""" Legendre Series (:mod: `numpy.polynomial.legendre`) =================================================== .. currentmodule:: numpy.polynomial.polynomial This module provides a number of objects (mostly functions) useful for dealing with Legendre series, including a `Legendre` class that encapsulates the usual arithmetic operations. (General information on how this module represents and works with such polynomials is in the docstring for its "parent" sub-package, `numpy.polynomial`). Constants --------- .. autosummary:: :toctree: generated/ legdomain Legendre series default domain, [-1,1]. legzero Legendre series that evaluates identically to 0. legone Legendre series that evaluates identically to 1. legx Legendre series for the identity map, ``f(x) = x``. Arithmetic ---------- .. autosummary:: :toctree: generated/ legmulx multiply a Legendre series in P_i(x) by x. legadd add two Legendre series. legsub subtract one Legendre series from another. legmul multiply two Legendre series. legdiv divide one Legendre series by another. legpow raise a Legendre series to an positive integer power legval evaluate a Legendre series at given points. Calculus -------- .. autosummary:: :toctree: generated/ legder differentiate a Legendre series. legint integrate a Legendre series. Misc Functions -------------- .. autosummary:: :toctree: generated/ legfromroots create a Legendre series with specified roots. legroots find the roots of a Legendre series. legvander Vandermonde-like matrix for Legendre polynomials. legfit least-squares fit returning a Legendre series. legtrim trim leading coefficients from a Legendre series. legline Legendre series representing given straight line. leg2poly convert a Legendre series to a polynomial. poly2leg convert a polynomial to a Legendre series. Classes ------- Legendre A Legendre series class. See also -------- numpy.polynomial.polynomial numpy.polynomial.chebyshev numpy.polynomial.laguerre numpy.polynomial.hermite numpy.polynomial.hermite_e """ from __future__ import division __all__ = ['legzero', 'legone', 'legx', 'legdomain', 'legline', 'legadd', 'legsub', 'legmulx', 'legmul', 'legdiv', 'legpow', 'legval', 'legder', 'legint', 'leg2poly', 'poly2leg', 'legfromroots', 'legvander', 'legfit', 'legtrim', 'legroots', 'Legendre'] import numpy as np import numpy.linalg as la import polyutils as pu import warnings from polytemplate import polytemplate legtrim = pu.trimcoef def poly2leg(pol) : """ Convert a polynomial to a Legendre series. Convert an array representing the coefficients of a polynomial (relative to the "standard" basis) ordered from lowest degree to highest, to an array of the coefficients of the equivalent Legendre series, ordered from lowest to highest degree. Parameters ---------- pol : array_like 1-d array containing the polynomial coefficients Returns ------- cs : ndarray 1-d array containing the coefficients of the equivalent Legendre series. See Also -------- leg2poly Notes ----- The easy way to do conversions between polynomial basis sets is to use the convert method of a class instance. Examples -------- >>> from numpy import polynomial as P >>> p = P.Polynomial(np.arange(4)) >>> p Polynomial([ 0., 1., 2., 3.], [-1., 1.]) >>> c = P.Legendre(P.poly2leg(p.coef)) >>> c Legendre([ 1. , 3.25, 1. , 0.75], [-1., 1.]) """ [pol] = pu.as_series([pol]) deg = len(pol) - 1 res = 0 for i in range(deg, -1, -1) : res = legadd(legmulx(res), pol[i]) return res def leg2poly(cs) : """ Convert a Legendre series to a polynomial. Convert an array representing the coefficients of a Legendre series, ordered from lowest degree to highest, to an array of the coefficients of the equivalent polynomial (relative to the "standard" basis) ordered from lowest to highest degree. Parameters ---------- cs : array_like 1-d array containing the Legendre