| | """ |
| | ============================================================== |
| | Hermite Series, "Physicists" (:mod:`numpy.polynomial.hermite`) |
| | ============================================================== |
| | |
| | This module provides a number of objects (mostly functions) useful for |
| | dealing with Hermite series, including a `Hermite` 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/ |
| | |
| | Hermite |
| | |
| | Constants |
| | --------- |
| | .. autosummary:: |
| | :toctree: generated/ |
| | |
| | hermdomain |
| | hermzero |
| | hermone |
| | hermx |
| | |
| | Arithmetic |
| | ---------- |
| | .. autosummary:: |
| | :toctree: generated/ |
| | |
| | hermadd |
| | hermsub |
| | hermmulx |
| | hermmul |
| | hermdiv |
| | hermpow |
| | hermval |
| | hermval2d |
| | hermval3d |
| | hermgrid2d |
| | hermgrid3d |
| | |
| | Calculus |
| | -------- |
| | .. autosummary:: |
| | :toctree: generated/ |
| | |
| | hermder |
| | hermint |
| | |
| | Misc Functions |
| | -------------- |
| | .. autosummary:: |
| | :toctree: generated/ |
| | |
| | hermfromroots |
| | hermroots |
| | hermvander |
| | hermvander2d |
| | hermvander3d |
| | hermgauss |
| | hermweight |
| | hermcompanion |
| | hermfit |
| | hermtrim |
| | hermline |
| | herm2poly |
| | poly2herm |
| | |
| | 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__ = [ |
| | 'hermzero', 'hermone', 'hermx', 'hermdomain', 'hermline', 'hermadd', |
| | 'hermsub', 'hermmulx', 'hermmul', 'hermdiv', 'hermpow', 'hermval', |
| | 'hermder', 'hermint', 'herm2poly', 'poly2herm', 'hermfromroots', |
| | 'hermvander', 'hermfit', 'hermtrim', 'hermroots', 'Hermite', |
| | 'hermval2d', 'hermval3d', 'hermgrid2d', 'hermgrid3d', 'hermvander2d', |
| | 'hermvander3d', 'hermcompanion', 'hermgauss', 'hermweight'] |
| |
|
| | hermtrim = pu.trimcoef |
| |
|
| |
|
| | def poly2herm(pol): |
| | """ |
| | poly2herm(pol) |
| | |
| | Convert a polynomial to a Hermite 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 Hermite 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 Hermite |
| | series. |
| | |
| | See Also |
| | -------- |
| | herm2poly |
| | |
| | Notes |
| | ----- |
| | The easy way to do conversions between polynomial basis sets |
| | is to use the convert method of a class instance. |
| | |
| | Examples |
| | -------- |
| | >>> from numpy.polynomial.hermite import poly2herm |
| | >>> poly2herm(np.arange(4)) |
| | array([1. , 2.75 , 0.5 , 0.375]) |
| | |
| | """ |
| | [pol] = pu.as_series([pol]) |
| | deg = len(pol) - 1 |
| | res = 0 |
| | for i in range(deg, -1, -1): |
| | res = hermadd(hermmulx(res), pol[i]) |
| | return res |
| |
|
| |
|
| | def herm2poly(c): |
| | """ |
| | Convert a Hermite series to a polynomial. |
| | |
| | Convert an array representing the coefficients of a Hermite 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 Hermite 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 |
| | -------- |
| | poly2herm |
| | |
| | Notes |
| | ----- |
| | The easy way to do conversions between polynomial basis sets |
| | is to use the convert method of a class instance. |
| | |
| | Examples |
| | -------- |
| | >>> from numpy.polynomial.hermite import herm2poly |
| | >>> herm2poly([ 1. , 2.75 , 0.5 , 0.375]) |
| | array([0., 1., 2., 3.]) |
| | |
| | """ |
| | from .polynomial import polyadd, polysub, polymulx |
| |
|
| | [c] = pu.as_series([c]) |
| | n = len(c) |
| | if n == 1: |
| | return c |
| | if n == 2: |
