/
polynomial.py
1556 lines (1270 loc) · 48 KB
/
polynomial.py
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"""
Objects for dealing with polynomials.
This module provides a number of objects (mostly functions) useful for
dealing with polynomials, including a `Polynomial` class that
encapsulates the usual arithmetic operations. (General information
on how this module represents and works with polynomial objects is in
the docstring for its "parent" sub-package, `numpy.polynomial`).
Constants
---------
- `polydomain` -- Polynomial default domain, [-1,1].
- `polyzero` -- (Coefficients of the) "zero polynomial."
- `polyone` -- (Coefficients of the) constant polynomial 1.
- `polyx` -- (Coefficients of the) identity map polynomial, ``f(x) = x``.
Arithmetic
----------
- `polyadd` -- add two polynomials.
- `polysub` -- subtract one polynomial from another.
- `polymul` -- multiply two polynomials.
- `polydiv` -- divide one polynomial by another.
- `polypow` -- raise a polynomial to an positive integer power
- `polyval` -- evaluate a polynomial at given points.
- `polyval2d` -- evaluate a 2D polynomial at given points.
- `polyval3d` -- evaluate a 3D polynomial at given points.
- `polygrid2d` -- evaluate a 2D polynomial on a Cartesian product.
- `polygrid3d` -- evaluate a 3D polynomial on a Cartesian product.
Calculus
--------
- `polyder` -- differentiate a polynomial.
- `polyint` -- integrate a polynomial.
Misc Functions
--------------
- `polyfromroots` -- create a polynomial with specified roots.
- `polyroots` -- find the roots of a polynomial.
- `polyvander` -- Vandermonde-like matrix for powers.
- `polyvander2d` -- Vandermonde-like matrix for 2D power series.
- `polyvander3d` -- Vandermonde-like matrix for 3D power series.
- `polycompanion` -- companion matrix in power series form.
- `polyfit` -- least-squares fit returning a polynomial.
- `polytrim` -- trim leading coefficients from a polynomial.
- `polyline` -- polynomial representing given straight line.
Classes
-------
- `Polynomial` -- polynomial class.
See Also
--------
`numpy.polynomial`
"""
from __future__ import division, absolute_import, print_function
__all__ = [
'polyzero', 'polyone', 'polyx', 'polydomain', 'polyline', 'polyadd',
'polysub', 'polymulx', 'polymul', 'polydiv', 'polypow', 'polyval',
'polyder', 'polyint', 'polyfromroots', 'polyvander', 'polyfit',
'polytrim', 'polyroots', 'Polynomial', 'polyval2d', 'polyval3d',
'polygrid2d', 'polygrid3d', 'polyvander2d', 'polyvander3d']
import warnings
import numpy as np
import numpy.linalg as la
from . import polyutils as pu
from ._polybase import ABCPolyBase
polytrim = pu.trimcoef
#
# These are constant arrays are of integer type so as to be compatible
# with the widest range of other types, such as Decimal.
#
# Polynomial default domain.
polydomain = np.array([-1, 1])
# Polynomial coefficients representing zero.
polyzero = np.array([0])
# Polynomial coefficients representing one.
polyone = np.array([1])
# Polynomial coefficients representing the identity x.
polyx = np.array([0, 1])
#
# Polynomial series functions
#
def polyline(off, scl):
"""
Returns an array representing a linear polynomial.
Parameters
----------
off, scl : scalars
The "y-intercept" and "slope" of the line, respectively.
Returns
-------
y : ndarray
This module's representation of the linear polynomial ``off +
scl*x``.
See Also
--------
chebline
Examples
--------
>>> from numpy.polynomial import polynomial as P
>>> P.polyline(1,-1)
array([ 1, -1])
>>> P.polyval(1, P.polyline(1,-1)) # should be 0
0.0
"""
if scl != 0:
return np.array([off, scl])
else:
return np.array([off])
def polyfromroots(roots):
"""
Generate a monic polynomial with given roots.
Return the coefficients of the polynomial
.. math:: p(x) = (x - r_0) * (x - r_1) * ... * (x - r_n),
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 * x + ... + x^n
The coefficient of the last term is 1 for monic polynomials in this
form.
Parameters
----------
roots : array_like
Sequence containing the roots.
Returns
-------
out : ndarray
1-D array of the polynomial's coefficients If all the roots are
real, then `out` is also real, otherwise it is complex. (see
Examples below).
