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| """ | |
| ================================================== | |
| Legendre Series (:mod:`numpy.polynomial.legendre`) | |
| ================================================== | |
| 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`). | |
| Classes | |
| ------- | |
| .. autosummary:: | |
| :toctree: generated/ | |
| Legendre | |
| Constants | |
| --------- | |
| .. autosummary:: | |
| :toctree: generated/ | |
| legdomain | |
| legzero | |
| legone | |
| legx | |
| Arithmetic | |
| ---------- | |
| .. autosummary:: | |
| :toctree: generated/ | |
| legadd | |
| legsub | |
| legmulx | |
| legmul | |
| legdiv | |
| legpow | |
| legval | |
| legval2d | |
| legval3d | |
| leggrid2d | |
| leggrid3d | |
| Calculus | |
| -------- | |
| .. autosummary:: | |
| :toctree: generated/ | |
| legder | |
| legint | |
| Misc Functions | |
| -------------- | |
| .. autosummary:: | |
| :toctree: generated/ | |
| legfromroots | |
| legroots | |
| legvander | |
| legvander2d | |
| legvander3d | |
| leggauss | |
| legweight | |
| legcompanion | |
| legfit | |
| legtrim | |
| legline | |
| leg2poly | |
| poly2leg | |
| See also | |
| -------- | |
| numpy.polynomial | |
| """ | |
| import numpy as np | |
| import numpy.linalg as la | |
| from numpy.core.multiarray import normalize_axis_index | |
| from . import polyutils as pu | |
| from ._polybase import ABCPolyBase | |
| __all__ = [ | |
| 'legzero', 'legone', 'legx', 'legdomain', 'legline', 'legadd', | |
| 'legsub', 'legmulx', 'legmul', 'legdiv', 'legpow', 'legval', 'legder', | |
| 'legint', 'leg2poly', 'poly2leg', 'legfromroots', 'legvander', | |
| 'legfit', 'legtrim', 'legroots', 'Legendre', 'legval2d', 'legval3d', | |
| 'leggrid2d', 'leggrid3d', 'legvander2d', 'legvander3d', 'legcompanion', | |
| 'leggauss', 'legweight'] | |
| 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 | |
| ------- | |
| c : 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.], domain=[-1, 1], window=[-1, 1]) | |
| >>> c = P.Legendre(P.legendre.poly2leg(p.coef)) | |
| >>> c | |
| Legendre([ 1. , 3.25, 1. , 0.75], domain=[-1, 1], window=[-1, 1]) # may vary | |
| """ | |
| [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(c): | |
| """ | |
| 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 | |
| ---------- | |
| c : 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 | |
| -------- | |
| >>> from numpy import polynomial as P | |
| >>> c = P.Legendre(range(4)) | |
| >>> c | |
| Legendre([0., 1., 2., 3.], domain=[-1, 1], window=[-1, 1]) | |
| >>> p = c.convert(kind=P.Polynomial) | |
| >>> p | |
| Polynomial([-1. , -3.5, 3. , 7.5], domain=[-1., 1.], window=[-1., 1.]) | |
| >>> P.leg2poly(range(4)) | |
| array([-1. , -3.5, 3. , 7.5]) | |
| """ | |
| from .polynomial import polyadd, polysub, polymulx | |
| [c] = pu.as_series([c]) | |
| n = len(c) | |
| if n < 3: | |
| return c | |
| else: | |
| c0 = c[-2] | |
| c1 = c[-1] | |
| # i is the current degree of c1 | |
| for i in range(n - 1, 1, -1): | |
| tmp = c0 | |
| c0 = polysub(c[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 | |
| -------- | |
| numpy.polynomial.polynomial.polyline | |
| numpy.polynomial.chebyshev.chebline | |
| numpy.polynomial.laguerre.lagline | |
| numpy.polynomial.hermite.hermline | |
| numpy.polynomial.hermite_e.hermeline | |
| 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 given roots. | |
| The function returns the coefficients of the polynomial | |
| .. math:: p(x) = (x - r_0) * (x - r_1) * ... * (x - r_n), | |
| in Legendre form, where the `r_n` are the roots specified in `roots`. | |
| If a zero has multiplicity n, then it must appear in `roots` n times. | |
| For instance, if 2 is a root of multiplicity three and 3 is a root of | |
| multiplicity 2, then `roots` looks something like [2, 2, 2, 3, 3]. The | |
| roots can appear in any order. | |
| If the returned coefficients are `c`, then | |
| .. math:: p(x) = c_0 + c_1 * L_1(x) + ... + c_n * L_n(x) | |
| The coefficient of the last term is not generally 1 for monic | |
| polynomials in Legendre form. | |
| Parameters | |
| ---------- | |
| roots : array_like | |
| Sequence containing the roots. | |
| Returns | |
| ------- | |
| out : ndarray | |
| 1-D array of coefficients. If all roots are real then `out` is a | |
| real array, if some of the roots are complex, then `out` is complex | |
| even if all the coefficients in the result are real (see Examples | |
| below). | |
| See Also | |
| -------- | |
| numpy.polynomial.polynomial.polyfromroots | |
| numpy.polynomial.chebyshev.chebfromroots | |
| numpy.polynomial.laguerre.lagfromroots | |
| numpy.polynomial.hermite.hermfromroots | |
| numpy.polynomial.hermite_e.hermefromroots | |