| | c[1] *= 2 |
| | return c |
| | else: |
| | c0 = c[-2] |
| | c1 = c[-1] |
| | |
| | for i in range(n - 1, 1, -1): |
| | tmp = c0 |
| | c0 = polysub(c[i - 2], c1*(2*(i - 1))) |
| | c1 = polyadd(tmp, polymulx(c1)*2) |
| | return polyadd(c0, polymulx(c1)*2) |
| |
|
| | |
| | |
| | |
| | |
| |
|
| | |
| | hermdomain = np.array([-1, 1]) |
| |
|
| | |
| | hermzero = np.array([0]) |
| |
|
| | |
| | hermone = np.array([1]) |
| |
|
| | |
| | hermx = np.array([0, 1/2]) |
| |
|
| |
|
| | def hermline(off, scl): |
| | """ |
| | Hermite 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 Hermite series for |
| | ``off + scl*x``. |
| | |
| | See Also |
| | -------- |
| | numpy.polynomial.polynomial.polyline |
| | numpy.polynomial.chebyshev.chebline |
| | numpy.polynomial.legendre.legline |
| | numpy.polynomial.laguerre.lagline |
| | numpy.polynomial.hermite_e.hermeline |
| | |
| | Examples |
| | -------- |
| | >>> from numpy.polynomial.hermite import hermline, hermval |
| | >>> hermval(0,hermline(3, 2)) |
| | 3.0 |
| | >>> hermval(1,hermline(3, 2)) |
| | 5.0 |
| | |
| | """ |
| | if scl != 0: |
| | return np.array([off, scl/2]) |
| | else: |
| | return np.array([off]) |
| |
|
| |
|
| | def hermfromroots(roots): |
| | """ |
| | Generate a Hermite 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 Hermite 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 * H_1(x) + ... + c_n * H_n(x) |
| | |
| | The coefficient of the last term is not generally 1 for monic |
| | polynomials in Hermite 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.legendre.legfromroots |
| | numpy.polynomial.laguerre.lagfromroots |
| | numpy.polynomial.chebyshev.chebfromroots |
| | numpy.polynomial.hermite_e.hermefromroots |
| | |
| | Examples |
| | -------- |
| | >>> from numpy.polynomial.hermite import hermfromroots, hermval |
| | >>> coef = hermfromroots((-1, 0, 1)) |
| | >>> hermval((-1, 0, 1), coef) |
| | array([0., 0., 0.]) |
| | >>> coef = hermfromroots((-1j, 1j)) |
| | >>> hermval((-1j, 1j), coef) |
| | array([0.+0.j, 0.+0.j]) |
| | |
| | """ |
| | return pu._fromroots(hermline, hermmul, roots) |
| |
|
| |
|
| | def hermadd(c1, c2): |
| | """ |
| | Add one Hermite series to another. |
| | |
| | Returns the sum of two Hermite 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 Hermite series coefficients ordered from low to |
| | high. |
| | |
| | Returns |
| | ------- |
| | out : ndarray |
| | Array representing the Hermite series of their sum. |
| | |
| | See Also |
| | -------- |
| | hermsub, hermmulx, hermmul, hermdiv, hermpow |
| | |
| | Notes |
| | ----- |
| | Unlike multiplication, division, etc., the sum of two Hermite series |
| | is a Hermite 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.hermite import hermadd |
| | >>> hermadd([1, 2, 3], [1, 2, 3, 4]) |
| | array([2., 4., 6., 4.]) |
| | |
| | """ |
| | return pu._add(c1, c2) |
| |
|
| |
|
| | def hermsub(c1, c2): |
| | """ |
| | Subtract one Hermite series from another. |
| | |
| | Returns the difference of two Hermite 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 Hermite series coefficients ordered from low to |
| | high. |
| | |
| | Returns |
| | ------- |
| | out : ndarray |
| | Of Hermite series coefficients representing their difference. |
| | |
| | See Also |
| | -------- |
| | hermadd, hermmulx, hermmul, hermdiv, hermpow |
| | |
| | Notes |
| | ----- |
| | Unlike multiplication, division, etc., the difference of two Hermite |