See Also
--------
chebfromroots, legfromroots, lagfromroots, hermfromroots
hermefromroots
Notes
-----
The coefficients are determined by multiplying together linear factors
of the form `(x - r_i)`, i.e.
.. math:: p(x) = (x - r_0) (x - r_1) ... (x - r_n)
where ``n == len(roots) - 1``; note that this implies that `1` is always
returned for :math:`a_n`.
Examples
--------
>>> from numpy.polynomial import polynomial as P
>>> P.polyfromroots((-1,0,1)) # x(x - 1)(x + 1) = x^3 - x
array([ 0., -1., 0., 1.])
>>> j = complex(0,1)
>>> P.polyfromroots((-j,j)) # complex returned, though values are real
array([ 1.+0.j, 0.+0.j, 1.+0.j])
"""
if len(roots) == 0:
return np.ones(1)
else:
[roots] = pu.as_series([roots], trim=False)
roots.sort()
p = [polyline(-r, 1) for r in roots]
n = len(p)
while n > 1:
m, r = divmod(n, 2)
tmp = [polymul(p[i], p[i+m]) for i in range(m)]
if r:
tmp[0] = polymul(tmp[0], p[-1])
p = tmp
n = m
return p[0]
def polyadd(c1, c2):
"""
Add one polynomial to another.
Returns the sum of two polynomials `c1` + `c2`. The arguments are
sequences of coefficients from lowest order term to highest, i.e.,
[1,2,3] represents the polynomial ``1 + 2*x + 3*x**2``.
Parameters
----------
c1, c2 : array_like
1-D arrays of polynomial coefficients ordered from low to high.
Returns
-------
out : ndarray
The coefficient array representing their sum.
See Also
--------
polysub, polymul, polydiv, polypow
Examples
--------
>>> from numpy.polynomial import polynomial as P
>>> c1 = (1,2,3)
>>> c2 = (3,2,1)
>>> sum = P.polyadd(c1,c2); sum
array([ 4., 4., 4.])
>>> P.polyval(2, sum) # 4 + 4(2) + 4(2**2)
28.0
"""
# c1, c2 are trimmed copies
[c1, c2] = pu.as_series([c1, c2])
if len(c1) > len(c2):
c1[:c2.size] += c2
ret = c1
else:
c2[:c1.size] += c1
ret = c2
return pu.trimseq(ret)
def polysub(c1, c2):
"""
Subtract one polynomial from another.
Returns the difference of two polynomials `c1` - `c2`. The arguments
are sequences of coefficients from lowest order term to highest, i.e.,
[1,2,3] represents the polynomial ``1 + 2*x + 3*x**2``.
Parameters
----------
c1, c2 : array_like
1-D arrays of polynomial coefficients ordered from low to
high.
Returns
-------
out : ndarray
Of coefficients representing their difference.
See Also
--------
polyadd, polymul, polydiv, polypow
Examples
--------
>>> from numpy.polynomial import polynomial as P
>>> c1 = (1,2,3)
>>> c2 = (3,2,1)
>>> P.polysub(c1,c2)
array([-2., 0., 2.])
>>> P.polysub(c2,c1) # -P.polysub(c1,c2)
array([ 2., 0., -2.])
"""
# c1, c2 are trimmed copies
[c1, c2] = pu.as_series([c1, c2])
if len(c1) > len(c2):
c1[:c2.size] -= c2
ret = c1
else:
c2 = -c2
c2[:c1.size] += c1
ret = c2
return pu.trimseq(ret)
def polymulx(c):
"""Multiply a polynomial by x.
Multiply the polynomial `c` by x, where x is the independent
variable.
Parameters
----------
c : array_like
1-D array of polynomial coefficients ordered from low to
high.
Returns
-------
out : ndarray
Array representing the result of the multiplication.
Notes
-----
.. versionadded:: 1.5.0
"""
# 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
return prd
def polymul(c1, c2):
"""
Multiply one polynomial by another.
Returns the product of two polynomials `c1` * `c2`. The arguments are
sequences of coefficients, from lowest order term to highest, e.g.,
[1,2,3] represents the polynomial ``1 + 2*x + 3*x**2.``
Parameters
----------
c1, c2 : array_like
1-D arrays of coefficients representing a polynomial, relative to the
"standard" basis, and ordered from lowest order term to highest.
Returns
-------
out : ndarray
Of the coefficients of their product.
