"""
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
Notes
-----
.. versionadded:: 1.7.0
"""
c = polyval(x, c)
c = polyval(y, c)
c = polyval(z, c)
return c
def polyvander(x, deg):
"""Vandermonde matrix of given degree.
Returns the Vandermonde matrix of degree `deg` and sample points
`x`. The Vandermonde matrix is defined by
.. math:: V[..., i] = x^i,
where `0 <= i <= deg`. The leading indices of `V` index the elements of
`x` and the last index is the power of `x`.
If `c` is a 1-D array of coefficients of length `n + 1` and `V` is the
matrix ``V = polyvander(x, n)``, then ``np.dot(V, c)`` and
``polyval(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 polynomials 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 Vandermonde matrix. The shape of the returned matrix is
``x.shape + (deg + 1,)``, where the last index is the power of `x`.
The dtype will be the same as the converted `x`.
See Also
--------
polyvander2d, polyvander3d
"""
ideg = int(deg)
if ideg != deg:
raise ValueError("deg must be integer")
if ideg < 0:
raise ValueError("deg must be non-negative")
x = np.array(x, copy=0, 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
return np.rollaxis(v, 0, v.ndim)
def polyvander2d(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]*i + j] = x^i * y^j,
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 powers of
`x` and `y`.
If ``V = polyvander2d(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 ``polyval2d(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 polynomials
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
--------
polyvander, polyvander3d. polyval2d, polyval3d
"""
ideg = [int(d) for d in deg]
is_valid = [id == d and id >= 0 for id, d in zip(ideg, deg)]
if is_valid != [1, 1]:
raise ValueError("degrees must be non-negative integers")
degx, degy = ideg
x, y = np.array((x, y), copy=0) + 0.0
vx = polyvander(x, degx)
vy = polyvander(y, degy)
v = vx[..., None]*vy[..., None,:]
# einsum bug
#v = np.einsum("...i,...j->...ij", vx, vy)
return v.reshape(v.shape[:-2] + (-1,))
def polyvander3d(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] = x^i * y^j * z^k,
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 powers of `x`, `y`, and `z`.
If ``V = polyvander3d(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 ``polyval3d(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 polynomials
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
--------
polyvander, polyvander3d. polyval2d, polyval3d
Notes
-----
.. versionadded:: 1.7.0
"""
ideg = [int(d) for d in deg]
is_valid = [id == d and id >= 0 for id, d in zip(ideg, deg)]
if is_valid != [1, 1, 1]:
raise ValueError("degrees must be non-negative integers")
degx, degy, degz = ideg
x, y, z = np.array((x, y, z), copy=0) + 0.0
vx = polyvander(x, degx)
vy = polyvander(y, degy)
vz = polyvander(z, degz)
v = vx[..., None, None]*vy[..., None,:, None]*vz[..., None, None,:]
# einsum bug
#v = np.einsum("...i, ...j, ...k->...ijk", vx, vy, vz)
return v.reshape(v.shape[:-3] + (-1,))
def polyfit(x, y, deg, rcond=None, full=False, w=None):
"""
Least-squares fit of a polynomial to data.
Return the coefficients of a polynomial 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 * x + ... + c_n * x^n,
where `n` is `deg`.
Parameters
----------
x : array_like, shape (`M`,)
x-coordinates of the `M` sample (data) points ``(x[i], y[i])``.
y : array_like, shape (`M`,) or (`M`, `K`)
y-coordinates of the sample points. Several sets of sample points
sharing the same x-coordinates can be (independently) fit with one
call to `polyfit` by passing in for `y` a 2-D array that contains
one data set 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 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 `rcond`, relative to the largest singular value, will be
ignored. The default value is ``len(x)*eps``, where `eps` is the
relative precision of the platform's float type, about 2e-16 in
most cases.
full : bool, optional
Switch determining the nature of the return value. When ``False``
(the default) just the coefficients are returned; when ``True``,
diagnostic information from the singular value decomposition (used
to solve the fit's matrix equation) is also returned.
w : array_like, shape (`M`,), optional
Weights. If not None, the contribution of each point
``(x[i],y[i])`` to the fit is weighted by `w[i]`. Ideally the
weights are chosen so that the errors of the products ``w[i]*y[i]``
all have the same variance. The default value is None.
