text: """
  Objects for dealing with Chebyshev series.
  
  This module provides a number of objects (mostly functions) useful for
  dealing with Chebyshev series, including a `Chebyshev` class that
  encapsulates the usual arithmetic operations.  (General information
  on how this module represents and works with such polynomials is in the
  docstring for its "parent" sub-package, `numpy.polynomial`).
  
  Constants
  ---------
  - `chebdomain` -- Chebyshev series default domain, [-1,1].
  - `chebzero` -- (Coefficients of the) Chebyshev series that evaluates
    identically to 0.
  - `chebone` -- (Coefficients of the) Chebyshev series that evaluates
    identically to 1.
  - `chebx` -- (Coefficients of the) Chebyshev series for the identity map,
    ``f(x) = x``.
  
  Arithmetic
  ----------
  - `chebadd` -- add two Chebyshev series.
  - `chebsub` -- subtract one Chebyshev series from another.
  - `chebmul` -- multiply two Chebyshev series.
  - `chebdiv` -- divide one Chebyshev series by another.
  - `chebpow` -- raise a Chebyshev series to an positive integer power
  - `chebval` -- evaluate a Chebyshev series at given points.
  - `chebval2d` -- evaluate a 2D Chebyshev series at given points.
  - `chebval3d` -- evaluate a 3D Chebyshev series at given points.
  - `chebgrid2d` -- evaluate a 2D Chebyshev series on a Cartesian product.
  - `chebgrid3d` -- evaluate a 3D Chebyshev series on a Cartesian product.
  
  Calculus
  --------
  - `chebder` -- differentiate a Chebyshev series.
  - `chebint` -- integrate a Chebyshev series.
  
  Misc Functions
  --------------
  - `chebfromroots` -- create a Chebyshev series with specified roots.
  - `chebroots` -- find the roots of a Chebyshev series.
  - `chebvander` -- Vandermonde-like matrix for Chebyshev polynomials.
  - `chebvander2d` -- Vandermonde-like matrix for 2D power series.
  - `chebvander3d` -- Vandermonde-like matrix for 3D power series.
  - `chebgauss` -- Gauss-Chebyshev quadrature, points and weights.
  - `chebweight` -- Chebyshev weight function.
  - `chebcompanion` -- symmetrized companion matrix in Chebyshev form.
  - `chebfit` -- least-squares fit returning a Chebyshev series.
  - `chebpts1` -- Chebyshev points of the first kind.
  - `chebpts2` -- Chebyshev points of the second kind.
  - `chebtrim` -- trim leading coefficients from a Chebyshev series.
  - `chebline` -- Chebyshev series representing given straight line.
  - `cheb2poly` -- convert a Chebyshev series to a polynomial.
  - `poly2cheb` -- convert a polynomial to a Chebyshev series.
  
  Classes
  -------
  - `Chebyshev` -- A Chebyshev series class.
  
  See also
  --------
  `numpy.polynomial`
  
  Notes
  -----
  The implementations of multiplication, division, integration, and
  differentiation use the algebraic identities [1]_:
  
  .. math ::
      T_n(x) = \\frac{z^n + z^{-n}}{2} \\\\
      z\\frac{dx}{dz} = \\frac{z - z^{-1}}{2}.
  
  where
  
  .. math :: x = \\frac{z + z^{-1}}{2}.
  
  These identities allow a Chebyshev series to be expressed as a finite,
  symmetric Laurent series.  In this module, this sort of Laurent series
  is referred to as a "z-series."
  
  References
  ----------
  .. [1] A. T. Benjamin, et al., "Combinatorial Trigonometry with Chebyshev
    Polynomials," *Journal of Statistical Planning and Inference 14*, 2008
    (preprint: http://www.math.hmc.edu/~benjamin/papers/CombTrig.pdf, pg. 4)
  
  """
  from __future__ import division, absolute_import
  
  import numpy as np
  import numpy.linalg as la
  from . import polyutils as pu
  import warnings
  from .polytemplate import polytemplate
  
  __all__ = ['chebzero', 'chebone', 'chebx', 'chebdomain', 'chebline',
      'chebadd', 'chebsub', 'chebmulx', 'chebmul', 'chebdiv', 'chebpow',
      'chebval', 'chebder', 'chebint', 'cheb2poly', 'poly2cheb',
      'chebfromroots', 'chebvander', 'chebfit', 'chebtrim', 'chebroots',
      'chebpts1', 'chebpts2', 'Chebyshev', 'chebval2d', 'chebval3d',
      'chebgrid2d', 'chebgrid3d', 'chebvander2d','chebvander3d',
      'chebcompanion', 'chebgauss', 'chebweight']
  
  chebtrim = pu.trimcoef
  
  #
  # A collection of functions for manipulating z-series. These are private
  # functions and do minimal error checking.
  #
  
  def _cseries_to_zseries(c) :
      """Covert Chebyshev series to z-series.
  
      Covert a Chebyshev series to the equivalent z-series. The result is
      never an empty array. The dtype of the return is the same as that of
      the input. No checks are run on the arguments as this routine is for
      internal use.
  
      Parameters
      ----------
      c : 1-D ndarray
          Chebyshev coefficients, ordered from low to high
  
      Returns
      -------
      zs : 1-D ndarray
          Odd length symmetric z-series, ordered from  low to high.
  
      """
      n = c.size
      zs = np.zeros(2*n-1, dtype=c.dtype)
      zs[n-1:] = c/2
      return zs + zs[::-1]
  
  
  def _zseries_to_cseries(zs) :
      """Covert z-series to a Chebyshev series.
  
      Covert a z series to the equivalent Chebyshev series. The result is
      never an empty array. The dtype of the return is the same as that of
      the input. No checks are run on the arguments as this routine is for
      internal use.
  
      Parameters
      ----------
      zs : 1-D ndarray
          Odd length symmetric z-series, ordered from  low to high.
  
      Returns
      -------
      c : 1-D ndarray
          Chebyshev coefficients, ordered from  low to high.
  
      """
      n = (zs.size + 1)//2
      c = zs[n-1:].copy()
      c[1:n] *= 2
      return c
  
  
  def _zseries_mul(z1, z2) :
      """Multiply two z-series.
  
      Multiply two z-series to produce a z-series.
  
      Parameters
      ----------
      z1, z2 : 1-D ndarray
          The arrays must be 1-D but this is not checked.
  
      Returns
      -------
      product : 1-D ndarray
          The product z-series.
  
      Notes
      -----
      This is simply convolution. If symmetric/anti-symmetric z-series are
      denoted by S/A then the following rules apply:
  
      S*S, A*A -> S
      S*A, A*S -> A
  
      """
      return np.convolve(z1, z2)
  
  
  def _zseries_div(z1, z2) :
      """Divide the first z-series by the second.
  
      Divide `z1` by `z2` and return the quotient and remainder as z-series.
      Warning: this implementation only applies when both z1 and z2 have the
      same symmetry, which is sufficient for present purposes.
  
      Parameters
      ----------
      z1, z2 : 1-D ndarray
          The arrays must be 1-D and have the same symmetry, but this is not
          checked.
  
      Returns
      -------
  
      (quotient, remainder) : 1-D ndarrays
          Quotient and remainder as z-series.
  
      Notes
      -----
      This is not the same as polynomial division on account of the desired form
      of the remainder. If symmetric/anti-symmetric z-series are denoted by S/A
      then the following rules apply:
  
      S/S -> S,S
      A/A -> S,A
  
      The restriction to types of the same symmetry could be fixed but seems like
      unneeded generality. There is no natural form for the remainder in the case
      where there is no symmetry.
  