| | series is a Hermite 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.hermite import hermsub |
| | >>> hermsub([1, 2, 3, 4], [1, 2, 3]) |
| | array([0., 0., 0., 4.]) |
| | |
| | """ |
| | return pu._sub(c1, c2) |
| |
|
| |
|
| | def hermmulx(c): |
| | """Multiply a Hermite series by x. |
| | |
| | Multiply the Hermite series `c` by x, where x is the independent |
| | variable. |
| | |
| | |
| | Parameters |
| | ---------- |
| | c : array_like |
| | 1-D array of Hermite series coefficients ordered from low to |
| | high. |
| | |
| | Returns |
| | ------- |
| | out : ndarray |
| | Array representing the result of the multiplication. |
| | |
| | See Also |
| | -------- |
| | hermadd, hermsub, hermmul, hermdiv, hermpow |
| | |
| | Notes |
| | ----- |
| | The multiplication uses the recursion relationship for Hermite |
| | polynomials in the form |
| | |
| | .. math:: |
| | |
| | xP_i(x) = (P_{i + 1}(x)/2 + i*P_{i - 1}(x)) |
| | |
| | Examples |
| | -------- |
| | >>> from numpy.polynomial.hermite import hermmulx |
| | >>> hermmulx([1, 2, 3]) |
| | array([2. , 6.5, 1. , 1.5]) |
| | |
| | """ |
| | |
| | [c] = pu.as_series([c]) |
| | |
| | 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]/2 |
| | for i in range(1, len(c)): |
| | prd[i + 1] = c[i]/2 |
| | prd[i - 1] += c[i]*i |
| | return prd |
| |
|
| |
|
| | def hermmul(c1, c2): |
| | """ |
| | Multiply one Hermite series by another. |
| | |
| | Returns the product of two Hermite 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 Hermite series coefficients ordered from low to |
| | high. |
| | |
| | Returns |
| | ------- |
| | out : ndarray |
| | Of Hermite series coefficients representing their product. |
| | |
| | See Also |
| | -------- |
| | hermadd, hermsub, hermmulx, hermdiv, hermpow |
| | |
| | Notes |
| | ----- |
| | In general, the (polynomial) product of two C-series results in terms |
| | that are not in the Hermite polynomial basis set. Thus, to express |
| | the product as a Hermite 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.hermite import hermmul |
| | >>> hermmul([1, 2, 3], [0, 1, 2]) |
| | array([52., 29., 52., 7., 6.]) |
| | |
| | """ |
| | |
| | [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 = hermsub(c[-i]*xs, c1*(2*(nd - 1))) |
| | c1 = hermadd(tmp, hermmulx(c1)*2) |
| | return hermadd(c0, hermmulx(c1)*2) |
| |
|
| |
|
| | def hermdiv(c1, c2): |
| | """ |
| | Divide one Hermite series by another. |
| | |
| | Returns the quotient-with-remainder of two Hermite 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 Hermite series coefficients ordered from low to |
| | high. |
| | |
| | Returns |
| | ------- |
| | [quo, rem] : ndarrays |
| | Of Hermite series coefficients representing the quotient and |
| | remainder. |
| | |
| | See Also |
| | -------- |
| | hermadd, hermsub, hermmulx, hermmul, hermpow |
| | |
| | Notes |
| | ----- |
| | In general, the (polynomial) division of one Hermite series by another |
| | results in quotient and remainder terms that are not in the Hermite |
| | polynomial basis set. Thus, to express these results as a Hermite |
| | series, it is necessary to "reproject" the results onto the Hermite |
| | basis set, which may produce "unintuitive" (but correct) results; see |
| | Examples section below. |
| | |
| | Examples |
| | -------- |
| | >>> from numpy.polynomial.hermite import hermdiv |
| | >>> hermdiv([ 52., 29., 52., 7., 6.], [0, 1, 2]) |
