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""" |
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================================================= |
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Power Series (:mod:`numpy.polynomial.polynomial`) |
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================================================= |
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|
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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`). |
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|
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Classes |
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------- |
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.. autosummary:: |
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:toctree: generated/ |
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|
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Polynomial |
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|
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Constants |
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--------- |
|
.. autosummary:: |
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:toctree: generated/ |
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|
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polydomain |
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polyzero |
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polyone |
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polyx |
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|
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Arithmetic |
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---------- |
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.. autosummary:: |
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:toctree: generated/ |
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polyadd |
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polysub |
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polymulx |
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polymul |
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polydiv |
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polypow |
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polyval |
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polyval2d |
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polyval3d |
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polygrid2d |
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polygrid3d |
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|
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Calculus |
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-------- |
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.. autosummary:: |
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:toctree: generated/ |
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polyder |
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polyint |
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|
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Misc Functions |
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-------------- |
|
.. autosummary:: |
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:toctree: generated/ |
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|
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polyfromroots |
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polyroots |
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polyvalfromroots |
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polyvander |
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polyvander2d |
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polyvander3d |
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polycompanion |
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polyfit |
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polytrim |
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polyline |
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See Also |
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-------- |
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`numpy.polynomial` |
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|
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""" |
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__all__ = [ |
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'polyzero', 'polyone', 'polyx', 'polydomain', 'polyline', 'polyadd', |
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'polysub', 'polymulx', 'polymul', 'polydiv', 'polypow', 'polyval', |
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'polyvalfromroots', 'polyder', 'polyint', 'polyfromroots', 'polyvander', |
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'polyfit', 'polytrim', 'polyroots', 'Polynomial', 'polyval2d', 'polyval3d', |
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'polygrid2d', 'polygrid3d', 'polyvander2d', 'polyvander3d'] |
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|
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import numpy as np |
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import numpy.linalg as la |
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from numpy.core.multiarray import normalize_axis_index |
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|
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from . import polyutils as pu |
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from ._polybase import ABCPolyBase |
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|
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polytrim = pu.trimcoef |
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polydomain = np.array([-1, 1]) |
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polyzero = np.array([0]) |
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polyone = np.array([1]) |
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polyx = np.array([0, 1]) |
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|
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def polyline(off, scl): |
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""" |
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Returns an array representing a linear polynomial. |
|
|
|
Parameters |
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---------- |
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off, scl : scalars |
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The "y-intercept" and "slope" of the line, respectively. |
|
|
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Returns |
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------- |
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y : ndarray |
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This module's representation of the linear polynomial ``off + |
|
scl*x``. |
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|