| 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]) # may vary | |
| """ | |
| return pu._fromroots(legline, legmul, roots) | |
| 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, legmulx, 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.]) | |
| """ | |
| return pu._add(c1, c2) | |
| 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, legmulx, 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.]) | |
| """ | |
| return pu._sub(c1, c2) | |
| def legmulx(c): | |
| """Multiply a Legendre series by x. | |
| Multiply the Legendre series `c` by x, where x is the independent | |
| variable. | |
| Parameters | |
| ---------- | |
| c : array_like | |
| 1-D array of Legendre series coefficients ordered from low to | |
| high. | |
| Returns | |
| ------- | |
| out : ndarray | |
| Array representing the result of the multiplication. | |
| See Also | |
| -------- | |
| legadd, legmul, legmul, legdiv, legpow | |
| 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) | |
| Examples | |
| -------- | |
| >>> from numpy.polynomial import legendre as L | |
| >>> L.legmulx([1,2,3]) | |
| array([ 0.66666667, 2.2, 1.33333333, 1.8]) # may vary | |
| """ | |
| # c is a trimmed copy | |
| [c] = pu.as_series([c]) | |
| # The zero series needs special treatment | |
| if len(c) == 1 and c[0] == 0: | |
| return c | |
| prd = np.empty(len(c) + 1, dtype=c.dtype) | |
| prd[0] = c[0]*0 | |
| prd[1] = c[0] | |
| for i in range(1, len(c)): | |
| j = i + 1 | |
| k = i - 1 | |
| s = i + j | |
| prd[j] = (c[i]*j)/s | |
| prd[k] += (c[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, legmulx, 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 "reproject" the | |
| product onto said basis set, which may produce "unintuitive" (but | |
| correct) results; see Examples section below. | |
| Examples | |
| -------- | |
| >>> from numpy.polynomial import legendre as L | |
| >>> c1 = (1,2,3) | |
| >>> c2 = (3,2) | |
| >>> L.legmul(c1,c2) # multiplication requires "reprojection" | |
| array([ 4.33333333, 10.4 , 11.66666667, 3.6 ]) # may vary | |
| """ | |
| # s1, s2 are trimmed copies | |
| [c1, c2] = pu.as_series([c1, c2]) | |
| if len(c1) > len(c2): | |
| c = c2 | |
| xs = c1 | |
| else: | |
| c = c1 | |
| xs = c2 | |
| if len(c) == 1: | |
| c0 = c[0]*xs | |
| c1 = 0 | |
| elif len(c) == 2: | |
| c0 = c[0]*xs | |
| c1 = c[1]*xs | |
| else: | |
| nd = len(c) | |
| c0 = c[-2]*xs | |
| c1 = c[-1]*xs | |
| for i in range(3, len(c) + 1): | |
| tmp = c0 | |
| nd = nd - 1 | |
| c0 = legsub(c[-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, legmulx, 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 "reproject" the results onto the Legendre | |
| basis set, which may produce "unintuitive" (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])) # may vary | |
| """ | |
| return pu._div(legmul, c1, c2) | |
| def legpow(c, pow, maxpower=16): | |
| """Raise a Legendre series to a power. | |
| Returns the Legendre series `c` raised to the power `pow`. The | |
| argument `c` 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 | |
| ---------- | |
| c : array_like | |
| 1-D 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 unmanageable size. Default is 16 | |
| Returns | |
| ------- | |
| coef : ndarray | |
| Legendre series of power. | |
| See Also | |
| -------- | |
| legadd, legsub, legmulx, legmul, legdiv | |
| """ | |
| return pu._pow(legmul, c, pow, maxpower) | |
| def legder(c, m=1, scl=1, axis=0): | |
| """ | |
| Differentiate a Legendre series. | |
| Returns the Legendre series coefficients `c` differentiated `m` times | |
| along `axis`. At each iteration the result is multiplied by `scl` (the | |
| scaling factor is for use in a linear change of variable). The argument | |
| `c` is an array of coefficients from low to high degree along each | |
| axis, e.g., [1,2,3] represents the series ``1*L_0 + 2*L_1 + 3*L_2`` | |
| while [[1,2],[1,2]] represents ``1*L_0(x)*L_0(y) + 1*L_1(x)*L_0(y) + | |
| 2*L_0(x)*L_1(y) + 2*L_1(x)*L_1(y)`` if axis=0 is ``x`` and axis=1 is | |
| ``y``. | |
| Parameters | |
| ---------- | |
| c : array_like | |
| Array of Legendre series coefficients. If c is multidimensional the | |
| different axis correspond to different variables with the degree in | |
| each axis given by the corresponding index. | |
| 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) | |
| axis : int, optional | |
| Axis over which the derivative is taken. (Default: 0). | |
| .. versionadded:: 1.7.0 | |
| 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 "unintuitive," albeit correct; see Examples section | |
| below. | |
| Examples | |
| -------- | |
| >>> from numpy.polynomial import legendre as L | |
