of 'last', 'alist', 'blist', 'rlist', and 'elist': 'lst' : The number of subintervals needed for the integration (call it K_f). 'rslst' : A rank-1 array of length M_f=limlst, whose first K_f elements contain the integral contribution over the interval (a+(k-1)c, a+kc) where c = (2*floor(|w|) + 1) * pi / |w| and k=1,2,...,K_f. 'erlst' : A rank-1 array of length M_f containing the error estimate corresponding to the interval in the same position in infodict['rslist']. 'ierlst' : A rank-1 integer array of length M_f containing an error flag corresponding to the interval in the same position in infodict['rslist']. See the explanation dictionary (last entry in the output tuple) for the meaning of the codes. """) return def quad(func, a, b, args=(), full_output=0, epsabs=1.49e-8, epsrel=1.49e-8, limit=50, points=None, weight=None, wvar=None, wopts=None, maxp1=50, limlst=50): """ Compute a definite integral. Integrate func from a to b (possibly infinite interval) using a technique from the Fortran library QUADPACK. If func takes many arguments, it is integrated along the axis corresponding to the first argument. Use the keyword argument `args` to pass the other arguments. Run scipy.integrate.quad_explain() for more information on the more esoteric inputs and outputs. Parameters ---------- func : function A Python function or method to integrate. a : float Lower limit of integration (use -numpy.inf for -infinity). b : float Upper limit of integration (use numpy.inf for +infinity). args : tuple, optional extra arguments to pass to func full_output : int Non-zero to return a dictionary of integration information. If non-zero, warning messages are also suppressed and the message is appended to the output tuple. Returns ------- y : float The integral of func from a to b. abserr : float an estimate of the absolute error in the result. infodict : dict a dictionary containing additional information. Run scipy.integrate.quad_explain() for more information. message : a convergence message. explain : appended only with 'cos' or 'sin' weighting and infinite integration limits, it contains an explanation of the codes in infodict['ierlst'] Other Parameters ---------------- epsabs : absolute error tolerance. epsrel : relative error tolerance. limit : an upper bound on the number of subintervals used in the adaptive algorithm. points : a sequence of break points in the bounded integration interval where local difficulties of the integrand may occur (e.g., singularities, discontinuities). The sequence does not have to be sorted. weight : string indicating weighting function. wvar : variables for use with weighting functions. limlst : Upper bound on the number of cylces (>=3) for use with a sinusoidal weighting and an infinite end-point. wopts : Optional input for reusing Chebyshev moments. maxp1 : An upper bound on the number of Chebyshev moments. See Also -------- dblquad, tplquad : double and triple integrals fixed_quad : fixed-order Gaussian quadrature quadrature : adaptive Gaussian quadrature odeint, ode : ODE integrators simps, trapz, romb : integrators for sampled data scipy.special : for coefficients and roots of orthogonal polynomials Examples -------- Calculate :math:`\\int^4_0 x^2 dx` and compare with an analytic result >>> from scipy import integrate >>> x2 = lambda x: x**2 >>> integrate.quad(x2,0.,4.) (21.333333333333332, 2.3684757858670003e-13) >> print 4.**3/3 21.3333333333 Calculate :math:`\\int^\\infty_0 e^{-x} dx` >>> invexp = lambda x: exp(-x) >>> integrate.quad(invexp,0,inf) (0.99999999999999989, 5.8426061711142159e-11) >>> f = lambda x,a : a*x >>> y, err = integrate.quad(f, 0, 1, args=(1,)) >>> y 0.5 >>> y, err = integrate.quad(f, 0, 1, args=(3,)) >>> y 1.5 """ if type(args) != type(()): args = (args,) if (weight is None): retval = _quad(func,a,b,args,full_output,epsabs,epsrel,limit,points) else: retval = _quad_weight(func,a,b,args,full_output,epsabs,epsrel,limlst,limit,maxp1,weight,wvar,wopts) ier = retval[-1] if ier == 0: return retval[:-1] msgs = {80: "A Python error occurred possibly while calling the function.", 1: "The maximum number of subdivisions (%d) has been achieved.