code: """
  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
  ---------
  - `hermdomain` -- Hermite series default domain, [-1,1].
  - `hermzero` -- Hermite series that evaluates identically to 0.
  - `hermone` -- Hermite series that evaluates identically to 1.
  - `hermx` -- Hermite series for the identity map, ``f(x) = x``.
  
  Arithmetic
  ----------
  - `hermmulx` -- multiply a Hermite series in ``P_i(x)`` by ``x``.
  - `hermadd` -- add two Hermite series.
  - `hermsub` -- subtract one Hermite series from another.
  - `hermmul` -- multiply two Hermite series.
  - `hermdiv` -- divide one Hermite series by another.
  - `hermval` -- evaluate a Hermite series at given points.
  - `hermval2d` -- evaluate a 2D Hermite series at given points.
  - `hermval3d` -- evaluate a 3D Hermite series at given points.
  - `hermgrid2d` -- evaluate a 2D Hermite series on a Cartesian product.
  - `hermgrid3d` -- evaluate a 3D Hermite series on a Cartesian product.
  
  Calculus
  --------
  - `hermder` -- differentiate a Hermite series.
  - `hermint` -- integrate a Hermite series.
  
  Misc Functions
  --------------
  - `hermfromroots` -- create a Hermite series with specified roots.
  - `hermroots` -- find the roots of a Hermite series.
  - `hermvander` -- Vandermonde-like matrix for Hermite polynomials.
  - `hermvander2d` -- Vandermonde-like matrix for 2D power series.
  - `hermvander3d` -- Vandermonde-like matrix for 3D power series.
  - `hermgauss` -- Gauss-Hermite quadrature, points and weights.
  - `hermweight` -- Hermite weight function.
  - `hermcompanion` -- symmetrized companion matrix in Hermite form.
  - `hermfit` -- least-squares fit returning a Hermite series.
  - `hermtrim` -- trim leading coefficients from a Hermite series.
  - `hermline` -- Hermite series of given straight line.
  - `herm2poly` -- convert a Hermite series to a polynomial.
  - `poly2herm` -- convert a polynomial to a Hermite series.
  
  Classes
  -------
  - `Hermite` -- A Hermite series class.
  
  See also
  --------
  `numpy.polynomial`
  
  """
  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__ = ['hermzero', 'hermone', 'hermx', 'hermdomain', 'hermline',
      'hermadd', 'hermsub', 'hermmulx', 'hermmul', 'hermdiv', 'hermpow',
      'hermval', 'hermder', 'hermint', 'herm2poly', 'poly2herm',
      'hermfromroots', 'hermvander', 'hermfit', 'hermtrim', 'hermroots',
      'Hermite', 'hermval2d', 'hermval3d', 'hermgrid2d', 'hermgrid3d',
      'hermvander2d', 'hermvander3d', 'hermcompanion', 'hermgauss',
      'hermweight']
  
  hermtrim = pu.trimcoef
  
  
  def poly2herm(pol) :
      """
      poly2herm(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
      -------
      c : ndarray
          1-D array containing the coefficients of the equivalent Hermite
          series.
  
      See Also
      --------
      herm2poly
  
      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
      >>> poly2herm(np.arange(4))
      array([ 1.   ,  2.75 ,  0.5  ,  0.375])
  
      """
      [pol] = pu.as_series([pol])
      deg = len(pol) - 1
      res = 0
      for i in range(deg, -1, -1) :
          res = hermadd(hermmulx(res), pol[i])
      return res
  
  
  def herm2poly(c) :
      """
      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
      ----------
      c : 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
      --------
      poly2herm
  
      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 import herm2poly
      >>> herm2poly([ 1.   ,  2.75 ,  0.5  ,  0.375])
      array([ 0.,  1.,  2.,  3.])
  
