text: """
  Objects for dealing with Hermite series.
  
  This module provides a number of objects (mostly functions) useful for
  dealing with Hermite series, including a `Hermite` 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
  ---------
  - `hermedomain` -- Hermite series default domain, [-1,1].
  - `hermezero` -- Hermite series that evaluates identically to 0.
  - `hermeone` -- Hermite series that evaluates identically to 1.
  - `hermex` -- Hermite series for the identity map, ``f(x) = x``.
  
  Arithmetic
  ----------
  - `hermemulx` -- multiply a Hermite series in ``P_i(x)`` by ``x``.
  - `hermeadd` -- add two Hermite series.
  - `hermesub` -- subtract one Hermite series from another.
  - `hermemul` -- multiply two Hermite series.
  - `hermediv` -- divide one Hermite series by another.
  - `hermeval` -- evaluate a Hermite series at given points.
  
  Calculus
  --------
  - `hermeder` -- differentiate a Hermite series.
  - `hermeint` -- integrate a Hermite series.
  
  Misc Functions
  --------------
  - `hermefromroots` -- create a Hermite series with specified roots.
  - `hermeroots` -- find the roots of a Hermite series.
  - `hermevander` -- Vandermonde-like matrix for Hermite polynomials.
  - `hermefit` -- least-squares fit returning a Hermite series.
  - `hermetrim` -- trim leading coefficients from a Hermite series.
  - `hermeline` -- Hermite series of given straight line.
  - `herme2poly` -- convert a Hermite series to a polynomial.
  - `poly2herme` -- convert a polynomial to a Hermite series.
  
  Classes
  -------
  - `Hermite` -- A Hermite series class.
  
  See also
  --------
  `numpy.polynomial`
  
  """
  
  
  import numpy as np
  import numpy.linalg as la
  from . import polyutils as pu
  import warnings
  from .polytemplate import polytemplate
  
  __all__ = ['hermezero', 'hermeone', 'hermex', 'hermedomain', 'hermeline',
          'hermeadd', 'hermesub', 'hermemulx', 'hermemul', 'hermediv', 'hermeval',
          'hermeder', 'hermeint', 'herme2poly', 'poly2herme', 'hermefromroots',
          'hermevander', 'hermefit', 'hermetrim', 'hermeroots', 'HermiteE']
  
  hermetrim = pu.trimcoef
  
  def poly2herme(pol) :
      """
      poly2herme(pol)
  
      Convert a polynomial to a Hermite 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 Hermite series, ordered
      from lowest to highest degree.
  
      Parameters
      ----------
      pol : array_like
          1-d array containing the polynomial coefficients
  
      Returns
      -------
      cs : ndarray
          1-d array containing the coefficients of the equivalent Hermite
          series.
  
      See Also
      --------
      herme2poly
  
      Notes
      -----
      The easy way to do conversions between polynomial basis sets
      is to use the convert method of a class instance.
  
      Examples
      --------
      >>> from numpy.polynomial.hermite_e import poly2herme
      >>> poly2herme(np.arange(4))
      array([  2.,  10.,   2.,   3.])
  
      """
      [pol] = pu.as_series([pol])
      deg = len(pol) - 1
      res = 0
      for i in range(deg, -1, -1) :
          res = hermeadd(hermemulx(res), pol[i])
      return res
  
  
  def herme2poly(cs) :
      """
      Convert a Hermite series to a polynomial.
  
      Convert an array representing the coefficients of a Hermite 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
      ----------
      cs : array_like
          1-d array containing the Hermite 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
      --------
      poly2herme
  
      Notes
      -----
      The easy way to do conversions between polynomial basis sets
      is to use the convert method of a class instance.
  
      Examples
      --------
      >>> from numpy.polynomial.hermite_e import herme2poly
      >>> herme2poly([  2.,  10.,   2.,   3.])
      array([ 0.,  1.,  2.,  3.])
  
