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763 lines
21 KiB
Python
763 lines
21 KiB
Python
"""
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Utility classes and functions for the polynomial modules.
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This module provides: error and warning objects; a polynomial base class;
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and some routines used in both the `polynomial` and `chebyshev` modules.
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Error objects
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-------------
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.. autosummary::
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:toctree: generated/
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PolyError base class for this sub-package's errors.
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PolyDomainError raised when domains are mismatched.
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Warning objects
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---------------
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.. autosummary::
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:toctree: generated/
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RankWarning raised in least-squares fit for rank-deficient matrix.
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Base class
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----------
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.. autosummary::
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:toctree: generated/
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PolyBase Obsolete base class for the polynomial classes. Do not use.
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Functions
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---------
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.. autosummary::
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:toctree: generated/
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as_series convert list of array_likes into 1-D arrays of common type.
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trimseq remove trailing zeros.
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trimcoef remove small trailing coefficients.
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getdomain return the domain appropriate for a given set of abscissae.
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mapdomain maps points between domains.
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mapparms parameters of the linear map between domains.
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"""
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from __future__ import division, absolute_import, print_function
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import operator
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import warnings
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import numpy as np
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__all__ = [
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'RankWarning', 'PolyError', 'PolyDomainError', 'as_series', 'trimseq',
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'trimcoef', 'getdomain', 'mapdomain', 'mapparms', 'PolyBase']
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#
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# Warnings and Exceptions
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#
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class RankWarning(UserWarning):
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"""Issued by chebfit when the design matrix is rank deficient."""
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pass
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class PolyError(Exception):
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"""Base class for errors in this module."""
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pass
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class PolyDomainError(PolyError):
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"""Issued by the generic Poly class when two domains don't match.
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This is raised when an binary operation is passed Poly objects with
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different domains.
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"""
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pass
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#
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# Base class for all polynomial types
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#
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class PolyBase(object):
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"""
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Base class for all polynomial types.
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Deprecated in numpy 1.9.0, use the abstract
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ABCPolyBase class instead. Note that the latter
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requires a number of virtual functions to be
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implemented.
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"""
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pass
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#
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# Helper functions to convert inputs to 1-D arrays
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#
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def trimseq(seq):
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"""Remove small Poly series coefficients.
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Parameters
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----------
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seq : sequence
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Sequence of Poly series coefficients. This routine fails for
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empty sequences.
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Returns
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-------
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series : sequence
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Subsequence with trailing zeros removed. If the resulting sequence
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would be empty, return the first element. The returned sequence may
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or may not be a view.
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Notes
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-----
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Do not lose the type info if the sequence contains unknown objects.
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"""
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if len(seq) == 0:
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return seq
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else:
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for i in range(len(seq) - 1, -1, -1):
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if seq[i] != 0:
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break
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return seq[:i+1]
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def as_series(alist, trim=True):
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"""
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Return argument as a list of 1-d arrays.
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The returned list contains array(s) of dtype double, complex double, or
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object. A 1-d argument of shape ``(N,)`` is parsed into ``N`` arrays of
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size one; a 2-d argument of shape ``(M,N)`` is parsed into ``M`` arrays
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of size ``N`` (i.e., is "parsed by row"); and a higher dimensional array
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raises a Value Error if it is not first reshaped into either a 1-d or 2-d
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array.
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Parameters
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----------
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alist : array_like
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A 1- or 2-d array_like
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trim : boolean, optional
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When True, trailing zeros are removed from the inputs.
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When False, the inputs are passed through intact.
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Returns
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-------
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[a1, a2,...] : list of 1-D arrays
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A copy of the input data as a list of 1-d arrays.
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Raises
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------
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ValueError
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Raised when `as_series` cannot convert its input to 1-d arrays, or at
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least one of the resulting arrays is empty.
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Examples
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--------
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>>> from numpy.polynomial import polyutils as pu
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>>> a = np.arange(4)
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>>> pu.as_series(a)
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[array([0.]), array([1.]), array([2.]), array([3.])]
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>>> b = np.arange(6).reshape((2,3))
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>>> pu.as_series(b)
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[array([0., 1., 2.]), array([3., 4., 5.])]
