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"""
==================================================
Legendre Series (:mod:`numpy.polynomial.legendre`)
==================================================
This module provides a number of objects (mostly functions) useful for
dealing with Legendre series, including a `Legendre` class that
encapsulates the usual arithmetic operations. (General information
on how this module represents and works with such polynomials is in the
docstring for its "parent" sub-package, `numpy.polynomial`).
Classes
-------
.. autosummary::
:toctree: generated/
Legendre
Constants
---------
.. autosummary::
:toctree: generated/
legdomain
legzero
legone
legx
Arithmetic
----------
.. autosummary::
:toctree: generated/
legadd
legsub
legmulx
legmul
legdiv
legpow
legval
legval2d
legval3d
leggrid2d
leggrid3d
Calculus
--------
.. autosummary::
:toctree: generated/
legder
legint
Misc Functions
--------------
.. autosummary::
:toctree: generated/
legfromroots
legroots
legvander
legvander2d
legvander3d
leggauss
legweight
legcompanion
legfit
legtrim
legline
leg2poly
poly2leg
See also
--------
numpy.polynomial
"""
import numpy as np
import numpy.linalg as la
from numpy.core.multiarray import normalize_axis_index
from . import polyutils as pu
from ._polybase import ABCPolyBase
__all__ = [
'legzero', 'legone', 'legx', 'legdomain', 'legline', 'legadd',
'legsub', 'legmulx', 'legmul', 'legdiv', 'legpow', 'legval', 'legder',
'legint', 'leg2poly', 'poly2leg', 'legfromroots', 'legvander',
'legfit', 'legtrim', 'legroots', 'Legendre', 'legval2d', 'legval3d',
'leggrid2d', 'leggrid3d', 'legvander2d', 'legvander3d', 'legcompanion',
'leggauss', 'legweight']
legtrim = pu.trimcoef
def poly2leg(pol):
"""
Convert a polynomial to a Legendre series.
Convert an array representing the coefficients of a polynomial (relative
to the "standard" basis) ordered from lowest degree to highest, to an
array of the coefficients of the equivalent Legendre series, ordered
from lowest to highest degree.
Parameters
----------
pol : array_like
1-D array containing the polynomial coefficients
Returns
-------
c : ndarray
1-D array containing the coefficients of the equivalent Legendre
series.
See Also
--------
leg2poly
Notes
-----
The easy way to do conversions between polynomial basis sets
is to use the convert method of a class instance.
Examples
--------
>>> from numpy import polynomial as P
>>> p = P.Polynomial(np.arange(4))
>>> p
Polynomial([0., 1., 2., 3.], domain=[-1, 1], window=[-1, 1])
>>> c = P.Legendre(P.legendre.poly2leg(p.coef))
>>> c
Legendre([ 1. , 3.25, 1. , 0.75], domain=[-1, 1], window=[-1, 1]) # may vary
"""
[pol] = pu.as_series([pol])
deg = len(pol) - 1
res = 0
for i in range(deg, -1, -1):
res = legadd(legmulx(res), pol[i])
return res
def leg2poly(c):
"""
Convert a Legendre series to a polynomial.
Convert an array representing the coefficients of a Legendre series,
ordered from lowest degree to highest, to an array of the coefficients
of the equivalent polynomial (relative to the "standard" basis) ordered
from lowest to highest degree.
Parameters
----------
c : array_like
1-D array containing the Legendre series coefficients, ordered
from lowest order term to highest.
Returns
-------
pol : ndarray
1-D array containing the coefficients of the equivalent polynomial
(relative to the "standard" basis) ordered from lowest order term
to highest.
See Also
--------
poly2leg
Notes
-----
The easy way to do conversions between polynomial basis sets
is to use the convert method of a class instance.
Examples
--------
>>> from numpy import polynomial as P
>>> c = P.Legendre(range(4))
>>> c
Legendre([0., 1., 2., 3.], domain=[-1, 1], window=[-1, 1])
>>> p = c.convert(kind=P.Polynomial)
>>> p
Polynomial([-1. , -3.5, 3. , 7.5], domain=[-1., 1.], window=[-1., 1.])
>>> P.legendre.leg2poly(range(4))
array([-1. , -3.5, 3. , 7.5])
"""
from .polynomial import polyadd, polysub, polymulx
[c] = pu.as_series([c])
n = len(c)
if n < 3:
return c
else:
c0 = c[-2]
c1 = c[-1]
# i is the current degree of c1
for i in range(n - 1, 1, -1):
tmp = c0
c0 = polysub(c[i - 2], (c1*(i - 1))/i)
c1 = polyadd(tmp, (polymulx(c1)*(2*i - 1))/i)
return polyadd(c0, polymulx(c1))
#
# These are constant arrays are of integer type so as to be compatible
# with the widest range of other types, such as Decimal.
#
# Legendre
legdomain = np.array([-1, 1])
# Legendre coefficients representing zero.
legzero = np.array([0])
# Legendre coefficients representing one.
legone = np.array([1])
# Legendre coefficients representing the identity x.
legx = np.array([0, 1])
def legline(off, scl):
"""
Legendre series whose graph is a straight line.
Parameters
----------
off, scl : scalars
The specified line is given by ``off + scl*x``.
Returns
-------
y : ndarray
This module's representation of the Legendre series for
``off + scl*x``.
See Also
--------
numpy.polynomial.polynomial.polyline
numpy.polynomial.chebyshev.chebline
numpy.polynomial.laguerre.lagline
numpy.polynomial.hermite.hermline
numpy.polynomial.hermite_e.hermeline
Examples
--------
>>> import numpy.polynomial.legendre as L
>>> L.legline(3,2)
array([3, 2])
>>> L.legval(-3, L.legline(3,2)) # should be -3
-3.0
"""
if scl != 0:
return np.array([off, scl])
else:
return np.array([off])
def legfromroots(roots):
"""
Generate a Legendre series with given roots.