See Also
--------
polyadd, polysub, polydiv, polypow
Examples
--------
>>> from numpy.polynomial import polynomial as P
>>> c1 = (1,2,3)
>>> c2 = (3,2,1)
>>> P.polymul(c1,c2)
array([ 3., 8., 14., 8., 3.])
"""
# c1, c2 are trimmed copies
[c1, c2] = pu.as_series([c1, c2])
ret = np.convolve(c1, c2)
return pu.trimseq(ret)
def polydiv(c1, c2):
"""
Divide one polynomial by another.
Returns the quotient-with-remainder of two polynomials `c1` / `c2`.
The arguments are sequences of coefficients, from lowest order term
to highest, e.g., [1,2,3] represents ``1 + 2*x + 3*x**2``.
Parameters
----------
c1, c2 : array_like
1-D arrays of polynomial coefficients ordered from low to high.
Returns
-------
[quo, rem] : ndarrays
Of coefficient series representing the quotient and remainder.
See Also
--------
polyadd, polysub, polymul, polypow
Examples
--------
>>> from numpy.polynomial import polynomial as P
>>> c1 = (1,2,3)
>>> c2 = (3,2,1)
>>> P.polydiv(c1,c2)
(array([ 3.]), array([-8., -4.]))
>>> P.polydiv(c2,c1)
(array([ 0.33333333]), array([ 2.66666667, 1.33333333]))
"""
# c1, c2 are trimmed copies
[c1, c2] = pu.as_series([c1, c2])
if c2[-1] == 0:
raise ZeroDivisionError()
len1 = len(c1)
len2 = len(c2)
if len2 == 1:
return c1/c2[-1], c1[:1]*0
elif len1 < len2:
return c1[:1]*0, c1
else:
dlen = len1 - len2
scl = c2[-1]
c2 = c2[:-1]/scl
i = dlen
j = len1 - 1
while i >= 0:
c1[i:j] -= c2*c1[j]
i -= 1
j -= 1
return c1[j+1:]/scl, pu.trimseq(c1[:j+1])
def polypow(c, pow, maxpower=None):
"""Raise a polynomial to a power.
Returns the polynomial `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 ``1 + 2*x + 3*x**2.``
Parameters
----------
c : array_like
1-D array of array of series coefficients ordered from low to
high degree.
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
Power series of power.
See Also
--------
polyadd, polysub, polymul, polydiv
Examples
--------
"""
# c is a trimmed copy
[c] = pu.as_series([c])
power = int(pow)
if power != pow or power < 0:
raise ValueError("Power must be a non-negative integer.")
elif maxpower is not None and power > maxpower:
raise ValueError("Power is too large")
elif power == 0:
return np.array([1], dtype=c.dtype)
elif power == 1:
return c
else:
# This can be made more efficient by using powers of two
# in the usual way.
prd = c
for i in range(2, power + 1):
prd = np.convolve(prd, c)
return prd
def polyder(c, m=1, scl=1, axis=0):
"""
Differentiate a polynomial.
Returns the polynomial 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 polynomial ``1 + 2*x + 3*x**2``
while [[1,2],[1,2]] represents ``1 + 1*x + 2*y + 2*x*y`` if axis=0 is
``x`` and axis=1 is ``y``.
Parameters
----------
c : array_like
Array of polynomial 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
Polynomial coefficients of the derivative.
See Also
--------
polyint
Examples
--------
>>> from numpy.polynomial import polynomial as P
>>> c = (1,2,3,4) # 1 + 2x + 3x**2 + 4x**3
>>> P.polyder(c) # (d/dx)(c) = 2 + 6x + 12x**2
array([ 2., 6., 12.])
>>> P.polyder(c,3) # (d**3/dx**3)(c) = 24
array([ 24.])
>>> P.polyder(c,scl=-1) # (d/d(-x))(c) = -2 - 6x - 12x**2
array([ -2., -6., -12.])
>>> P.polyder(c,2,-1) # (d**2/d(-x)**2)(c) = 6 + 24x
array([ 6., 24.])
"""
c = np.array(c, ndmin=1, copy=1)
if c.dtype.char in '?bBhHiIlLqQpP':
# astype fails with NA
c = c + 0.0
cdt = c.dtype
cnt, iaxis = [int(t) for t in [m, axis]]
if cnt != m:
raise ValueError("The order of derivation must be integer")
if cnt < 0:
raise ValueError("The order of derivation must be non-negative")
if iaxis != axis:
raise ValueError("The axis must be integer")
if not -c.ndim <= iaxis < c.ndim:
raise ValueError("The axis is out of range")
if iaxis < 0:
iaxis += c.ndim
if cnt == 0:
return c
c = np.rollaxis(c, iaxis)
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=cdt)
for j in range(n, 0, -1):
der[j - 1] = j*c[j]
c = der
c = np.rollaxis(c, 0, iaxis + 1)
return c
def polyint(c, m=1, k=[], lbnd=0, scl=1, axis=0):
"""
Integrate a polynomial.