series coefficients, ordered from lowest order term to highest. Returns ------- pol : ndarray 1-d array containing the coefficients of the equivalent polynomial (relative to the "standard" basis) ordered from lowest order term to highest. See Also -------- poly2leg Notes ----- The easy way to do conversions between polynomial basis sets is to use the convert method of a class instance. Examples -------- >>> c = P.Legendre(range(4)) >>> c Legendre([ 0., 1., 2., 3.], [-1., 1.]) >>> p = c.convert(kind=P.Polynomial) >>> p Polynomial([-1. , -3.5, 3. , 7.5], [-1., 1.]) >>> P.leg2poly(range(4)) array([-1. , -3.5, 3. , 7.5]) """ from polynomial import polyadd, polysub, polymulx [cs] = pu.as_series([cs]) n = len(cs) if n < 3: return cs else: c0 = cs[-2] c1 = cs[-1] # i is the current degree of c1 for i in range(n - 1, 1, -1) : tmp = c0 c0 = polysub(cs[i - 2], (c1*(i - 1))/i) c1 = polyadd(tmp, (polymulx(c1)*(2*i - 1))/i) return polyadd(c0, polymulx(c1)) # # These are constant arrays are of integer type so as to be compatible # with the widest range of other types, such as Decimal. # # Legendre legdomain = np.array([-1,1]) # Legendre coefficients representing zero. legzero = np.array([0]) # Legendre coefficients representing one. legone = np.array([1]) # Legendre coefficients representing the identity x. legx = np.array([0,1]) def legline(off, scl) : """ Legendre series whose graph is a straight line. Parameters ---------- off, scl : scalars The specified line is given by ``off + scl*x``. Returns ------- y : ndarray This module's representation of the Legendre series for ``off + scl*x``. See Also -------- polyline, chebline Examples -------- >>> import numpy.polynomial.legendre as L >>> L.legline(3,2) array([3, 2]) >>> L.legval(-3, L.legline(3,2)) # should be -3 -3.0 """ if scl != 0 : return np.array([off,scl]) else : return np.array([off]) def legfromroots(roots) : """ Generate a Legendre series with the given roots. Return the array of coefficients for the P-series whose roots (a.k.a. "zeros") are given by *roots*. The returned array of coefficients is ordered from lowest order "term" to highest, and zeros of multiplicity greater than one must be included in *roots* a number of times equal to their multiplicity (e.g., if `2` is a root of multiplicity three, then [2,2,2] must be in *roots*). Parameters ---------- roots : array_like Sequence containing the roots. Returns ------- out : ndarray 1-d array of the Legendre series coefficients, ordered from low to high. If all roots are real, ``out.dtype`` is a float type; otherwise, ``out.dtype`` is a complex type, even if all the coefficients in the result are real (see Examples below). See Also -------- polyfromroots, chebfromroots Notes ----- What is returned are the :math:`c_i` such that: .. math:: \\sum_{i=0}^{n} c_i*P_i(x) = \\prod_{i=0}^{n} (x - roots[i]) where ``n == len(roots)`` and :math:`P_i(x)` is the `i`-th Legendre (basis) polynomial over the domain `[-1,1]`. Note that, unlike `polyfromroots`, due to the nature of the Legendre basis set, the above identity *does not* imply :math:`c_n = 1` identically (see Examples). Examples -------- >>> import numpy.polynomial.legendre as L >>> L.legfromroots((-1,0,1)) # x^3 - x relative to the standard basis array([ 0. , -0.4, 0. , 0.4]) >>> j = complex(0,1) >>> L.legfromroots((-j,j)) # x^2 + 1 relative to the standard basis array([ 1.33333333+0.j, 0.00000000+0.j, 0.66666667+0.j]) """ if len(roots) == 0 : return np.ones(1) else : [roots] = pu.as_series([roots], trim=False) roots.sort() p = [legline(-r, 1) for r in roots] n = len(p) while n > 1: m, r = divmod(n, 2) tmp = [legmul(p[i], p[i+m]) for i in range(m)] if r: tmp[0] = legmul(tmp[0], p[-1]) p = tmp n = m return p[0] def legadd(c1, c2): """ Add one Legendre series to another. Returns the sum of two Legendre series `c1` + `c2`. The arguments are sequences of coefficients ordered from lowest order term to highest, i.e., [1,2,3] represents the series ``P_0 + 2*P_1 + 3*P_2``. Parameters ---------- c1, c2 : array_like 1-d arrays of Legendre series coefficients ordered from low to high. Returns ------- out : ndarray Array representing the Legendre series of their sum. See Also -------- legsub, legmul, legdiv, legpow Notes ----- Unlike multiplication, division, etc., the sum of two Legendre series is a Legendre series (without having to "reproject" the result onto the basis set) so addition, just like that of "standard" polynomials, is simply "component-wise." Examples -------- >>> from numpy.polynomial import legendre as L >>> c1 = (1,2,3) >>> c2 = (3,2,1) >>> L.legadd(c1,c2) array([ 4., 4., 4.]) """ # c1, c2 are trimmed copies [c1, c2] = pu.as_series([c1, c2]) if len(c1) > len(c2) : c1[:c2.size] += c2 ret = c1 else : c2[:c1.size] += c1 ret = c2 return pu.trimseq(ret) def legsub(c1, c2): """ Subtract one Legendre series from another. Returns the difference of two Legendre series `c1` - `c2`. The sequences of coefficients are from lowest order term to highest, i.e., [1,2,3] represents the series ``P_0 + 2*P_1 + 3*P_2``. Parameters ---------- c1, c2 : array_like 1-d arrays of Legendre series coefficients ordered from low to high. Returns ------- out : ndarray Of Legendre series coefficients representing their difference. See Also -------- legadd, legmul, legdiv, legpow Notes ----- Unlike multiplication, division, etc., the difference of two Legendre series is a Legendre series (without having to "reproject" the result onto the basis set) so subtraction, just like that of "standard" polynomials, is simply "component-wise." Examples -------- >>> from numpy.polynomial import legendre as L >>> c1 = (1,2,3) >>> c2 = (3,2,1) >>> L.legsub(c1,c2) array([-2., 0., 2.]) >>> L.legsub(c2,c1) # -C.legsub(c1,c2) array([ 2., 0., -2.]) """ # c1, c2 are trimmed copies [c1, c2] = pu.as_series([c1, c2]) if len(c1) > len(c2) : c1[:c2.size] -= c2 ret = c1 else : c2 = -c2 c2[:c1.size] += c1 ret = c2 return pu.trimseq(ret) def legmulx(cs): """Multiply a Legendre series by x. Multiply the Legendre series `cs` by x, where x is the independent variable. Parameters ---------- cs : array_like 1-d array of Legendre series coefficients ordered from low to high. Returns ------- out : ndarray Array representing the result of the multiplication. Notes ----- The multiplication uses the recursion relationship for Legendre polynomials in the form .. math:: xP_i(x) = ((i + 1)*P_{i + 1}(x) + i*P_{i - 1}(x))/(2i + 1) """ # cs is a trimmed copy [cs] = pu.as_series([cs]) # The zero series needs special treatment if len(cs) == 1 and cs[0] == 0: return cs prd = np.empty(len(cs) + 1, dtype=cs.dtype) prd[0] = cs[0]*0 prd[1] = cs[0] for i in range(1, len(cs)): j = i + 1 k = i - 1 s = i + j prd[j] = (cs[i]*j)/s prd[k] += (cs[i]*i)/s return prd def legmul(c1, c2): """ Multiply one Legendre series by another. Returns the product of two Legendre series `c1` * `c2`. The arguments are sequences of coefficients, from lowest order "term" to highest, e.g., [1,2,3] represents the series ``P_0 + 2*P_1 + 3*P_2``. Parameters ---------- c1, c2 : array_like 1-d arrays of Legendre series coefficients ordered from low to high. Returns ------- out : ndarray Of Legendre series coefficients representing their product. See Also -------- legadd, legsub, legdiv, legpow Notes ----- In general, the (polynomial) product of two C-series results in terms that are not in the Legendre polynomial basis set. Thus, to express the product as a Legendre series, it is necessary to "re-project" the product onto said basis set, which may produce "un-intuitive" (but correct) results; see Examples section below. Examples -------- >>> from numpy.polynomial import