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See Also |
|
-------- |
|
numpy.polynomial.chebyshev.chebline |
|
numpy.polynomial.legendre.legline |
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numpy.polynomial.laguerre.lagline |
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numpy.polynomial.hermite.hermline |
|
numpy.polynomial.hermite_e.hermeline |
|
|
|
Examples |
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-------- |
|
>>> from numpy.polynomial import polynomial as P |
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>>> P.polyline(1,-1) |
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array([ 1, -1]) |
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>>> P.polyval(1, P.polyline(1,-1)) # should be 0 |
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0.0 |
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|
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""" |
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if scl != 0: |
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return np.array([off, scl]) |
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else: |
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return np.array([off]) |
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|
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def polyfromroots(roots): |
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""" |
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Generate a monic polynomial with given roots. |
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|
|
Return the coefficients of the polynomial |
|
|
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.. math:: p(x) = (x - r_0) * (x - r_1) * ... * (x - r_n), |
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|
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where the ``r_n`` are the roots specified in `roots`. If a zero has |
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multiplicity n, then it must appear in `roots` n times. For instance, |
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if 2 is a root of multiplicity three and 3 is a root of multiplicity 2, |
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then `roots` looks something like [2, 2, 2, 3, 3]. The roots can appear |
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in any order. |
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|
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If the returned coefficients are `c`, then |
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|
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.. math:: p(x) = c_0 + c_1 * x + ... + x^n |
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|
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The coefficient of the last term is 1 for monic polynomials in this |
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form. |
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|
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Parameters |
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---------- |
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roots : array_like |
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Sequence containing the roots. |
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|
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Returns |
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------- |
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out : ndarray |
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1-D array of the polynomial's coefficients If all the roots are |
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real, then `out` is also real, otherwise it is complex. (see |
|
Examples below). |
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|
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See Also |
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-------- |
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numpy.polynomial.chebyshev.chebfromroots |
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numpy.polynomial.legendre.legfromroots |
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numpy.polynomial.laguerre.lagfromroots |
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numpy.polynomial.hermite.hermfromroots |
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numpy.polynomial.hermite_e.hermefromroots |
|
|
|
Notes |
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----- |
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The coefficients are determined by multiplying together linear factors |
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of the form ``(x - r_i)``, i.e. |
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|
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.. math:: p(x) = (x - r_0) (x - r_1) ... (x - r_n) |
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|
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where ``n == len(roots) - 1``; note that this implies that ``1`` is always |
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returned for :math:`a_n`. |
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|
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Examples |
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-------- |
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>>> from numpy.polynomial import polynomial as P |
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>>> P.polyfromroots((-1,0,1)) # x(x - 1)(x + 1) = x^3 - x |
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array([ 0., -1., 0., 1.]) |
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>>> j = complex(0,1) |
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>>> P.polyfromroots((-j,j)) # complex returned, though values are real |
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array([1.+0.j, 0.+0.j, 1.+0.j]) |
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|
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""" |
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return pu._fromroots(polyline, polymul, roots) |
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def polyadd(c1, c2): |
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""" |
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Add one polynomial to another. |
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|
|
Returns the sum of two polynomials `c1` + `c2`. The arguments are |
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sequences of coefficients from lowest order term to highest, i.e., |
|
[1,2,3] represents the polynomial ``1 + 2*x + 3*x**2``. |
|
|
|
Parameters |
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---------- |
|
c1, c2 : array_like |
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1-D arrays of polynomial coefficients ordered from low to high. |
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|
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Returns |