Returns the polynomial 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 polynomial ``1 + 2*x + 3*x**2`` while [[1,2],[1,2]]
represents ``1 + 1*x + 2*y + 2*x*y`` if axis=0 is ``x`` and axis=1 is
``y``.
Parameters
----------
c : array_like
1-D array of polynomial coefficients, ordered from low to high.
m : int, optional
Order of integration, must be positive. (Default: 1)
k : {[], list, scalar}, optional
Integration constant(s). The value of the first integral at zero
is the first value in the list, the value of the second integral
at zero 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
Coefficient array of the integral.
Raises
------
ValueError
If ``m < 1``, ``len(k) > m``.
See Also
--------
polyder
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.
Examples
--------
>>> from numpy.polynomial import polynomial as P
>>> c = (1,2,3)
>>> P.polyint(c) # should return array([0, 1, 1, 1])
array([ 0., 1., 1., 1.])
>>> P.polyint(c,3) # should return array([0, 0, 0, 1/6, 1/12, 1/20])
array([ 0. , 0. , 0. , 0.16666667, 0.08333333,
0.05 ])
>>> P.polyint(c,k=3) # should return array([3, 1, 1, 1])
array([ 3., 1., 1., 1.])
>>> P.polyint(c,lbnd=-2) # should return array([6, 1, 1, 1])
array([ 6., 1., 1., 1.])
>>> P.polyint(c,scl=-2) # should return array([0, -2, -2, -2])
array([ 0., -2., -2., -2.])
"""
c = np.array(c, ndmin=1, copy=1)
if c.dtype.char in '?bBhHiIlLqQpP':
# astype doesn't preserve mask attribute.
c = c + 0.0
cdt = c.dtype
if not np.iterable(k):
k = [k]
cnt, iaxis = [int(t) for t in [m, axis]]
if cnt != m:
raise ValueError("The order of integration must be integer")
if cnt < 0:
raise ValueError("The order of integration must be non-negative")
if len(k) > cnt:
raise ValueError("Too many integration constants")
if iaxis != axis:
raise ValueError("The axis must be integer")
if not -c.ndim <= iaxis < c.ndim:
raise ValueError("The axis is out of range")
if iaxis < 0:
iaxis += c.ndim
if cnt == 0:
return c
k = list(k) + [0]*(cnt - len(k))
c = np.rollaxis(c, iaxis)
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=cdt)
tmp[0] = c[0]*0
tmp[1] = c[0]
for j in range(1, n):
tmp[j + 1] = c[j]/(j + 1)
tmp[0] += k[i] - polyval(lbnd, tmp)
c = tmp
c = np.rollaxis(c, 0, iaxis + 1)
return c
def polyval(x, c, tensor=True):
"""
Evaluate a polynomial at points x.
If `c` is of length `n + 1`, this function returns the value
.. math:: p(x) = c_0 + c_1 * x + ... + c_n * x^n
The parameter `x` is converted to an array only if it is a tuple or a
list, otherwise it is treated as a scalar. In either case, either `x`
or its elements must support multiplication and addition both with
themselves and with the elements of `c`.
If `c` is a 1-D array, then `p(x)` will have the same shape as `x`. If
`c` is multidimensional, then the shape of the result depends on the
value of `tensor`. If `tensor` is true the shape will be c.shape[1:] +
x.shape. If `tensor` is false the shape will be c.shape[1:]. Note that
scalars have shape (,).
Trailing zeros in the coefficients will be used in the evaluation, so
they should be avoided if efficiency is a concern.
Parameters
----------
x : array_like, compatible object
If `x` is a list or tuple, it is converted to an ndarray, otherwise
it is left unchanged and treated as a scalar. In either case, `x`
or its elements must support addition and multiplication with
with themselves and with the elements of `c`.
c : array_like
Array of coefficients ordered so that the coefficients for terms of
degree n are contained in c[n]. If `c` is multidimensional the
remaining indices enumerate multiple polynomials. In the two
dimensional case the coefficients may be thought of as stored in
the columns of `c`.
tensor : boolean, optional
If True, the shape of the coefficient array is extended with ones
on the right, one for each dimension of `x`. Scalars have dimension 0
for this action. The result is that every column of coefficients in
`c` is evaluated for every element of `x`. If False, `x` is broadcast
over the columns of `c` for the evaluation. This keyword is useful
when `c` is multidimensional. The default value is True.