      """
      z1 = z1.copy()
      z2 = z2.copy()
      len1 = len(z1)
      len2 = len(z2)
      if len2 == 1 :
          z1 /= z2
          return z1, z1[:1]*0
      elif len1 < len2 :
          return z1[:1]*0, z1
      else :
          dlen = len1 - len2
          scl = z2[0]
          z2 /= scl
          quo = np.empty(dlen + 1, dtype=z1.dtype)
          i = 0
          j = dlen
          while i < j :
              r = z1[i]
              quo[i] = z1[i]
              quo[dlen - i] = r
              tmp = r*z2
              z1[i:i+len2] -= tmp
              z1[j:j+len2] -= tmp
              i += 1
              j -= 1
          r = z1[i]
          quo[i] = r
          tmp = r*z2
          z1[i:i+len2] -= tmp
          quo /= scl
          rem = z1[i+1:i-1+len2].copy()
          return quo, rem
  
  
  def _zseries_der(zs) :
      """Differentiate a z-series.
  
      The derivative is with respect to x, not z. This is achieved using the
      chain rule and the value of dx/dz given in the module notes.
  
      Parameters
      ----------
      zs : z-series
          The z-series to differentiate.
  
      Returns
      -------
      derivative : z-series
          The derivative
  
      Notes
      -----
      The zseries for x (ns) has been multiplied by two in order to avoid
      using floats that are incompatible with Decimal and likely other
      specialized scalar types. This scaling has been compensated by
      multiplying the value of zs by two also so that the two cancels in the
      division.
  
      """
      n = len(zs)//2
      ns = np.array([-1, 0, 1], dtype=zs.dtype)
      zs *= np.arange(-n, n+1)*2
      d, r = _zseries_div(zs, ns)
      return d
  
  
  def _zseries_int(zs) :
      """Integrate a z-series.
  
      The integral is with respect to x, not z. This is achieved by a change
      of variable using dx/dz given in the module notes.
  
      Parameters
      ----------
      zs : z-series
          The z-series to integrate
  
      Returns
      -------
      integral : z-series
          The indefinite integral
  
      Notes
      -----
      The zseries for x (ns) has been multiplied by two in order to avoid
      using floats that are incompatible with Decimal and likely other
      specialized scalar types. This scaling has been compensated by
      dividing the resulting zs by two.
  
      """
      n = 1 + len(zs)//2
      ns = np.array([-1, 0, 1], dtype=zs.dtype)
      zs = _zseries_mul(zs, ns)
      div = np.arange(-n, n+1)*2
      zs[:n] /= div[:n]
      zs[n+1:] /= div[n+1:]
      zs[n] = 0
      return zs
  
  #
  # Chebyshev series functions
  #
  
  
  def poly2cheb(pol) :
      """
      Convert a polynomial to a Chebyshev series.
  
      Convert an array representing the coefficients of a polynomial (relative
      to the "standard" basis) ordered from lowest degree to highest, to an
      array of the coefficients of the equivalent Chebyshev series, ordered
      from lowest to highest degree.
  
      Parameters
      ----------
      pol : array_like
          1-D array containing the polynomial coefficients
  
      Returns
      -------
      c : ndarray
          1-D array containing the coefficients of the equivalent Chebyshev
          series.
  
      See Also
      --------
      cheb2poly
  
      Notes
      -----
      The easy way to do conversions between polynomial basis sets
      is to use the convert method of a class instance.
  
      Examples
      --------
      >>> from numpy import polynomial as P
      >>> p = P.Polynomial(range(4))
      >>> p
      Polynomial([ 0.,  1.,  2.,  3.], [-1.,  1.])
      >>> c = p.convert(kind=P.Chebyshev)
      >>> c
      Chebyshev([ 1.  ,  3.25,  1.  ,  0.75], [-1.,  1.])
      >>> P.poly2cheb(range(4))
      array([ 1.  ,  3.25,  1.  ,  0.75])
  
      """
      [pol] = pu.as_series([pol])
      deg = len(pol) - 1
      res = 0
      for i in range(deg, -1, -1) :
          res = chebadd(chebmulx(res), pol[i])
      return res
  
  
  def cheb2poly(c) :
      """
      Convert a Chebyshev series to a polynomial.
  
      Convert an array representing the coefficients of a Chebyshev series,
      ordered from lowest degree to highest, to an array of the coefficients
      of the equivalent polynomial (relative to the "standard" basis) ordered
      from lowest to highest degree.
  
      Parameters
      ----------
      c : array_like
          1-D array containing the Chebyshev series coefficients, ordered
          from lowest order term to highest.
  
      Returns
      -------
      pol : ndarray
          1-D array containing the coefficients of the equivalent polynomial
          (relative to the "standard" basis) ordered from lowest order term
          to highest.
  
      See Also
      --------
      poly2cheb
  
      Notes
      -----
      The easy way to do conversions between polynomial basis sets
      is to use the convert method of a class instance.
  
      Examples
      --------
      >>> from numpy import polynomial as P
      >>> c = P.Chebyshev(range(4))
      >>> c
      Chebyshev([ 0.,  1.,  2.,  3.], [-1.,  1.])
      >>> p = c.convert(kind=P.Polynomial)
      >>> p
      Polynomial([ -2.,  -8.,   4.,  12.], [-1.,  1.])
      >>> P.cheb2poly(range(4))
      array([ -2.,  -8.,   4.,  12.])
  
      """
      from .polynomial import polyadd, polysub, polymulx
  
      [c] = pu.as_series([c])
      n = len(c)
      if n < 3:
          return c
      else:
          c0 = c[-2]
          c1 = c[-1]
          # i is the current degree of c1
          for i in range(n - 1, 1, -1) :
              tmp = c0
              c0 = polysub(c[i - 2], c1)
              c1 = polyadd(tmp, polymulx(c1)*2)
          return polyadd(c0, polymulx(c1))
  
  
  #
  # These are constant arrays are of integer type so as to be compatible
  # with the widest range of other types, such as Decimal.
  #
  
  # Chebyshev default domain.
  chebdomain = np.array([-1,1])
  
  # Chebyshev coefficients representing zero.
  chebzero = np.array([0])
  
  # Chebyshev coefficients representing one.
  chebone = np.array([1])
  
  # Chebyshev coefficients representing the identity x.
  chebx = np.array([0,1])
  
  
  def chebline(off, scl) :
      """
      Chebyshev series whose graph is a straight line.
  
  
  
      Parameters
      ----------
      off, scl : scalars
          The specified line is given by ``off + scl*x``.
  
      Returns
      -------
      y : ndarray
          This module's representation of the Chebyshev series for
          ``off + scl*x``.
  
      See Also
      --------
      polyline
  
      Examples
      --------
      >>> import numpy.polynomial.chebyshev as C
      >>> C.chebline(3,2)
      array([3, 2])
      >>> C.chebval(-3, C.chebline(3,2)) # should be -3
      -3.0
  
      """
      if scl != 0 :
          return np.array([off,scl])
      else :
          return np.array([off])
  
  
  def chebfromroots(roots) :
      """
      Generate a Chebyshev series with given roots.
  
      The function returns the coefficients of the polynomial
  
      .. math:: p(x) = (x - r_0) * (x - r_1) * ... * (x - r_n),
  
      in Chebyshev form, where the `r_n` are the roots specified in `roots`.
      If a zero has multiplicity n, then it must appear in `roots` n times.
      For instance, if 2 is a root of multiplicity three and 3 is a root of
      multiplicity 2, then `roots` looks something like [2, 2, 2, 3, 3]. The
      roots can appear in any order.
  
      If the returned coefficients are `c`, then
  
      .. math:: p(x) = c_0 + c_1 * T_1(x) + ... +  c_n * T_n(x)
  
      The coefficient of the last term is not generally 1 for monic
      polynomials in Chebyshev form.
  
      Parameters
      ----------
      roots : array_like
          Sequence containing the roots.
  
      Returns
      -------
      out : ndarray
          1-D array of coefficients.  If all roots are real then `out` is a
          real array, if some of the roots are complex, then `out` is complex
          even if all the coefficients in the result are real (see Examples
          below).
  