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------- |
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out : ndarray |
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The coefficient array representing their sum. |
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|
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See Also |
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-------- |
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polysub, polymulx, polymul, polydiv, polypow |
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|
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Examples |
|
-------- |
|
>>> from numpy.polynomial import polynomial as P |
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>>> c1 = (1,2,3) |
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>>> c2 = (3,2,1) |
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>>> sum = P.polyadd(c1,c2); sum |
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array([4., 4., 4.]) |
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>>> P.polyval(2, sum) # 4 + 4(2) + 4(2**2) |
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28.0 |
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|
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""" |
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return pu._add(c1, c2) |
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def polysub(c1, c2): |
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""" |
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Subtract one polynomial from another. |
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|
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Returns the difference of two polynomials `c1` - `c2`. The arguments |
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are sequences of coefficients from lowest order term to highest, i.e., |
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[1,2,3] represents the polynomial ``1 + 2*x + 3*x**2``. |
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|
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Parameters |
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---------- |
|
c1, c2 : array_like |
|
1-D arrays of polynomial coefficients ordered from low to |
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high. |
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|
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Returns |
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------- |
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out : ndarray |
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Of coefficients representing their difference. |
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|
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See Also |
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-------- |
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polyadd, polymulx, polymul, polydiv, polypow |
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|
|
Examples |
|
-------- |
|
>>> from numpy.polynomial import polynomial as P |
|
>>> c1 = (1,2,3) |
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>>> c2 = (3,2,1) |
|
>>> P.polysub(c1,c2) |
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array([-2., 0., 2.]) |
|
>>> P.polysub(c2,c1) # -P.polysub(c1,c2) |
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array([ 2., 0., -2.]) |
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|
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""" |
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return pu._sub(c1, c2) |
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def polymulx(c): |
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"""Multiply a polynomial by x. |
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|
|
Multiply the polynomial `c` by x, where x is the independent |
|
variable. |
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Parameters |
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---------- |
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c : array_like |
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1-D array of polynomial coefficients ordered from low to |
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high. |
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|
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Returns |
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------- |
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out : ndarray |
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Array representing the result of the multiplication. |
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|
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See Also |
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-------- |
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polyadd, polysub, polymul, polydiv, polypow |
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|
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Notes |
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----- |
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|
|
.. versionadded:: 1.5.0 |
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|
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""" |
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|
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[c] = pu.as_series([c]) |
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|
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if len(c) == 1 and c[0] == 0: |
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return c |
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|
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prd = np.empty(len(c) + 1, dtype=c.dtype) |
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prd[0] = c[0]*0 |
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prd[1:] = c |
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return prd |
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def polymul(c1, c2): |
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""" |
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Multiply one polynomial by another. |
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|
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Returns the product of two polynomials `c1` * `c2`. The arguments are |
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sequences of coefficients, from lowest order term to highest, e.g., |
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[1,2,3] represents the polynomial ``1 + 2*x + 3*x**2.`` |
|
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Parameters |
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---------- |
|
c1, c2 : array_like |
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1-D arrays of coefficients representing a polynomial, relative to the |
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"standard" basis, and ordered from lowest order term to highest. |
