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