.. versionadded:: 1.5.0
Returns
-------
coef : ndarray, shape (`deg` + 1,) or (`deg` + 1, `K`)
Polynomial coefficients ordered from low to high. If `y` was 2-D,
the coefficients in column `k` of `coef` represent the polynomial
fit to the data in `y`'s `k`-th column.
[residuals, rank, singular_values, rcond] : list
These values are only returned if `full` = True
resid -- sum of squared residuals of the least squares fit
rank -- the numerical rank of the scaled Vandermonde matrix
sv -- singular values of the scaled Vandermonde matrix
rcond -- value of `rcond`.
For more details, see `linalg.lstsq`.
Raises
------
RankWarning
Raised if the matrix in the least-squares fit is rank deficient.
The warning is only raised if `full` == False. The warnings can
be turned off by:
>>> import warnings
>>> warnings.simplefilter('ignore', RankWarning)
See Also
--------
chebfit, legfit, lagfit, hermfit, hermefit
polyval : Evaluates a polynomial.
polyvander : Vandermonde matrix for powers.
linalg.lstsq : Computes a least-squares fit from the matrix.
scipy.interpolate.UnivariateSpline : Computes spline fits.
Notes
-----
The solution is the coefficients of the polynomial `p` that minimizes
the sum of the weighted squared errors
.. math :: E = \\sum_j w_j^2 * |y_j - p(x_j)|^2,
where the :math:`w_j` are the weights. This problem is solved by
setting up the (typically) over-determined 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 (and `full` == ``False``), a `RankWarning` will be raised.
This means that the coefficient values may be poorly determined.
Fitting to a lower order polynomial will usually get rid of the warning
(but may not be what you want, of course; if you have independent
reason(s) for choosing the degree which isn't working, you may have to:
a) reconsider those reasons, and/or b) reconsider the quality of your
data). 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.
Polynomial fits using double precision tend to "fail" at about
(polynomial) degree 20. Fits using Chebyshev or Legendre series are
generally better conditioned, but much can still 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.
Examples
--------
>>> from numpy.polynomial import polynomial as P
>>> x = np.linspace(-1,1,51) # x "data": [-1, -0.96, ..., 0.96, 1]
>>> y = x**3 - x + np.random.randn(len(x)) # x^3 - x + N(0,1) "noise"
>>> c, stats = P.polyfit(x,y,3,full=True)
>>> c # c[0], c[2] should be approx. 0, c[1] approx. -1, c[3] approx. 1
array([ 0.01909725, -1.30598256, -0.00577963, 1.02644286])
>>> stats # note the large SSR, explaining the rather poor results
[array([ 38.06116253]), 4, array([ 1.38446749, 1.32119158, 0.50443316,
0.28853036]), 1.1324274851176597e-014]
Same thing without the added noise
>>> y = x**3 - x
>>> c, stats = P.polyfit(x,y,3,full=True)
>>> c # c[0], c[2] should be "very close to 0", c[1] ~= -1, c[3] ~= 1
array([ -1.73362882e-17, -1.00000000e+00, -2.67471909e-16,
1.00000000e+00])
>>> stats # note the minuscule SSR
[array([ 7.46346754e-31]), 4, array([ 1.38446749, 1.32119158,
0.50443316, 0.28853036]), 1.1324274851176597e-014]
"""
x = np.asarray(x) + 0.0
y = np.asarray(y) + 0.0
deg = np.asarray(deg)
# check arguments.
if deg.ndim > 1 or deg.dtype.kind not in 'iu' or deg.size == 0:
raise TypeError("deg must be an int or non-empty 1-D array of int")
if deg.min() < 0:
raise ValueError("expected deg >= 0")
if x.ndim != 1:
raise TypeError("expected 1D vector for x")
if x.size == 0:
raise TypeError("expected non-empty vector for x")
if y.ndim < 1 or y.ndim > 2:
raise TypeError("expected 1D or 2D array for y")
if len(x) != len(y):
raise TypeError("expected x and y to have same length")
if deg.ndim == 0:
lmax = deg
order = lmax + 1
van = polyvander(x, lmax)
else:
deg = np.sort(deg)
lmax = deg[-1]
order = len(deg)
van = polyvander(x, lmax)[:, deg]
# set up the least squares matrices in transposed form
lhs = van.T
rhs = y.T
if w is not None:
w = np.asarray(w) + 0.0
if w.ndim != 1:
raise TypeError("expected 1D vector for w")
if len(x) != len(w):
raise TypeError("expected x and w to have same length")