\n If increasing the limit yields no improvement it is advised to analyze \n the integrand in order to determine the difficulties. If the position of a \n local difficulty can be determined (singularity, discontinuity) one will \n probably gain from splitting up the interval and calling the integrator \n on the subranges. Perhaps a special-purpose integrator should be used." % limit, 2: "The occurrence of roundoff error is detected, which prevents \n the requested tolerance from being achieved. The error may be \n underestimated.", 3: "Extremely bad integrand behavior occurs at some points of the\n integration interval.", 4: "The algorithm does not converge. Roundoff error is detected\n in the extrapolation table. It is assumed that the requested tolerance\n cannot be achieved, and that the returned result (if full_output = 1) is \n the best which can be obtained.", 5: "The integral is probably divergent, or slowly convergent.", 6: "The input is invalid.", 7: "Abnormal termination of the routine. The estimates for result\n and error are less reliable. It is assumed that the requested accuracy\n has not been achieved.", 'unknown': "Unknown error."} if weight in ['cos','sin'] and (b == Inf or a == -Inf): msgs[1] = "The maximum number of cycles allowed has been achieved., e.e.\n of subintervals (a+(k-1)c, a+kc) where c = (2*int(abs(omega)+1))\n *pi/abs(omega), for k = 1, 2, ..., lst. One can allow more cycles by increasing the value of limlst. Look at info['ierlst'] with full_output=1." msgs[4] = "The extrapolation table constructed for convergence acceleration\n of the series formed by the integral contributions over the cycles, \n does not converge to within the requested accuracy. Look at \n info['ierlst'] with full_output=1." msgs[7] = "Bad integrand behavior occurs within one or more of the cycles.\n Location and type of the difficulty involved can be determined from \n the vector info['ierlist'] obtained with full_output=1." explain = {1: "The maximum number of subdivisions (= limit) has been \n achieved on this cycle.", 2: "The occurrence of roundoff error is detected and prevents\n the tolerance imposed on this cycle from being achieved.", 3: "Extremely bad integrand behavior occurs at some points of\n this cycle.", 4: "The integral over this cycle does not converge (to within the required accuracy) due ot roundoff in the extrapolation procedure invoked on this cycle. It is assumed that the result on this interval is the best which can be obtained.", 5: "The integral over this cycle is probably divergent or slowly convergent."} try: msg = msgs[ier] except KeyError: msg = msgs['unknown'] if ier in [1,2,3,4,5,7]: if full_output: if weight in ['cos','sin'] and (b == Inf or a == Inf): return retval[:-1] + (msg, explain) else: return retval[:-1] + (msg,) else: import warnings warnings.warn(msg) return retval[:-1] else: raise ValueError(msg) def _quad(func,a,b,args,full_output,epsabs,epsrel,limit,points): infbounds = 0 if (b != Inf and a != -Inf): pass # standard integration elif (b == Inf and a != -Inf): infbounds = 1 bound = a elif (b == Inf and a == -Inf): infbounds = 2 bound = 0 # ignored elif (b != Inf and a == -Inf): infbounds = -1 bound = b else: raise RuntimeError("Infinity comparisons don't work for you.") if points is None: if infbounds == 0: return _quadpack._qagse(func,a,b,args,full_output,epsabs,epsrel,limit) else: return _quadpack._qagie(func,bound,infbounds,args,full_output,epsabs,epsrel,limit) else: if infbounds !=0: raise ValueError("Infinity inputs cannot be used with break points.") else: nl = len(points) the_points = numpy.zeros((nl+2,), float) the_points[:nl] = points return _quadpack._qagpe(func,a,b,the_points,args,full_output,epsabs,epsrel,limit) def _quad_weight(func,a,b,args,full_output,epsabs,epsrel,limlst,limit,maxp1,weight,wvar,wopts): if weight not in ['cos','sin','alg','alg-loga','alg-logb','alg-log','cauchy']: raise ValueError("%s not a recognized weighting function." % weight) strdict = {'cos':1,'sin':2,'alg':1,'alg-loga':2,'alg-logb':3,'alg-log':4} if weight in ['cos','sin']: integr = strdict[weight] if (b != Inf and a != -Inf): # finite limits if wopts is None: # no precomputed chebyshev moments return _quadpack._qawoe(func,a,b,wvar,integr,args,full_output,epsabs,epsrel,limit,maxp1,1) else: # precomputed chebyshev moments momcom = wopts[0] chebcom = wopts[1] return _quadpack._qawoe(func,a,b,wvar,integr,args,full_output,epsabs,epsrel,limit,maxp1,2,momcom,chebcom) elif (b == Inf and a != -Inf): return _quadpack._qawfe(func,a,wvar,integr,args,full_output,epsabs,limlst,limit,maxp1) elif (b != Inf and a == -Inf): # remap function and interval if weight == 'cos': def thefunc(x,*myargs): y = -x func = myargs[0] myargs = (y,) + myargs[1:] return apply(func,myargs) else: def thefunc(x,*myargs): y = -x func = myargs[0] myargs = (y,) + myargs[1:] return -apply(func,myargs) args = (func,) + args return _quadpack._qawfe(thefunc,-b,wvar,integr,args,full_output,epsabs,limlst,limit,maxp1) else: raise ValueError("Cannot integrate with this weight from -Inf to +Inf.") else: if a in [-Inf,Inf] or b in [-Inf,Inf]: raise ValueError("Cannot integrate with this weight over an infinite interval.") if weight[:3] == 'alg': integr = strdict[weight] return _quadpack._qawse(func,a,b,wvar,integr,args,full_output,epsabs,epsrel,limit) else: # weight == 'cauchy' return _quadpack._qawce(func,a,b,wvar,args,full_output,epsabs,epsrel,limit) def _infunc(x,func,gfun,hfun,more_args): a = gfun(x) b = hfun(x) myargs = (x,) + more_args return quad(func,a,b,args=myargs)[0] def dblquad(func, a, b, gfun, hfun, args=(), epsabs=1.49e-8, epsrel=1.49e-8): """ Compute a double integral. Return the double (definite) integral of func(y,x) from x=a..b and y=gfun(x)..hfun(x). Parameters ---------- func : callable A Python function or method of at least two variables: y must be the first argument and x the second argument. (a,b) : tuple The limits of integration in x: a < b gfun : callable The lower boundary curve in y which is a function taking a single floating point argument (x) and returning a floating point result: a lambda function can be useful here. hfun : callable The upper boundary curve in y (same requirements as `gfun`). args : sequence, optional Extra arguments to pass to `func2d`. epsabs : float, optional Absolute tolerance passed directly to the inner 1-D quadrature integration. Default is 1.49e-8. epsrel : float Relative tolerance of the inner 1-D integrals. Default is 1.49e-8. Returns ------- y : float The resultant integral. abserr : float An estimate of the error. See also -------- quad : single integral tplquad : triple integral fixed_quad : fixed-order Gaussian quadrature quadrature : adaptive Gaussian quadrature odeint : ODE integrator ode : ODE integrator simps : integrator for sampled data romb : integrator for sampled data scipy.special : for coefficients and roots of orthogonal polynomials """ return quad(_infunc,a,b,(func,gfun,hfun,args),epsabs=epsabs,epsrel=epsrel) def _infunc2(y,x,func,qfun,rfun,more_args): a2 = qfun(x,y) b2 = rfun(x,y) myargs = (y,x) + more_args return quad(func,a2,b2,args=myargs)[0] def tplquad(func, a, b, gfun, hfun, qfun, rfun, args=(), epsabs=1.49e-8, epsrel=1.49e-8): """ Compute a triple (definite) integral. Return the triple integral of func(z, y, x) from x=a..b, y=gfun(x)..hfun(x), and z=qfun(x,y)..rfun(x,y) Parameters ---------- func : function A Python function or method of at least three variables in the order (z, y, x). (a,b) : tuple The limits of integration in x: a < b gfun : function The lower boundary curve in y which is a function taking a single floating point argument (x) and returning a floating point result: a lambda function can be useful here. hfun : function The upper boundary curve in y (same requirements as gfun). qfun : function The lower boundary surface in z. It must be a function that takes two floats in the order (x, y) and returns a float. rfun : function The upper boundary surface in z. (Same requirements as qfun.) args : Arguments Extra arguments to pass to func3d. epsabs : float, optional Absolute tolerance passed directly to the innermost 1-D quadrature integration. Default is 1.49e-8. epsrel : float, optional Relative tolerance of the innermost 1-D integrals. Default is 1.49e-8. Returns ------- y : float The resultant integral. abserr : float An estimate of the error. See Also -------- quad: Adaptive quadrature using QUADPACK quadrature: Adaptive Gaussian quadrature fixed_quad: Fixed-order Gaussian quadrature dblquad: Double integrals romb: Integrators for sampled data simps: Integrators for sampled data ode: ODE integrators odeint: ODE integrators scipy.special: For coefficients and roots of orthogonal polynomials """ return dblquad(_infunc2,a,b,gfun,hfun,(func,qfun,rfun,args),epsabs=epsabs,epsrel=epsrel)