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>>> pu.as_series((1, np.arange(3), np.arange(2, dtype=np.float16)))
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[array([1.]), array([0., 1., 2.]), array([0., 1.])]
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>>> pu.as_series([2, [1.1, 0.]])
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[array([2.]), array([1.1])]
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>>> pu.as_series([2, [1.1, 0.]], trim=False)
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[array([2.]), array([1.1, 0. ])]
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"""
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arrays = [np.array(a, ndmin=1, copy=0) for a in alist]
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if min([a.size for a in arrays]) == 0:
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raise ValueError("Coefficient array is empty")
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if any([a.ndim != 1 for a in arrays]):
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raise ValueError("Coefficient array is not 1-d")
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if trim:
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arrays = [trimseq(a) for a in arrays]
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if any([a.dtype == np.dtype(object) for a in arrays]):
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ret = []
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for a in arrays:
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if a.dtype != np.dtype(object):
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tmp = np.empty(len(a), dtype=np.dtype(object))
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tmp[:] = a[:]
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ret.append(tmp)
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else:
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ret.append(a.copy())
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else:
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try:
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dtype = np.common_type(*arrays)
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except Exception:
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raise ValueError("Coefficient arrays have no common type")
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ret = [np.array(a, copy=1, dtype=dtype) for a in arrays]
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return ret
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def trimcoef(c, tol=0):
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"""
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Remove "small" "trailing" coefficients from a polynomial.
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"Small" means "small in absolute value" and is controlled by the
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parameter `tol`; "trailing" means highest order coefficient(s), e.g., in
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``[0, 1, 1, 0, 0]`` (which represents ``0 + x + x**2 + 0*x**3 + 0*x**4``)
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both the 3-rd and 4-th order coefficients would be "trimmed."
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Parameters
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----------
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c : array_like
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1-d array of coefficients, ordered from lowest order to highest.
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tol : number, optional
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Trailing (i.e., highest order) elements with absolute value less
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than or equal to `tol` (default value is zero) are removed.
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Returns
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-------
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trimmed : ndarray
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1-d array with trailing zeros removed. If the resulting series
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would be empty, a series containing a single zero is returned.
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Raises
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------
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ValueError
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If `tol` < 0
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See Also
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--------
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trimseq
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Examples
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--------
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>>> from numpy.polynomial import polyutils as pu
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>>> pu.trimcoef((0,0,3,0,5,0,0))
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array([0., 0., 3., 0., 5.])
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>>> pu.trimcoef((0,0,1e-3,0,1e-5,0,0),1e-3) # item == tol is trimmed
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array([0.])
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>>> i = complex(0,1) # works for complex
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>>> pu.trimcoef((3e-4,1e-3*(1-i),5e-4,2e-5*(1+i)), 1e-3)
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array([0.0003+0.j , 0.001 -0.001j])
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"""
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if tol < 0:
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raise ValueError("tol must be non-negative")
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[c] = as_series([c])
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[ind] = np.nonzero(np.abs(c) > tol)
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if len(ind) == 0:
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return c[:1]*0
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else:
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return c[:ind[-1] + 1].copy()
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def getdomain(x):
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"""
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Return a domain suitable for given abscissae.
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Find a domain suitable for a polynomial or Chebyshev series
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defined at the values supplied.
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Parameters
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----------
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x : array_like
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1-d array of abscissae whose domain will be determined.
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Returns
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-------
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domain : ndarray
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1-d array containing two values. If the inputs are complex, then
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the two returned points are the lower left and upper right corners
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of the smallest rectangle (aligned with the axes) in the complex
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plane containing the points `x`. If the inputs are real, then the
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two points are the ends of the smallest interval containing the
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points `x`.
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See Also
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--------
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mapparms, mapdomain
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Examples
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--------
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>>> from numpy.polynomial import polyutils as pu
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>>> points = np.arange(4)**2 - 5; points
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array([-5, -4, -1, 4])
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>>> pu.getdomain(points)
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array([-5., 4.])