| | (array([1., 2., 3.]), array([0.])) |
| | >>> hermdiv([ 54., 31., 52., 7., 6.], [0, 1, 2]) |
| | (array([1., 2., 3.]), array([2., 2.])) |
| | >>> hermdiv([ 53., 30., 52., 7., 6.], [0, 1, 2]) |
| | (array([1., 2., 3.]), array([1., 1.])) |
| | |
| | """ |
| | return pu._div(hermmul, c1, c2) |
| |
|
| |
|
| | def hermpow(c, pow, maxpower=16): |
| | """Raise a Hermite series to a power. |
| | |
| | Returns the Hermite 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 Hermite 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 |
| | Hermite series of power. |
| | |
| | See Also |
| | -------- |
| | hermadd, hermsub, hermmulx, hermmul, hermdiv |
| | |
| | Examples |
| | -------- |
| | >>> from numpy.polynomial.hermite import hermpow |
| | >>> hermpow([1, 2, 3], 2) |
| | array([81., 52., 82., 12., 9.]) |
| | |
| | """ |
| | return pu._pow(hermmul, c, pow, maxpower) |
| |
|
| |
|
| | def hermder(c, m=1, scl=1, axis=0): |
| | """ |
| | Differentiate a Hermite series. |
| | |
| | Returns the Hermite 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*H_0 + 2*H_1 + 3*H_2`` |
| | while [[1,2],[1,2]] represents ``1*H_0(x)*H_0(y) + 1*H_1(x)*H_0(y) + |
| | 2*H_0(x)*H_1(y) + 2*H_1(x)*H_1(y)`` if axis=0 is ``x`` and axis=1 is |
| | ``y``. |
| | |
| | Parameters |
| | ---------- |
| | c : array_like |
| | Array of Hermite 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 |
| | Hermite series of the derivative. |
| | |
| | See Also |
| | -------- |
| | hermint |
| | |
| | Notes |
| | ----- |
| | In general, the result of differentiating a Hermite 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.hermite import hermder |
| | >>> hermder([ 1. , 0.5, 0.5, 0.5]) |
| | array([1., 2., 3.]) |
| | >>> hermder([-0.5, 1./2., 1./8., 1./12., 1./16.], m=2) |
| | array([1., 2., 3.]) |
| | |
| | """ |
| | 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, 0, -1): |
| | der[j - 1] = (2*j)*c[j] |
| | c = der |
| | c = np.moveaxis(c, 0, iaxis) |
| | return c |
| |
|
| |
|
| | def hermint(c, m=1, k=[], lbnd=0, scl=1, axis=0): |
| | """ |
| | Integrate a Hermite series. |
| | |
| | Returns the Hermite 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 ``H_0 + 2*H_1 + 3*H_2`` while [[1,2],[1,2]] |
| | represents ``1*H_0(x)*H_0(y) + 1*H_1(x)*H_0(y) + 2*H_0(x)*H_1(y) + |
| | 2*H_1(x)*H_1(y)`` if axis=0 is ``x`` and axis=1 is ``y``. |
| | |
| | Parameters |
| | ---------- |
| | c : array_like |
| | Array of Hermite 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 |
| | Hermite series coefficients of the integral. |
| | |
| | Raises |
| | ------ |
| | ValueError |
| | If ``m < 0``, ``len(k) > m``, ``np.ndim(lbnd) != 0``, or |
| | ``np.ndim(scl) != 0``. |
| | |
| | See Also |
| | -------- |
| | hermder |
| | |
| | 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.hermite import hermint |
| | >>> hermint([1,2,3]) # integrate once, value 0 at 0. |
| | array([1. , 0.5, 0.5, 0.5]) |
| | >>> hermint([1,2,3], m=2) # integrate twice, value & deriv 0 at 0 |
| | array([-0.5 , 0.5 , 0.125 , 0.08333333, 0.0625 ]) # may vary |
| | >>> hermint([1,2,3], k=1) # integrate once, value 1 at 0. |
| | array([2. , 0.5, 0.5, 0.5]) |
| | >>> hermint([1,2,3], lbnd=-1) # integrate once, value 0 at -1 |
| | array([-2. , 0.5, 0.5, 0.5]) |
| | >>> hermint([1,2,3], m=2, k=[1,2], lbnd=-1) |
| | array([ 1.66666667, -0.5 , 0.125 , 0.08333333, 0.0625 ]) # 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]/2 |
| | for j in range(1, n): |