| >>> c = (1,2,3,4) | |
| >>> L.legder(c) | |
| array([ 6., 9., 20.]) | |
| >>> L.legder(c, 3) | |
| array([60.]) | |
| >>> L.legder(c, scl=-1) | |
| array([ -6., -9., -20.]) | |
| >>> L.legder(c, 2,-1) | |
| array([ 9., 60.]) | |
| """ | |
| c = np.array(c, ndmin=1, copy=True) | |
| if c.dtype.char in '?bBhHiIlLqQpP': | |
| c = c.astype(np.double) | |
| cnt = pu._deprecate_as_int(m, "the order of derivation") | |
| iaxis = pu._deprecate_as_int(axis, "the axis") | |
| if cnt < 0: | |
| raise ValueError("The order of derivation must be non-negative") | |
| iaxis = normalize_axis_index(iaxis, c.ndim) | |
| if cnt == 0: | |
| return c | |
| c = np.moveaxis(c, iaxis, 0) | |
| n = len(c) | |
| if cnt >= n: | |
| c = c[:1]*0 | |
| else: | |
| for i in range(cnt): | |
| n = n - 1 | |
| c *= scl | |
| der = np.empty((n,) + c.shape[1:], dtype=c.dtype) | |
| for j in range(n, 2, -1): | |
| der[j - 1] = (2*j - 1)*c[j] | |
| c[j - 2] += c[j] | |
| if n > 1: | |
| der[1] = 3*c[2] | |
| der[0] = c[1] | |
| c = der | |
| c = np.moveaxis(c, 0, iaxis) | |
| return c | |
| def legint(c, m=1, k=[], lbnd=0, scl=1, axis=0): | |
| """ | |
| Integrate a Legendre series. | |
| Returns the Legendre series coefficients `c` integrated `m` times from | |
| `lbnd` along `axis`. 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 `c` is an array of | |
| coefficients from low to high degree along each axis, e.g., [1,2,3] | |
| represents the series ``L_0 + 2*L_1 + 3*L_2`` while [[1,2],[1,2]] | |
| represents ``1*L_0(x)*L_0(y) + 1*L_1(x)*L_0(y) + 2*L_0(x)*L_1(y) + | |
| 2*L_1(x)*L_1(y)`` if axis=0 is ``x`` and axis=1 is ``y``. | |
| Parameters | |
| ---------- | |
| c : array_like | |
| Array of Legendre series coefficients. If c is multidimensional the | |
| different axis correspond to different variables with the degree in | |
| each axis given by the corresponding index. | |
| 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) | |
| axis : int, optional | |
| Axis over which the integral is taken. (Default: 0). | |
| .. versionadded:: 1.7.0 | |
| Returns | |
| ------- | |
| S : ndarray | |
| Legendre series coefficient array of the integral. | |
| Raises | |
| ------ | |
| ValueError | |
| If ``m < 0``, ``len(k) > m``, ``np.ndim(lbnd) != 0``, or | |
| ``np.ndim(scl) != 0``. | |
| 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 "reprojected" onto the C-series basis set. Thus, typically, | |
| the result of this function is "unintuitive," albeit correct; see | |
| Examples section below. | |
| Examples | |
| -------- | |
| >>> from numpy.polynomial import legendre as L | |
| >>> c = (1,2,3) | |
| >>> L.legint(c) | |
| array([ 0.33333333, 0.4 , 0.66666667, 0.6 ]) # may vary | |
| >>> L.legint(c, 3) | |
| array([ 1.66666667e-02, -1.78571429e-02, 4.76190476e-02, # may vary | |
| -1.73472348e-18, 1.90476190e-02, 9.52380952e-03]) | |
| >>> L.legint(c, k=3) | |
| array([ 3.33333333, 0.4 , 0.66666667, 0.6 ]) # may vary | |
| >>> L.legint(c, lbnd=-2) | |
| array([ 7.33333333, 0.4 , 0.66666667, 0.6 ]) # may vary | |
| >>> L.legint(c, scl=2) | |
| array([ 0.66666667, 0.8 , 1.33333333, 1.2 ]) # may vary | |
| """ | |
| c = np.array(c, ndmin=1, copy=True) | |
| if c.dtype.char in '?bBhHiIlLqQpP': | |
| c = c.astype(np.double) | |
| if not np.iterable(k): | |
| k = [k] | |
| cnt = pu._deprecate_as_int(m, "the order of integration") | |
| iaxis = pu._deprecate_as_int(axis, "the axis") | |
| if cnt < 0: | |
| raise ValueError("The order of integration must be non-negative") | |
| if len(k) > cnt: | |
| raise ValueError("Too many integration constants") | |
| if np.ndim(lbnd) != 0: | |
| raise ValueError("lbnd must be a scalar.") | |
| if np.ndim(scl) != 0: | |
| raise ValueError("scl must be a scalar.") | |
| iaxis = normalize_axis_index(iaxis, c.ndim) | |
| if cnt == 0: | |
| return c | |
| c = np.moveaxis(c, iaxis, 0) | |
| k = list(k) + [0]*(cnt - len(k)) | |
| for i in range(cnt): | |
| n = len(c) | |
| c *= scl | |
| if n == 1 and np.all(c[0] == 0): | |
| c[0] += k[i] | |
| else: | |
| tmp = np.empty((n + 1,) + c.shape[1:], dtype=c.dtype) | |
| tmp[0] = c[0]*0 | |
| tmp[1] = c[0] | |
| if n > 1: | |
| tmp[2] = c[1]/3 | |
| for j in range(2, n): | |
| t = c[j]/(2*j + 1) | |
| tmp[j + 1] = t | |
| tmp[j - 1] -= t | |
| tmp[0] += k[i] - legval(lbnd, tmp) | |
| c = tmp | |
| c = np.moveaxis(c, 0, iaxis) | |
| return c | |
| def legval(x, c, tensor=True): | |
| """ | |
| Evaluate a Legendre series at points x. | |
| If `c` is of length `n + 1`, this function returns the value: | |
| .. math:: p(x) = c_0 * L_0(x) + c_1 * L_1(x) + ... + c_n * L_n(x) | |
| The parameter `x` is converted to an array only if it is a tuple or a | |
| list, otherwise it is treated as a scalar. In either case, either `x` | |