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>>> c = np.exp(complex(0,1)*np.pi*np.arange(12)/6) # unit circle
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>>> pu.getdomain(c)
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array([-1.-1.j, 1.+1.j])
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"""
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[x] = as_series([x], trim=False)
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if x.dtype.char in np.typecodes['Complex']:
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rmin, rmax = x.real.min(), x.real.max()
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imin, imax = x.imag.min(), x.imag.max()
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return np.array((complex(rmin, imin), complex(rmax, imax)))
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else:
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return np.array((x.min(), x.max()))
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def mapparms(old, new):
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"""
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Linear map parameters between domains.
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Return the parameters of the linear map ``offset + scale*x`` that maps
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`old` to `new` such that ``old[i] -> new[i]``, ``i = 0, 1``.
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Parameters
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----------
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old, new : array_like
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Domains. Each domain must (successfully) convert to a 1-d array
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containing precisely two values.
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Returns
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-------
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offset, scale : scalars
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The map ``L(x) = offset + scale*x`` maps the first domain to the
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second.
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See Also
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--------
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getdomain, mapdomain
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Notes
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-----
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Also works for complex numbers, and thus can be used to calculate the
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parameters required to map any line in the complex plane to any other
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line therein.
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Examples
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--------
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>>> from numpy.polynomial import polyutils as pu
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>>> pu.mapparms((-1,1),(-1,1))
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(0.0, 1.0)
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>>> pu.mapparms((1,-1),(-1,1))
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(-0.0, -1.0)
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>>> i = complex(0,1)
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>>> pu.mapparms((-i,-1),(1,i))
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((1+1j), (1-0j))
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"""
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oldlen = old[1] - old[0]
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newlen = new[1] - new[0]
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off = (old[1]*new[0] - old[0]*new[1])/oldlen
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scl = newlen/oldlen
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return off, scl
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def mapdomain(x, old, new):
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"""
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Apply linear map to input points.
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The linear map ``offset + scale*x`` that maps the domain `old` to
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the domain `new` is applied to the points `x`.
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Parameters
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----------
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x : array_like
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Points to be mapped. If `x` is a subtype of ndarray the subtype
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will be preserved.
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old, new : array_like
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The two domains that determine the map. Each must (successfully)
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convert to 1-d arrays containing precisely two values.
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Returns
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-------
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x_out : ndarray
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Array of points of the same shape as `x`, after application of the
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linear map between the two domains.
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See Also
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--------
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getdomain, mapparms
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Notes
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-----
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Effectively, this implements:
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.. math ::
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x\\_out = new[0] + m(x - old[0])
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where
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.. math ::
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m = \\frac{new[1]-new[0]}{old[1]-old[0]}
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Examples
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--------
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>>> from numpy.polynomial import polyutils as pu
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>>> old_domain = (-1,1)
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>>> new_domain = (0,2*np.pi)
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>>> x = np.linspace(-1,1,6); x
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array([-1. , -0.6, -0.2, 0.2, 0.6, 1. ])
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>>> x_out = pu.mapdomain(x, old_domain, new_domain); x_out
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array([ 0. , 1.25663706, 2.51327412, 3.76991118, 5.02654825, # may vary
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6.28318531])
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>>> x - pu.mapdomain(x_out, new_domain, old_domain)
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array([0., 0., 0., 0., 0., 0.])
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Also works for complex numbers (and thus can be used to map any line in
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the complex plane to any other line therein).
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>>> i = complex(0,1)
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>>> old = (-1 - i, 1 + i)
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>>> new = (-1 + i, 1 - i)
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>>> z = np.linspace(old[0], old[1], 6); z
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array([-1. -1.j , -0.6-0.6j, -0.2-0.2j, 0.2+0.2j, 0.6+0.6j, 1. +1.j ])
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>>> new_z = pu.mapdomain(z, old, new); new_z
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array([-1.0+1.j , -0.6+0.6j, -0.2+0.2j, 0.2-0.2j, 0.6-0.6j, 1.0-1.j ]) # may vary
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"""
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x = np.asanyarray(x)
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off, scl = mapparms(old, new)
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return off + scl*x
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def _vander2d(vander_f, x, y, deg):
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"""
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Helper function used to implement the ``<type>vander2d`` functions.