The function returns the coefficients of the polynomial
.. math:: p(x) = (x - r_0) * (x - r_1) * ... * (x - r_n),
in Legendre form, where the `r_n` are the roots specified in `roots`.
If a zero has multiplicity n, then it must appear in `roots` n times.
For instance, if 2 is a root of multiplicity three and 3 is a root of
multiplicity 2, then `roots` looks something like [2, 2, 2, 3, 3]. The
roots can appear in any order.
If the returned coefficients are `c`, then
.. math:: p(x) = c_0 + c_1 * L_1(x) + ... + c_n * L_n(x)
The coefficient of the last term is not generally 1 for monic
polynomials in Legendre form.
Parameters
----------
roots : array_like
Sequence containing the roots.
Returns
-------
out : ndarray
1-D array of coefficients. If all roots are real then `out` is a
real array, if some of the roots are complex, then `out` is complex
even if all the coefficients in the result are real (see Examples
below).
See Also
--------
numpy.polynomial.polynomial.polyfromroots
numpy.polynomial.chebyshev.chebfromroots
numpy.polynomial.laguerre.lagfromroots
numpy.polynomial.hermite.hermfromroots
numpy.polynomial.hermite_e.hermefromroots
Examples
--------
>>> import numpy.polynomial.legendre as L
>>> L.legfromroots((-1,0,1)) # x^3 - x relative to the standard basis
array([ 0. , -0.4, 0. , 0.4])
>>> j = complex(0,1)
>>> L.legfromroots((-j,j)) # x^2 + 1 relative to the standard basis
array([ 1.33333333+0.j, 0.00000000+0.j, 0.66666667+0.j]) # may vary
"""
return pu._fromroots(legline, legmul, roots)
def legadd(c1, c2):
"""
Add one Legendre series to another.
Returns the sum of two Legendre series `c1` + `c2`. The arguments
are sequences of coefficients ordered from lowest order term to
highest, i.e., [1,2,3] represents the series ``P_0 + 2*P_1 + 3*P_2``.
Parameters
----------
c1, c2 : array_like
1-D arrays of Legendre series coefficients ordered from low to
high.
Returns
-------
out : ndarray
Array representing the Legendre series of their sum.
See Also
--------
legsub, legmulx, legmul, legdiv, legpow
Notes
-----
Unlike multiplication, division, etc., the sum of two Legendre series
is a Legendre series (without having to "reproject" the result onto
the basis set) so addition, just like that of "standard" polynomials,
is simply "component-wise."
Examples
--------
>>> from numpy.polynomial import legendre as L
>>> c1 = (1,2,3)
>>> c2 = (3,2,1)
>>> L.legadd(c1,c2)
array([4., 4., 4.])
"""
return pu._add(c1, c2)
def legsub(c1, c2):
"""
Subtract one Legendre series from another.
Returns the difference of two Legendre series `c1` - `c2`. The
sequences of coefficients are from lowest order term to highest, i.e.,
[1,2,3] represents the series ``P_0 + 2*P_1 + 3*P_2``.
Parameters
----------
c1, c2 : array_like
1-D arrays of Legendre series coefficients ordered from low to
high.
Returns
-------
out : ndarray
Of Legendre series coefficients representing their difference.
See Also
--------
legadd, legmulx, legmul, legdiv, legpow
Notes
-----
Unlike multiplication, division, etc., the difference of two Legendre
series is a Legendre series (without having to "reproject" the result
onto the basis set) so subtraction, just like that of "standard"
polynomials, is simply "component-wise."
Examples
--------
>>> from numpy.polynomial import legendre as L
>>> c1 = (1,2,3)
>>> c2 = (3,2,1)
>>> L.legsub(c1,c2)
array([-2., 0., 2.])
>>> L.legsub(c2,c1) # -C.legsub(c1,c2)
array([ 2., 0., -2.])
"""
return pu._sub(c1, c2)
def legmulx(c):
"""Multiply a Legendre series by x.
Multiply the Legendre series `c` by x, where x is the independent
variable.
Parameters
----------
c : array_like
1-D array of Legendre series coefficients ordered from low to
high.
Returns
-------
out : ndarray
Array representing the result of the multiplication.
See Also
--------
legadd, legmul, legdiv, legpow
Notes
-----
The multiplication uses the recursion relationship for Legendre
polynomials in the form
.. math::
xP_i(x) = ((i + 1)*P_{i + 1}(x) + i*P_{i - 1}(x))/(2i + 1)
Examples
--------
>>> from numpy.polynomial import legendre as L
>>> L.legmulx([1,2,3])
array([ 0.66666667, 2.2, 1.33333333, 1.8]) # may vary
"""
# c is a trimmed copy
[c] = pu.as_series([c])
# The zero series needs special treatment
if len(c) == 1 and c[0] == 0:
return c
prd = np.empty(len(c) + 1, dtype=c.dtype)
prd[0] = c[0]*0
prd[1] = c[0]
for i in range(1, len(c)):
j = i + 1
k = i - 1
s = i + j
prd[j] = (c[i]*j)/s
prd[k] += (c[i]*i)/s
return prd
def legmul(c1, c2):
"""
Multiply one Legendre series by another.
Returns the product of two Legendre series `c1` * `c2`. The arguments
are sequences of coefficients, from lowest order "term" to highest,
e.g., [1,2,3] represents the series ``P_0 + 2*P_1 + 3*P_2``.
Parameters
----------
c1, c2 : array_like
1-D arrays of Legendre series coefficients ordered from low to
high.