| or its elements must support multiplication and addition both with | |
| themselves and with the elements of `c`. | |
| If `c` is a 1-D array, then `p(x)` will have the same shape as `x`. If | |
| `c` is multidimensional, then the shape of the result depends on the | |
| value of `tensor`. If `tensor` is true the shape will be c.shape[1:] + | |
| x.shape. If `tensor` is false the shape will be c.shape[1:]. Note that | |
| scalars have shape (,). | |
| Trailing zeros in the coefficients will be used in the evaluation, so | |
| they should be avoided if efficiency is a concern. | |
| Parameters | |
| ---------- | |
| x : array_like, compatible object | |
| If `x` is a list or tuple, it is converted to an ndarray, otherwise | |
| it is left unchanged and treated as a scalar. In either case, `x` | |
| or its elements must support addition and multiplication with | |
| with themselves and with the elements of `c`. | |
| c : array_like | |
| Array of coefficients ordered so that the coefficients for terms of | |
| degree n are contained in c[n]. If `c` is multidimensional the | |
| remaining indices enumerate multiple polynomials. In the two | |
| dimensional case the coefficients may be thought of as stored in | |
| the columns of `c`. | |
| tensor : boolean, optional | |
| If True, the shape of the coefficient array is extended with ones | |
| on the right, one for each dimension of `x`. Scalars have dimension 0 | |
| for this action. The result is that every column of coefficients in | |
| `c` is evaluated for every element of `x`. If False, `x` is broadcast | |
| over the columns of `c` for the evaluation. This keyword is useful | |
| when `c` is multidimensional. The default value is True. | |
| .. versionadded:: 1.7.0 | |
| Returns | |
| ------- | |
| values : ndarray, algebra_like | |
| The shape of the return value is described above. | |
| See Also | |
| -------- | |
| legval2d, leggrid2d, legval3d, leggrid3d | |
| Notes | |
| ----- | |
| The evaluation uses Clenshaw recursion, aka synthetic division. | |
| """ | |
| c = np.array(c, ndmin=1, copy=False) | |
| if c.dtype.char in '?bBhHiIlLqQpP': | |
| c = c.astype(np.double) | |
| if isinstance(x, (tuple, list)): | |
| x = np.asarray(x) | |
| if isinstance(x, np.ndarray) and tensor: | |
| c = c.reshape(c.shape + (1,)*x.ndim) | |
| if len(c) == 1: | |
| c0 = c[0] | |
| c1 = 0 | |
| elif len(c) == 2: | |
| c0 = c[0] | |
| c1 = c[1] | |
| else: | |
| nd = len(c) | |
| c0 = c[-2] | |
| c1 = c[-1] | |
| for i in range(3, len(c) + 1): | |
| tmp = c0 | |
| nd = nd - 1 | |
| c0 = c[-i] - (c1*(nd - 1))/nd | |
| c1 = tmp + (c1*x*(2*nd - 1))/nd | |
| return c0 + c1*x | |
| def legval2d(x, y, c): | |
| """ | |
| Evaluate a 2-D Legendre series at points (x, y). | |
| This function returns the values: | |
| .. math:: p(x,y) = \\sum_{i,j} c_{i,j} * L_i(x) * L_j(y) | |
| The parameters `x` and `y` are converted to arrays only if they are | |
| tuples or a lists, otherwise they are treated as a scalars and they | |
| must have the same shape after conversion. In either case, either `x` | |
| and `y` or their elements must support multiplication and addition both | |
| with themselves and with the elements of `c`. | |
| If `c` is a 1-D array a one is implicitly appended to its shape to make | |
| it 2-D. The shape of the result will be c.shape[2:] + x.shape. | |
| Parameters | |
| ---------- | |
| x, y : array_like, compatible objects | |
| The two dimensional series is evaluated at the points `(x, y)`, | |
| where `x` and `y` must have the same shape. If `x` or `y` is a list | |
| or tuple, it is first converted to an ndarray, otherwise it is left | |
| unchanged and if it isn't an ndarray it is treated as a scalar. | |
| c : array_like | |
| Array of coefficients ordered so that the coefficient of the term | |
| of multi-degree i,j is contained in ``c[i,j]``. If `c` has | |
| dimension greater than two the remaining indices enumerate multiple | |
| sets of coefficients. | |
| Returns | |
| ------- | |
| values : ndarray, compatible object | |
| The values of the two dimensional Legendre series at points formed | |
| from pairs of corresponding values from `x` and `y`. | |
| See Also | |
| -------- | |
| legval, leggrid2d, legval3d, leggrid3d | |
| Notes | |
| ----- | |
| .. versionadded:: 1.7.0 | |
| """ | |
| return pu._valnd(legval, c, x, y) | |
| def leggrid2d(x, y, c): | |
| """ | |
| Evaluate a 2-D Legendre series on the Cartesian product of x and y. | |
| This function returns the values: | |
| .. math:: p(a,b) = \\sum_{i,j} c_{i,j} * L_i(a) * L_j(b) | |
| where the points `(a, b)` consist of all pairs formed by taking | |