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Parameters
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----------
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vander_f : function(array_like, int) -> ndarray
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The 1d vander function, such as ``polyvander``
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x, y, deg :
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See the ``<type>vander2d`` functions for more detail
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"""
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degx, degy = [
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_deprecate_as_int(d, "degrees")
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for d in deg
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]
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x, y = np.array((x, y), copy=0) + 0.0
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vx = vander_f(x, degx)
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vy = vander_f(y, degy)
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v = vx[..., None]*vy[..., None,:]
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return v.reshape(v.shape[:-2] + (-1,))
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def _vander3d(vander_f, x, y, z, deg):
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"""
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Helper function used to implement the ``<type>vander3d`` functions.
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Parameters
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----------
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vander_f : function(array_like, int) -> ndarray
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The 1d vander function, such as ``polyvander``
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x, y, z, deg :
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See the ``<type>vander3d`` functions for more detail
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"""
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degx, degy, degz = [
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_deprecate_as_int(d, "degrees")
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for d in deg
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]
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x, y, z = np.array((x, y, z), copy=0) + 0.0
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vx = vander_f(x, degx)
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vy = vander_f(y, degy)
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vz = vander_f(z, degz)
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v = vx[..., None, None]*vy[..., None,:, None]*vz[..., None, None,:]
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return v.reshape(v.shape[:-3] + (-1,))
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def _fromroots(line_f, mul_f, roots):
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"""
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Helper function used to implement the ``<type>fromroots`` functions.
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Parameters
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----------
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line_f : function(float, float) -> ndarray
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The ``<type>line`` function, such as ``polyline``
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mul_f : function(array_like, array_like) -> ndarray
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The ``<type>mul`` function, such as ``polymul``
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roots :
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See the ``<type>fromroots`` functions for more detail
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"""
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if len(roots) == 0:
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return np.ones(1)
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else:
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[roots] = as_series([roots], trim=False)
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roots.sort()
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p = [line_f(-r, 1) for r in roots]
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n = len(p)
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while n > 1:
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m, r = divmod(n, 2)
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tmp = [mul_f(p[i], p[i+m]) for i in range(m)]
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if r:
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tmp[0] = mul_f(tmp[0], p[-1])
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p = tmp
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n = m
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return p[0]
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def _valnd(val_f, c, *args):
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"""
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Helper function used to implement the ``<type>val<n>d`` functions.
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Parameters
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----------
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val_f : function(array_like, array_like, tensor: bool) -> array_like
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The ``<type>val`` function, such as ``polyval``
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c, args :
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See the ``<type>val<n>d`` functions for more detail
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"""
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try:
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args = tuple(np.array(args, copy=False))
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except Exception:
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# preserve the old error message
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if len(args) == 2:
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raise ValueError('x, y, z are incompatible')
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elif len(args) == 3:
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raise ValueError('x, y are incompatible')
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else:
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raise ValueError('ordinates are incompatible')
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it = iter(args)
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x0 = next(it)
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# use tensor on only the first
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c = val_f(x0, c)
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for xi in it:
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c = val_f(xi, c, tensor=False)
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return c
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def _gridnd(val_f, c, *args):
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"""
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Helper function used to implement the ``<type>grid<n>d`` functions.
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Parameters
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----------
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val_f : function(array_like, array_like, tensor: bool) -> array_like
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The ``<type>val`` function, such as ``polyval``
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c, args :
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See the ``<type>grid<n>d`` functions for more detail
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"""
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for xi in args:
|
|
c = val_f(xi, c)
|
|
return c
|
|
|
|
|
|
def _div(mul_f, c1, c2):
|
|
"""
|
|
Helper function used to implement the ``<type>div`` functions.
|
|
|
|
Implementation uses repeated subtraction of c2 multiplied by the nth basis.
|
|
For some polynomial types, a more efficient approach may be possible.