Returns
-------
out : ndarray
Of Legendre series coefficients representing their product.
See Also
--------
legadd, legsub, legmulx, legdiv, legpow
Notes
-----
In general, the (polynomial) product of two C-series results in terms
that are not in the Legendre polynomial basis set. Thus, to express
the product as a Legendre series, it is necessary to "reproject" the
product onto said basis set, which may produce "unintuitive" (but
correct) results; see Examples section below.
Examples
--------
>>> from numpy.polynomial import legendre as L
>>> c1 = (1,2,3)
>>> c2 = (3,2)
>>> L.legmul(c1,c2) # multiplication requires "reprojection"
array([ 4.33333333, 10.4 , 11.66666667, 3.6 ]) # may vary
"""
# s1, s2 are trimmed copies
[c1, c2] = pu.as_series([c1, c2])
if len(c1) > len(c2):
c = c2
xs = c1
else:
c = c1
xs = c2
if len(c) == 1:
c0 = c[0]*xs
c1 = 0
elif len(c) == 2:
c0 = c[0]*xs
c1 = c[1]*xs
else:
nd = len(c)
c0 = c[-2]*xs
c1 = c[-1]*xs
for i in range(3, len(c) + 1):
tmp = c0
nd = nd - 1
c0 = legsub(c[-i]*xs, (c1*(nd - 1))/nd)
c1 = legadd(tmp, (legmulx(c1)*(2*nd - 1))/nd)
return legadd(c0, legmulx(c1))
def legdiv(c1, c2):
"""
Divide one Legendre series by another.
Returns the quotient-with-remainder of two Legendre series
`c1` / `c2`. The arguments are sequences of coefficients from lowest
order "term" to highest, e.g., [1,2,3] represents the series
``P_0 + 2*P_1 + 3*P_2``.
Parameters
----------
c1, c2 : array_like
1-D arrays of Legendre series coefficients ordered from low to
high.
Returns
-------
quo, rem : ndarrays
Of Legendre series coefficients representing the quotient and
remainder.
See Also
--------
legadd, legsub, legmulx, legmul, legpow
Notes
-----
In general, the (polynomial) division of one Legendre series by another
results in quotient and remainder terms that are not in the Legendre
polynomial basis set. Thus, to express these results as a Legendre
series, it is necessary to "reproject" the results onto the Legendre
basis set, which may produce "unintuitive" (but correct) results; see
Examples section below.
Examples
--------
>>> from numpy.polynomial import legendre as L
>>> c1 = (1,2,3)
>>> c2 = (3,2,1)
>>> L.legdiv(c1,c2) # quotient "intuitive," remainder not
(array([3.]), array([-8., -4.]))
>>> c2 = (0,1,2,3)
>>> L.legdiv(c2,c1) # neither "intuitive"
(array([-0.07407407, 1.66666667]), array([-1.03703704, -2.51851852])) # may vary
"""
return pu._div(legmul, c1, c2)
def legpow(c, pow, maxpower=16):
"""Raise a Legendre series to a power.
Returns the Legendre series `c` raised to the power `pow`. The
argument `c` is a sequence of coefficients ordered from low to high.
i.e., [1,2,3] is the series ``P_0 + 2*P_1 + 3*P_2.``
Parameters
----------
c : array_like
1-D array of Legendre series coefficients ordered from low to
high.
pow : integer
Power to which the series will be raised
maxpower : integer, optional
Maximum power allowed. This is mainly to limit growth of the series
to unmanageable size. Default is 16
Returns
-------
coef : ndarray
Legendre series of power.
See Also
--------
legadd, legsub, legmulx, legmul, legdiv
"""
return pu._pow(legmul, c, pow, maxpower)
def legder(c, m=1, scl=1, axis=0):
"""
Differentiate a Legendre series.
Returns the Legendre series coefficients `c` differentiated `m` times
along `axis`. At each iteration the result is multiplied by `scl` (the
scaling factor is for use in a linear change of variable). The argument
`c` is an array of coefficients from low to high degree along each
axis, e.g., [1,2,3] represents the series ``1*L_0 + 2*L_1 + 3*L_2``
while [[1,2],[1,2]] represents ``1*L_0(x)*L_0(y) + 1*L_1(x)*L_0(y) +
2*L_0(x)*L_1(y) + 2*L_1(x)*L_1(y)`` if axis=0 is ``x`` and axis=1 is
``y``.
Parameters
----------
c : array_like
Array of Legendre series coefficients. If c is multidimensional the
different axis correspond to different variables with the degree in
each axis given by the corresponding index.
m : int, optional
Number of derivatives taken, must be non-negative. (Default: 1)
scl : scalar, optional
Each differentiation is multiplied by `scl`. The end result is
multiplication by ``scl**m``. This is for use in a linear change of
variable. (Default: 1)
axis : int, optional
Axis over which the derivative is taken. (Default: 0).
.. versionadded:: 1.7.0
Returns
-------
der : ndarray
Legendre series of the derivative.
See Also
--------
legint
Notes
-----
In general, the result of differentiating a Legendre series does not
resemble the same operation on a power series. Thus the result of this
function may be "unintuitive," albeit correct; see Examples section
below.
Examples
--------
>>> from numpy.polynomial import legendre as L
>>> c = (1,2,3,4)
>>> L.legder(c)
array([ 6., 9., 20.])
>>> L.legder(c, 3)
array([60.])
>>> L.legder(c, scl=-1)
array([ -6., -9., -20.])
>>> L.legder(c, 2,-1)
array([ 9., 60.])