      """
      from .polynomial import polyadd, polysub, polymulx
  
      [cs] = pu.as_series([cs])
      n = len(cs)
      if n == 1:
          return cs
      if n == 2:
          return cs
      else:
          c0 = cs[-2]
          c1 = cs[-1]
          # i is the current degree of c1
          for i in range(n - 1, 1, -1) :
              tmp = c0
              c0 = polysub(cs[i - 2], c1*(i - 1))
              c1 = polyadd(tmp, polymulx(c1))
          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.
  #
  
  # Hermite
  hermedomain = np.array([-1,1])
  
  # Hermite coefficients representing zero.
  hermezero = np.array([0])
  
  # Hermite coefficients representing one.
  hermeone = np.array([1])
  
  # Hermite coefficients representing the identity x.
  hermex = np.array([0, 1])
  
  
  def hermeline(off, scl) :
      """
      Hermite 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 Hermite series for
          ``off + scl*x``.
  
      See Also
      --------
      polyline, chebline
  
      Examples
      --------
      >>> from numpy.polynomial.hermite_e import hermeline
      >>> from numpy.polynomial.hermite_e import hermeline, hermeval
      >>> hermeval(0,hermeline(3, 2))
      3.0
      >>> hermeval(1,hermeline(3, 2))
      5.0
  
      """
      if scl != 0 :
          return np.array([off,scl])
      else :
          return np.array([off])
  
  
  def hermefromroots(roots) :
      """
      Generate a Hermite series with the given roots.
  
      Return the array of coefficients for the P-series whose roots (a.k.a.
      "zeros") are given by *roots*.  The returned array of coefficients is
      ordered from lowest order "term" to highest, and zeros of multiplicity
      greater than one must be included in *roots* a number of times equal
      to their multiplicity (e.g., if `2` is a root of multiplicity three,
      then [2,2,2] must be in *roots*).
  
      Parameters
      ----------
      roots : array_like
          Sequence containing the roots.
  
      Returns
      -------
      out : ndarray
          1-d array of the Hermite series coefficients, ordered from low to
          high.  If all roots are real, ``out.dtype`` is a float type;
          otherwise, ``out.dtype`` is a complex type, even if all the
          coefficients in the result are real (see Examples below).
  
      See Also
      --------
      polyfromroots, chebfromroots
  
      Notes
      -----
      What is returned are the :math:`c_i` such that:
  
      .. math::
  
          \\sum_{i=0}^{n} c_i*P_i(x) = \\prod_{i=0}^{n} (x - roots[i])
  
      where ``n == len(roots)`` and :math:`P_i(x)` is the `i`-th Hermite
      (basis) polynomial over the domain `[-1,1]`.  Note that, unlike
      `polyfromroots`, due to the nature of the Hermite basis set, the
      above identity *does not* imply :math:`c_n = 1` identically (see
      Examples).
  
      Examples
      --------
      >>> from numpy.polynomial.hermite_e import hermefromroots, hermeval
      >>> coef = hermefromroots((-1, 0, 1))
      >>> hermeval((-1, 0, 1), coef)
      array([ 0.,  0.,  0.])
      >>> coef = hermefromroots((-1j, 1j))
      >>> hermeval((-1j, 1j), coef)
      array([ 0.+0.j,  0.+0.j])
  
      """
      if len(roots) == 0 :
          return np.ones(1)
      else :
          [roots] = pu.as_series([roots], trim=False)
          roots.sort()
          p = [hermeline(-r, 1) for r in roots]
          n = len(p)
          while n > 1:
              m, r = divmod(n, 2)
              tmp = [hermemul(p[i], p[i+m]) for i in range(m)]
              if r:
                  tmp[0] = hermemul(tmp[0], p[-1])
              p = tmp
              n = m
          return p[0]
  
  
  def hermeadd(c1, c2):
      """
      Add one Hermite series to another.
  
      Returns the sum of two Hermite series `c1` + `c2`.  The arguments
      are sequences of coefficients ordered from lowest order term to
      highest, i.e., [1,2,3] represents the series ``P_0 + 2*P_1 + 3*P_2``.
  
      Parameters
      ----------
      c1, c2 : array_like
          1-d arrays of Hermite series coefficients ordered from low to
          high.
  
      Returns
      -------
      out : ndarray
          Array representing the Hermite series of their sum.
  
      See Also
      --------
      hermesub, hermemul, hermediv, hermepow
  
      Notes
      -----
      Unlike multiplication, division, etc., the sum of two Hermite series
      is a Hermite 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.hermite_e import hermeadd
      >>> hermeadd([1, 2, 3], [1, 2, 3, 4])
      array([ 2.,  4.,  6.,  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 hermesub(c1, c2):
      """
      Subtract one Hermite series from another.
  