| | tmp[j + 1] = c[j]/(2*(j + 1)) |
| | tmp[0] += k[i] - hermval(lbnd, tmp) |
| | c = tmp |
| | c = np.moveaxis(c, 0, iaxis) |
| | return c |
| |
|
| |
|
| | def hermval(x, c, tensor=True): |
| | """ |
| | Evaluate an Hermite series at points x. |
| | |
| | If `c` is of length `n + 1`, this function returns the value: |
| | |
| | .. math:: p(x) = c_0 * H_0(x) + c_1 * H_1(x) + ... + c_n * H_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 |
| | 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 |
| | -------- |
| | hermval2d, hermgrid2d, hermval3d, hermgrid3d |
| | |
| | Notes |
| | ----- |
| | The evaluation uses Clenshaw recursion, aka synthetic division. |
| | |
| | Examples |
| | -------- |
| | >>> from numpy.polynomial.hermite import hermval |
| | >>> coef = [1,2,3] |
| | >>> hermval(1, coef) |
| | 11.0 |
| | >>> hermval([[1,2],[3,4]], coef) |
| | array([[ 11., 51.], |
| | [115., 203.]]) |
| | |
| | """ |
| | 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) |
| |
|
| | x2 = x*2 |
| | 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*(2*(nd - 1)) |
| | c1 = tmp + c1*x2 |
| | return c0 + c1*x2 |
| |
|
| |
|
| | def hermval2d(x, y, c): |
| | """ |
| | Evaluate a 2-D Hermite series at points (x, y). |
| | |
| | This function returns the values: |
| | |
| | .. math:: p(x,y) = \\sum_{i,j} c_{i,j} * H_i(x) * H_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 polynomial at points formed with |
| | pairs of corresponding values from `x` and `y`. |
| | |
| | See Also |
| | -------- |
| | hermval, hermgrid2d, hermval3d, hermgrid3d |
| | |
| | Notes |
| | ----- |
| | |
| | .. versionadded:: 1.7.0 |
| | |
| | """ |
| | return pu._valnd(hermval, c, x, y) |
| |
|
| |
|
| | def hermgrid2d(x, y, c): |
| | """ |
| | Evaluate a 2-D Hermite series on the Cartesian product of x and y. |
| | |
| | This function returns the values: |
| | |
| | .. math:: p(a,b) = \\sum_{i,j} c_{i,j} * H_i(a) * H_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. |
| | |
| | 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 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 |
| | -------- |
| | hermval, hermval2d, hermval3d, hermgrid3d |
| | |
| | Notes |
| | ----- |
| | |
| | .. versionadded:: 1.7.0 |
| | |
| | """ |
| | return pu._gridnd(hermval, c, x, y) |
| |
|
| |
|
| | def hermval3d(x, y, z, c): |
| | """ |
| | Evaluate a 3-D Hermite series at points (x, y, z). |
| | |
| | This function returns the values: |
| | |
| | .. math:: p(x,y,z) = \\sum_{i,j,k} c_{i,j,k} * H_i(x) * H_j(y) * H_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 |
| | -------- |
| | hermval, hermval2d, hermgrid2d, hermgrid3d |
| | |
| | Notes |
| | ----- |
| | |
| | .. versionadded:: 1.7.0 |
| | |
| | """ |
| | return pu._valnd(hermval, c, x, y, z) |
| |
|
| |
|
| | def hermgrid3d(x, y, z, c): |
| | """ |
| | Evaluate a 3-D Hermite 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} * H_i(a) * H_j(b) * H_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 |
| | -------- |
| | hermval, hermval2d, hermgrid2d, hermval3d |
| | |
| | Notes |
| | ----- |
| | |
| | .. versionadded:: 1.7.0 |
| | |
| | """ |
| | return pu._gridnd(hermval, c, x, y, z) |
| |
|
| |
|
| | def hermvander(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] = H_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 Hermite polynomial. |
| | |
| | If `c` is a 1-D array of coefficients of length `n + 1` and `V` is the |