| `a` from `x` and `b` from `y`. The resulting points form a grid with | |
| `x` in the first dimension and `y` in the second. | |
| The parameters `x` and `y` are converted to arrays only if they are | |
| tuples or a lists, otherwise they are treated as a scalars. In either | |
| case, either `x` and `y` or their elements must support multiplication | |
| and addition both with themselves and with the elements of `c`. | |
| If `c` has fewer than two dimensions, ones are implicitly appended to | |
| its shape to make it 2-D. The shape of the result will be c.shape[2:] + | |
| x.shape + y.shape. | |
| Parameters | |
| ---------- | |
| x, y : array_like, compatible objects | |
| The two dimensional series is evaluated at the points in the | |
| Cartesian product of `x` and `y`. If `x` or `y` is a list or | |
| tuple, it is first converted to an ndarray, otherwise it is left | |
| unchanged and, if it isn't an ndarray, it is treated as a scalar. | |
| c : array_like | |
| Array of coefficients ordered so that the coefficient of the term of | |
| multi-degree i,j is contained in `c[i,j]`. If `c` has dimension | |
| greater than two the remaining indices enumerate multiple sets of | |
| coefficients. | |
| Returns | |
| ------- | |
| values : ndarray, compatible object | |
| The values of the two dimensional Chebyshev series at points in the | |
| Cartesian product of `x` and `y`. | |
| See Also | |
| -------- | |
| legval, legval2d, legval3d, leggrid3d | |
| Notes | |
| ----- | |
| .. versionadded:: 1.7.0 | |
| """ | |
| return pu._gridnd(legval, c, x, y) | |
| def legval3d(x, y, z, c): | |
| """ | |
| Evaluate a 3-D Legendre series at points (x, y, z). | |
| This function returns the values: | |
| .. math:: p(x,y,z) = \\sum_{i,j,k} c_{i,j,k} * L_i(x) * L_j(y) * L_k(z) | |
| The parameters `x`, `y`, and `z` are converted to arrays only if | |
| they are tuples or a lists, otherwise they are treated as a scalars and | |
| they must have the same shape after conversion. In either case, either | |
| `x`, `y`, and `z` or their elements must support multiplication and | |
| addition both with themselves and with the elements of `c`. | |
| If `c` has fewer than 3 dimensions, ones are implicitly appended to its | |
| shape to make it 3-D. The shape of the result will be c.shape[3:] + | |
| x.shape. | |
| Parameters | |
| ---------- | |
| x, y, z : array_like, compatible object | |
| The three dimensional series is evaluated at the points | |
| `(x, y, z)`, where `x`, `y`, and `z` must have the same shape. If | |
| any of `x`, `y`, or `z` is a list or tuple, it is first converted | |
| to an ndarray, otherwise it is left unchanged and if it isn't an | |
| ndarray it is treated as a scalar. | |
| c : array_like | |
| Array of coefficients ordered so that the coefficient of the term of | |
| multi-degree i,j,k is contained in ``c[i,j,k]``. If `c` has dimension | |
| greater than 3 the remaining indices enumerate multiple sets of | |
| coefficients. | |
| Returns | |
| ------- | |
| values : ndarray, compatible object | |
| The values of the multidimensional polynomial on points formed with | |
| triples of corresponding values from `x`, `y`, and `z`. | |
| See Also | |
| -------- | |
| legval, legval2d, leggrid2d, leggrid3d | |
| Notes | |
| ----- | |
| .. versionadded:: 1.7.0 | |
| """ | |
| return pu._valnd(legval, c, x, y, z) | |
| def leggrid3d(x, y, z, c): | |
| """ | |
| Evaluate a 3-D Legendre series on the Cartesian product of x, y, and z. | |
| This function returns the values: | |
| .. math:: p(a,b,c) = \\sum_{i,j,k} c_{i,j,k} * L_i(a) * L_j(b) * L_k(c) | |
| where the points `(a, b, c)` consist of all triples formed by taking | |
| `a` from `x`, `b` from `y`, and `c` from `z`. The resulting points form | |
| a grid with `x` in the first dimension, `y` in the second, and `z` in | |
| the third. | |
| The parameters `x`, `y`, and `z` are converted to arrays only if they | |
| are tuples or a lists, otherwise they are treated as a scalars. In | |
| either case, either `x`, `y`, and `z` or their elements must support | |
| multiplication and addition both with themselves and with the elements | |
| of `c`. | |
| If `c` has fewer than three dimensions, ones are implicitly appended to | |
| its shape to make it 3-D. The shape of the result will be c.shape[3:] + | |
| x.shape + y.shape + z.shape. | |
| Parameters | |
| ---------- | |
| x, y, z : array_like, compatible objects | |
| The three dimensional series is evaluated at the points in the | |
| Cartesian product of `x`, `y`, and `z`. If `x`,`y`, or `z` is a | |
| list or tuple, it is first converted to an ndarray, otherwise it is | |
| left unchanged and, if it isn't an ndarray, it is treated as a | |