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Returns |
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------- |
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out : ndarray |
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Of the coefficients of their product. |
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See Also |
|
-------- |
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polyadd, polysub, polymulx, polydiv, polypow |
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|
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Examples |
|
-------- |
|
>>> from numpy.polynomial import polynomial as P |
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>>> c1 = (1,2,3) |
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>>> c2 = (3,2,1) |
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>>> P.polymul(c1,c2) |
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array([ 3., 8., 14., 8., 3.]) |
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|
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""" |
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|
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[c1, c2] = pu.as_series([c1, c2]) |
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ret = np.convolve(c1, c2) |
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return pu.trimseq(ret) |
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|
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def polydiv(c1, c2): |
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""" |
|
Divide one polynomial by another. |
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|
|
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 |
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1-D arrays of polynomial coefficients ordered from low to high. |
|
|
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Returns |
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------- |
|
[quo, rem] : ndarrays |
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Of coefficient series representing the quotient and remainder. |
|
|
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See Also |
|
-------- |
|
polyadd, polysub, polymulx, polymul, polypow |
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|
|
Examples |
|
-------- |
|
>>> from numpy.polynomial import polynomial as P |
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>>> c1 = (1,2,3) |
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>>> c2 = (3,2,1) |
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>>> P.polydiv(c1,c2) |
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(array([3.]), array([-8., -4.])) |
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>>> P.polydiv(c2,c1) |
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(array([ 0.33333333]), array([ 2.66666667, 1.33333333])) # may vary |
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""" |
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[c1, c2] = pu.as_series([c1, c2]) |
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if c2[-1] == 0: |
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raise ZeroDivisionError() |
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lc1 = len(c1) |
|
lc2 = len(c2) |
|
if lc1 < lc2: |
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return c1[:1]*0, c1 |
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elif lc2 == 1: |
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return c1/c2[-1], c1[:1]*0 |
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else: |
|
dlen = lc1 - lc2 |
|
scl = c2[-1] |
|
c2 = c2[:-1]/scl |
|
i = dlen |
|
j = lc1 - 1 |
|
while i >= 0: |
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c1[i:j] -= c2*c1[j] |
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i -= 1 |
|
j -= 1 |
|
return c1[j+1:]/scl, pu.trimseq(c1[:j+1]) |
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def polypow(c, pow, maxpower=None): |
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"""Raise a polynomial to a power. |
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|
|
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 |
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1-D array of array of series coefficients ordered from low to |
|
high degree. |
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pow : integer |
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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 |
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|
|
Returns |
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------- |
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coef : ndarray |
|
Power series of power. |
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|
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See Also |
|
-------- |
|
polyadd, polysub, polymulx, polymul, polydiv |
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|
|
Examples |
|
-------- |
|
>>> from numpy.polynomial import polynomial as P |
|
>>> P.polypow([1,2,3], 2) |
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array([ 1., 4., 10., 12., 9.]) |
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|
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""" |
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return pu._pow(np.convolve, c, pow, maxpower) |
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def polyder(c, m=1, scl=1, axis=0): |
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""" |
|
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.]) |
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|
|
""" |
|
c = np.array(c, ndmin=1, copy=True) |
|
if c.dtype.char in '?bBhHiIlLqQpP': |
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|
|
c = c + 0.0 |
|
cdt = c.dtype |
|
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=cdt) |
|
for j in range(n, 0, -1): |
|
der[j - 1] = j*c[j] |
|
c = der |
|
c = np.moveaxis(c, 0, iaxis) |
|
return c |
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|
|
|
|
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``, ``np.ndim(lbnd) != 0``, or |
|
``np.ndim(scl) != 0``. |
|
|
|
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, # may vary |
|
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=True) |
|
if c.dtype.char in '?bBhHiIlLqQpP': |
|
|
|
c = c + 0.0 |
|
cdt = c.dtype |
|
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 |
|
|
|
k = list(k) + [0]*(cnt - len(k)) |
|
c = np.moveaxis(c, iaxis, 0) |
|
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.moveaxis(c, 0, iaxis) |
|
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=False) |
|
if c.dtype.char in '?bBhHiIlLqQpP': |
|
|
|
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 polyvalfromroots(x, r, tensor=True): |
|
""" |
|
Evaluate a polynomial specified by its roots at points x. |
|
|
|
If `r` is of length `N`, this function returns the value |
|
|
|
.. math:: p(x) = \\prod_{n=1}^{N} (x - r_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 `r`. |
|
|
|