|
|
|
|
Parameters
|
|
----------
|
|
mul_f : function(array_like, array_like) -> array_like
|
|
The ``<type>mul`` function, such as ``polymul``
|
|
c1, c2 :
|
|
See the ``<type>div`` functions for more detail
|
|
"""
|
|
# c1, c2 are trimmed copies
|
|
[c1, c2] = as_series([c1, c2])
|
|
if c2[-1] == 0:
|
|
raise ZeroDivisionError()
|
|
|
|
lc1 = len(c1)
|
|
lc2 = len(c2)
|
|
if lc1 < lc2:
|
|
return c1[:1]*0, c1
|
|
elif lc2 == 1:
|
|
return c1/c2[-1], c1[:1]*0
|
|
else:
|
|
quo = np.empty(lc1 - lc2 + 1, dtype=c1.dtype)
|
|
rem = c1
|
|
for i in range(lc1 - lc2, - 1, -1):
|
|
p = mul_f([0]*i + [1], c2)
|
|
q = rem[-1]/p[-1]
|
|
rem = rem[:-1] - q*p[:-1]
|
|
quo[i] = q
|
|
return quo, trimseq(rem)
|
|
|
|
|
|
def _add(c1, c2):
|
|
""" Helper function used to implement the ``<type>add`` functions. """
|
|
# c1, c2 are trimmed copies
|
|
[c1, c2] = as_series([c1, c2])
|
|
if len(c1) > len(c2):
|
|
c1[:c2.size] += c2
|
|
ret = c1
|
|
else:
|
|
c2[:c1.size] += c1
|
|
ret = c2
|
|
return trimseq(ret)
|
|
|
|
|
|
def _sub(c1, c2):
|
|
""" Helper function used to implement the ``<type>sub`` functions. """
|
|
# c1, c2 are trimmed copies
|
|
[c1, c2] = 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 trimseq(ret)
|
|
|
|
|
|
def _fit(vander_f, x, y, deg, rcond=None, full=False, w=None):
|
|
"""
|
|
Helper function used to implement the ``<type>fit`` functions.
|
|
|
|
Parameters
|
|
----------
|
|
vander_f : function(array_like, int) -> ndarray
|
|
The 1d vander function, such as ``polyvander``
|
|
c1, c2 :
|
|
See the ``<type>fit`` functions for more detail
|
|
"""
|
|
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 = vander_f(x, lmax)
|
|
else:
|
|
deg = np.sort(deg)
|
|
lmax = deg[-1]
|
|
order = len(deg)
|
|
van = vander_f(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 = np.linalg.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 > 0:
|
|
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, RankWarning, stacklevel=2)
|
|
|
|
if full:
|
|
return c, [resids, rank, s, rcond]
|
|
else:
|
|
return c
|
|
|
|
|
|
def _pow(mul_f, c, pow, maxpower):
|
|
"""
|
|
Helper function used to implement the ``<type>pow`` functions.
|
|
|
|
Parameters
|
|
----------
|
|
vander_f : function(array_like, int) -> ndarray
|
|
The 1d vander function, such as ``polyvander``
|
|
pow, maxpower :
|
|
See the ``<type>pow`` functions for more detail
|
|
mul_f : function(array_like, array_like) -> ndarray
|
|
The ``<type>mul`` function, such as ``polymul``
|
|
"""
|
|
# c is a trimmed copy
|
|
[c] = 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 = mul_f(prd, c)
|
|
return prd
|
|
|
|
|
|
def _deprecate_as_int(x, desc):
|
|
"""
|
|
Like `operator.index`, but emits a deprecation warning when passed a float
|
|
|
|
Parameters
|
|
----------
|
|
x : int-like, or float with integral value
|
|
Value to interpret as an integer
|
|
desc : str
|
|
description to include in any error message
|
|
|
|
Raises
|
|
------
|
|
TypeError : if x is a non-integral float or non-numeric
|
|
DeprecationWarning : if x is an integral float
|
|
"""
|
|
try:
|
|
return operator.index(x)
|
|
except TypeError:
|
|
# Numpy 1.17.0, 2019-03-11
|
|
try:
|
|
ix = int(x)
|
|
except TypeError:
|
|
pass
|
|
else:
|
|
if ix == x:
|
|
warnings.warn(
|
|
"In future, this will raise TypeError, as {} will need to "
|
|
"be an integer not just an integral float."
|
|
.format(desc),
|
|
DeprecationWarning,
|
|
stacklevel=3
|
|
)
|
|
return ix
|
|
|
|
raise TypeError("{} must be an integer".format(desc))
|