# apply weights. Don't use inplace operations as they
# can cause problems with NA.
lhs = lhs * w
rhs = rhs * w
# set rcond
if rcond is None:
rcond = len(x)*np.finfo(x.dtype).eps
# Determine the norms of the design matrix columns.
if issubclass(lhs.dtype.type, np.complexfloating):
scl = np.sqrt((np.square(lhs.real) + np.square(lhs.imag)).sum(1))
else:
scl = np.sqrt(np.square(lhs).sum(1))
scl[scl == 0] = 1
# Solve the least squares problem.
c, resids, rank, s = la.lstsq(lhs.T/scl, rhs.T, rcond)
c = (c.T/scl).T
# Expand c to include non-fitted coefficients which are set to zero
if deg.ndim == 1:
if c.ndim == 2:
cc = np.zeros((lmax + 1, c.shape[1]), dtype=c.dtype)
else:
cc = np.zeros(lmax + 1, dtype=c.dtype)
cc[deg] = c
c = cc
# warn on rank reduction
if rank != order and not full:
msg = "The fit may be poorly conditioned"
warnings.warn(msg, pu.RankWarning)
if full:
return c, [resids, rank, s, rcond]
else:
return c
def polycompanion(c):
"""
Return the companion matrix of c.
The companion matrix for power series cannot be made symmetric by
scaling the basis, so this function differs from those for the
orthogonal polynomials.
Parameters
----------
c : array_like
1-D array of polynomial coefficients ordered from low to high
degree.
Returns
-------
mat : ndarray
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)
bot = mat.reshape(-1)[n::n+1]
bot[...] = 1
mat[:, -1] -= c[:-1]/c[-1]
return mat
def polyroots(c):
"""
Compute the roots of a polynomial.
Return the roots (a.k.a. "zeros") of the polynomial
.. math:: p(x) = \\sum_i c[i] * x^i.
Parameters
----------
c : 1-D array_like
1-D array of polynomial coefficients.
Returns
-------
out : ndarray
Array of the roots of the polynomial. If all the roots are real,
then `out` is also real, otherwise it is complex.
See Also
--------
chebroots
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 power 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.
Examples
--------
>>> import numpy.polynomial.polynomial as poly
>>> poly.polyroots(poly.polyfromroots((-1,0,1)))
array([-1., 0., 1.])
>>> poly.polyroots(poly.polyfromroots((-1,0,1))).dtype
dtype('float64')
>>> j = complex(0,1)
>>> poly.polyroots(poly.polyfromroots((-j,0,j)))
array([ 0.00000000e+00+0.j, 0.00000000e+00+1.j, 2.77555756e-17-1.j])
"""
# 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]])
m = polycompanion(c)
r = la.eigvals(m)
r.sort()
return r
#
# polynomial class
#
class Polynomial(ABCPolyBase):
"""A power series class.
The Polynomial 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
Polynomial coefficients in order of increasing degree, i.e.,
``(1, 2, 3)`` give ``1 + 2*x + 3*x**2``.
domain : (2,) array_like, optional
Domain to use. The interval ``[domain[0], domain[1]]`` is mapped
to the interval ``[window[0], window[1]]`` by shifting and scaling.
The default value is [-1, 1].
window : (2,) array_like, optional
Window, see `domain` for its use. The default value is [-1, 1].
.. versionadded:: 1.6.0
"""
# Virtual Functions
_add = staticmethod(polyadd)
_sub = staticmethod(polysub)
_mul = staticmethod(polymul)
_div = staticmethod(polydiv)
_pow = staticmethod(polypow)
_val = staticmethod(polyval)
_int = staticmethod(polyint)
_der = staticmethod(polyder)
_fit = staticmethod(polyfit)
_line = staticmethod(polyline)
_roots = staticmethod(polyroots)
_fromroots = staticmethod(polyfromroots)
# Virtual properties
nickname = 'poly'
domain = np.array(polydomain)
window = np.array(polydomain)