If `r` is a 1-D array, then `p(x)` will have the same shape as `x`. If `r` |
|
is multidimensional, then the shape of the result depends on the value of |
|
`tensor`. If `tensor` is ``True`` the shape will be r.shape[1:] + x.shape; |
|
that is, each polynomial is evaluated at every value of `x`. If `tensor` is |
|
``False``, the shape will be r.shape[1:]; that is, each polynomial is |
|
evaluated only for the corresponding broadcast value of `x`. Note that |
|
scalars have shape (,). |
|
|
|
.. versionadded:: 1.12 |
|
|
|
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 `r`. |
|
r : array_like |
|
Array of roots. If `r` is multidimensional the first index is the |
|
root index, while the remaining indices enumerate multiple |
|
polynomials. For instance, in the two dimensional case the roots |
|
of each polynomial may be thought of as stored in the columns of `r`. |
|
tensor : boolean, optional |
|
If True, the shape of the roots 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 `r` is |
|
evaluated for every element of `x`. If False, `x` is broadcast over the |
|
columns of `r` for the evaluation. This keyword is useful when `r` is |
|
multidimensional. The default value is True. |
|
|
|
Returns |
|
------- |
|
values : ndarray, compatible object |
|
The shape of the returned array is described above. |
|
|
|
See Also |
|
-------- |
|
polyroots, polyfromroots, polyval |
|
|
|
Examples |
|
-------- |
|
>>> from numpy.polynomial.polynomial import polyvalfromroots |
|
>>> polyvalfromroots(1, [1,2,3]) |
|
0.0 |
|
>>> a = np.arange(4).reshape(2,2) |
|
>>> a |
|
array([[0, 1], |
|
[2, 3]]) |
|
>>> polyvalfromroots(a, [-1, 0, 1]) |
|
array([[-0., 0.], |
|
[ 6., 24.]]) |
|
>>> r = np.arange(-2, 2).reshape(2,2) # multidimensional coefficients |
|
>>> r # each column of r defines one polynomial |
|
array([[-2, -1], |
|
[ 0, 1]]) |
|
>>> b = [-2, 1] |
|
>>> polyvalfromroots(b, r, tensor=True) |
|
array([[-0., 3.], |
|
[ 3., 0.]]) |
|
>>> polyvalfromroots(b, r, tensor=False) |
|
array([-0., 0.]) |
|
""" |
|
r = np.array(r, ndmin=1, copy=False) |
|
if r.dtype.char in '?bBhHiIlLqQpP': |
|
r = r.astype(np.double) |
|
if isinstance(x, (tuple, list)): |
|
x = np.asarray(x) |
|
if isinstance(x, np.ndarray): |
|
if tensor: |
|
r = r.reshape(r.shape + (1,)*x.ndim) |
|
elif x.ndim >= r.ndim: |
|
raise ValueError("x.ndim must be < r.ndim when tensor == False") |
|
return np.prod(x - r, axis=0) |
|
|
|
|
|
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 |
|
|
|
""" |
|
return pu._valnd(polyval, c, x, y) |
|
|
|
|
|
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 |
|
|
|
""" |
|
return pu._gridnd(polyval, c, x, y) |
|
|
|
|
|
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 |
|
|
|
""" |
|
return pu._valnd(polyval, c, x, y, z) |
|
|
|
|
|
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 |
|
|
|
""" |
|
return pu._gridnd(polyval, c, x, y, z) |
|
|
|
|
|
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 = pu._deprecate_as_int(deg, "deg") |
|
if ideg < 0: |
|
raise ValueError("deg must be non-negative") |
|
|
|
x = np.array(x, copy=False, ndmin=1) + 0.0 |
|
dims = (ideg + 1,) + x.shape |
|
dtyp = x.dtype |
|
v = np.empty(dims, dtype=dtyp) |
|
v[0] = x*0 + 1 |
|
if ideg > 0: |
|
v[1] = x |
|
for i in range(2, ideg + 1): |
|
v[i] = v[i-1]*x |
|
return np.moveaxis(v, 0, -1) |
|
|
|
|
|
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] + 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 |
|
|
|
""" |
|
return pu._vander_nd_flat((polyvander, polyvander), (x, y), deg) |
|
|
|
|
|
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 |
|
|
|
""" |
|
return pu._vander_nd_flat((polyvander, polyvander, polyvander), (x, y, z), deg) |
|
|
|
|
|
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 >= 1.11.0 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 `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 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 (`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`` |
|
|
|
- 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`. |
|
|
|
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', np.RankWarning) |
|
|
|
See Also |
|
-------- |
|
numpy.polynomial.chebyshev.chebfit |
|
numpy.polynomial.legendre.legfit |
|
numpy.polynomial.laguerre.lagfit |
|
numpy.polynomial.hermite.hermfit |
|
numpy.polynomial.hermite_e.hermefit |
|
polyval : Evaluates a polynomial. |
|
polyvander : Vandermonde matrix for powers. |
|
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 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 |
|
-------- |
|
>>> np.random.seed(123) |
|
>>> 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 + Gaussian noise |
|
>>> c, stats = P.polyfit(x,y,3,full=True) |
|
>>> np.random.seed(123) |
|
>>> 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]) # may vary |
|
>>> stats # note the large SSR, explaining the rather poor results |
|
[array([ 38.06116253]), 4, array([ 1.38446749, 1.32119158, 0.50443316, # may vary |
|
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([-6.36925336e-18, -1.00000000e+00, -4.08053781e-16, 1.00000000e+00]) |
|
>>> stats # note the minuscule SSR |
|
[array([ 7.46346754e-31]), 4, array([ 1.38446749, 1.32119158, # may vary |
|
0.50443316, 0.28853036]), 1.1324274851176597e-014] |
|
|
|
""" |
|
return pu._fit(polyvander, x, y, deg, rcond, full, w) |
|
|
|
|
|
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] = 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 |
|
-------- |
|
numpy.polynomial.chebyshev.chebroots |
|
numpy.polynomial.legendre.legroots |
|
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 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]) # may vary |
|
|
|
""" |
|
|
|
[c] = pu.as_series([c]) |
|
if len(c) < 2: |
|
return np.array([], dtype=c.dtype) |
|
if len(c) == 2: |
|
return np.array([-c[0]/c[1]]) |
|
|
|
|
|
m = polycompanion(c)[::-1,::-1] |
|
r = la.eigvals(m) |
|
r.sort() |
|
return r |
|
|
|
|
|
|
|
|
|
|
|
|
|
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 |
|
symbol : str, optional |
|
Symbol used to represent the independent variable in string |
|
representations of the polynomial expression, e.g. for printing. |
|
The symbol must be a valid Python identifier. Default value is 'x'. |
|
|
|
.. versionadded:: 1.24 |
|
|
|
""" |
|
|
|
_add = staticmethod(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) |
|
|
|
|
|
domain = np.array(polydomain) |
|
window = np.array(polydomain) |
|
basis_name = None |
|
|
|
@classmethod |
|
def _str_term_unicode(cls, i, arg_str): |
|
if i == '1': |
|
return f"·{arg_str}" |
|
else: |
|
return f"·{arg_str}{i.translate(cls._superscript_mapping)}" |
|
|
|
@staticmethod |
|
def _str_term_ascii(i, arg_str): |
|
if i == '1': |
|
return f" {arg_str}" |
|
else: |
|
return f" {arg_str}**{i}" |
|
|
|
@staticmethod |
|
def _repr_latex_term(i, arg_str, needs_parens): |
|
if needs_parens: |
|
arg_str = rf"\left({arg_str}\right)" |
|
if i == 0: |
|
return '1' |
|
elif i == 1: |
|
return arg_str |
|
else: |
|
return f"{arg_str}^{{{i}}}" |
|
|
