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""" 

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. 

 

""" 

import operator 

import functools 

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: 

""" 

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=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 

---------- 

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 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