| | array ``V = hermvander(x, n)``, then ``np.dot(V, c)`` and |
| | ``hermval(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 Hermite 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 Hermite polynomial. The dtype will be the same as |
| | the converted `x`. |
| | |
| | Examples |
| | -------- |
| | >>> from numpy.polynomial.hermite import hermvander |
| | >>> x = np.array([-1, 0, 1]) |
| | >>> hermvander(x, 3) |
| | array([[ 1., -2., 2., 4.], |
| | [ 1., 0., -2., -0.], |
| | [ 1., 2., 2., -4.]]) |
| | |
| | """ |
| | 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) |
| | v[0] = x*0 + 1 |
| | if ideg > 0: |
| | x2 = x*2 |
| | v[1] = x2 |
| | for i in range(2, ideg + 1): |
| | v[i] = (v[i-1]*x2 - v[i-2]*(2*(i - 1))) |
| | return np.moveaxis(v, 0, -1) |
| |
|
| |
|
| | def hermvander2d(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] = H_i(x) * H_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 Hermite polynomials. |
| | |
| | If ``V = hermvander2d(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 ``hermval2d(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 Hermite |
| | 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 |
| | -------- |
| | hermvander, hermvander3d, hermval2d, hermval3d |
| | |
| | Notes |
| | ----- |
| | |
| | .. versionadded:: 1.7.0 |
| | |
| | """ |
| | return pu._vander_nd_flat((hermvander, hermvander), (x, y), deg) |
| |
|
| |
|
| | def hermvander3d(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] = H_i(x)*H_j(y)*H_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 Hermite polynomials. |
| | |
| | If ``V = hermvander3d(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 ``hermval3d(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 Hermite |
| | 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 |
| | -------- |
| | hermvander, hermvander3d, hermval2d, hermval3d |
| | |
| | Notes |
| | ----- |
| | |
| | .. versionadded:: 1.7.0 |
| | |
| | """ |
| | return pu._vander_nd_flat((hermvander, hermvander, hermvander), (x, y, z), deg) |
| |
|
| |
|
| | def hermfit(x, y, deg, rcond=None, full=False, w=None): |
| | """ |
| | Least squares fit of Hermite series to data. |
| | |
| | Return the coefficients of a Hermite 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 * H_1(x) + ... + c_n * H_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 weight ``w[i]`` applies to the unsquared |
| | residual ``y[i] - y_hat[i]`` at ``x[i]``. Ideally the weights are |
| | chosen so that the errors of the products ``w[i]*y[i]`` all have the |
| | same variance. When using inverse-variance weighting, use |
| | ``w[i] = 1/sigma(y[i])``. The default value is None. |
| | |
| | Returns |
| | ------- |
| | coef : ndarray, shape (M,) or (M, K) |
| | Hermite 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] : list |
| | These values are only returned if ``full == True`` |
| | |
| | - residuals -- sum of squared residuals of the least squares fit |
| | - rank -- the numerical rank of the scaled Vandermonde matrix |
| | - singular_values -- 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.chebyshev.chebfit |
| | numpy.polynomial.legendre.legfit |
| | numpy.polynomial.laguerre.lagfit |
| | numpy.polynomial.polynomial.polyfit |
| | numpy.polynomial.hermite_e.hermefit |
| | hermval : Evaluates a Hermite series. |
| | hermvander : Vandermonde matrix of Hermite series. |
| | hermweight : Hermite weight function |