legendre as L >>> c1 = (1,2,3) >>> c2 = (3,2) >>> P.legmul(c1,c2) # multiplication requires "reprojection" array([ 4.33333333, 10.4 , 11.66666667, 3.6 ]) """ # s1, s2 are trimmed copies [c1, c2] = pu.as_series([c1, c2]) if len(c1) > len(c2): cs = c2 xs = c1 else: cs = c1 xs = c2 if len(cs) == 1: c0 = cs[0]*xs c1 = 0 elif len(cs) == 2: c0 = cs[0]*xs c1 = cs[1]*xs else : nd = len(cs) c0 = cs[-2]*xs c1 = cs[-1]*xs for i in range(3, len(cs) + 1) : tmp = c0 nd = nd - 1 c0 = legsub(cs[-i]*xs, (c1*(nd - 1))/nd) c1 = legadd(tmp, (legmulx(c1)*(2*nd - 1))/nd) return legadd(c0, legmulx(c1)) def legdiv(c1, c2): """ Divide one Legendre series by another. Returns the quotient-with-remainder of two Legendre series `c1` / `c2`. The arguments are sequences of coefficients from lowest order "term" to highest, e.g., [1,2,3] represents the series ``P_0 + 2*P_1 + 3*P_2``. Parameters ---------- c1, c2 : array_like 1-D arrays of Legendre series coefficients ordered from low to high. Returns ------- quo, rem : ndarrays Of Legendre series coefficients representing the quotient and remainder. See Also -------- legadd, legsub, legmul, legpow Notes ----- In general, the (polynomial) division of one Legendre series by another results in quotient and remainder terms that are not in the Legendre polynomial basis set. Thus, to express these results as a Legendre series, it is necessary to "re-project" the results onto the Legendre basis set, which may produce "un-intuitive" (but correct) results; see Examples section below. Examples -------- >>> from numpy.polynomial import legendre as L >>> c1 = (1,2,3) >>> c2 = (3,2,1) >>> L.legdiv(c1,c2) # quotient "intuitive," remainder not (array([ 3.]), array([-8., -4.])) >>> c2 = (0,1,2,3) >>> L.legdiv(c2,c1) # neither "intuitive" (array([-0.07407407, 1.66666667]), array([-1.03703704, -2.51851852])) """ # c1, c2 are trimmed copies [c1, c2] = pu.as_series([c1, c2]) if c2[-1] == 0 : raise ZeroDivisionError() lc1 = len(c1) lc2 = len(c2) if lc1 < lc2 : return c1[:1]*0, c1 elif lc2 == 1 : return c1/c2[-1], c1[:1]*0 else : quo = np.empty(lc1 - lc2 + 1, dtype=c1.dtype) rem = c1 for i in range(lc1 - lc2, - 1, -1): p = legmul([0]*i + [1], c2) q = rem[-1]/p[-1] rem = rem[:-1] - q*p[:-1] quo[i] = q return quo, pu.trimseq(rem) def legpow(cs, pow, maxpower=16) : """Raise a Legendre series to a power. Returns the Legendre series `cs` raised to the power `pow`. The arguement `cs` is a sequence of coefficients ordered from low to high. i.e., [1,2,3] is the series ``P_0 + 2*P_1 + 3*P_2.`` Parameters ---------- cs : array_like 1d array of Legendre series coefficients ordered from low to high. pow : integer Power to which the series will be raised maxpower : integer, optional Maximum power allowed. This is mainly to limit growth of the series to umanageable size. Default is 16 Returns ------- coef : ndarray Legendre series of power. See Also -------- legadd, legsub, legmul, legdiv Examples -------- """ # cs is a trimmed copy [cs] = pu.as_series([cs]) power = int(pow) if power != pow or power < 0 : raise ValueError("Power must be a non-negative integer.") elif maxpower is not None and power > maxpower : raise ValueError("Power is too large") elif power == 0 : return np.array([1], dtype=cs.dtype) elif power == 1 : return cs else : # This can be made more efficient by using powers of two # in the usual way. prd = cs for i in range(2, power + 1) : prd = legmul(prd, cs) return prd def legder(cs, m=1, scl=1) : """ Differentiate a Legendre series. Returns the series `cs` differentiated `m` times. At each iteration the result is multiplied by `scl` (the scaling factor is for use in a linear change of variable). The argument `cs` is the sequence of coefficients from lowest order "term" to highest, e.g., [1,2,3] represents the series ``P_0 + 2*P_1 + 3*P_2``. Parameters ---------- cs : array_like 1-D array of Legendre series coefficients ordered from low to high. m : int, optional Number of derivatives taken, must be non-negative. (Default: 1) scl : scalar, optional Each differentiation is multiplied by `scl`. The end result is multiplication by ``scl**m``. This is for use in a linear change of variable. (Default: 1) Returns ------- der : ndarray Legendre series of the derivative. See Also -------- legint Notes ----- In general, the result of differentiating a Legendre series does not resemble the same operation on a power series. Thus the result of this function may be "un-intuitive," albeit correct; see Examples section below. Examples -------- >>> from numpy.polynomial import legendre as L >>> cs = (1,2,3,4) >>> L.legder(cs) array([ 6., 9., 20.]) >>> L.legder(cs,3) array([ 60.]) >>> L.legder(cs,scl=-1) array([ -6., -9., -20.]) >>> L.legder(cs,2,-1) array([ 9., 60.]) """ cnt = int(m) if cnt != m: raise ValueError, "The order of derivation must be integer" if cnt < 0 : raise ValueError, "The order of derivation must be non-negative" # cs is a trimmed copy [cs] = pu.as_series([cs]) if cnt == 0: return cs elif cnt >= len(cs): return cs[:1]*0 else : for i in range(cnt): n = len(cs) - 1 cs *= scl der = np.empty(n, dtype=cs.dtype) for j in range(n, 0, -1): der[j - 1] = (2*j - 1)*cs[j] cs[j - 2] += cs[j] cs = der return cs def legint(cs, m=1, k=[], lbnd=0, scl=1): """ Integrate a Legendre series. Returns a Legendre series that is the Legendre series `cs`, integrated `m` times from `lbnd` to `x`. At each iteration the resulting series is **multiplied** by `scl` and an integration constant, `k`, is added. The scaling factor is for use in a linear change of variable. ("Buyer beware": note that, depending on what one is doing, one may want `scl` to be the reciprocal of what one might expect; for more information, see the Notes section below.) The argument `cs` is a sequence of coefficients, from lowest order Legendre series "term" to highest, e.g., [1,2,3] represents the series :math:`P_0(x) + 2P_1(x) + 3P_2(x)`. Parameters ---------- cs : array_like 1-d array of Legendre series coefficients, ordered from low to high. m : int, optional Order of integration, must be positive. (Default: 1) k : {[], list, scalar}, optional Integration constant(s). The value of the first integral at ``lbnd`` is the first value in the list, the value of the second integral at ``lbnd`` is the second value, etc. If ``k == []`` (the default), all constants are set to zero. If ``m == 1``, a single scalar can be given instead of a list. lbnd : scalar, optional The lower bound of the integral. (Default: 0) scl : scalar, optional Following each integration the result is *multiplied* by `scl` before the integration constant is added. (Default: 1) Returns ------- S : ndarray Legendre series coefficients of the integral. Raises ------ ValueError If ``m < 0``, ``len(k) > m``, ``np.isscalar(lbnd) == False``, or ``np.isscalar(scl) == False``. See Also -------- legder Notes ----- Note that the result of each integration is *multiplied* by `scl`. Why is this important to note? Say one is making a linear change of variable :math:`u = ax + b` in an integral relative to `x`. Then :math:`dx = du/a`, so one will need to set `scl` equal to :math:`1/a` - perhaps not what one would have first thought. Also note that, in general, the result of integrating a C-series needs to be "re-projected" onto the C-series basis set. Thus, typically, the result of this function is "un-intuitive," albeit correct; see Examples section below. Examples -------- >>> from numpy.polynomial import legendre as L >>> cs = (1,2,3) >>> L.legint(cs) array([ 0.33333333, 0.4 , 0.66666667, 0.6 ]) >>> L.legint(cs,3) array([ 1.66666667e-02, -1.78571429e-02, 4.76190476e-02, -1.73472348e-18, 1.90476190e-02, 