790 lines
23 KiB
Python
790 lines
23 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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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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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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import operator
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import functools
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import warnings
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import numpy as np
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from numpy.core.multiarray import dragon4_positional, dragon4_scientific
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from numpy.core.umath import absolute
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__all__ = [
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'RankWarning', 'as_series', 'trimseq',
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'trimcoef', 'getdomain', 'mapdomain', 'mapparms',
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'format_float']
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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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#
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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=False) 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 as e:
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raise ValueError("Coefficient arrays have no common type") from e
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ret = [np.array(a, copy=True, 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 _nth_slice(i, ndim):
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sl = [np.newaxis] * ndim
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sl[i] = slice(None)
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return tuple(sl)
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def _vander_nd(vander_fs, points, degrees):
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r"""
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A generalization of the Vandermonde matrix for N dimensions
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The result is built by combining the results of 1d Vandermonde matrices,
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.. math::
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W[i_0, \ldots, i_M, j_0, \ldots, j_N] = \prod_{k=0}^N{V_k(x_k)[i_0, \ldots, i_M, j_k]}
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where
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.. math::
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N &= \texttt{len(points)} = \texttt{len(degrees)} = \texttt{len(vander\_fs)} \\
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M &= \texttt{points[k].ndim} \\
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V_k &= \texttt{vander\_fs[k]} \\
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x_k &= \texttt{points[k]} \\
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0 \le j_k &\le \texttt{degrees[k]}
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Expanding the one-dimensional :math:`V_k` functions gives:
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.. math::
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W[i_0, \ldots, i_M, j_0, \ldots, j_N] = \prod_{k=0}^N{B_{k, j_k}(x_k[i_0, \ldots, i_M])}
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where :math:`B_{k,m}` is the m'th basis of the polynomial construction used along
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dimension :math:`k`. For a regular polynomial, :math:`B_{k, m}(x) = P_m(x) = x^m`.
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Parameters
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----------
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vander_fs : Sequence[function(array_like, int) -> ndarray]
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The 1d vander function to use for each axis, such as ``polyvander``
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points : Sequence[array_like]
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Arrays of point coordinates, all of the same shape. The dtypes
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will be converted to either float64 or complex128 depending on
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whether any of the elements are complex. Scalars are converted to
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1-D arrays.
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This must be the same length as `vander_fs`.
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degrees : Sequence[int]
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The maximum degree (inclusive) to use for each axis.
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This must be the same length as `vander_fs`.
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Returns
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-------
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vander_nd : ndarray
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An array of shape ``points[0].shape + tuple(d + 1 for d in degrees)``.
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"""
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n_dims = len(vander_fs)
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if n_dims != len(points):
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raise ValueError(
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f"Expected {n_dims} dimensions of sample points, got {len(points)}")
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if n_dims != len(degrees):
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raise ValueError(
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f"Expected {n_dims} dimensions of degrees, got {len(degrees)}")
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if n_dims == 0:
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raise ValueError("Unable to guess a dtype or shape when no points are given")
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# convert to the same shape and type
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points = tuple(np.array(tuple(points), copy=False) + 0.0)
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# produce the vandermonde matrix for each dimension, placing the last
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# axis of each in an independent trailing axis of the output
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vander_arrays = (
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vander_fs[i](points[i], degrees[i])[(...,) + _nth_slice(i, n_dims)]
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for i in range(n_dims)
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)
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# we checked this wasn't empty already, so no `initial` needed
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return functools.reduce(operator.mul, vander_arrays)
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def _vander_nd_flat(vander_fs, points, degrees):
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"""
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Like `_vander_nd`, but flattens the last ``len(degrees)`` axes into a single axis
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Used to implement the public ``<type>vander<n>d`` functions.
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"""
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v = _vander_nd(vander_fs, points, degrees)
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return v.reshape(v.shape[:-len(degrees)] + (-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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args = [np.asanyarray(a) for a in args]
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shape0 = args[0].shape
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if not all((a.shape == shape0 for a in args[1:])):
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if len(args) == 3:
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raise ValueError('x, y, z are incompatible')
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elif len(args) == 2:
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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)
|
|
return c
|
|
|
|
|
|
def _gridnd(val_f, c, *args):
|
|
"""
|
|
Helper function used to implement the ``<type>grid<n>d`` functions.
|
|
|
|
Parameters
|
|
----------
|
|
val_f : function(array_like, array_like, tensor: bool) -> array_like
|
|
The ``<type>val`` function, such as ``polyval``
|
|
c, args
|
|
See the ``<type>grid<n>d`` functions for more detail
|
|
"""
|
|
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
|
|
----------
|
|
mul_f : function(array_like, array_like) -> ndarray
|
|
The ``<type>mul`` function, such as ``polymul``
|
|
c : array_like
|
|
1-D array of array of series coefficients
|
|
pow, maxpower
|
|
See the ``<type>pow`` functions for more detail
|
|
"""
|
|
# 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 as e:
|
|
# Numpy 1.17.0, 2019-03-11
|
|
try:
|
|
ix = int(x)
|
|
except TypeError:
|
|
pass
|
|
else:
|
|
if ix == x:
|
|
warnings.warn(
|
|
f"In future, this will raise TypeError, as {desc} will "
|
|
"need to be an integer not just an integral float.",
|
|
DeprecationWarning,
|
|
stacklevel=3
|
|
)
|
|
return ix
|
|
|
|
raise TypeError(f"{desc} must be an integer") from e
|
|
|
|
|
|
def format_float(x, parens=False):
|
|
if not np.issubdtype(type(x), np.floating):
|
|
return str(x)
|
|
|
|
opts = np.get_printoptions()
|
|
|
|
if np.isnan(x):
|
|
return opts['nanstr']
|
|
elif np.isinf(x):
|
|
return opts['infstr']
|
|
|
|
exp_format = False
|
|
if x != 0:
|
|
a = absolute(x)
|
|
if a >= 1.e8 or a < 10**min(0, -(opts['precision']-1)//2):
|
|
exp_format = True
|
|
|
|
trim, unique = '0', True
|
|
if opts['floatmode'] == 'fixed':
|
|
trim, unique = 'k', False
|
|
|
|
if exp_format:
|
|
s = dragon4_scientific(x, precision=opts['precision'],
|
|
unique=unique, trim=trim,
|
|
sign=opts['sign'] == '+')
|
|
if parens:
|
|
s = '(' + s + ')'
|
|
else:
|
|
s = dragon4_positional(x, precision=opts['precision'],
|
|
fractional=True,
|
|
unique=unique, trim=trim,
|
|
sign=opts['sign'] == '+')
|
|
return s
|