"""
c = np.array(c, ndmin=1, copy=True)
if c.dtype.char in '?bBhHiIlLqQpP':
c = c.astype(np.double)
cnt = pu._deprecate_as_int(m, "the order of derivation")
iaxis = pu._deprecate_as_int(axis, "the axis")
if cnt < 0:
raise ValueError("The order of derivation must be non-negative")
iaxis = normalize_axis_index(iaxis, c.ndim)
if cnt == 0:
return c
c = np.moveaxis(c, iaxis, 0)
n = len(c)
if cnt >= n:
c = c[:1]*0
else:
for i in range(cnt):
n = n - 1
c *= scl
der = np.empty((n,) + c.shape[1:], dtype=c.dtype)
for j in range(n, 2, -1):
der[j - 1] = (2*j - 1)*c[j]
c[j - 2] += c[j]
if n > 1:
der[1] = 3*c[2]
der[0] = c[1]
c = der
c = np.moveaxis(c, 0, iaxis)
return c
def legint(c, m=1, k=[], lbnd=0, scl=1, axis=0):
"""
Integrate a Legendre series.
Returns the Legendre series coefficients `c` integrated `m` times from
`lbnd` along `axis`. At each iteration the resulting series is
**multiplied** by `scl` and an integration constant, `k`, is added.
The scaling factor is for use in a linear change of variable. ("Buyer
beware": note that, depending on what one is doing, one may want `scl`
to be the reciprocal of what one might expect; for more information,
see the Notes section below.) The argument `c` is an array of
coefficients from low to high degree along each axis, e.g., [1,2,3]
represents the series ``L_0 + 2*L_1 + 3*L_2`` while [[1,2],[1,2]]
represents ``1*L_0(x)*L_0(y) + 1*L_1(x)*L_0(y) + 2*L_0(x)*L_1(y) +
2*L_1(x)*L_1(y)`` if axis=0 is ``x`` and axis=1 is ``y``.
Parameters
----------
c : array_like
Array of Legendre series coefficients. If c is multidimensional the
different axis correspond to different variables with the degree in
each axis given by the corresponding index.
m : int, optional
Order of integration, must be positive. (Default: 1)
k : {[], list, scalar}, optional
Integration constant(s). The value of the first integral at
``lbnd`` is the first value in the list, the value of the second
integral at ``lbnd`` is the second value, etc. If ``k == []`` (the
default), all constants are set to zero. If ``m == 1``, a single
scalar can be given instead of a list.
lbnd : scalar, optional
The lower bound of the integral. (Default: 0)
scl : scalar, optional
Following each integration the result is *multiplied* by `scl`
before the integration constant is added. (Default: 1)
axis : int, optional
Axis over which the integral is taken. (Default: 0).
.. versionadded:: 1.7.0
Returns
-------
S : ndarray
Legendre series coefficient array of the integral.
Raises
------
ValueError
If ``m < 0``, ``len(k) > m``, ``np.ndim(lbnd) != 0``, or
``np.ndim(scl) != 0``.
See Also
--------
legder
Notes
-----
Note that the result of each integration is *multiplied* by `scl`.
Why is this important to note? Say one is making a linear change of
variable :math:`u = ax + b` in an integral relative to `x`. Then
:math:`dx = du/a`, so one will need to set `scl` equal to
:math:`1/a` - perhaps not what one would have first thought.
Also note that, in general, the result of integrating a C-series needs
to be "reprojected" onto the C-series basis set. Thus, typically,
the result of this function is "unintuitive," albeit correct; see
Examples section below.
Examples
--------
>>> from numpy.polynomial import legendre as L
>>> c = (1,2,3)
>>> L.legint(c)
array([ 0.33333333, 0.4 , 0.66666667, 0.6 ]) # may vary
>>> L.legint(c, 3)
array([ 1.66666667e-02, -1.78571429e-02, 4.76190476e-02, # may vary
-1.73472348e-18, 1.90476190e-02, 9.52380952e-03])
>>> L.legint(c, k=3)
array([ 3.33333333, 0.4 , 0.66666667, 0.6 ]) # may vary
>>> L.legint(c, lbnd=-2)
array([ 7.33333333, 0.4 , 0.66666667, 0.6 ]) # may vary
>>> L.legint(c, scl=2)
array([ 0.66666667, 0.8 , 1.33333333, 1.2 ]) # may vary
"""
c = np.array(c, ndmin=1, copy=True)
if c.dtype.char in '?bBhHiIlLqQpP':
c = c.astype(np.double)
if not np.iterable(k):
k = [k]
cnt = pu._deprecate_as_int(m, "the order of integration")
iaxis = pu._deprecate_as_int(axis, "the axis")
if cnt < 0:
raise ValueError("The order of integration must be non-negative")
if len(k) > cnt:
raise ValueError("Too many integration constants")
if np.ndim(lbnd) != 0:
raise ValueError("lbnd must be a scalar.")
if np.ndim(scl) != 0:
raise ValueError("scl must be a scalar.")
iaxis = normalize_axis_index(iaxis, c.ndim)
if cnt == 0:
return c
c = np.moveaxis(c, iaxis, 0)
k = list(k) + [0]*(cnt - len(k))
for i in range(cnt):
n = len(c)
c *= scl
if n == 1 and np.all(c[0] == 0):
c[0] += k[i]
else:
tmp = np.empty((n + 1,) + c.shape[1:], dtype=c.dtype)
tmp[0] = c[0]*0
tmp[1] = c[0]
if n > 1:
tmp[2] = c[1]/3
for j in range(2, n):
t = c[j]/(2*j + 1)
tmp[j + 1] = t
tmp[j - 1] -= t
tmp[0] += k[i] - legval(lbnd, tmp)
c = tmp
c = np.moveaxis(c, 0, iaxis)
return c
def legval(x, c, tensor=True):
"""
Evaluate a Legendre series at points x.