| scalar. | |
| c : array_like | |
| Array of coefficients ordered so that the coefficients for terms of | |
| degree i,j are contained in ``c[i,j]``. If `c` has dimension | |
| greater than two the remaining indices enumerate multiple sets of | |
| coefficients. | |
| Returns | |
| ------- | |
| values : ndarray, compatible object | |
| The values of the two dimensional polynomial at points in the Cartesian | |
| product of `x` and `y`. | |
| See Also | |
| -------- | |
| legval, legval2d, leggrid2d, legval3d | |
| Notes | |
| ----- | |
| .. versionadded:: 1.7.0 | |
| """ | |
| return pu._gridnd(legval, c, x, y, z) | |
| def legvander(x, deg): | |
| """Pseudo-Vandermonde matrix of given degree. | |
| Returns the pseudo-Vandermonde matrix of degree `deg` and sample points | |
| `x`. The pseudo-Vandermonde matrix is defined by | |
| .. math:: V[..., i] = L_i(x) | |
| where `0 <= i <= deg`. The leading indices of `V` index the elements of | |
| `x` and the last index is the degree of the Legendre polynomial. | |
| If `c` is a 1-D array of coefficients of length `n + 1` and `V` is the | |
| array ``V = legvander(x, n)``, then ``np.dot(V, c)`` and | |
| ``legval(x, c)`` are the same up to roundoff. This equivalence is | |
| useful both for least squares fitting and for the evaluation of a large | |
| number of Legendre series of the same degree and sample points. | |
| Parameters | |
| ---------- | |
| x : array_like | |
| Array of points. The dtype is converted to float64 or complex128 | |
| depending on whether any of the elements are complex. If `x` is | |
| scalar it is converted to a 1-D array. | |
| deg : int | |
| Degree of the resulting matrix. | |
| Returns | |
| ------- | |
| vander : ndarray | |
| The pseudo-Vandermonde matrix. The shape of the returned matrix is | |
| ``x.shape + (deg + 1,)``, where The last index is the degree of the | |
| corresponding Legendre polynomial. The dtype will be the same as | |
| the converted `x`. | |
| """ | |
| ideg = pu._deprecate_as_int(deg, "deg") | |
| if ideg < 0: | |
| raise ValueError("deg must be non-negative") | |
| x = np.array(x, copy=False, ndmin=1) + 0.0 | |
| dims = (ideg + 1,) + x.shape | |
| dtyp = x.dtype | |
| v = np.empty(dims, dtype=dtyp) | |
| # 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.moveaxis(v, 0, -1) | |
| def legvander2d(x, y, deg): | |
| """Pseudo-Vandermonde matrix of given degrees. | |
| Returns the pseudo-Vandermonde matrix of degrees `deg` and sample | |
| points `(x, y)`. The pseudo-Vandermonde matrix is defined by | |
| .. math:: V[..., (deg[1] + 1)*i + j] = L_i(x) * L_j(y), | |
| where `0 <= i <= deg[0]` and `0 <= j <= deg[1]`. The leading indices of | |
| `V` index the points `(x, y)` and the last index encodes the degrees of | |
| the Legendre polynomials. | |
| If ``V = legvander2d(x, y, [xdeg, ydeg])``, then the columns of `V` | |
| correspond to the elements of a 2-D coefficient array `c` of shape | |
| (xdeg + 1, ydeg + 1) in the order | |
| .. math:: c_{00}, c_{01}, c_{02} ... , c_{10}, c_{11}, c_{12} ... | |
| and ``np.dot(V, c.flat)`` and ``legval2d(x, y, c)`` will be the same | |
| up to roundoff. This equivalence is useful both for least squares | |
| fitting and for the evaluation of a large number of 2-D Legendre | |
| series of the same degrees and sample points. | |
| Parameters | |
| ---------- | |
| x, y : array_like | |
| Arrays of point coordinates, all of the same shape. The dtypes | |
| will be converted to either float64 or complex128 depending on | |
| whether any of the elements are complex. Scalars are converted to | |
| 1-D arrays. | |
| deg : list of ints | |
| List of maximum degrees of the form [x_deg, y_deg]. | |
| Returns | |
| ------- | |
| vander2d : ndarray | |
| The shape of the returned matrix is ``x.shape + (order,)``, where | |
| :math:`order = (deg[0]+1)*(deg[1]+1)`. The dtype will be the same | |
| as the converted `x` and `y`. | |
| See Also | |
| -------- | |
| legvander, legvander3d, legval2d, legval3d | |
| Notes | |
| ----- | |
| .. versionadded:: 1.7.0 | |
| """ | |
| return pu._vander_nd_flat((legvander, legvander), (x, y), deg) | |
| def legvander3d(x, y, z, deg): | |
| """Pseudo-Vandermonde matrix of given degrees. | |
| Returns the pseudo-Vandermonde matrix of degrees `deg` and sample | |
| points `(x, y, z)`. If `l, m, n` are the given degrees in `x, y, z`, | |
| then The pseudo-Vandermonde matrix is defined by | |
| .. math:: V[..., (m+1)(n+1)i + (n+1)j + k] = L_i(x)*L_j(y)*L_k(z), | |
| where `0 <= i <= l`, `0 <= j <= m`, and `0 <= j <= n`. The leading | |
| indices of `V` index the points `(x, y, z)` and the last index encodes | |
| the degrees of the Legendre polynomials. | |