      Returns the difference of two Hermite series `c1` - `c2`.  The
      sequences of coefficients are from lowest order term to highest, i.e.,
      [1,2,3] represents the series ``P_0 + 2*P_1 + 3*P_2``.
  
      Parameters
      ----------
      c1, c2 : array_like
          1-d arrays of Hermite series coefficients ordered from low to
          high.
  
      Returns
      -------
      out : ndarray
          Of Hermite series coefficients representing their difference.
  
      See Also
      --------
      hermeadd, hermemul, hermediv, hermepow
  
      Notes
      -----
      Unlike multiplication, division, etc., the difference of two Hermite
      series is a Hermite 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.hermite_e import hermesub
      >>> hermesub([1, 2, 3, 4], [1, 2, 3])
      array([ 0.,  0.,  0.,  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 = -c2
          c2[:c1.size] += c1
          ret = c2
      return pu.trimseq(ret)
  
  
  def hermemulx(cs):
      """Multiply a Hermite series by x.
  
      Multiply the Hermite series `cs` by x, where x is the independent
      variable.
  
  
      Parameters
      ----------
      cs : array_like
          1-d array of Hermite series coefficients ordered from low to
          high.
  
      Returns
      -------
      out : ndarray
          Array representing the result of the multiplication.
  
      Notes
      -----
      The multiplication uses the recursion relationship for Hermite
      polynomials in the form
  
      .. math::
  
      xP_i(x) = (P_{i + 1}(x) + iP_{i - 1}(x)))
  
      Examples
      --------
      >>> from numpy.polynomial.hermite_e import hermemulx
      >>> hermemulx([1, 2, 3])
      array([ 2.,  7.,  2.,  3.])
  
      """
      # cs is a trimmed copy
      [cs] = pu.as_series([cs])
      # The zero series needs special treatment
      if len(cs) == 1 and cs[0] == 0:
          return cs
  
      prd = np.empty(len(cs) + 1, dtype=cs.dtype)
      prd[0] = cs[0]*0
      prd[1] = cs[0]
      for i in range(1, len(cs)):
          prd[i + 1] = cs[i]
          prd[i - 1] += cs[i]*i
      return prd
  
  
  def hermemul(c1, c2):
      """
      Multiply one Hermite series by another.
  
      Returns the product of two Hermite series `c1` * `c2`.  The arguments
      are sequences of coefficients, from lowest order "term" to highest,
      e.g., [1,2,3] represents the series ``P_0 + 2*P_1 + 3*P_2``.
  
      Parameters
      ----------
      c1, c2 : array_like
          1-d arrays of Hermite series coefficients ordered from low to
          high.
  
      Returns
      -------
      out : ndarray
          Of Hermite series coefficients representing their product.
  
      See Also
      --------
      hermeadd, hermesub, hermediv, hermepow
  
      Notes
      -----
      In general, the (polynomial) product of two C-series results in terms
      that are not in the Hermite polynomial basis set.  Thus, to express
      the product as a Hermite series, it is necessary to "re-project" the
      product onto said basis set, which may produce "un-intuitive" (but
      correct) results; see Examples section below.
  
      Examples
      --------
      >>> from numpy.polynomial.hermite_e import hermemul
      >>> hermemul([1, 2, 3], [0, 1, 2])
      array([ 14.,  15.,  28.,   7.,   6.])
  
      """
      # s1, s2 are trimmed copies
      [c1, c2] = pu.as_series([c1, c2])
  
      if len(c1) > len(c2):
          cs = c2
          xs = c1
      else:
          cs = c1
          xs = c2
  
      if len(cs) == 1:
          c0 = cs[0]*xs
          c1 = 0
      elif len(cs) == 2:
          c0 = cs[0]*xs
          c1 = cs[1]*xs
      else :
          nd = len(cs)
          c0 = cs[-2]*xs
          c1 = cs[-1]*xs
          for i in range(3, len(cs) + 1) :
              tmp = c0
              nd =  nd - 1
              c0 = hermesub(cs[-i]*xs, c1*(nd - 1))
              c1 = hermeadd(tmp, hermemulx(c1))
      return hermeadd(c0, hermemulx(c1))
  
  
  def hermediv(c1, c2):
      """
      Divide one Hermite series by another.
  
