text: """
  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.
  
  """
  
  
  __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
  
  #
  # We need the any function for python < 2.5
  #
  if sys.version_info[:2] < (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 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 1d-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
  