"""
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.
Warning objects
---------------
.. autosummary::
:toctree: generated/
RankWarning raised in least-squares fit for rank-deficient matrix.
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.
"""
import operator
import functools
import warnings
import numpy as np
__all__ = [
'RankWarning', 'as_series', 'trimseq',
'trimcoef', 'getdomain', 'mapdomain', 'mapparms']
#
# Warnings and Exceptions
#
class RankWarning(UserWarning):
"""Issued by chebfit when the design matrix is rank deficient."""
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=False) 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 as e:
raise ValueError("Coefficient arrays have no common type") from e
ret = [np.array(a, copy=True, 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 _nth_slice(i, ndim):
sl = [np.newaxis] * ndim
sl[i] = slice(None)
return tuple(sl)
def _vander_nd(vander_fs, points, degrees):
r"""
A generalization of the Vandermonde matrix for N dimensions
The result is built by combining the results of 1d Vandermonde matrices,
.. math::
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]}
where
.. math::
N &= \texttt{len(points)} = \texttt{len(degrees)} = \texttt{len(vander\_fs)} \\
M &= \texttt{points[k].ndim} \\
V_k &= \texttt{vander\_fs[k]} \\
x_k &= \texttt{points[k]} \\
0 \le j_k &\le \texttt{degrees[k]}
Expanding the one-dimensional :math:`V_k` functions gives:
.. math::
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])}
where :math:`B_{k,m}` is the m'th basis of the polynomial construction used along
dimension :math:`k`. For a regular polynomial, :math:`B_{k, m}(x) = P_m(x) = x^m`.
Parameters
----------
vander_fs : Sequence[function(array_like, int) -> ndarray]
The 1d vander function to use for each axis, such as ``polyvander``
points : Sequence[array_like]
Arrays of point coordinates, all of the same shape. The dtypes
will be converted to either float64 or complex128 depending on
whether any of the elements are complex. Scalars are converted to
1-D arrays.
This must be the same length as `vander_fs`.
degrees : Sequence[int]
The maximum degree (inclusive) to use for each axis.
This must be the same length as `vander_fs`.
Returns
-------
vander_nd : ndarray
An array of shape ``points[0].shape + tuple(d + 1 for d in degrees)``.
"""
n_dims = len(vander_fs)
if n_dims != len(points):
raise ValueError(
f"Expected {n_dims} dimensions of sample points, got {len(points)}")
if n_dims != len(degrees):
raise ValueError(
f"Expected {n_dims} dimensions of degrees, got {len(degrees)}")
if n_dims == 0:
raise ValueError("Unable to guess a dtype or shape when no points are given")
# convert to the same shape and type
points = tuple(np.array(tuple(points), copy=False) + 0.0)
# produce the vandermonde matrix for each dimension, placing the last
# axis of each in an independent trailing axis of the output
vander_arrays = (
vander_fs[i](points[i], degrees[i])[(...,) + _nth_slice(i, n_dims)]
for i in range(n_dims)
)
# we checked this wasn't empty already, so no `initial` needed
return functools.reduce(operator.mul, vander_arrays)
def _vander_nd_flat(vander_fs, points, degrees):
"""
Like `_vander_nd`, but flattens the last ``len(degrees)`` axes into a single axis
Used to implement the public ``<type>vander<n>d`` functions.
"""
v = _vander_nd(vander_fs, points, degrees)
return v.reshape(v.shape[:-len(degrees)] + (-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
"""
args = [np.asanyarray(a) for a in args]
shape0 = args[0].shape
if not all((a.shape == shape0 for a in args[1:])):
if len(args) == 3:
raise ValueError('x, y, z are incompatible')
elif len(args) == 2:
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
----------
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