      Returns the quotient-with-remainder of two Hermite series
      `c1` / `c2`.  The arguments are sequences of coefficients from lowest
      order "term" to highest, e.g., [1,2,3] represents the series
      ``P_0 + 2*P_1 + 3*P_2``.
  
      Parameters
      ----------
      c1, c2 : array_like
          1-d arrays of Hermite series coefficients ordered from low to
          high.
  
      Returns
      -------
      [quo, rem] : ndarrays
          Of Hermite series coefficients representing the quotient and
          remainder.
  
      See Also
      --------
      hermeadd, hermesub, hermemul, hermepow
  
      Notes
      -----
      In general, the (polynomial) division of one Hermite series by another
      results in quotient and remainder terms that are not in the Hermite
      polynomial basis set.  Thus, to express these results as a Hermite
      series, it is necessary to "re-project" the results onto the Hermite
      basis set, which may produce "un-intuitive" (but correct) results; see
      Examples section below.
  
      Examples
      --------
      >>> from numpy.polynomial.hermite_e import hermediv
      >>> hermediv([ 14.,  15.,  28.,   7.,   6.], [0, 1, 2])
      (array([ 1.,  2.,  3.]), array([ 0.]))
      >>> hermediv([ 15.,  17.,  28.,   7.,   6.], [0, 1, 2])
      (array([ 1.,  2.,  3.]), array([ 1.,  2.]))
  
      """
      # 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 :
          quo = np.empty(lc1 - lc2 + 1, dtype=c1.dtype)
          rem = c1
          for i in range(lc1 - lc2, - 1, -1):
              p = hermemul([0]*i + [1], c2)
              q = rem[-1]/p[-1]
              rem = rem[:-1] - q*p[:-1]
              quo[i] = q
          return quo, pu.trimseq(rem)
  
  
  def hermepow(cs, pow, maxpower=16) :
      """Raise a Hermite series to a power.
  
      Returns the Hermite series `cs` raised to the power `pow`. The
      arguement `cs` is a sequence of coefficients ordered from low to high.
      i.e., [1,2,3] is the series  ``P_0 + 2*P_1 + 3*P_2.``
  
      Parameters
      ----------
      cs : array_like
          1d array of Hermite 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 umanageable size. Default is 16
  
      Returns
      -------
      coef : ndarray
          Hermite series of power.
  
      See Also
      --------
      hermeadd, hermesub, hermemul, hermediv
  
      Examples
      --------
      >>> from numpy.polynomial.hermite_e import hermepow
      >>> hermepow([1, 2, 3], 2)
      array([ 23.,  28.,  46.,  12.,   9.])
  
      """
      # cs is a trimmed copy
      [cs] = pu.as_series([cs])
      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=cs.dtype)
      elif power == 1 :
          return cs
      else :
          # This can be made more efficient by using powers of two
          # in the usual way.
          prd = cs
          for i in range(2, power + 1) :
              prd = hermemul(prd, cs)
          return prd
  
  
  def hermeder(cs, m=1, scl=1) :
      """
      Differentiate a Hermite series.
  
      Returns the series `cs` differentiated `m` times.  At each iteration the
      result is multiplied by `scl` (the scaling factor is for use in a linear
      change of variable).  The argument `cs` is the sequence of coefficients
      from lowest order "term" to highest, e.g., [1,2,3] represents the series
      ``P_0 + 2*P_1 + 3*P_2``.
  
      Parameters
      ----------
      cs: array_like
          1-d array of Hermite series coefficients ordered from low to high.
      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)
  
      Returns
      -------
      der : ndarray
          Hermite series of the derivative.
  
      See Also
      --------
      hermeint
  
      Notes
      -----
      In general, the result of differentiating a Hermite series does not
      resemble the same operation on a power series. Thus the result of this
      function may be "un-intuitive," albeit correct; see Examples section
      below.
  