9.52380952e-03]) >>> L.legint(cs, k=3) array([ 3.33333333, 0.4 , 0.66666667, 0.6 ]) >>> L.legint(cs, lbnd=-2) array([ 7.33333333, 0.4 , 0.66666667, 0.6 ]) >>> L.legint(cs, scl=2) array([ 0.66666667, 0.8 , 1.33333333, 1.2 ]) """ cnt = int(m) if np.isscalar(k) : k = [k] if cnt != m: raise ValueError, "The order of integration must be integer" if cnt < 0 : raise ValueError, "The order of integration must be non-negative" if len(k) > cnt : raise ValueError, "Too many integration constants" # cs is a trimmed copy [cs] = pu.as_series([cs]) if cnt == 0: return cs k = list(k) + [0]*(cnt - len(k)) for i in range(cnt) : n = len(cs) cs *= scl if n == 1 and cs[0] == 0: cs[0] += k[i] else: tmp = np.empty(n + 1, dtype=cs.dtype) tmp[0] = cs[0]*0 tmp[1] = cs[0] for j in range(1, n): t = cs[j]/(2*j + 1) tmp[j + 1] = t tmp[j - 1] -= t tmp[0] += k[i] - legval(lbnd, tmp) cs = tmp return cs def legval(x, cs): """Evaluate a Legendre series. If `cs` is of length `n`, this function returns : ``p(x) = cs[0]*P_0(x) + cs[1]*P_1(x) + ... + cs[n-1]*P_{n-1}(x)`` If x is a sequence or array then p(x) will have the same shape as x. If r is a ring_like object that supports multiplication and addition by the values in `cs`, then an object of the same type is returned. Parameters ---------- x : array_like, ring_like Array of numbers or objects that support multiplication and addition with themselves and with the elements of `cs`. cs : array_like 1-d array of Legendre coefficients ordered from low to high. Returns ------- values : ndarray, ring_like If the return is an ndarray then it has the same shape as `x`. See Also -------- legfit Notes ----- The evaluation uses Clenshaw recursion, aka synthetic division. Examples -------- """ # cs is a trimmed copy [cs] = pu.as_series([cs]) if isinstance(x, tuple) or isinstance(x, list) : x = np.asarray(x) if len(cs) == 1 : c0 = cs[0] c1 = 0 elif len(cs) == 2 : c0 = cs[0] c1 = cs[1] else : nd = len(cs) c0 = cs[-2] c1 = cs[-1] for i in range(3, len(cs) + 1) : tmp = c0 nd = nd - 1 c0 = cs[-i] - (c1*(nd - 1))/nd c1 = tmp + (c1*x*(2*nd - 1))/nd return c0 + c1*x def legvander(x, deg) : """Vandermonde matrix of given degree. Returns the Vandermonde matrix of degree `deg` and sample points `x`. This isn't a true Vandermonde matrix because `x` can be an arbitrary ndarray and the Legendre polynomials aren't powers. If ``V`` is the returned matrix and `x` is a 2d array, then the elements of ``V`` are ``V[i,j,k] = P_k(x[i,j])``, where ``P_k`` is the Legendre polynomial of degree ``k``. Parameters ---------- x : array_like Array of points. The values are converted to double or complex doubles. If x is scalar it is converted to a 1D array. deg : integer Degree of the resulting matrix. Returns ------- vander : Vandermonde matrix. The shape of the returned matrix is ``x.shape + (deg+1,)``. The last index is the degree. """ ideg = int(deg) if ideg != deg: raise ValueError("deg must be integer") if ideg < 0: raise ValueError("deg must be non-negative") x = np.array(x, copy=0, ndmin=1) + 0.0 v = np.empty((ideg + 1,) + x.shape, dtype=x.dtype) # Use forward recursion to generate the entries. This is not as accurate # as reverse recursion in this application but it is more efficient. v[0] = x*0 + 1 if ideg > 0 : v[1] = x for i in range(2, ideg + 1) : v[i] = (v[i-1]*x*(2*i - 1) - v[i-2]*(i - 1))/i return np.rollaxis(v, 0, v.ndim) def legfit(x, y, deg, rcond=None, full=False, w=None): """ Least squares fit of Legendre series to data. Return the coefficients of a Legendre series of degree `deg` that is the least squares fit to the data values `y` given at points `x`. If `y` is 1-D the returned coefficients will also be 1-D. If `y` is 2-D multiple fits are done, one for each column of `y`, and the resulting coefficients are stored in the corresponding columns of a 2-D return. The