      See Also
      --------
      polyfromroots, legfromroots, lagfromroots, hermfromroots,
      hermefromroots.
  
      Examples
      --------
      >>> import numpy.polynomial.chebyshev as C
      >>> C.chebfromroots((-1,0,1)) # x^3 - x relative to the standard basis
      array([ 0.  , -0.25,  0.  ,  0.25])
      >>> j = complex(0,1)
      >>> C.chebfromroots((-j,j)) # x^2 + 1 relative to the standard basis
      array([ 1.5+0.j,  0.0+0.j,  0.5+0.j])
  
      """
      if len(roots) == 0 :
          return np.ones(1)
      else :
          [roots] = pu.as_series([roots], trim=False)
          roots.sort()
          p = [chebline(-r, 1) for r in roots]
          n = len(p)
          while n > 1:
              m, r = divmod(n, 2)
              tmp = [chebmul(p[i], p[i+m]) for i in range(m)]
              if r:
                  tmp[0] = chebmul(tmp[0], p[-1])
              p = tmp
              n = m
          return p[0]
  
  
  def chebadd(c1, c2):
      """
      Add one Chebyshev series to another.
  
      Returns the sum of two Chebyshev series `c1` + `c2`.  The arguments
      are sequences of coefficients ordered from lowest order term to
      highest, i.e., [1,2,3] represents the series ``T_0 + 2*T_1 + 3*T_2``.
  
      Parameters
      ----------
      c1, c2 : array_like
          1-D arrays of Chebyshev series coefficients ordered from low to
          high.
  
      Returns
      -------
      out : ndarray
          Array representing the Chebyshev series of their sum.
  
      See Also
      --------
      chebsub, chebmul, chebdiv, chebpow
  
      Notes
      -----
      Unlike multiplication, division, etc., the sum of two Chebyshev series
      is a Chebyshev series (without having to "reproject" the result onto
      the basis set) so addition, just like that of "standard" polynomials,
      is simply "component-wise."
  
      Examples
      --------
      >>> from numpy.polynomial import chebyshev as C
      >>> c1 = (1,2,3)
      >>> c2 = (3,2,1)
      >>> C.chebadd(c1,c2)
      array([ 4.,  4.,  4.])
  
      """
      # c1, c2 are trimmed copies
      [c1, c2] = pu.as_series([c1, c2])
      if len(c1) > len(c2) :
          c1[:c2.size] += c2
          ret = c1
      else :
          c2[:c1.size] += c1
          ret = c2
      return pu.trimseq(ret)
  
  
  def chebsub(c1, c2):
      """
      Subtract one Chebyshev series from another.
  
      Returns the difference of two Chebyshev series `c1` - `c2`.  The
      sequences of coefficients are from lowest order term to highest, i.e.,
      [1,2,3] represents the series ``T_0 + 2*T_1 + 3*T_2``.
  
      Parameters
      ----------
      c1, c2 : array_like
          1-D arrays of Chebyshev series coefficients ordered from low to
          high.
  
      Returns
      -------
      out : ndarray
          Of Chebyshev series coefficients representing their difference.
  
      See Also
      --------
      chebadd, chebmul, chebdiv, chebpow
  
      Notes
      -----
      Unlike multiplication, division, etc., the difference of two Chebyshev
      series is a Chebyshev series (without having to "reproject" the result
      onto the basis set) so subtraction, just like that of "standard"
      polynomials, is simply "component-wise."
  
      Examples
      --------
      >>> from numpy.polynomial import chebyshev as C
      >>> c1 = (1,2,3)
      >>> c2 = (3,2,1)
      >>> C.chebsub(c1,c2)
      array([-2.,  0.,  2.])
      >>> C.chebsub(c2,c1) # -C.chebsub(c1,c2)
      array([ 2.,  0., -2.])
  
      """
      # c1, c2 are trimmed copies
      [c1, c2] = pu.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 pu.trimseq(ret)
  
  
  def chebmulx(c):
      """Multiply a Chebyshev series by x.
  
      Multiply the polynomial `c` by x, where x is the independent
      variable.
  
  
      Parameters
      ----------
      c : array_like
          1-D array of Chebyshev series coefficients ordered from low to
          high.
  
      Returns
      -------
      out : ndarray
          Array representing the result of the multiplication.
  
      Notes
      -----
  
      .. versionadded:: 1.5.0
  
      """
      # c is a trimmed copy
      [c] = pu.as_series([c])
      # The zero series needs special treatment
      if len(c) == 1 and c[0] == 0:
          return c
  
      prd = np.empty(len(c) + 1, dtype=c.dtype)
      prd[0] = c[0]*0
      prd[1] = c[0]
      if len(c) > 1:
          tmp = c[1:]/2
          prd[2:] = tmp
          prd[0:-2] += tmp
      return prd
  
  
  def chebmul(c1, c2):
      """
      Multiply one Chebyshev series by another.
  
      Returns the product of two Chebyshev series `c1` * `c2`.  The arguments
      are sequences of coefficients, from lowest order "term" to highest,
      e.g., [1,2,3] represents the series ``T_0 + 2*T_1 + 3*T_2``.
  
      Parameters
      ----------
      c1, c2 : array_like
          1-D arrays of Chebyshev series coefficients ordered from low to
          high.
  
      Returns
      -------
      out : ndarray
          Of Chebyshev series coefficients representing their product.
  
      See Also
      --------
      chebadd, chebsub, chebdiv, chebpow
  
      Notes
      -----
      In general, the (polynomial) product of two C-series results in terms
      that are not in the Chebyshev polynomial basis set.  Thus, to express
      the product as a C-series, it is typically necessary to "reproject"
      the product onto said basis set, which typically produces
      "unintuitive live" (but correct) results; see Examples section below.
  
      Examples
      --------
      >>> from numpy.polynomial import chebyshev as C
      >>> c1 = (1,2,3)
      >>> c2 = (3,2,1)
      >>> C.chebmul(c1,c2) # multiplication requires "reprojection"
      array([  6.5,  12. ,  12. ,   4. ,   1.5])
  
      """
      # c1, c2 are trimmed copies
      [c1, c2] = pu.as_series([c1, c2])
      z1 = _cseries_to_zseries(c1)
      z2 = _cseries_to_zseries(c2)
      prd = _zseries_mul(z1, z2)
      ret = _zseries_to_cseries(prd)
      return pu.trimseq(ret)
  
  
  def chebdiv(c1, c2):
      """
      Divide one Chebyshev series by another.
  
      Returns the quotient-with-remainder of two Chebyshev series
      `c1` / `c2`.  The arguments are sequences of coefficients from lowest
      order "term" to highest, e.g., [1,2,3] represents the series
      ``T_0 + 2*T_1 + 3*T_2``.
  
      Parameters
      ----------
      c1, c2 : array_like
          1-D arrays of Chebyshev series coefficients ordered from low to
          high.
  
      Returns
      -------
      [quo, rem] : ndarrays
          Of Chebyshev series coefficients representing the quotient and
          remainder.
  
      See Also
      --------
      chebadd, chebsub, chebmul, chebpow
  
      Notes
      -----
      In general, the (polynomial) division of one C-series by another
      results in quotient and remainder terms that are not in the Chebyshev
      polynomial basis set.  Thus, to express these results as C-series, it
      is typically necessary to "reproject" the results onto said basis
      set, which typically produces "unintuitive" (but correct) results;
      see Examples section below.
  
      Examples
      --------
      >>> from numpy.polynomial import chebyshev as C
      >>> c1 = (1,2,3)
      >>> c2 = (3,2,1)
      >>> C.chebdiv(c1,c2) # quotient "intuitive," remainder not
      (array([ 3.]), array([-8., -4.]))
      >>> c2 = (0,1,2,3)
      >>> C.chebdiv(c2,c1) # neither "intuitive"
      (array([ 0.,  2.]), array([-2., -4.]))
  