If `c` is of length `n + 1`, this function returns the value:
.. math:: p(x) = c_0 * L_0(x) + c_1 * L_1(x) + ... + c_n * L_n(x)
The parameter `x` is converted to an array only if it is a tuple or a
list, otherwise it is treated as a scalar. In either case, either `x`
or its elements must support multiplication and addition both with
themselves and with the elements of `c`.
If `c` is a 1-D array, then `p(x)` will have the same shape as `x`. If
`c` is multidimensional, then the shape of the result depends on the
value of `tensor`. If `tensor` is true the shape will be c.shape[1:] +
x.shape. If `tensor` is false the shape will be c.shape[1:]. Note that
scalars have shape (,).
Trailing zeros in the coefficients will be used in the evaluation, so
they should be avoided if efficiency is a concern.
Parameters
----------
x : array_like, compatible object
If `x` is a list or tuple, it is converted to an ndarray, otherwise
it is left unchanged and treated as a scalar. In either case, `x`
or its elements must support addition and multiplication with
themselves and with the elements of `c`.
c : array_like
Array of coefficients ordered so that the coefficients for terms of
degree n are contained in c[n]. If `c` is multidimensional the
remaining indices enumerate multiple polynomials. In the two
dimensional case the coefficients may be thought of as stored in
the columns of `c`.
tensor : boolean, optional
If True, the shape of the coefficient array is extended with ones
on the right, one for each dimension of `x`. Scalars have dimension 0
for this action. The result is that every column of coefficients in
`c` is evaluated for every element of `x`. If False, `x` is broadcast
over the columns of `c` for the evaluation. This keyword is useful
when `c` is multidimensional. The default value is True.
.. versionadded:: 1.7.0
Returns
-------
values : ndarray, algebra_like
The shape of the return value is described above.
See Also
--------
legval2d, leggrid2d, legval3d, leggrid3d
Notes
-----
The evaluation uses Clenshaw recursion, aka synthetic division.
"""
c = np.array(c, ndmin=1, copy=False)
if c.dtype.char in '?bBhHiIlLqQpP':
c = c.astype(np.double)
if isinstance(x, (tuple, list)):
x = np.asarray(x)
if isinstance(x, np.ndarray) and tensor:
c = c.reshape(c.shape + (1,)*x.ndim)
if len(c) == 1:
c0 = c[0]
c1 = 0
elif len(c) == 2:
c0 = c[0]
c1 = c[1]
else:
nd = len(c)
c0 = c[-2]
c1 = c[-1]
for i in range(3, len(c) + 1):
tmp = c0
nd = nd - 1
c0 = c[-i] - (c1*(nd - 1))/nd
c1 = tmp + (c1*x*(2*nd - 1))/nd
return c0 + c1*x
def legval2d(x, y, c):
"""
Evaluate a 2-D Legendre series at points (x, y).
This function returns the values:
.. math:: p(x,y) = \\sum_{i,j} c_{i,j} * L_i(x) * L_j(y)
The parameters `x` and `y` are converted to arrays only if they are
tuples or a lists, otherwise they are treated as a scalars and they
must have the same shape after conversion. In either case, either `x`
and `y` or their elements must support multiplication and addition both
with themselves and with the elements of `c`.
If `c` is a 1-D array a one is implicitly appended to its shape to make
it 2-D. The shape of the result will be c.shape[2:] + x.shape.
Parameters
----------
x, y : array_like, compatible objects
The two dimensional series is evaluated at the points `(x, y)`,
where `x` and `y` must have the same shape. If `x` or `y` is a list
or tuple, it is first converted to an ndarray, otherwise it is left
unchanged and if it isn't an ndarray it is treated as a scalar.
c : array_like
Array of coefficients ordered so that the coefficient of the term
of multi-degree i,j is contained in ``c[i,j]``. If `c` has
dimension greater than two the remaining indices enumerate multiple
sets of coefficients.
Returns
-------
values : ndarray, compatible object
The values of the two dimensional Legendre series at points formed
from pairs of corresponding values from `x` and `y`.
See Also
--------
legval, leggrid2d, legval3d, leggrid3d
Notes
-----
.. versionadded:: 1.7.0
"""
return pu._valnd(legval, c, x, y)
def leggrid2d(x, y, c):
"""
Evaluate a 2-D Legendre series on the Cartesian product of x and y.
This function returns the values:
.. math:: p(a,b) = \\sum_{i,j} c_{i,j} * L_i(a) * L_j(b)
where the points `(a, b)` consist of all pairs formed by taking
`a` from `x` and `b` from `y`. The resulting points form a grid with
`x` in the first dimension and `y` in the second.
The parameters `x` and `y` are converted to arrays only if they are
tuples or a lists, otherwise they are treated as a scalars. In either
case, either `x` and `y` or their elements must support multiplication
and addition both with themselves and with the elements of `c`.
If `c` has fewer than two dimensions, ones are implicitly appended to
its shape to make it 2-D. The shape of the result will be c.shape[2:] +
x.shape + y.shape.
Parameters
----------
x, y : array_like, compatible objects
The two dimensional series is evaluated at the points in the
Cartesian product of `x` and `y`. If `x` or `y` is a list or
tuple, it is first converted to an ndarray, otherwise it is left
unchanged and, if it isn't an ndarray, it is treated as a scalar.
c : array_like
Array of coefficients ordered so that the coefficient of the term of
multi-degree i,j is contained in `c[i,j]`. If `c` has dimension
greater than two the remaining indices enumerate multiple sets of
coefficients.
Returns
-------
values : ndarray, compatible object
The values of the two dimensional Chebyshev series at points in the
Cartesian product of `x` and `y`.