fitted polynomial(s) are in the form .. math:: p(x) = c_0 + c_1 * L_1(x) + ... + c_n * L_n(x), where `n` is `deg`. Parameters ---------- x : array_like, shape (M,) x-coordinates of the M sample points ``(x[i], y[i])``. y : array_like, shape (M,) or (M, K) y-coordinates of the sample points. Several data sets of sample points sharing the same x-coordinates can be fitted at once by passing in a 2D-array that contains one dataset per column. deg : int Degree of the fitting polynomial rcond : float, optional Relative condition number of the fit. Singular values smaller than this relative to the largest singular value will be ignored. The default value is len(x)*eps, where eps is the relative precision of the float type, about 2e-16 in most cases. full : bool, optional Switch determining nature of return value. When it is False (the default) just the coefficients are returned, when True diagnostic information from the singular value decomposition is also returned. w : array_like, shape (`M`,), optional Weights. If not None, the contribution of each point ``(x[i],y[i])`` to the fit is weighted by `w[i]`. Ideally the weights are chosen so that the errors of the products ``w[i]*y[i]`` all have the same variance. The default value is None. .. versionadded:: 1.5.0 Returns ------- coef : ndarray, shape (M,) or (M, K) Legendre coefficients ordered from low to high. If `y` was 2-D, the coefficients for the data in column k of `y` are in column `k`. [residuals, rank, singular_values, rcond] : present when `full` = True Residuals of the least-squares fit, the effective rank of the scaled Vandermonde matrix and its singular values, and the specified value of `rcond`. For more details, see `linalg.lstsq`. Warns ----- RankWarning The rank of the coefficient matrix in the least-squares fit is deficient. The warning is only raised if `full` = False. The warnings can be turned off by >>> import warnings >>> warnings.simplefilter('ignore', RankWarning) See Also -------- chebfit, polyfit, lagfit, hermfit, hermefit legval : Evaluates a Legendre series. legvander : Vandermonde matrix of Legendre series. legweight : Legendre weight function (= 1). linalg.lstsq : Computes a least-squares fit from the matrix. scipy.interpolate.UnivariateSpline : Computes spline fits. Notes ----- The solution is the coefficients of the Legendre series `p` that minimizes the sum of the weighted squared errors .. math:: E = \\sum_j w_j^2 * |y_j - p(x_j)|^2, where :math:`w_j` are the weights. This problem is solved by setting up as the (typically) overdetermined matrix equation .. math:: V(x) * c = w * y, where `V` is the weighted pseudo Vandermonde matrix of `x`, `c` are the coefficients to be solved for, `w` are the weights, and `y` are the observed values. This equation is then solved using the singular value decomposition of `V`. If some of the singular values of `V` are so small that they are neglected, then a `RankWarning` will be issued. This means that the coeficient values may be poorly determined. Using a lower order fit will usually get rid of the warning. The `rcond` parameter can also be set to a value smaller than its default, but the resulting fit may be spurious and have large contributions from roundoff error. Fits using Legendre series are usually better conditioned than fits using power series, but much can depend on the distribution of the sample points and the smoothness of the data. If the quality of the fit is inadequate splines may be a good alternative. References ---------- .. [1] Wikipedia, "Curve fitting", http://en.wikipedia.org/wiki/Curve_fitting Examples -------- """ order = int(deg) + 1 x = np.asarray(x) + 0.0 y = np.asarray(y) + 0.0 # check arguments. if deg < 0 : raise ValueError, "expected deg >= 0" if x.ndim != 1: raise TypeError, "expected 1D vector for x" if x.size == 