.. versionadded:: 1.7.0
Returns
-------
values : ndarray, compatible object
The shape of the returned array is described above.
See Also
--------
polyval2d, polygrid2d, polyval3d, polygrid3d
Notes
-----
The evaluation uses Horner's method.
Examples
--------
>>> from numpy.polynomial.polynomial import polyval
>>> polyval(1, [1,2,3])
6.0
>>> a = np.arange(4).reshape(2,2)
>>> a
array([[0, 1],
[2, 3]])
>>> polyval(a, [1,2,3])
array([[ 1., 6.],
[ 17., 34.]])
>>> coef = np.arange(4).reshape(2,2) # multidimensional coefficients
>>> coef
array([[0, 1],
[2, 3]])
>>> polyval([1,2], coef, tensor=True)
array([[ 2., 4.],
[ 4., 7.]])
>>> polyval([1,2], coef, tensor=False)
array([ 2., 7.])
"""
c = np.array(c, ndmin=1, copy=0)
if c.dtype.char in '?bBhHiIlLqQpP':
# astype fails with NA
c = c + 0.0
if isinstance(x, (tuple, list)):
x = np.asarray(x)
if isinstance(x, np.ndarray) and tensor:
c = c.reshape(c.shape + (1,)*x.ndim)
c0 = c[-1] + x*0
for i in range(2, len(c) + 1):
c0 = c[-i] + c0*x
return c0
def polyval2d(x, y, c):
"""
Evaluate a 2-D polynomial at points (x, y).
This function returns the value
.. math:: p(x,y) = \\sum_{i,j} c_{i,j} * x^i * y^j
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` has fewer than two dimensions, ones are implicitly appended to
its shape to make it 2-D. The shape of the result will be c.shape[2:] +
x.shape.
Parameters
----------
x, y : array_like, compatible objects
The two dimensional series is evaluated at the points `(x, y)`,
where `x` and `y` must have the same shape. If `x` or `y` is a list
or tuple, it is first converted to an ndarray, otherwise it is left
unchanged and, if it isn't an ndarray, it is treated as a scalar.
c : array_like
Array of coefficients ordered so that the coefficient of the term
of multi-degree i,j is contained in `c[i,j]`. If `c` has
dimension greater than two the remaining indices enumerate multiple
sets of coefficients.
Returns
-------
values : ndarray, compatible object
The values of the two dimensional polynomial at points formed with
pairs of corresponding values from `x` and `y`.
See Also
--------
polyval, polygrid2d, polyval3d, polygrid3d
Notes
-----
.. versionadded:: 1.7.0
"""
try:
x, y = np.array((x, y), copy=0)
except:
raise ValueError('x, y are incompatible')
c = polyval(x, c)
c = polyval(y, c, tensor=False)
return c
def polygrid2d(x, y, c):
"""
Evaluate a 2-D polynomial on the Cartesian product of x and y.
This function returns the values:
.. math:: p(a,b) = \\sum_{i,j} c_{i,j} * a^i * b^j
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 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
--------
polyval, polyval2d, polyval3d, polygrid3d
Notes
-----
.. versionadded:: 1.7.0
"""
c = polyval(x, c)
c = polyval(y, c)
return c
def polyval3d(x, y, z, c):
"""
Evaluate a 3-D polynomial at points (x, y, z).
This function returns the values:
.. math:: p(x,y,z) = \\sum_{i,j,k} c_{i,j,k} * x^i * y^j * z^k
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
--------
polyval, polyval2d, polygrid2d, polygrid3d
Notes
-----
.. versionadded:: 1.7.0
"""
try:
x, y, z = np.array((x, y, z), copy=0)
except:
raise ValueError('x, y, z are incompatible')
c = polyval(x, c)
c = polyval(y, c, tensor=False)
c = polyval(z, c, tensor=False)
return c
def polygrid3d(x, y, z, c):
"""
Evaluate a 3-D polynomial 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} * a^i * b^j * c^k
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
--------
polyval, polyval2d, polygrid2d, polyval3d