|
""" |
|
================================================== |
|
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/ |
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|
|
Legendre |
|
|
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Constants |
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--------- |
|
|
|
.. autosummary:: |
|
:toctree: generated/ |
|
|
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legdomain |
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legzero |
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legone |
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legx |
|
|
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Arithmetic |
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---------- |
|
|
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.. autosummary:: |
|
:toctree: generated/ |
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|
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legadd |
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legsub |
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legmulx |
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legmul |
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legdiv |
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legpow |
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legval |
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legval2d |
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legval3d |
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leggrid2d |
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leggrid3d |
|
|
|
Calculus |
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-------- |
|
|
|
.. autosummary:: |
|
:toctree: generated/ |
|
|
|
legder |
|
legint |
|
|
|
Misc Functions |
|
-------------- |
|
|
|
.. autosummary:: |
|
:toctree: generated/ |
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|
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legfromroots |
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legroots |
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legvander |
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legvander2d |
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legvander3d |
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leggauss |
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legweight |
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legcompanion |
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legfit |
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legtrim |
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legline |
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leg2poly |
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poly2leg |
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|
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See also |
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-------- |
|
numpy.polynomial |
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|
|
""" |
|
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__ = [ |
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'legzero', 'legone', 'legx', 'legdomain', 'legline', 'legadd', |
|
'legsub', 'legmulx', 'legmul', 'legdiv', 'legpow', 'legval', 'legder', |
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'legint', 'leg2poly', 'poly2leg', 'legfromroots', 'legvander', |
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'legfit', 'legtrim', 'legroots', 'Legendre', 'legval2d', 'legval3d', |
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'leggrid2d', 'leggrid3d', 'legvander2d', 'legvander3d', 'legcompanion', |
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'leggauss', 'legweight'] |
|
|
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legtrim = pu.trimcoef |
|
|
|
|
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def poly2leg(pol): |
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""" |
|
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)) |
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>>> p |
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Polynomial([0., 1., 2., 3.], domain=[-1, 1], window=[-1, 1]) |
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>>> c = P.Legendre(P.legendre.poly2leg(p.coef)) |
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>>> c |
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Legendre([ 1. , 3.25, 1. , 0.75], domain=[-1, 1], window=[-1, 1]) # may vary |
|
|
|
""" |
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[pol] = pu.as_series([pol]) |
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deg = len(pol) - 1 |
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res = 0 |
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for i in range(deg, -1, -1): |
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res = legadd(legmulx(res), pol[i]) |
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return res |
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|
|
|
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def leg2poly(c): |
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""" |
|
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 |
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Polynomial([-1. , -3.5, 3. , 7.5], domain=[-1., 1.], window=[-1., 1.]) |
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>>> P.legendre.leg2poly(range(4)) |
|
array([-1. , -3.5, 3. , 7.5]) |
|
|
|
|
|
""" |
|
from .polynomial import polyadd, polysub, polymulx |
|
|
|
[c] = pu.as_series([c]) |
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n = len(c) |
|
if n < 3: |
|
return c |
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else: |
|
c0 = c[-2] |
|
c1 = c[-1] |
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|
|
for i in range(n - 1, 1, -1): |
|
tmp = c0 |
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c0 = polysub(c[i - 2], (c1*(i - 1))/i) |
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c1 = polyadd(tmp, (polymulx(c1)*(2*i - 1))/i) |
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return polyadd(c0, polymulx(c1)) |
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|
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legdomain = np.array([-1, 1]) |
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|
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legzero = np.array([0]) |
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legone = np.array([1]) |
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legx = np.array([0, 1]) |
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|
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def legline(off, scl): |
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""" |
|
Legendre series whose graph is a straight line. |
|
|
|
|
|
|
|
Parameters |
|
---------- |
|
off, scl : scalars |
|
The specified line is given by ``off + scl*x``. |
|
|
|
Returns |
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------- |
|
y : ndarray |
|
This module's representation of the Legendre series for |
|
``off + scl*x``. |
|
|
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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]) |
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else: |
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return np.array([off]) |
|
|
|
|
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def legfromroots(roots): |
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""" |
|
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) |
|
|
|
|
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def legadd(c1, c2): |
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""" |
|
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, 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] = 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] |
|
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 |
|
|
|
""" |
|
|
|
[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 |
|
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) |
|
|
|
|
|
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 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. |
|
|
|
.. 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`` |
|
|
|
- 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.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] = 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] = 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]]) |
|
|
|
|
|
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") |
|
|
|
|
|
|
|
c = np.array([0]*deg + [1]) |
|
m = legcompanion(c) |
|
x = la.eigvalsh(m) |
|
|
|
|
|
dy = legval(x, c) |
|
df = legval(x, legder(c)) |
|
x -= dy/df |
|
|
|
|
|
|
|
fm = legval(x, c[1:]) |
|
fm /= np.abs(fm).max() |
|
df /= np.abs(df).max() |
|
w = 1/(fm * df) |
|
|
|
|
|
w = (w + w[::-1])/2 |
|
x = (x - x[::-1])/2 |
|
|
|
|
|
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 |
|
|
|
|
|
|
|
|
|
|
|
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 |
|
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(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) |
|
|
|
|
|
domain = np.array(legdomain) |
|
window = np.array(legdomain) |
|
basis_name = 'P' |
|
|