      """
      # c1, c2 are trimmed copies
      [c1, c2] = pu.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 :
          z1 = _cseries_to_zseries(c1)
          z2 = _cseries_to_zseries(c2)
          quo, rem = _zseries_div(z1, z2)
          quo = pu.trimseq(_zseries_to_cseries(quo))
          rem = pu.trimseq(_zseries_to_cseries(rem))
          return quo, rem
  
  
  def chebpow(c, pow, maxpower=16) :
      """Raise a Chebyshev series to a power.
  
      Returns the Chebyshev series `c` raised to the power `pow`. The
      argument `c` is a sequence of coefficients ordered from low to high.
      i.e., [1,2,3] is the series  ``T_0 + 2*T_1 + 3*T_2.``
  
      Parameters
      ----------
      c : array_like
          1-D array of Chebyshev series coefficients ordered from low to
          high.
      pow : integer
          Power to which the series will be raised
      maxpower : integer, optional
          Maximum power allowed. This is mainly to limit growth of the series
          to unmanageable size. Default is 16
  
      Returns
      -------
      coef : ndarray
          Chebyshev series of power.
  
      See Also
      --------
      chebadd, chebsub, chebmul, chebdiv
  
      Examples
      --------
  
      """
      # c is a trimmed copy
      [c] = pu.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.
          zs = _cseries_to_zseries(c)
          prd = zs
          for i in range(2, power + 1) :
              prd = np.convolve(prd, zs)
          return _zseries_to_cseries(prd)
  
  
  def chebder(c, m=1, scl=1, axis=0) :
      """
      Differentiate a Chebyshev series.
  
      Returns the Chebyshev series coefficients `c` differentiated `m` times
      along `axis`.  At each iteration the result is multiplied by `scl` (the
      scaling factor is for use in a linear change of variable). The argument
      `c` is an array of coefficients from low to high degree along each
      axis, e.g., [1,2,3] represents the series ``1*T_0 + 2*T_1 + 3*T_2``
      while [[1,2],[1,2]] represents ``1*T_0(x)*T_0(y) + 1*T_1(x)*T_0(y) +
      2*T_0(x)*T_1(y) + 2*T_1(x)*T_1(y)`` if axis=0 is ``x`` and axis=1 is
      ``y``.
  
      Parameters
      ----------
      c : array_like
          Array of Chebyshev series coefficients. If c is multidimensional
          the different axis correspond to different variables with the
          degree in each axis given by the corresponding index.
      m : int, optional
          Number of derivatives taken, must be non-negative. (Default: 1)
      scl : scalar, optional
          Each differentiation is multiplied by `scl`.  The end result is
          multiplication by ``scl**m``.  This is for use in a linear change of
          variable. (Default: 1)
      axis : int, optional
          Axis over which the derivative is taken. (Default: 0).
  
          .. versionadded:: 1.7.0
  
      Returns
      -------
      der : ndarray
          Chebyshev series of the derivative.
  
      See Also
      --------
      chebint
  
      Notes
      -----
      In general, the result of differentiating a C-series needs to be
      "reprojected" onto the C-series basis set. Thus, typically, the
      result of this function is "unintuitive," albeit correct; see Examples
      section below.
  
      Examples
      --------
      >>> from numpy.polynomial import chebyshev as C
      >>> c = (1,2,3,4)
      >>> C.chebder(c)
      array([ 14.,  12.,  24.])
      >>> C.chebder(c,3)
      array([ 96.])
      >>> C.chebder(c,scl=-1)
      array([-14., -12., -24.])
      >>> C.chebder(c,2,-1)
      array([ 12.,  96.])
  
      """
      c = np.array(c, ndmin=1, copy=1)
      if c.dtype.char in '?bBhHiIlLqQpP':
          c = c.astype(np.double)
      cnt, iaxis = [int(t) for t in [m, axis]]
  
      if cnt != m:
          raise ValueError("The order of derivation must be integer")
      if cnt < 0:
          raise ValueError("The order of derivation must be non-negative")
      if iaxis != axis:
          raise ValueError("The axis must be integer")
      if not -c.ndim <= iaxis < c.ndim:
          raise ValueError("The axis is out of range")
      if iaxis < 0:
          iaxis += c.ndim
  
      if cnt == 0:
          return c
  
      c = np.rollaxis(c, iaxis)
      n = len(c)
      if cnt >= n:
          c = c[:1]*0
      else:
          for i in range(cnt):
              n = n - 1
              c *= scl
              der = np.empty((n,) + c.shape[1:], dtype=c.dtype)
              for j in range(n, 2, -1):
                  der[j - 1] = (2*j)*c[j]
                  c[j - 2] += (j*c[j])/(j - 2)
              if n > 1:
                  der[1] = 4*c[2]
              der[0] = c[1]
              c = der
      c = np.rollaxis(c, 0, iaxis + 1)
      return c
  
  
  def chebint(c, m=1, k=[], lbnd=0, scl=1, axis=0):
      """
      Integrate a Chebyshev series.
  
      Returns the Chebyshev series coefficients `c` integrated `m` times from
      `lbnd` along `axis`. At each iteration the resulting series is
      **multiplied** by `scl` and an integration constant, `k`, is added.
      The scaling factor is for use in a linear change of variable.  ("Buyer
      beware": note that, depending on what one is doing, one may want `scl`
      to be the reciprocal of what one might expect; for more information,
      see the Notes section below.)  The argument `c` is an array of
      coefficients from low to high degree along each axis, e.g., [1,2,3]
      represents the series ``T_0 + 2*T_1 + 3*T_2`` while [[1,2],[1,2]]
      represents ``1*T_0(x)*T_0(y) + 1*T_1(x)*T_0(y) + 2*T_0(x)*T_1(y) +
      2*T_1(x)*T_1(y)`` if axis=0 is ``x`` and axis=1 is ``y``.
  
      Parameters
      ----------
      c : array_like
          Array of Chebyshev series coefficients. If c is multidimensional
          the different axis correspond to different variables with the
          degree in each axis given by the corresponding index.
      m : int, optional
          Order of integration, must be positive. (Default: 1)
      k : {[], list, scalar}, optional
          Integration constant(s).  The value of the first integral at zero
          is the first value in the list, the value of the second integral
          at zero is the second value, etc.  If ``k == []`` (the default),
          all constants are set to zero.  If ``m == 1``, a single scalar can
          be given instead of a list.
      lbnd : scalar, optional
          The lower bound of the integral. (Default: 0)
      scl : scalar, optional
          Following each integration the result is *multiplied* by `scl`
          before the integration constant is added. (Default: 1)
      axis : int, optional
          Axis over which the integral is taken. (Default: 0).
  
          .. versionadded:: 1.7.0
  
      Returns
      -------
      S : ndarray
          C-series coefficients of the integral.
  
      Raises
      ------
      ValueError
          If ``m < 1``, ``len(k) > m``, ``np.isscalar(lbnd) == False``, or
          ``np.isscalar(scl) == False``.
  
      See Also
      --------
      chebder
  
      Notes
      -----
      Note that the result of each integration is *multiplied* by `scl`.
      Why is this important to note?  Say one is making a linear change of
      variable :math:`u = ax + b` in an integral relative to `x`.  Then
      .. math::`dx = du/a`, so one will need to set `scl` equal to
      :math:`1/a`- perhaps not what one would have first thought.
  
      Also note that, in general, the result of integrating a C-series needs
      to be "reprojected" onto the C-series basis set.  Thus, typically,
      the result of this function is "unintuitive," albeit correct; see
      Examples section below.
  