      Examples
      --------
      >>> from numpy.polynomial.hermite_e import hermeder
      >>> hermeder([ 1.,  1.,  1.,  1.])
      array([ 1.,  2.,  3.])
      >>> hermeder([-0.25,  1.,  1./2.,  1./3.,  1./4 ], m=2)
      array([ 1.,  2.,  3.])
  
      """
      cnt = int(m)
  
      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")
  
      # cs is a trimmed copy
      [cs] = pu.as_series([cs])
      if cnt == 0:
          return cs
      elif cnt >= len(cs):
          return cs[:1]*0
      else :
          for i in range(cnt):
              n = len(cs) - 1
              cs *= scl
              der = np.empty(n, dtype=cs.dtype)
              for j in range(n, 0, -1):
                  der[j - 1] = j*cs[j]
              cs = der
          return cs
  
  
  def hermeint(cs, m=1, k=[], lbnd=0, scl=1):
      """
      Integrate a Hermite series.
  
      Returns a Hermite series that is the Hermite series `cs`, integrated
      `m` times from `lbnd` to `x`.  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 `cs` is a sequence of
      coefficients, from lowest order Hermite series "term" to highest,
      e.g., [1,2,3] represents the series :math:`P_0(x) + 2P_1(x) + 3P_2(x)`.
  
      Parameters
      ----------
      cs : array_like
          1-d array of Hermite series coefficients, ordered from low to high.
      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
          ``lbnd`` is the first value in the list, the value of the second
          integral at ``lbnd`` 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)
  
      Returns
      -------
      S : ndarray
          Hermite series coefficients of the integral.
  
      Raises
      ------
      ValueError
          If ``m < 0``, ``len(k) > m``, ``np.isscalar(lbnd) == False``, or
          ``np.isscalar(scl) == False``.
  
      See Also
      --------
      hermeder
  
      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 "re-projected" onto the C-series basis set.  Thus, typically,
      the result of this function is "un-intuitive," albeit correct; see
      Examples section below.
  
      Examples
      --------
      >>> from numpy.polynomial.hermite_e import hermeint
      >>> hermeint([1, 2, 3]) # integrate once, value 0 at 0.
      array([ 1.,  1.,  1.,  1.])
      >>> hermeint([1, 2, 3], m=2) # integrate twice, value & deriv 0 at 0 
      array([-0.25      ,  1.        ,  0.5       ,  0.33333333,  0.25      ])
      >>> hermeint([1, 2, 3], k=1) # integrate once, value 1 at 0.
      array([ 2.,  1.,  1.,  1.])
      >>> hermeint([1, 2, 3], lbnd=-1) # integrate once, value 0 at -1
      array([-1.,  1.,  1.,  1.])
      >>> hermeint([1, 2, 3], m=2, k=[1,2], lbnd=-1)
      array([ 1.83333333,  0.        ,  0.5       ,  0.33333333,  0.25      ])
  
      """
      cnt = int(m)
      if np.isscalar(k) :
          k = [k]
  
      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")
  
      # cs is a trimmed copy
      [cs] = pu.as_series([cs])
      if cnt == 0:
          return cs
  
      k = list(k) + [0]*(cnt - len(k))
      for i in range(cnt) :
          n = len(cs)
          cs *= scl
          if n == 1 and cs[0] == 0:
              cs[0] += k[i]
          else:
              tmp = np.empty(n + 1, dtype=cs.dtype)
              tmp[0] = cs[0]*0
              tmp[1] = cs[0]
              for j in range(1, n):
                  tmp[j + 1] = cs[j]/(j + 1)
              tmp[0] += k[i] - hermeval(lbnd, tmp)
              cs = tmp
      return cs
  
  
  def hermeval(x, cs):
      """Evaluate a Hermite series.
  
      If `cs` is of length `n`, this function returns :
  
      ``p(x) = cs[0]*P_0(x) + cs[1]*P_1(x) + ... + cs[n-1]*P_{n-1}(x)``
  
      If x is a sequence or array then p(x) will have the same shape as x.
      If r is a ring_like object that supports multiplication and addition
      by the values in `cs`, then an object of the same type is returned.
  
      Parameters
      ----------
      x : array_like, ring_like
          Array of numbers or objects that support multiplication and
          addition with themselves and with the elements of `cs`.
      cs : array_like
          1-d array of Hermite coefficients ordered from low to high.
  
