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"""Utililty functions for polynomial modules.
This modules provides errors, warnings, and a polynomial base class along
with some common routines that are used in both the polynomial and
chebyshev modules.
Errors
------
- PolyError -- base class for errors
- PolyDomainError -- mismatched domains
Warnings
--------
- RankWarning -- issued by least squares fits to warn of deficient rank
Base Class
----------
- PolyBase -- Base class for the Polynomial and Chebyshev classes.
Functions
---------
- as_series -- turns list of array_like into 1d arrays of common type
- trimseq -- removes trailing zeros
- trimcoef -- removes trailing coefficients less than given magnitude
- getdomain -- finds appropriate domain for collection of points
- mapdomain -- maps points between domains
- mapparms -- parameters of the linear map between domains
"""
from __future__ import division
__all__ = ['RankWarning', 'PolyError', 'PolyDomainError', 'PolyBase',
'as_series', 'trimseq', 'trimcoef', 'getdomain', 'mapdomain',
'mapparms']
import warnings, exceptions
import numpy as np
#
# 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) :
pass
#
# We need the any function for python < 2.5
#
def any(iterable) :
for element in iterable:
if element :
return True
return False
#
# Helper functions to convert inputs to 1d 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 arguments as a list of 1d arrays.
The return type will always be an array of double, complex double. or
object.
Parameters
----------
[a1, a2,...] : list of array_like.
The arrays must have no more than one dimension when converted.
trim : boolean
When True, trailing zeros are removed from the inputs.
When False, the inputs are passed through intact.
Returns
-------
[a1, a2,...] : list of 1d-arrays
A copy of the input data as a 1d-arrays.
Raises
------
ValueError :
Raised when an input can not be coverted to 1-d array or the
resulting array is empty.
"""
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 max([a.ndim for a in arrays]) > 1 :
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 :
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 series.
Parameters
----------
c : array_like
1-d array of coefficients, ordered from low to high.
tol : number
Trailing elements with absolute value less than tol are removed.
Returns
-------
trimmed : ndarray
1_d array with tailing zeros removed. If the resulting series would
be empty, a series containing a singel zero is returned.
Raises
------
ValueError : if tol < 0
"""
if tol < 0 :
raise ValueError("tol must be non-negative")
[c] = as_series([c])
[ind] = np.where(np.abs(c) > tol)
if len(ind) == 0 :
return c[:1]*0
else :
return c[:ind[-1] + 1].copy()
def getdomain(x) :
"""Determine suitable domain for given points.
Find a suitable domain in which to fit a function defined at the points
`x` with a polynomial or Chebyshev series.
Parameters
----------
x : array_like
1D array of points whose domain will be determined.
Returns
-------
domain : ndarray
1D ndarray containing two values. If the inputs are complex, then
the two points are the corners of the smallest rectangle alinged
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
"""
[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 between domains.
Return the parameters of the linear map ``off + scl*x`` that maps the
`old` domain to the `new` domain. The map is defined by the requirement
that the left end of the old domain map to the left end of the new
domain, and similarly for the right ends.
Parameters
----------
old, new : array_like
The two domains should convert as 1D arrays containing two values.
Returns
-------
off, scl : scalars
The map `=``off + scl*x`` maps the first domain to the second.
See Also
--------
getdomain, mapdomain
"""
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 of the form ``off + scl*x`` that takes the `old` domain
to the `new` domain is applied to the points `x`.
Parameters
----------
x : array_like
Points to be mapped
old, new : array_like
The two domains that determin the map. They should both convert as
1D arrays containing two values.
Returns
-------
new_x : ndarray
Array of points of the same shape as the input `x` after the linear
map defined by the two domains is applied.
See Also
--------
getdomain, mapparms
"""
[x] = as_series([x], trim=False)
off, scl = mapparms(old, new)
return off + scl*x
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