      Examples
      --------
      >>> from numpy.polynomial import chebyshev as C
      >>> c = (1,2,3)
      >>> C.chebint(c)
      array([ 0.5, -0.5,  0.5,  0.5])
      >>> C.chebint(c,3)
      array([ 0.03125   , -0.1875    ,  0.04166667, -0.05208333,  0.01041667,
              0.00625   ])
      >>> C.chebint(c, k=3)
      array([ 3.5, -0.5,  0.5,  0.5])
      >>> C.chebint(c,lbnd=-2)
      array([ 8.5, -0.5,  0.5,  0.5])
      >>> C.chebint(c,scl=-2)
      array([-1.,  1., -1., -1.])
  
      """
      c = np.array(c, ndmin=1, copy=1)
      if c.dtype.char in '?bBhHiIlLqQpP':
          c = c.astype(np.double)
      if not np.iterable(k):
          k = [k]
      cnt, iaxis = [int(t) for t in [m, axis]]
  
      if cnt != m:
          raise ValueError("The order of integration must be integer")
      if cnt < 0 :
          raise ValueError("The order of integration must be non-negative")
      if len(k) > cnt :
          raise ValueError("Too many integration constants")
      if iaxis != axis:
          raise ValueError("The axis must be integer")
      if not -c.ndim <= iaxis < c.ndim:
          raise ValueError("The axis is out of range")
      if iaxis < 0:
          iaxis += c.ndim
  
      if cnt == 0:
          return c
  
      c = np.rollaxis(c, iaxis)
      k = list(k) + [0]*(cnt - len(k))
      for i in range(cnt) :
          n = len(c)
          c *= scl
          if n == 1 and np.all(c[0] == 0):
              c[0] += k[i]
          else:
              tmp = np.empty((n + 1,) + c.shape[1:], dtype=c.dtype)
              tmp[0] = c[0]*0
              tmp[1] = c[0]
              if n > 1:
                  tmp[2] = c[1]/4
              for j in range(2, n):
                  t = c[j]/(2*j + 1)
                  tmp[j + 1] = c[j]/(2*(j + 1))
                  tmp[j - 1] -= c[j]/(2*(j - 1))
              tmp[0] += k[i] - chebval(lbnd, tmp)
              c = tmp
      c = np.rollaxis(c, 0, iaxis + 1)
      return c
  
  
  def chebval(x, c, tensor=True):
      """
      Evaluate a Chebyshev series at points x.
  
      If `c` is of length `n + 1`, this function returns the value:
  
      .. math:: p(x) = c_0 * T_0(x) + c_1 * T_1(x) + ... + c_n * T_n(x)
  
      The parameter `x` is converted to an array only if it is a tuple or a
      list, otherwise it is treated as a scalar. In either case, either `x`
      or its elements must support multiplication and addition both with
      themselves and with the elements of `c`.
  
      If `c` is a 1-D array, then `p(x)` will have the same shape as `x`.  If
      `c` is multidimensional, then the shape of the result depends on the
      value of `tensor`. If `tensor` is true the shape will be c.shape[1:] +
      x.shape. If `tensor` is false the shape will be c.shape[1:]. Note that
      scalars have shape (,).
  
      Trailing zeros in the coefficients will be used in the evaluation, so
      they should be avoided if efficiency is a concern.
  
      Parameters
      ----------
      x : array_like, compatible object
          If `x` is a list or tuple, it is converted to an ndarray, otherwise
          it is left unchanged and treated as a scalar. In either case, `x`
          or its elements must support addition and multiplication with
          with themselves and with the elements of `c`.
      c : array_like
          Array of coefficients ordered so that the coefficients for terms of
          degree n are contained in c[n]. If `c` is multidimensional the
          remaining indices enumerate multiple polynomials. In the two
          dimensional case the coefficients may be thought of as stored in
          the columns of `c`.
      tensor : boolean, optional
          If True, the shape of the coefficient array is extended with ones
          on the right, one for each dimension of `x`. Scalars have dimension 0
          for this action. The result is that every column of coefficients in
          `c` is evaluated for every element of `x`. If False, `x` is broadcast
          over the columns of `c` for the evaluation.  This keyword is useful
          when `c` is multidimensional. The default value is True.
  
          .. versionadded:: 1.7.0
  
      Returns
      -------
      values : ndarray, algebra_like
          The shape of the return value is described above.
  
      See Also
      --------
      chebval2d, chebgrid2d, chebval3d, chebgrid3d
  
      Notes
      -----
      The evaluation uses Clenshaw recursion, aka synthetic division.
  
      Examples
      --------
  
      """
      c = np.array(c, ndmin=1, copy=1)
      if c.dtype.char in '?bBhHiIlLqQpP':
          c = c.astype(np.double)
      if isinstance(x, (tuple, list)):
          x = np.asarray(x)
      if isinstance(x, np.ndarray) and tensor:
         c = c.reshape(c.shape + (1,)*x.ndim)
  
      if len(c) == 1 :
          c0 = c[0]
          c1 = 0
      elif len(c) == 2 :
          c0 = c[0]
          c1 = c[1]
      else :
          x2 = 2*x
          c0 = c[-2]
          c1 = c[-1]
          for i in range(3, len(c) + 1) :
              tmp = c0
              c0 = c[-i] - c1
              c1 = tmp + c1*x2
      return c0 + c1*x
  
  
  def chebval2d(x, y, c):
      """
      Evaluate a 2-D Chebyshev series at points (x, y).
  
      This function returns the values:
  
      .. math:: p(x,y) = \\sum_{i,j} c_{i,j} * T_i(x) * T_j(y)
  
      The parameters `x` and `y` are converted to arrays only if they are
      tuples or a lists, otherwise they are treated as a scalars and they
      must have the same shape after conversion. In either case, either `x`
      and `y` or their elements must support multiplication and addition both
      with themselves and with the elements of `c`.
  
      If `c` is a 1-D array a one is implicitly appended to its shape to make
      it 2-D. The shape of the result will be c.shape[2:] + x.shape.
  
      Parameters
      ----------
      x, y : array_like, compatible objects
          The two dimensional series is evaluated at the points `(x, y)`,
          where `x` and `y` must have the same shape. If `x` or `y` is a list
          or tuple, it is first converted to an ndarray, otherwise it is left
          unchanged and if it isn't an ndarray it is treated as a scalar.
      c : array_like
          Array of coefficients ordered so that the coefficient of the term
          of multi-degree i,j is contained in ``c[i,j]``. If `c` has
          dimension greater than 2 the remaining indices enumerate multiple
          sets of coefficients.
  
      Returns
      -------
      values : ndarray, compatible object
          The values of the two dimensional Chebyshev series at points formed
          from pairs of corresponding values from `x` and `y`.
  
      See Also
      --------
      chebval, chebgrid2d, chebval3d, chebgrid3d
  
      Notes
      -----
  
      .. versionadded::1.7.0
  
      """
      try:
          x, y = np.array((x, y), copy=0)
      except:
          raise ValueError('x, y are incompatible')
  
      c = chebval(x, c)
      c = chebval(y, c, tensor=False)
      return c
  
  
  def chebgrid2d(x, y, c):
      """
      Evaluate a 2-D Chebyshev series on the Cartesian product of x and y.
  
      This function returns the values:
  
      .. math:: p(a,b) = \sum_{i,j} c_{i,j} * T_i(a) * T_j(b),
  
      where the points `(a, b)` consist of all pairs formed by taking
      `a` from `x` and `b` from `y`. The resulting points form a grid with
      `x` in the first dimension and `y` in the second.
  
      The parameters `x` and `y` are converted to arrays only if they are
      tuples or a lists, otherwise they are treated as a scalars. In either
      case, either `x` and `y` or their elements must support multiplication
      and addition both with themselves and with the elements of `c`.
  
      If `c` has fewer than two dimensions, ones are implicitly appended to
      its shape to make it 2-D. The shape of the result will be c.shape[2:] +
      x.shape + y.shape.
  