      Returns
      -------
      values : ndarray, ring_like
          If the return is an ndarray then it has the same shape as `x`.
  
      See Also
      --------
      hermefit
  
      Examples
      --------
  
      Notes
      -----
      The evaluation uses Clenshaw recursion, aka synthetic division.
  
      Examples
      --------
      >>> from numpy.polynomial.hermite_e import hermeval
      >>> coef = [1,2,3]
      >>> hermeval(1, coef)
      3.0
      >>> hermeval([[1,2],[3,4]], coef)
      array([[  3.,  14.],
             [ 31.,  54.]])
  
      """
      # cs is a trimmed copy
      [cs] = pu.as_series([cs])
      if isinstance(x, tuple) or isinstance(x, list) :
          x = np.asarray(x)
  
      if len(cs) == 1 :
          c0 = cs[0]
          c1 = 0
      elif len(cs) == 2 :
          c0 = cs[0]
          c1 = cs[1]
      else :
          nd = len(cs)
          c0 = cs[-2]
          c1 = cs[-1]
          for i in range(3, len(cs) + 1) :
              tmp = c0
              nd =  nd - 1
              c0 = cs[-i] - c1*(nd - 1)
              c1 = tmp + c1*x
      return c0 + c1*x
  
  
  def hermevander(x, deg) :
      """Vandermonde matrix of given degree.
  
      Returns the Vandermonde matrix of degree `deg` and sample points `x`.
      This isn't a true Vandermonde matrix because `x` can be an arbitrary
      ndarray and the Hermite polynomials aren't powers. If ``V`` is the
      returned matrix and `x` is a 2d array, then the elements of ``V`` are
      ``V[i,j,k] = P_k(x[i,j])``, where ``P_k`` is the Hermite polynomial
      of degree ``k``.
  
      Parameters
      ----------
      x : array_like
          Array of points. The values are converted to double or complex
          doubles. If x is scalar it is converted to a 1D array.
      deg : integer
          Degree of the resulting matrix.
  
      Returns
      -------
      vander : Vandermonde matrix.
          The shape of the returned matrix is ``x.shape + (deg+1,)``. The last
          index is the degree.
  
      Examples
      --------
      >>> from numpy.polynomial.hermite_e import hermevander
      >>> x = np.array([-1, 0, 1])
      >>> hermevander(x, 3)
      array([[ 1., -1.,  0.,  2.],
             [ 1.,  0., -1., -0.],
             [ 1.,  1.,  0., -2.]])
  
      """
      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
      v = np.empty((ideg + 1,) + x.shape, dtype=x.dtype)
      v[0] = x*0 + 1
      if ideg > 0 :
          v[1] = x
          for i in range(2, ideg + 1) :
              v[i] = (v[i-1]*x - v[i-2]*(i - 1))
      return np.rollaxis(v, 0, v.ndim)
  
  
  def hermefit(x, y, deg, rcond=None, full=False, w=None):
      """
      Least squares fit of Hermite series to data.
  
      Return the coefficients of a HermiteE 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 * He_1(x) + ... + c_n * He_n(x),
  
      where `n` is `deg`.
  
      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 polynomial
      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.
  
      Returns
      -------
      coef : ndarray, shape (M,) or (M, K)
          Hermite 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
      --------
      chebfit, legfit, polyfit, hermfit, polyfit
      hermeval : Evaluates a Hermite series.
      hermevander : pseudo Vandermonde matrix of Hermite series.
      hermeweight : HermiteE 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 HermiteE 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 the :math:`w_j` are the weights. This problem is solved by
      setting up the (typically) overdetermined matrix equation
  
      .. math:: V(x) * c = w * y,
  
      where `V` is the pseudo Vandermonde matrix of `x`, the elements of `c`
      are the coefficients to be solved for, and the elements of `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
      coeficient 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 HermiteE series are probably most useful when the data can
      be approximated by ``sqrt(w(x)) * p(x)``, where `w(x)` is the HermiteE
      weight. In that case the wieght ``sqrt(w(x[i])`` should be used
      together with data values ``y[i]/sqrt(w(x[i])``. The weight function is
      available as `hermeweight`.
  