| | 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 Hermite 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 the :math:`w_j` are the weights. This problem is solved by |
| | setting up 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, `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 Hermite series are probably most useful when the data can be |
| | approximated by ``sqrt(w(x)) * p(x)``, where `w(x)` is the Hermite |
| | weight. In that case the weight ``sqrt(w(x[i]))`` should be used |
| | together with data values ``y[i]/sqrt(w(x[i]))``. The weight function is |
| | available as `hermweight`. |
| | |
| | References |
| | ---------- |
| | .. [1] Wikipedia, "Curve fitting", |
| | https://en.wikipedia.org/wiki/Curve_fitting |
| | |
| | Examples |
| | -------- |
| | >>> from numpy.polynomial.hermite import hermfit, hermval |
| | >>> x = np.linspace(-10, 10) |
| | >>> err = np.random.randn(len(x))/10 |
| | >>> y = hermval(x, [1, 2, 3]) + err |
| | >>> hermfit(x, y, 2) |
| | array([1.0218, 1.9986, 2.9999]) # may vary |
| | |
| | """ |
| | return pu._fit(hermvander, x, y, deg, rcond, full, w) |
| |
|
| |
|
| | def hermcompanion(c): |
| | """Return the scaled companion matrix of c. |
| | |
| | The basis polynomials are scaled so that the companion matrix is |
| | symmetric when `c` is an Hermite 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 Hermite series coefficients ordered from low to high |
| | degree. |
| | |
| | Returns |
| | ------- |
| | mat : ndarray |
| | Scaled companion matrix of dimensions (deg, deg). |
| | |
| | Notes |
| | ----- |
| | |
| | .. versionadded:: 1.7.0 |
| | |
| | """ |
| | |
| | [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([[-.5*c[0]/c[1]]]) |
| |
|
| | n = len(c) - 1 |
| | mat = np.zeros((n, n), dtype=c.dtype) |
| | scl = np.hstack((1., 1./np.sqrt(2.*np.arange(n - 1, 0, -1)))) |
| | scl = np.multiply.accumulate(scl)[::-1] |
| | top = mat.reshape(-1)[1::n+1] |
| | bot = mat.reshape(-1)[n::n+1] |
| | top[...] = np.sqrt(.5*np.arange(1, n)) |
| | bot[...] = top |
| | mat[:, -1] -= scl*c[:-1]/(2.0*c[-1]) |
| | return mat |
| |
|
| |
|
| | def hermroots(c): |
| | """ |
| | Compute the roots of a Hermite series. |
| | |
| | Return the roots (a.k.a. "zeros") of the polynomial |
| | |
| | .. math:: p(x) = \\sum_i c[i] * H_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.legendre.legroots |
| | numpy.polynomial.laguerre.lagroots |
| | numpy.polynomial.chebyshev.chebroots |
| | 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 Hermite series basis polynomials aren't powers of `x` so the |
| | results of this function may seem unintuitive. |
| | |
| | Examples |
| | -------- |
| | >>> from numpy.polynomial.hermite import hermroots, hermfromroots |
| | >>> coef = hermfromroots([-1, 0, 1]) |
| | >>> coef |
| | array([0. , 0.25 , 0. , 0.125]) |
| | >>> hermroots(coef) |
| | array([-1.00000000e+00, -1.38777878e-17, 1.00000000e+00]) |
| | |
| | """ |
| | |
| | [c] = pu.as_series([c]) |
| | if len(c) <= 1: |
| | return np.array([], dtype=c.dtype) |
| | if len(c) == 2: |
| | return np.array([-.5*c[0]/c[1]]) |
| |
|
| | |
| | m = hermcompanion(c)[::-1,::-1] |
| | r = la.eigvals(m) |
| | r.sort() |
| | return r |
| |
|
| |
|
| | def _normed_hermite_n(x, n): |
| | """ |
| | Evaluate a normalized Hermite polynomial. |
| | |
| | Compute the value of the normalized Hermite polynomial of degree ``n`` |
| | at the points ``x``. |
| | |
| | |
| | Parameters |
| | ---------- |