      Parameters
      ----------
      x, y : array_like, compatible objects
          The two dimensional series is evaluated at the points in the
          Cartesian product of `x` and `y`.  If `x` or `y` is a list or
          tuple, it is first converted to an ndarray, otherwise it is left
          unchanged and, if it isn't an ndarray, it is treated as a scalar.
      c : array_like
          Array of coefficients ordered so that the coefficient of the term of
          multi-degree i,j is contained in `c[i,j]`. If `c` has dimension
          greater than two the remaining indices enumerate multiple sets of
          coefficients.
  
      Returns
      -------
      values : ndarray, compatible object
          The values of the two dimensional Chebyshev series at points in the
          Cartesian product of `x` and `y`.
  
      See Also
      --------
      chebval, chebval2d, chebval3d, chebgrid3d
  
      Notes
      -----
  
      .. versionadded::1.7.0
  
      """
      c = chebval(x, c)
      c = chebval(y, c)
      return c
  
  
  def chebval3d(x, y, z, c):
      """
      Evaluate a 3-D Chebyshev series at points (x, y, z).
  
      This function returns the values:
  
      .. math:: p(x,y,z) = \\sum_{i,j,k} c_{i,j,k} * T_i(x) * T_j(y) * T_k(z)
  
      The parameters `x`, `y`, and `z` are converted to arrays only if
      they are tuples or a lists, otherwise they are treated as a scalars and
      they must have the same shape after conversion. In either case, either
      `x`, `y`, and `z` or their elements must support multiplication and
      addition both with themselves and with the elements of `c`.
  
      If `c` has fewer than 3 dimensions, ones are implicitly appended to its
      shape to make it 3-D. The shape of the result will be c.shape[3:] +
      x.shape.
  
      Parameters
      ----------
      x, y, z : array_like, compatible object
          The three dimensional series is evaluated at the points
          `(x, y, z)`, where `x`, `y`, and `z` must have the same shape.  If
          any of `x`, `y`, or `z` is a list or tuple, it is first converted
          to an ndarray, otherwise it is left unchanged and if it isn't an
          ndarray it is  treated as a scalar.
      c : array_like
          Array of coefficients ordered so that the coefficient of the term of
          multi-degree i,j,k is contained in ``c[i,j,k]``. If `c` has dimension
          greater than 3 the remaining indices enumerate multiple sets of
          coefficients.
  
      Returns
      -------
      values : ndarray, compatible object
          The values of the multidimensional polynomial on points formed with
          triples of corresponding values from `x`, `y`, and `z`.
  
      See Also
      --------
      chebval, chebval2d, chebgrid2d, chebgrid3d
  
      Notes
      -----
  
      .. versionadded::1.7.0
  
      """
      try:
          x, y, z = np.array((x, y, z), copy=0)
      except:
          raise ValueError('x, y, z are incompatible')
  
      c = chebval(x, c)
      c = chebval(y, c, tensor=False)
      c = chebval(z, c, tensor=False)
      return c
  
  
  def chebgrid3d(x, y, z, c):
      """
      Evaluate a 3-D Chebyshev series on the Cartesian product of x, y, and z.
  
      This function returns the values:
  
      .. math:: p(a,b,c) = \\sum_{i,j,k} c_{i,j,k} * T_i(a) * T_j(b) * T_k(c)
  
      where the points `(a, b, c)` consist of all triples formed by taking
      `a` from `x`, `b` from `y`, and `c` from `z`. The resulting points form
      a grid with `x` in the first dimension, `y` in the second, and `z` in
      the third.
  
      The parameters `x`, `y`, and `z` are converted to arrays only if they
      are tuples or a lists, otherwise they are treated as a scalars. In
      either case, either `x`, `y`, and `z` or their elements must support
      multiplication and addition both with themselves and with the elements
      of `c`.
  
      If `c` has fewer than three dimensions, ones are implicitly appended to
      its shape to make it 3-D. The shape of the result will be c.shape[3:] +
      x.shape + y.shape + z.shape.
  
      Parameters
      ----------
      x, y, z : array_like, compatible objects
          The three dimensional series is evaluated at the points in the
          Cartesian product of `x`, `y`, and `z`.  If `x`,`y`, or `z` is a
          list or tuple, it is first converted to an ndarray, otherwise it is
          left unchanged and, if it isn't an ndarray, it is treated as a
          scalar.
      c : array_like
          Array of coefficients ordered so that the coefficients for terms of
          degree i,j are contained in ``c[i,j]``. If `c` has dimension
          greater than two the remaining indices enumerate multiple sets of
          coefficients.
  
      Returns
      -------
      values : ndarray, compatible object
          The values of the two dimensional polynomial at points in the Cartesian
          product of `x` and `y`.
  
      See Also
      --------
      chebval, chebval2d, chebgrid2d, chebval3d
  
      Notes
      -----
  
      .. versionadded::1.7.0
  
      """
      c = chebval(x, c)
      c = chebval(y, c)
      c = chebval(z, c)
      return c
  
  
  def chebvander(x, deg) :
      """Pseudo-Vandermonde matrix of given degree.
  
      Returns the pseudo-Vandermonde matrix of degree `deg` and sample points
      `x`. The pseudo-Vandermonde matrix is defined by
  
      .. math:: V[..., i] = T_i(x),
  
      where `0 <= i <= deg`. The leading indices of `V` index the elements of
      `x` and the last index is the degree of the Chebyshev polynomial.
  
      If `c` is a 1-D array of coefficients of length `n + 1` and `V` is the
      matrix ``V = chebvander(x, n)``, then ``np.dot(V, c)`` and
      ``chebval(x, c)`` are the same up to roundoff.  This equivalence is
      useful both for least squares fitting and for the evaluation of a large
      number of Chebyshev series of the same degree and sample points.
  
      Parameters
      ----------
      x : array_like
          Array of points. The dtype is converted to float64 or complex128
          depending on whether any of the elements are complex. If `x` is
          scalar it is converted to a 1-D array.
      deg : int
          Degree of the resulting matrix.
  
      Returns
      -------
      vander : ndarray
          The pseudo Vandermonde matrix. The shape of the returned matrix is
          ``x.shape + (deg + 1,)``, where The last index is the degree of the
          corresponding Chebyshev polynomial.  The dtype will be the same as
          the converted `x`.
  
      """
      ideg = int(deg)
      if ideg != deg:
          raise ValueError("deg must be integer")
      if ideg < 0:
          raise ValueError("deg must be non-negative")
  
      x = np.array(x, copy=0, ndmin=1) + 0.0
      dims = (ideg + 1,) + x.shape
      dtyp = x.dtype
      v = np.empty(dims, dtype=dtyp)
      # Use forward recursion to generate the entries.
      v[0] = x*0 + 1
      if ideg > 0 :
          x2 = 2*x
          v[1] = x
          for i in range(2, ideg + 1) :
              v[i] = v[i-1]*x2 - v[i-2]
      return np.rollaxis(v, 0, v.ndim)
  
  
  def chebvander2d(x, y, deg) :
      """Pseudo-Vandermonde matrix of given degrees.
  
      Returns the pseudo-Vandermonde matrix of degrees `deg` and sample
      points `(x, y)`. The pseudo-Vandermonde matrix is defined by
  
      .. math:: V[..., deg[1]*i + j] = T_i(x) * T_j(y),
  
      where `0 <= i <= deg[0]` and `0 <= j <= deg[1]`. The leading indices of
      `V` index the points `(x, y)` and the last index encodes the degrees of
      the Chebyshev polynomials.
  
      If ``V = chebvander2d(x, y, [xdeg, ydeg])``, then the columns of `V`
      correspond to the elements of a 2-D coefficient array `c` of shape
      (xdeg + 1, ydeg + 1) in the order
  
      .. math:: c_{00}, c_{01}, c_{02} ... , c_{10}, c_{11}, c_{12} ...
  
      and ``np.dot(V, c.flat)`` and ``chebval2d(x, y, c)`` will be the same
      up to roundoff. This equivalence is useful both for least squares
      fitting and for the evaluation of a large number of 2-D Chebyshev
      series of the same degrees and sample points.
  