      References
      ----------
      .. [1] Wikipedia, "Curve fitting",
             http://en.wikipedia.org/wiki/Curve_fitting
  
      Examples
      --------
      >>> from numpy.polynomial.hermite_e import hermefik, hermeval
      >>> x = np.linspace(-10, 10)
      >>> err = np.random.randn(len(x))/10
      >>> y = hermeval(x, [1, 2, 3]) + err
      >>> hermefit(x, y, 2)
      array([ 1.01690445,  1.99951418,  2.99948696])
  
      """
      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
      lhs = hermevander(x, deg)
      rhs = y
      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
          if rhs.ndim == 2:
              lhs *= w[:, np.newaxis]
              rhs *= w[:, np.newaxis]
          else:
              lhs *= w[:, np.newaxis]
              rhs *= w
  
      # set rcond
      if rcond is None :
          rcond = len(x)*np.finfo(x.dtype).eps
  
      # scale the design matrix and solve the least squares equation
      scl = np.sqrt((lhs*lhs).sum(0))
      c, resids, rank, s = la.lstsq(lhs/scl, rhs, 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 hermecompanion(cs):
      """Return the scaled companion matrix of cs.
  
      The basis polynomials are scaled so that the companion matrix is
      symmetric when `cs` represents a single HermiteE polynomial. This
      provides better eigenvalue estimates than the unscaled case and in the
      single polynomial case the eigenvalues are guaranteed to be real if
      `numpy.linalg.eigvalsh` is used to obtain them.
  
      Parameters
      ----------
      cs : array_like
          1-d array of Legendre series coefficients ordered from low to high
          degree.
  
      Returns
      -------
      mat : ndarray
          Scaled companion matrix of dimensions (deg, deg).
  
      """
      accprod = np.multiply.accumulate
      # cs is a trimmed copy
      [cs] = pu.as_series([cs])
      if len(cs) < 2:
          raise ValueError('Series must have maximum degree of at least 1.')
      if len(cs) == 2:
          return np.array(-cs[0]/cs[1])
  
      n = len(cs) - 1
      mat = np.zeros((n, n), dtype=cs.dtype)
      scl = np.hstack((1., np.sqrt(np.arange(1,n))))
      scl = np.multiply.accumulate(scl)
      top = mat.reshape(-1)[1::n+1]
      bot = mat.reshape(-1)[n::n+1]
      top[...] = np.sqrt(np.arange(1,n))
      bot[...] = top
      mat[:,-1] -= (cs[:-1]/cs[-1])*(scl/scl[-1])
      return mat
  
  
  def hermeroots(cs):
      """
      Compute the roots of a Hermite series.
  
      Return the roots (a.k.a "zeros") of the HermiteE series represented by
      `cs`, which is the sequence of coefficients from lowest order "term"
      to highest, e.g., [1,2,3] is the series ``L_0 + 2*L_1 + 3*L_2``.
  
      Parameters
      ----------
      cs : array_like
          1-d array of HermiteE series coefficients ordered from low to high.
  
      Returns
      -------
      out : ndarray
          Array of the roots.  If all the roots are real, then so is the
          dtype of ``out``; otherwise, ``out``'s dtype is complex.
  
      See Also
      --------
      polyroots
      chebroots
  
      Notes
      -----
      Algorithm(s) used:
  
      Remember: because the Hermite series basis set is different from the
      "standard" basis set, the results of this function *may* not be what
      one is expecting.
  
      Examples
      --------
      >>> from numpy.polynomial.hermite_e import hermeroots, hermefromroots
      >>> coef = hermefromroots([-1, 0, 1])
      >>> coef
      array([ 0.,  2.,  0.,  1.])
      >>> hermeroots(coef)
      array([-1.,  0.,  1.])
  
      """
      # cs is a trimmed copy
      [cs] = pu.as_series([cs])
      if len(cs) <= 1 :
          return np.array([], dtype=cs.dtype)
      if len(cs) == 2 :
          return np.array([-cs[0]/cs[1]])
  
      m = hermecompanion(cs)
      r = la.eigvals(m)
      r.sort()
      return r
  
  
  #
  # HermiteE series class
  #
  
  exec(polytemplate.substitute(name='HermiteE', nick='herme', domain='[-1,1]'))
  