| If ``V = legvander3d(x, y, z, [xdeg, ydeg, zdeg])``, then the columns | |
| of `V` correspond to the elements of a 3-D coefficient array `c` of | |
| shape (xdeg + 1, ydeg + 1, zdeg + 1) in the order | |
| .. math:: c_{000}, c_{001}, c_{002},... , c_{010}, c_{011}, c_{012},... | |
| and ``np.dot(V, c.flat)`` and ``legval3d(x, y, z, c)`` will be the | |
| same up to roundoff. This equivalence is useful both for least squares | |
| fitting and for the evaluation of a large number of 3-D Legendre | |
| series of the same degrees and sample points. | |
| Parameters | |
| ---------- | |
| x, y, z : array_like | |
| Arrays of point coordinates, all of the same shape. The dtypes will | |
| be converted to either float64 or complex128 depending on whether | |
| any of the elements are complex. Scalars are converted to 1-D | |
| arrays. | |
| deg : list of ints | |
| List of maximum degrees of the form [x_deg, y_deg, z_deg]. | |
| Returns | |
| ------- | |
| vander3d : ndarray | |
| The shape of the returned matrix is ``x.shape + (order,)``, where | |
| :math:`order = (deg[0]+1)*(deg[1]+1)*(deg[2]+1)`. The dtype will | |
| be the same as the converted `x`, `y`, and `z`. | |
| See Also | |
| -------- | |
| legvander, legvander3d, legval2d, legval3d | |
| Notes | |
| ----- | |
| .. versionadded:: 1.7.0 | |
| """ | |
| return pu._vander_nd_flat((legvander, legvander, legvander), (x, y, z), deg) | |
| 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 or 1-D array_like | |
| Degree(s) of the fitting polynomials. If `deg` is a single integer | |
| all terms up to and including the `deg`'th term are included in the | |
| fit. For NumPy versions >= 1.11.0 a list of integers specifying the | |
| degrees of the terms to include may be used instead. | |
| 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`. If `deg` is specified as a list, coefficients for | |
| terms not included in the fit are set equal to zero in the | |
| returned `coef`. | |
| [residuals, rank, singular_values, rcond] : list | |
| These values are only returned if `full` = True | |
| resid -- sum of squared residuals of the least squares fit | |
| rank -- the numerical rank of the scaled Vandermonde matrix | |
| sv -- singular values of the scaled Vandermonde matrix | |
| rcond -- value of `rcond`. | |
| For more details, see `numpy.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', np.RankWarning) | |
| See Also | |
| -------- | |
| numpy.polynomial.polynomial.polyfit | |
| numpy.polynomial.chebyshev.chebfit | |
| numpy.polynomial.laguerre.lagfit | |
| numpy.polynomial.hermite.hermfit | |
| numpy.polynomial.hermite_e.hermefit | |
| legval : Evaluates a Legendre series. | |
| legvander : Vandermonde matrix of Legendre series. | |
| legweight : Legendre weight function (= 1). | |
| numpy.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 | |
| coefficient 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", | |
| https://en.wikipedia.org/wiki/Curve_fitting | |
| Examples | |
| -------- | |
| """ | |
| return pu._fit(legvander, x, y, deg, rcond, full, w) | |
| def legcompanion(c): | |
| """Return the scaled companion matrix of c. | |
| The basis polynomials are scaled so that the companion matrix is | |
| symmetric when `c` is an Legendre basis polynomial. This provides | |
| better eigenvalue estimates than the unscaled case and for basis | |
| polynomials the eigenvalues are guaranteed to be real if | |
| `numpy.linalg.eigvalsh` is used to obtain them. | |
| Parameters | |
| ---------- | |
| c : 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). | |
| Notes | |
| ----- | |
| .. versionadded:: 1.7.0 | |
| """ | |
| # c is a trimmed copy | |
| [c] = pu.as_series([c]) | |
| if len(c) < 2: | |
| raise ValueError('Series must have maximum degree of at least 1.') | |
| if len(c) == 2: | |
| return np.array([[-c[0]/c[1]]]) | |
| n = len(c) - 1 | |
| mat = np.zeros((n, n), dtype=c.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] -= (c[:-1]/c[-1])*(scl/scl[-1])*(n/(2*n - 1)) | |
| return mat | |
| def legroots(c): | |
| """ | |
| Compute the roots of a Legendre series. | |
| Return the roots (a.k.a. "zeros") of the polynomial | |
| .. math:: p(x) = \\sum_i c[i] * L_i(x). | |
| Parameters | |
| ---------- | |
| c : 1-D array_like | |
| 1-D array of coefficients. | |
| Returns | |
| ------- | |
| out : ndarray | |
| Array of the roots of the series. If all the roots are real, | |
| then `out` is also real, otherwise it is complex. | |
| See Also | |
| -------- | |
| numpy.polynomial.polynomial.polyroots | |
| numpy.polynomial.chebyshev.chebroots | |
| numpy.polynomial.laguerre.lagroots | |
| numpy.polynomial.hermite.hermroots | |
| numpy.polynomial.hermite_e.hermeroots | |
