polyutils.py

"""
Utililty objects 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
-------------
- `PolyError` -- base class for this sub-package's errors.
- `PolyDomainError` -- raised when domains are "mismatched."

Warning objects
---------------
- `RankWarning` -- raised by a least-squares fit when a rank-deficient
  matrix is encountered.

Base class
----------
- `PolyBase` -- The base class for the `Polynomial` and `Chebyshev`
  classes.

Functions
---------
- `as_series` -- turns a list of array_likes into 1-D arrays of common
  type.
- `trimseq` -- removes trailing zeros.
- `trimcoef` -- removes trailing coefficients that are less than a given
  magnitude (thereby removing the corresponding terms).
- `getdomain` -- returns a 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

__all__ = ['RankWarning', 'PolyError', 'PolyDomainError', 'PolyBase',
           'as_series', 'trimseq', 'trimcoef', 'getdomain', 'mapdomain',
           'mapparms']

import warnings
import numpy as np
import sys

#
# 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

#
# 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
    ----------
    a : 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 import polynomial as P
    >>> a = np.arange(4)
    >>> P.as_series(a)
    [array([ 0.]), array([ 1.]), array([ 2.]), array([ 3.])]
    >>> b = np.arange(6).reshape((2,3))
    >>> P.as_series(b)
    [array([ 0.,  1.,  2.]), array([ 3.,  4.,  5.])]

    """
    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 :
            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 import polynomial as P
    >>> P.trimcoef((0,0,3,0,5,0,0))
    array([ 0.,  0.,  3.,  0.,  5.])
    >>> P.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
    >>> P.trimcoef((3e-4,1e-3*(1-i),5e-4,2e-5*(1+i)), 1e-3)
    array([ 0.0003+0.j   ,  0.0010-0.001j])

    """
    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) :
    """
    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 import polynomial as P
    >>> P.mapparms((-1,1),(-1,1))
    (0.0, 1.0)
    >>> P.mapparms((1,-1),(-1,1))
    (0.0, -1.0)
    >>> i = complex(0,1)
    >>> P.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 import polynomial as P
    >>> 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 = P.mapdomain(x, old_domain, new_domain); x_out
    array([ 0.        ,  1.25663706,  2.51327412,  3.76991118,  5.02654825,
            6.28318531])
    >>> x - P.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([-1.0-1.j , -0.6-0.6j, -0.2-0.2j,  0.2+0.2j,  0.6+0.6j,  1.0+1.j ])
    >>> new_z = P.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 ])

    """
    x = np.asanyarray(x)
    off, scl = mapparms(old, new)
    return off + scl*x