      Parameters
      ----------
      x, y : 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.
      deg : list of ints
          List of maximum degrees of the form [x_deg, y_deg].
  
      Returns
      -------
      vander2d : ndarray
          The shape of the returned matrix is ``x.shape + (order,)``, where
          :math:`order = (deg[0]+1)*(deg([1]+1)`.  The dtype will be the same
          as the converted `x` and `y`.
  
      See Also
      --------
      chebvander, chebvander3d. chebval2d, chebval3d
  
      Notes
      -----
  
      .. versionadded::1.7.0
  
      """
      ideg = [int(d) for d in deg]
      is_valid = [id == d and id >= 0 for id, d in zip(ideg, deg)]
      if is_valid != [1, 1]:
          raise ValueError("degrees must be non-negative integers")
      degx, degy = ideg
      x, y = np.array((x, y), copy=0) + 0.0
  
      vx = chebvander(x, degx)
      vy = chebvander(y, degy)
      v = vx[..., None]*vy[..., None, :]
      return v.reshape(v.shape[:-2] + (-1,))
  
  
  def chebvander3d(x, y, z, deg) :
      """Pseudo-Vandermonde matrix of given degrees.
  
      Returns the pseudo-Vandermonde matrix of degrees `deg` and sample
      points `(x, y, z)`. If `l, m, n` are the given degrees in `x, y, z`,
      then The pseudo-Vandermonde matrix is defined by
  
      .. math:: V[..., (m+1)(n+1)i + (n+1)j + k] = T_i(x)*T_j(y)*T_k(z),
  
      where `0 <= i <= l`, `0 <= j <= m`, and `0 <= j <= n`.  The leading
      indices of `V` index the points `(x, y, z)` and the last index encodes
      the degrees of the Chebyshev polynomials.
  
      If ``V = chebvander3d(x, y, z, [xdeg, ydeg, zdeg])``, then the columns
      of `V` correspond to the elements of a 3-D coefficient array `c` of
      shape (xdeg + 1, ydeg + 1, zdeg + 1) in the order
  
      .. math:: c_{000}, c_{001}, c_{002},... , c_{010}, c_{011}, c_{012},...
  
      and ``np.dot(V, c.flat)`` and ``chebval3d(x, y, z, c)`` will be the
      same up to roundoff. This equivalence is useful both for least squares
      fitting and for the evaluation of a large number of 3-D Chebyshev
      series of the same degrees and sample points.
  
      Parameters
      ----------
      x, y, z : 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.
      deg : list of ints
          List of maximum degrees of the form [x_deg, y_deg, z_deg].
  
      Returns
      -------
      vander3d : ndarray
          The shape of the returned matrix is ``x.shape + (order,)``, where
          :math:`order = (deg[0]+1)*(deg([1]+1)*(deg[2]+1)`.  The dtype will
          be the same as the converted `x`, `y`, and `z`.
  
      See Also
      --------
      chebvander, chebvander3d. chebval2d, chebval3d
  
      Notes
      -----
  
      .. versionadded::1.7.0
  
      """
      ideg = [int(d) for d in deg]
      is_valid = [id == d and id >= 0 for id, d in zip(ideg, deg)]
      if is_valid != [1, 1, 1]:
          raise ValueError("degrees must be non-negative integers")
      degx, degy, degz = ideg
      x, y, z = np.array((x, y, z), copy=0) + 0.0
  
      vx = chebvander(x, degx)
      vy = chebvander(y, degy)
      vz = chebvander(z, degz)
      v = vx[..., None, None]*vy[..., None, :, None]*vz[..., None, None, :]
      return v.reshape(v.shape[:-3] + (-1,))
  
  
  def chebfit(x, y, deg, rcond=None, full=False, w=None):
      """
      Least squares fit of Chebyshev series to data.
  
      Return the coefficients of a Legendre series of degree `deg` that is the
      least squares fit to the data values `y` given at points `x`. If `y` is
      1-D the returned coefficients will also be 1-D. If `y` is 2-D multiple
      fits are done, one for each column of `y`, and the resulting
      coefficients are stored in the corresponding columns of a 2-D return.
      The fitted polynomial(s) are in the form
  
      .. math::  p(x) = c_0 + c_1 * T_1(x) + ... + c_n * T_n(x),
  
      where `n` is `deg`.
  
      Since numpy version 1.7.0, chebfit also supports NA. If any of the
      elements of `x`, `y`, or `w` are NA, then the corresponding rows of the
      linear least squares problem (see Notes) are set to 0. If `y` is 2-D,
      then an NA in any row of `y` invalidates that whole row.
  
      Parameters
      ----------
      x : array_like, shape (M,)
          x-coordinates of the M sample points ``(x[i], y[i])``.
      y : array_like, shape (M,) or (M, K)
          y-coordinates of the sample points. Several data sets of sample
          points sharing the same x-coordinates can be fitted at once by
          passing in a 2D-array that contains one dataset per column.
      deg : int
          Degree of the fitting series
      rcond : float, optional
          Relative condition number of the fit. Singular values smaller than
          this relative to the largest singular value will be ignored. The
          default value is len(x)*eps, where eps is the relative precision of
          the float type, about 2e-16 in most cases.
      full : bool, optional
          Switch determining nature of return value. When it is False (the
          default) just the coefficients are returned, when True diagnostic
          information from the singular value decomposition is also returned.
      w : array_like, shape (`M`,), optional
          Weights. If not None, the contribution of each point
          ``(x[i],y[i])`` to the fit is weighted by `w[i]`. Ideally the
          weights are chosen so that the errors of the products ``w[i]*y[i]``
          all have the same variance.  The default value is None.
  
          .. versionadded:: 1.5.0
  
      Returns
      -------
      coef : ndarray, shape (M,) or (M, K)
          Chebyshev coefficients ordered from low to high. If `y` was 2-D,
          the coefficients for the data in column k  of `y` are in column
          `k`.
  
      [residuals, rank, singular_values, rcond] : present when `full` = True
          Residuals of the least-squares fit, the effective rank of the
          scaled Vandermonde matrix and its singular values, and the
          specified value of `rcond`. For more details, see `linalg.lstsq`.
  
      Warns
      -----
      RankWarning
          The rank of the coefficient matrix in the least-squares fit is
          deficient. The warning is only raised if `full` = False.  The
          warnings can be turned off by
  
          >>> import warnings
          >>> warnings.simplefilter('ignore', RankWarning)
  
      See Also
      --------
      polyfit, legfit, lagfit, hermfit, hermefit
      chebval : Evaluates a Chebyshev series.
      chebvander : Vandermonde matrix of Chebyshev series.
      chebweight : Chebyshev weight function.
      linalg.lstsq : Computes a least-squares fit from the matrix.
      scipy.interpolate.UnivariateSpline : Computes spline fits.
  
      Notes
      -----
      The solution is the coefficients of the Chebyshev series `p` that
      minimizes the sum of the weighted squared errors
  
      .. math:: E = \\sum_j w_j^2 * |y_j - p(x_j)|^2,
  
      where :math:`w_j` are the weights. This problem is solved by setting up
      as the (typically) overdetermined matrix equation
  
      .. math:: V(x) * c = w * y,
  
      where `V` is the weighted pseudo Vandermonde matrix of `x`, `c` are the
      coefficients to be solved for, `w` are the weights, and `y` are the
      observed values.  This equation is then solved using the singular value
      decomposition of `V`.
  
      If some of the singular values of `V` are so small that they are
      neglected, then a `RankWarning` will be issued. This means that the
      coefficient values may be poorly determined. Using a lower order fit
      will usually get rid of the warning.  The `rcond` parameter can also be
      set to a value smaller than its default, but the resulting fit may be
      spurious and have large contributions from roundoff error.
  