| Notes | |
| ----- | |
| The root estimates are obtained as the eigenvalues of the companion | |
| matrix, Roots far from the origin of the complex plane may have large | |
| errors due to the numerical instability of the 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 origin 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, all real roots | |
| array([-0.85099543, -0.11407192, 0.51506735]) # may vary | |
| """ | |
| # c is a trimmed copy | |
| [c] = pu.as_series([c]) | |
| if len(c) < 2: | |
| return np.array([], dtype=c.dtype) | |
| if len(c) == 2: | |
| return np.array([-c[0]/c[1]]) | |
| # rotated companion matrix reduces error | |
| m = legcompanion(c)[::-1,::-1] | |
| r = la.eigvals(m) | |
| r.sort() | |
| return r | |
| def leggauss(deg): | |
| """ | |
| Gauss-Legendre quadrature. | |
| Computes the sample points and weights for Gauss-Legendre quadrature. | |
| These sample points and weights will correctly integrate polynomials of | |
| degree :math:`2*deg - 1` or less over the interval :math:`[-1, 1]` with | |
| the weight function :math:`f(x) = 1`. | |
| Parameters | |
| ---------- | |
| deg : int | |
| Number of sample points and weights. It must be >= 1. | |
| Returns | |
| ------- | |
| x : ndarray | |
| 1-D ndarray containing the sample points. | |
| y : ndarray | |
| 1-D ndarray containing the weights. | |
| Notes | |
| ----- | |
| .. versionadded:: 1.7.0 | |
| The results have only been tested up to degree 100, higher degrees may | |
| be problematic. The weights are determined by using the fact that | |
| .. math:: w_k = c / (L'_n(x_k) * L_{n-1}(x_k)) | |
| where :math:`c` is a constant independent of :math:`k` and :math:`x_k` | |
| is the k'th root of :math:`L_n`, and then scaling the results to get | |
| the right value when integrating 1. | |
| """ | |
| ideg = pu._deprecate_as_int(deg, "deg") | |
| if ideg <= 0: | |
| raise ValueError("deg must be a positive integer") | |
| # first approximation of roots. We use the fact that the companion | |
| # matrix is symmetric in this case in order to obtain better zeros. | |
| c = np.array([0]*deg + [1]) | |
| m = legcompanion(c) | |
| x = la.eigvalsh(m) | |
| # improve roots by one application of Newton | |
| dy = legval(x, c) | |
| df = legval(x, legder(c)) | |
| x -= dy/df | |
| # compute the weights. We scale the factor to avoid possible numerical | |
| # overflow. | |
| fm = legval(x, c[1:]) | |
| fm /= np.abs(fm).max() | |
| df /= np.abs(df).max() | |
| w = 1/(fm * df) | |
| # for Legendre we can also symmetrize | |
| w = (w + w[::-1])/2 | |
| x = (x - x[::-1])/2 | |
| # scale w to get the right value | |
| w *= 2. / w.sum() | |
| return x, w | |
| def legweight(x): | |
| """ | |
| Weight function of the Legendre polynomials. | |
| The weight function is :math:`1` and the interval of integration is | |
| :math:`[-1, 1]`. The Legendre polynomials are orthogonal, but not | |
| normalized, with respect to this weight function. | |
| Parameters | |
| ---------- | |
| x : array_like | |
| Values at which the weight function will be computed. | |
| Returns | |
| ------- | |
| w : ndarray | |
| The weight function at `x`. | |
| Notes | |
| ----- | |
| .. versionadded:: 1.7.0 | |
| """ | |
| w = x*0.0 + 1.0 | |
| return w | |
| # | |
| # Legendre series class | |
| # | |
| class Legendre(ABCPolyBase): | |
| """A Legendre series class. | |
| The Legendre class provides the standard Python numerical methods | |
| '+', '-', '*', '//', '%', 'divmod', '**', and '()' as well as the | |
| attributes and methods listed in the `ABCPolyBase` documentation. | |
| Parameters | |
| ---------- | |
| coef : array_like | |
| Legendre coefficients in order of increasing degree, i.e., | |
| ``(1, 2, 3)`` gives ``1*P_0(x) + 2*P_1(x) + 3*P_2(x)``. | |
| domain : (2,) array_like, optional | |
| Domain to use. The interval ``[domain[0], domain[1]]`` is mapped | |
| to the interval ``[window[0], window[1]]`` by shifting and scaling. | |
| The default value is [-1, 1]. | |
| window : (2,) array_like, optional | |
| Window, see `domain` for its use. The default value is [-1, 1]. | |
| .. versionadded:: 1.6.0 | |
| """ | |
| # Virtual Functions | |
| _add = staticmethod(legadd) | |
| _sub = staticmethod(legsub) | |
| _mul = staticmethod(legmul) | |
| _div = staticmethod(legdiv) | |
| _pow = staticmethod(legpow) | |
| _val = staticmethod(legval) | |
| _int = staticmethod(legint) | |
| _der = staticmethod(legder) | |
| _fit = staticmethod(legfit) | |
| _line = staticmethod(legline) | |
| _roots = staticmethod(legroots) | |
| _fromroots = staticmethod(legfromroots) | |
| # Virtual properties | |
| domain = np.array(legdomain) | |
| window = np.array(legdomain) | |
| basis_name = 'P' | |