See Also
--------
legval, legval2d, legval3d, leggrid3d
Notes
-----
.. versionadded:: 1.7.0
"""
return pu._gridnd(legval, c, x, y)
def legval3d(x, y, z, c):
"""
Evaluate a 3-D Legendre series at points (x, y, z).
This function returns the values:
.. math:: p(x,y,z) = \\sum_{i,j,k} c_{i,j,k} * L_i(x) * L_j(y) * L_k(z)
The parameters `x`, `y`, and `z` are converted to arrays only if
they are tuples or a lists, otherwise they are treated as a scalars and
they must have the same shape after conversion. In either case, either
`x`, `y`, and `z` or their elements must support multiplication and
addition both with themselves and with the elements of `c`.
If `c` has fewer than 3 dimensions, ones are implicitly appended to its
shape to make it 3-D. The shape of the result will be c.shape[3:] +
x.shape.
Parameters
----------
x, y, z : array_like, compatible object
The three dimensional series is evaluated at the points
`(x, y, z)`, where `x`, `y`, and `z` must have the same shape. If
any of `x`, `y`, or `z` is a list or tuple, it is first converted
to an ndarray, otherwise it is left unchanged and if it isn't an
ndarray it is treated as a scalar.
c : array_like
Array of coefficients ordered so that the coefficient of the term of
multi-degree i,j,k is contained in ``c[i,j,k]``. If `c` has dimension
greater than 3 the remaining indices enumerate multiple sets of
coefficients.
Returns
-------
values : ndarray, compatible object
The values of the multidimensional polynomial on points formed with
triples of corresponding values from `x`, `y`, and `z`.
See Also
--------
legval, legval2d, leggrid2d, leggrid3d
Notes
-----
.. versionadded:: 1.7.0
"""
return pu._valnd(legval, c, x, y, z)
def leggrid3d(x, y, z, c):
"""
Evaluate a 3-D Legendre series on the Cartesian product of x, y, and z.
This function returns the values:
.. math:: p(a,b,c) = \\sum_{i,j,k} c_{i,j,k} * L_i(a) * L_j(b) * L_k(c)
where the points `(a, b, c)` consist of all triples formed by taking
`a` from `x`, `b` from `y`, and `c` from `z`. The resulting points form
a grid with `x` in the first dimension, `y` in the second, and `z` in
the third.
The parameters `x`, `y`, and `z` are converted to arrays only if they
are tuples or a lists, otherwise they are treated as a scalars. In
either case, either `x`, `y`, and `z` or their elements must support
multiplication and addition both with themselves and with the elements
of `c`.
If `c` has fewer than three dimensions, ones are implicitly appended to
its shape to make it 3-D. The shape of the result will be c.shape[3:] +
x.shape + y.shape + z.shape.
Parameters
----------
x, y, z : array_like, compatible objects
The three dimensional series is evaluated at the points in the
Cartesian product of `x`, `y`, and `z`. If `x`,`y`, or `z` is a
list or tuple, it is first converted to an ndarray, otherwise it is
left unchanged and, if it isn't an ndarray, it is treated as a
scalar.
c : array_like
Array of coefficients ordered so that the coefficients for terms of
degree i,j are contained in ``c[i,j]``. If `c` has dimension
greater than two the remaining indices enumerate multiple sets of
coefficients.
Returns
-------
values : ndarray, compatible object
The values of the two dimensional polynomial at points in the Cartesian
product of `x` and `y`.
See Also
--------
legval, legval2d, leggrid2d, legval3d
Notes
-----
.. versionadded:: 1.7.0
"""
return pu._gridnd(legval, c, x, y, z)
def legvander(x, deg):
"""Pseudo-Vandermonde matrix of given degree.
Returns the pseudo-Vandermonde matrix of degree `deg` and sample points
`x`. The pseudo-Vandermonde matrix is defined by
.. math:: V[..., i] = L_i(x)
where `0 <= i <= deg`. The leading indices of `V` index the elements of
`x` and the last index is the degree of the Legendre polynomial.
If `c` is a 1-D array of coefficients of length `n + 1` and `V` is the
array ``V = legvander(x, n)``, then ``np.dot(V, c)`` and
``legval(x, c)`` are the same up to roundoff. This equivalence is
useful both for least squares fitting and for the evaluation of a large
number of Legendre series of the same degree and sample points.
Parameters
----------
x : array_like
Array of points. The dtype is converted to float64 or complex128
depending on whether any of the elements are complex. If `x` is
scalar it is converted to a 1-D array.
deg : int
Degree of the resulting matrix.
Returns
-------
vander : ndarray
The pseudo-Vandermonde matrix. The shape of the returned matrix is
``x.shape + (deg + 1,)``, where The last index is the degree of the
corresponding Legendre polynomial. The dtype will be the same as
the converted `x`.
"""
ideg = pu._deprecate_as_int(deg, "deg")
if ideg < 0:
raise ValueError("deg must be non-negative")
x = np.array(x, copy=False, ndmin=1) + 0.0
dims = (ideg + 1,) + x.shape
dtyp = x.dtype
v = np.empty(dims, dtype=dtyp)
# Use forward recursion to generate the entries. This is not as accurate
# as reverse recursion in this application but it is more efficient.
v[0] = x*0 + 1
if ideg > 0:
v[1] = x
for i in range(2, ideg + 1):
v[i] = (v[i-1]*x*(2*i - 1) - v[i-2]*(i - 1))/i
return np.moveaxis(v, 0, -1)
def legvander2d(x, y, deg):
"""Pseudo-Vandermonde matrix of given degrees.