0: raise TypeError, "expected non-empty vector for x" if y.ndim < 1 or y.ndim > 2 : raise TypeError, "expected 1D or 2D array for y" if len(x) != len(y): raise TypeError, "expected x and y to have same length" # set up the least squares matrices lhs = legvander(x, deg) rhs = y if w is not None: w = np.asarray(w) + 0.0 if w.ndim != 1: raise TypeError, "expected 1D vector for w" if len(x) != len(w): raise TypeError, "expected x and w to have same length" # apply weights if rhs.ndim == 2: lhs *= w[:, np.newaxis] rhs *= w[:, np.newaxis] else: lhs *= w[:, np.newaxis] rhs *= w # set rcond if rcond is None : rcond = len(x)*np.finfo(x.dtype).eps # scale the design matrix and solve the least squares equation scl = np.sqrt((lhs*lhs).sum(0)) c, resids, rank, s = la.lstsq(lhs/scl, rhs, rcond) c = (c.T/scl).T # warn on rank reduction if rank != order and not full: msg = "The fit may be poorly conditioned" warnings.warn(msg, pu.RankWarning) if full : return c, [resids, rank, s, rcond] else : return c def legcompanion(cs): """Return the scaled companion matrix of cs. The basis polynomials are scaled so that the companion matrix is symmetric when `cs` represents a single Legendre polynomial. This provides better eigenvalue estimates than the unscaled case and in the single polynomial case the eigenvalues are guaranteed to be real if `numpy.linalg.eigvalsh` is used to obtain them. Parameters ---------- cs : array_like 1-d array of Legendre series coefficients ordered from low to high degree. Returns ------- mat : ndarray Scaled companion matrix of dimensions (deg, deg). """ # cs is a trimmed copy [cs] = pu.as_series([cs]) if len(cs) < 2: raise ValueError('Series must have maximum degree of at least 1.') if len(cs) == 2: return np.array(-cs[0]/cs[1]) n = len(cs) - 1 mat = np.zeros((n, n), dtype=cs.dtype) scl = 1./np.sqrt(2*np.arange(n) + 1) top = mat.reshape(-1)[1::n+1] bot = mat.reshape(-1)[n::n+1] top[...] = np.arange(1, n)*scl[:n-1]*scl[1:n] bot[...] = top mat[:,-1] -= (cs[:-1]/cs[-1])*(scl/scl[-1])*(n/(2*n - 1)) return mat def legroots(cs): """ Compute the roots of a Legendre series. Returns the roots (a.k.a "zeros") of the Legendre series represented by `cs`, which is the sequence of coefficients from lowest order "term" to highest, e.g., [1,2,3] is the series ``L_0 + 2*L_1 + 3*L_2``. Parameters ---------- cs : array_like 1-d array of Legendre series coefficients ordered from low to high. maxiter : int, optional Maximum number of iterations of Newton to use in refining the roots. Returns ------- out : ndarray Sorted array of the roots. If all the roots are real, then so is the dtype of ``out``; otherwise, ``out``'s dtype is complex. See Also -------- polyroots chebroots Notes ----- The root estimates are obtained as the eigenvalues of the companion matrix, Roots far from the real interval [-1, 1] in the complex plane may have large errors due to the numerical instability of the Lengendre series for such values. Roots with multiplicity greater than 1 will also show larger errors as the value of the series near such points is relatively insensitive to errors in the roots. Isolated roots near the interval [-1, 1] can be improved by a few iterations of Newton's method. The Legendre series basis polynomials aren't powers of ``x`` so the results of this function may seem unintuitive. Examples -------- >>> import numpy.polynomial.legendre as leg >>> leg.legroots((1, 2, 3, 4)) # 4L_3 + 3L_2 + 2L_1 + 1L_0 has only real roots array([-0.85099543, -0.11407192, 0.51506735]) """ # cs is a trimmed copy [cs] = pu.as_series([cs]) if len(cs) < 2: return np.array([], dtype=cs.dtype) if len(cs) == 2: return np.array([-cs[0]/cs[1]]) m = legcompanion(cs) r = la.eigvals(m) r.sort() return r # # Legendre series class # exec polytemplate.substitute(name='Legendre', nick='leg', domain='[-1,1]')