| | x : ndarray of double. |
| | Points at which to evaluate the function |
| | n : int |
| | Degree of the normalized Hermite function to be evaluated. |
| | |
| | Returns |
| | ------- |
| | values : ndarray |
| | The shape of the return value is described above. |
| | |
| | Notes |
| | ----- |
| | .. versionadded:: 1.10.0 |
| | |
| | This function is needed for finding the Gauss points and integration |
| | weights for high degrees. The values of the standard Hermite functions |
| | overflow when n >= 207. |
| | |
| | """ |
| | if n == 0: |
| | return np.full(x.shape, 1/np.sqrt(np.sqrt(np.pi))) |
| |
|
| | c0 = 0. |
| | c1 = 1./np.sqrt(np.sqrt(np.pi)) |
| | nd = float(n) |
| | for i in range(n - 1): |
| | tmp = c0 |
| | c0 = -c1*np.sqrt((nd - 1.)/nd) |
| | c1 = tmp + c1*x*np.sqrt(2./nd) |
| | nd = nd - 1.0 |
| | return c0 + c1*x*np.sqrt(2) |
| |
|
| |
|
| | def hermgauss(deg): |
| | """ |
| | Gauss-Hermite quadrature. |
| | |
| | Computes the sample points and weights for Gauss-Hermite quadrature. |
| | These sample points and weights will correctly integrate polynomials of |
| | degree :math:`2*deg - 1` or less over the interval :math:`[-\\inf, \\inf]` |
| | with the weight function :math:`f(x) = \\exp(-x^2)`. |
| | |
| | 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 / (H'_n(x_k) * H_{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:`H_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") |
| |
|
| | |
| | |
| | c = np.array([0]*deg + [1], dtype=np.float64) |
| | m = hermcompanion(c) |
| | x = la.eigvalsh(m) |
| |
|
| | |
| | dy = _normed_hermite_n(x, ideg) |
| | df = _normed_hermite_n(x, ideg - 1) * np.sqrt(2*ideg) |
| | x -= dy/df |
| |
|
| | |
| | |
| | fm = _normed_hermite_n(x, ideg - 1) |
| | fm /= np.abs(fm).max() |
| | w = 1/(fm * fm) |
| |
|
| | |
| | w = (w + w[::-1])/2 |
| | x = (x - x[::-1])/2 |
| |
|
| | |
| | w *= np.sqrt(np.pi) / w.sum() |
| |
|
| | return x, w |
| |
|
| |
|
| | def hermweight(x): |
| | """ |
| | Weight function of the Hermite polynomials. |
| | |
| | The weight function is :math:`\\exp(-x^2)` and the interval of |
| | integration is :math:`[-\\inf, \\inf]`. the Hermite 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 = np.exp(-x**2) |
| | return w |
| |
|
| |
|
| | |
| | |
| | |
| |
|
| | class Hermite(ABCPolyBase): |
| | """An Hermite series class. |
| | |
| | The Hermite 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 |
| | Hermite coefficients in order of increasing degree, i.e, |
| | ``(1, 2, 3)`` gives ``1*H_0(x) + 2*H_1(X) + 3*H_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 |
| | symbol : str, optional |
| | Symbol used to represent the independent variable in string |
| | representations of the polynomial expression, e.g. for printing. |
| | The symbol must be a valid Python identifier. Default value is 'x'. |
| | |
| | .. versionadded:: 1.24 |
| | |
| | """ |
| | |
| | _add = staticmethod(hermadd) |
| | _sub = staticmethod(hermsub) |
| | _mul = staticmethod(hermmul) |
| | _div = staticmethod(hermdiv) |
| | _pow = staticmethod(hermpow) |
| | _val = staticmethod(hermval) |
| | _int = staticmethod(hermint) |
| | _der = staticmethod(hermder) |
| | _fit = staticmethod(hermfit) |
| | _line = staticmethod(hermline) |
| | _roots = staticmethod(hermroots) |
| | _fromroots = staticmethod(hermfromroots) |
| |
|
| | |
| | domain = np.array(hermdomain) |
| | window = np.array(hermdomain) |
| | basis_name = 'H' |
| |
|