Returns the pseudo-Vandermonde matrix of degrees `deg` and sample
points `(x, y)`. The pseudo-Vandermonde matrix is defined by
.. math:: V[..., (deg[1] + 1)*i + j] = L_i(x) * L_j(y),
where `0 <= i <= deg[0]` and `0 <= j <= deg[1]`. The leading indices of
`V` index the points `(x, y)` and the last index encodes the degrees of
the Legendre polynomials.
If ``V = legvander2d(x, y, [xdeg, ydeg])``, then the columns of `V`
correspond to the elements of a 2-D coefficient array `c` of shape
(xdeg + 1, ydeg + 1) in the order
.. math:: c_{00}, c_{01}, c_{02} ... , c_{10}, c_{11}, c_{12} ...
and ``np.dot(V, c.flat)`` and ``legval2d(x, y, c)`` will be the same
up to roundoff. This equivalence is useful both for least squares
fitting and for the evaluation of a large number of 2-D Legendre
series of the same degrees and sample points.
Parameters
----------
x, y : array_like
Arrays of point coordinates, all of the same shape. The dtypes
will be converted to either float64 or complex128 depending on
whether any of the elements are complex. Scalars are converted to
1-D arrays.
deg : list of ints
List of maximum degrees of the form [x_deg, y_deg].
Returns
-------
vander2d : ndarray
The shape of the returned matrix is ``x.shape + (order,)``, where
:math:`order = (deg[0]+1)*(deg[1]+1)`. The dtype will be the same
as the converted `x` and `y`.
See Also
--------
legvander, legvander3d, legval2d, legval3d
Notes
-----
.. versionadded:: 1.7.0
"""
return pu._vander_nd_flat((legvander, legvander), (x, y), deg)
def legvander3d(x, y, z, deg):
"""Pseudo-Vandermonde matrix of given degrees.
Returns the pseudo-Vandermonde matrix of degrees `deg` and sample
points `(x, y, z)`. If `l, m, n` are the given degrees in `x, y, z`,
then The pseudo-Vandermonde matrix is defined by
.. math:: V[..., (m+1)(n+1)i + (n+1)j + k] = L_i(x)*L_j(y)*L_k(z),
where `0 <= i <= l`, `0 <= j <= m`, and `0 <= j <= n`. The leading
indices of `V` index the points `(x, y, z)` and the last index encodes
the degrees of the Legendre polynomials.
If ``V = legvander3d(x, y, z, [xdeg, ydeg, zdeg])``, then the columns
of `V` correspond to the elements of a 3-D coefficient array `c` of
shape (xdeg + 1, ydeg + 1, zdeg + 1) in the order
.. math:: c_{000}, c_{001}, c_{002},... , c_{010}, c_{011}, c_{012},...
and ``np.dot(V, c.flat)`` and ``legval3d(x, y, z, c)`` will be the
same up to roundoff. This equivalence is useful both for least squares
fitting and for the evaluation of a large number of 3-D Legendre
series of the same degrees and sample points.
Parameters
----------
x, y, z : array_like
Arrays of point coordinates, all of the same shape. The dtypes will
be converted to either float64 or complex128 depending on whether
any of the elements are complex. Scalars are converted to 1-D
arrays.
deg : list of ints
List of maximum degrees of the form [x_deg, y_deg, z_deg].
Returns
-------
vander3d : ndarray
The shape of the returned matrix is ``x.shape + (order,)``, where
:math:`order = (deg[0]+1)*(deg[1]+1)*(deg[2]+1)`. The dtype will
be the same as the converted `x`, `y`, and `z`.
See Also
--------
legvander, legvander3d, legval2d, legval3d
Notes
-----
.. versionadded:: 1.7.0
"""
return pu._vander_nd_flat((legvander, legvander, legvander), (x, y, z), deg)
def legfit(x, y, deg, rcond=None, full=False, w=None):
"""
Least squares fit of Legendre series to data.
Return the coefficients of a Legendre series of degree `deg` that is the
least squares fit to the data values `y` given at points `x`. If `y` is
1-D the returned coefficients will also be 1-D. If `y` is 2-D multiple
fits are done, one for each column of `y`, and the resulting
coefficients are stored in the corresponding columns of a 2-D return.
The fitted polynomial(s) are in the form
.. math:: p(x) = c_0 + c_1 * L_1(x) + ... + c_n * L_n(x),
where `n` is `deg`.
Parameters
----------
x : array_like, shape (M,)
x-coordinates of the M sample points ``(x[i], y[i])``.
y : array_like, shape (M,) or (M, K)
y-coordinates of the sample points. Several data sets of sample
points sharing the same x-coordinates can be fitted at once by
passing in a 2D-array that contains one dataset per column.
deg : int or 1-D array_like
Degree(s) of the fitting polynomials. If `deg` is a single integer
all terms up to and including the `deg`'th term are included in the
fit. For NumPy versions >= 1.11.0 a list of integers specifying the
degrees of the terms to include may be used instead.
rcond : float, optional
Relative condition number of the fit. Singular values smaller than
this relative to the largest singular value will be ignored. The
default value is len(x)*eps, where eps is the relative precision of
the float type, about 2e-16 in most cases.
full : bool, optional
Switch determining nature of return value. When it is False (the
default) just the coefficients are returned, when True diagnostic
information from the singular value decomposition is also returned.
w : array_like, shape (`M`,), optional
Weights. If not None, the 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 is a trimmed copy
[c] = pu.as_series([c])
if len(c) < 2:
raise ValueError('Series must have maximum degree of at least 1.')
if len(c) == 2:
return np.array([[-c[0]/c[1]]])
n = len(c) - 1
mat = np.zeros((n, n), dtype=c.dtype)
scl = 1./np.sqrt(2*np.arange(n) + 1)
top = mat.reshape(-1)[1::n+1]
bot = mat.reshape(-1)[n::n+1]
top[...] = np.arange(1, n)*scl[:n-1]*scl[1:n]
bot[...] = top
mat[:, -1] -= (c[:-1]/c[-1])*(scl/scl[-1])*(n/(2*n - 1))
return mat
def legroots(c):
"""
Compute the roots of a Legendre series.