      Fits using Chebyshev series are usually better conditioned than fits
      using power series, but much can depend on the distribution of the
      sample points and the smoothness of the data. If the quality of the fit
      is inadequate splines may be a good alternative.
  
      References
      ----------
      .. [1] Wikipedia, "Curve fitting",
             http://en.wikipedia.org/wiki/Curve_fitting
  
      Examples
      --------
  
      """
      order = int(deg) + 1
      x = np.asarray(x) + 0.0
      y = np.asarray(y) + 0.0
  
      # check arguments.
      if deg < 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")
  
      # set up the least squares matrices in transposed form
      lhs = chebvander(x, deg).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 = la.lstsq(lhs.T/scl, rhs.T, rcond)
      c = (c.T/scl).T
  
      # warn on rank reduction
      if rank != order and not full:
          msg = "The fit may be poorly conditioned"
          warnings.warn(msg, pu.RankWarning)
  
      if full :
          return c, [resids, rank, s, rcond]
      else :
          return c
  
  
  def chebcompanion(c):
      """Return the scaled companion matrix of c.
  
      The basis polynomials are scaled so that the companion matrix is
      symmetric when `c` is aa Chebyshev basis polynomial. This provides
      better eigenvalue estimates than the unscaled case and for basis
      polynomials the eigenvalues are guaranteed to be real if
      `numpy.linalg.eigvalsh` is used to obtain them.
  
      Parameters
      ----------
      c : array_like
          1-D array of Chebyshev series coefficients ordered from low to high
          degree.
  
      Returns
      -------
      mat : ndarray
          Scaled companion matrix of dimensions (deg, deg).
  
      Notes
      -----
  
      .. versionadded::1.7.0
  
      """
      # c is a trimmed copy
      [c] = pu.as_series([c])
      if len(c) < 2:
          raise ValueError('Series must have maximum degree of at least 1.')
      if len(c) == 2:
          return np.array(-c[0]/c[1])
  
      n = len(c) - 1
      mat = np.zeros((n, n), dtype=c.dtype)
      scl = np.array([1.] + [np.sqrt(.5)]*(n-1))
      top = mat.reshape(-1)[1::n+1]
      bot = mat.reshape(-1)[n::n+1]
      top[0] = np.sqrt(.5)
      top[1:] = 1/2
      bot[...] = top
      mat[:,-1] -= (c[:-1]/c[-1])*(scl/scl[-1])*.5
      return mat
  
  
  def chebroots(c):
      """
      Compute the roots of a Chebyshev series.
  
      Return the roots (a.k.a. "zeros") of the polynomial
  
      .. math:: p(x) = \\sum_i c[i] * T_i(x).
  
      Parameters
      ----------
      c : 1-D array_like
          1-D array of coefficients.
  
      Returns
      -------
      out : ndarray
          Array of the roots of the series. If all the roots are real,
          then `out` is also real, otherwise it is complex.
  
      See Also
      --------
      polyroots, legroots, lagroots, hermroots, hermeroots
  
      Notes
      -----
      The root estimates are obtained as the eigenvalues of the companion
      matrix, Roots far from the origin of the complex plane may have large
      errors due to the numerical instability of the series for such
      values. Roots with multiplicity greater than 1 will also show larger
      errors as the value of the series near such points is relatively
      insensitive to errors in the roots. Isolated roots near the origin can
      be improved by a few iterations of Newton's method.
  
      The Chebyshev series basis polynomials aren't powers of `x` so the
      results of this function may seem unintuitive.
  
      Examples
      --------
      >>> import numpy.polynomial.chebyshev as cheb
      >>> cheb.chebroots((-1, 1,-1, 1)) # T3 - T2 + T1 - T0 has real roots
      array([ -5.00000000e-01,   2.60860684e-17,   1.00000000e+00])
  
      """
      # c is a trimmed copy
      [c] = pu.as_series([c])
      if len(c) < 2:
          return np.array([], dtype=c.dtype)
      if len(c) == 2:
          return np.array([-c[0]/c[1]])
  
      m = chebcompanion(c)
      r = la.eigvals(m)
      r.sort()
      return r
  
  
  def chebgauss(deg):
      """
      Gauss-Chebyshev quadrature.
  
      Computes the sample points and weights for Gauss-Chebyshev quadrature.
      These sample points and weights will correctly integrate polynomials of
      degree :math:`2*deg - 1` or less over the interval :math:`[-1, 1]` with
      the weight function :math:`f(x) = 1/\sqrt{1 - x^2}`.
  
      Parameters
      ----------
      deg : int
          Number of sample points and weights. It must be >= 1.
  
      Returns
      -------
      x : ndarray
          1-D ndarray containing the sample points.
      y : ndarray
          1-D ndarray containing the weights.
  
      Notes
      -----
  
      .. versionadded:: 1.7.0
  
      The results have only been tested up to degree 100, higher degrees may
      be problematic. For Gauss-Chebyshev there are closed form solutions for
      the sample points and weights. If n = `deg`, then
  
      .. math:: x_i = \cos(\pi (2 i - 1) / (2 n))
  
      .. math:: w_i = \pi / n
  
      """
      ideg = int(deg)
      if ideg != deg or ideg < 1:
          raise ValueError("deg must be a non-negative integer")
  
      x = np.cos(np.pi * np.arange(1, 2*ideg, 2) / (2.0*ideg))
      w = np.ones(ideg)*(np.pi/ideg)
  
      return x, w
  
  
  def chebweight(x):
      """
      The weight function of the Chebyshev polynomials.
  
      The weight function is :math:`1/\sqrt{1 - x^2}` and the interval of
      integration is :math:`[-1, 1]`. The Chebyshev polynomials are orthogonal, but
      not normalized, with respect to this weight function.
  
      Parameters
      ----------
      x : array_like
         Values at which the weight function will be computed.
  
      Returns
      -------
      w : ndarray
         The weight function at `x`.
  
      Notes
      -----
  
      .. versionadded:: 1.7.0
  
      """
      w = 1./(np.sqrt(1. + x) * np.sqrt(1. - x))
      return w
  
  
  def chebpts1(npts):
      """
      Chebyshev points of the first kind.
  
      The Chebyshev points of the first kind are the points ``cos(x)``,
      where ``x = [pi*(k + .5)/npts for k in range(npts)]``.
  
      Parameters
      ----------
      npts : int
          Number of sample points desired.
  
      Returns
      -------
      pts : ndarray
          The Chebyshev points of the first kind.
  
      See Also
      --------
      chebpts2
  
      Notes
      -----
  
      .. versionadded:: 1.5.0
  
      """
      _npts = int(npts)
      if _npts != npts:
          raise ValueError("npts must be integer")
      if _npts < 1:
          raise ValueError("npts must be >= 1")
  
      x = np.linspace(-np.pi, 0, _npts, endpoint=False) + np.pi/(2*_npts)
      return np.cos(x)
  
  
  def chebpts2(npts):
      """
      Chebyshev points of the second kind.
  
      The Chebyshev points of the second kind are the points ``cos(x)``,
      where ``x = [pi*k/(npts - 1) for k in range(npts)]``.
  
      Parameters
      ----------
      npts : int
          Number of sample points desired.
  
      Returns
      -------
      pts : ndarray
          The Chebyshev points of the second kind.
  
      Notes
      -----
  
      .. versionadded:: 1.5.0
  
      """
      _npts = int(npts)
      if _npts != npts:
          raise ValueError("npts must be integer")
      if _npts < 2:
          raise ValueError("npts must be >= 2")
  
      x = np.linspace(-np.pi, 0, _npts)
      return np.cos(x)
  
  
  #
  # Chebyshev series class
  #
  
  exec(polytemplate.substitute(name='Chebyshev', nick='cheb', domain='[-1,1]'))
  