Return the roots (a.k.a. "zeros") of the polynomial
.. math:: p(x) = \\sum_i c[i] * L_i(x).
Parameters
----------
c : 1-D array_like
1-D array of coefficients.
Returns
-------
out : ndarray
Array of the roots of the series. If all the roots are real,
then `out` is also real, otherwise it is complex.
See Also
--------
numpy.polynomial.polynomial.polyroots
numpy.polynomial.chebyshev.chebroots
numpy.polynomial.laguerre.lagroots
numpy.polynomial.hermite.hermroots
numpy.polynomial.hermite_e.hermeroots
Notes
-----
The root estimates are obtained as the eigenvalues of the companion
matrix, Roots far from the origin of the complex plane may have large
errors due to the numerical instability of the series for such values.
Roots with multiplicity greater than 1 will also show larger errors as
the value of the series near such points is relatively insensitive to
errors in the roots. Isolated roots near the origin can be improved by
a few iterations of Newton's method.
The Legendre series basis polynomials aren't powers of ``x`` so the
results of this function may seem unintuitive.
Examples
--------
>>> import numpy.polynomial.legendre as leg
>>> leg.legroots((1, 2, 3, 4)) # 4L_3 + 3L_2 + 2L_1 + 1L_0, all real roots
array([-0.85099543, -0.11407192, 0.51506735]) # may vary
"""
# c is a trimmed copy
[c] = pu.as_series([c])
if len(c) < 2:
return np.array([], dtype=c.dtype)
if len(c) == 2:
return np.array([-c[0]/c[1]])
# rotated companion matrix reduces error
m = legcompanion(c)[::-1,::-1]
r = la.eigvals(m)
r.sort()
return r
def leggauss(deg):
"""
Gauss-Legendre quadrature.
Computes the sample points and weights for Gauss-Legendre quadrature.
These sample points and weights will correctly integrate polynomials of
degree :math:`2*deg - 1` or less over the interval :math:`[-1, 1]` with
the weight function :math:`f(x) = 1`.
Parameters
----------
deg : int
Number of sample points and weights. It must be >= 1.
Returns
-------
x : ndarray
1-D ndarray containing the sample points.
y : ndarray
1-D ndarray containing the weights.
Notes
-----
.. versionadded:: 1.7.0
The results have only been tested up to degree 100, higher degrees may
be problematic. The weights are determined by using the fact that
.. math:: w_k = c / (L'_n(x_k) * L_{n-1}(x_k))
where :math:`c` is a constant independent of :math:`k` and :math:`x_k`
is the k'th root of :math:`L_n`, and then scaling the results to get
the right value when integrating 1.
"""
ideg = pu._deprecate_as_int(deg, "deg")
if ideg <= 0:
raise ValueError("deg must be a positive integer")
# first approximation of roots. We use the fact that the companion
# matrix is symmetric in this case in order to obtain better zeros.
c = np.array([0]*deg + [1])
m = legcompanion(c)
x = la.eigvalsh(m)
# improve roots by one application of Newton
dy = legval(x, c)
df = legval(x, legder(c))
x -= dy/df
# compute the weights. We scale the factor to avoid possible numerical
# overflow.
fm = legval(x, c[1:])
fm /= np.abs(fm).max()
df /= np.abs(df).max()
w = 1/(fm * df)
# for Legendre we can also symmetrize
w = (w + w[::-1])/2
x = (x - x[::-1])/2
# scale w to get the right value
w *= 2. / w.sum()
return x, w
def legweight(x):
"""
Weight function of the Legendre polynomials.
The weight function is :math:`1` and the interval of integration is
:math:`[-1, 1]`. The Legendre polynomials are orthogonal, but not
normalized, with respect to this weight function.
Parameters
----------
x : array_like
Values at which the weight function will be computed.
Returns
-------
w : ndarray
The weight function at `x`.
Notes
-----
.. versionadded:: 1.7.0
"""
w = x*0.0 + 1.0
return w
#
# Legendre series class
#
class Legendre(ABCPolyBase):
"""A Legendre series class.
The Legendre class provides the standard Python numerical methods
'+', '-', '*', '//', '%', 'divmod', '**', and '()' as well as the
attributes and methods listed in the `ABCPolyBase` documentation.
Parameters
----------
coef : array_like
Legendre coefficients in order of increasing degree, i.e.,
``(1, 2, 3)`` gives ``1*P_0(x) + 2*P_1(x) + 3*P_2(x)``.
domain : (2,) array_like, optional
Domain to use. The interval ``[domain[0], domain[1]]`` is mapped
to the interval ``[window[0], window[1]]`` by shifting and scaling.
The default value is [-1, 1].
window : (2,) array_like, optional
Window, see `domain` for its use. The default value is [-1, 1].
.. versionadded:: 1.6.0
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
"""
# Virtual Functions
_add = staticmethod(legadd)
_sub = staticmethod(legsub)
_mul = staticmethod(legmul)
_div = staticmethod(legdiv)
_pow = staticmethod(legpow)
_val = staticmethod(legval)
_int = staticmethod(legint)
_der = staticmethod(legder)
_fit = staticmethod(legfit)
_line = staticmethod(legline)
_roots = staticmethod(legroots)
_fromroots = staticmethod(legfromroots)
# Virtual properties
domain = np.array(legdomain)
window = np.array(legdomain)
basis_name = 'P'