      """
      from .polynomial import polyadd, polysub, polymulx
  
      [c] = pu.as_series([c])
      n = len(c)
      if n == 1:
          return c
      if n == 2:
          c[1] *= 2
          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*(2*(i - 1)))
              c1 = polyadd(tmp, polymulx(c1)*2)
          return polyadd(c0, polymulx(c1)*2)
  
  #
  # These are constant arrays are of integer type so as to be compatible
  # with the widest range of other types, such as Decimal.
  #
  
  # Hermite
  hermdomain = np.array([-1,1])
  
  # Hermite coefficients representing zero.
  hermzero = np.array([0])
  
  # Hermite coefficients representing one.
  hermone = np.array([1])
  
  # Hermite coefficients representing the identity x.
  hermx = np.array([0, 1/2])
  
  
  def hermline(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 import hermline, hermval
      >>> hermval(0,hermline(3, 2))
      3.0
      >>> hermval(1,hermline(3, 2))
      5.0
  
      """
      if scl != 0 :
          return np.array([off,scl/2])
      else :
          return np.array([off])
  
  
  def hermfromroots(roots) :
      """
      Generate a Hermite 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 Hermite 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 * H_1(x) + ... +  c_n * H_n(x)
  
      The coefficient of the last term is not generally 1 for monic
      polynomials in Hermite 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, chebfromroots,
      hermefromroots.
  
      Examples
      --------
      >>> from numpy.polynomial.hermite import hermfromroots, hermval
      >>> coef = hermfromroots((-1, 0, 1))
      >>> hermval((-1, 0, 1), coef)
      array([ 0.,  0.,  0.])
      >>> coef = hermfromroots((-1j, 1j))
      >>> hermval((-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 = [hermline(-r, 1) for r in roots]
          n = len(p)
          while n > 1:
              m, r = divmod(n, 2)
              tmp = [hermmul(p[i], p[i+m]) for i in range(m)]
              if r:
                  tmp[0] = hermmul(tmp[0], p[-1])
              p = tmp
              n = m
          return p[0]
  
  
  def hermadd(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
      --------
      hermsub, hermmul, hermdiv, hermpow
  
      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 import hermadd
      >>> hermadd([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 hermsub(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
      --------
      hermadd, hermmul, hermdiv, hermpow
  
      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 import hermsub
      >>> hermsub([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 hermmulx(c):
      """Multiply a Hermite series by x.
  
      Multiply the Hermite series `c` by x, where x is the independent
      variable.
  
  
      Parameters
      ----------
      c : 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)/2 + i*P_{i - 1}(x))
  
      Examples
      --------
      >>> from numpy.polynomial.hermite import hermmulx
      >>> hermmulx([1, 2, 3])
      array([ 2. ,  6.5,  1. ,  1.5])
  
      """
      # 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]/2
      for i in range(1, len(c)):
          prd[i + 1] = c[i]/2
          prd[i - 1] += c[i]*i
      return prd
  
  
  def hermmul(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
      --------
      hermadd, hermsub, hermdiv, hermpow
  
      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 "reproject" the
      product onto said basis set, which may produce "unintuitive" (but
      correct) results; see Examples section below.
  
      Examples
      --------
      >>> from numpy.polynomial.hermite import hermmul
      >>> hermmul([1, 2, 3], [0, 1, 2])
      array([ 52.,  29.,  52.,   7.,   6.])
  
      """
      # s1, s2 are trimmed copies
      [c1, c2] = pu.as_series([c1, c2])
  
      if len(c1) > len(c2):
          c = c2
          xs = c1
      else:
          c = c1
          xs = c2
  
      if len(c) == 1:
          c0 = c[0]*xs
          c1 = 0
      elif len(c) == 2:
          c0 = c[0]*xs
          c1 = c[1]*xs
      else :
          nd = len(c)
          c0 = c[-2]*xs
          c1 = c[-1]*xs
          for i in range(3, len(c) + 1) :
              tmp = c0
              nd =  nd - 1
              c0 = hermsub(c[-i]*xs, c1*(2*(nd - 1)))
              c1 = hermadd(tmp, hermmulx(c1)*2)
      return hermadd(c0, hermmulx(c1)*2)
  
  
  def hermdiv(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
      --------
      hermadd, hermsub, hermmul, hermpow
  
      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 "reproject" the results onto the Hermite
      basis set, which may produce "unintuitive" (but correct) results; see
      Examples section below.
  
      Examples
      --------
      >>> from numpy.polynomial.hermite import hermdiv
      >>> hermdiv([ 52.,  29.,  52.,   7.,   6.], [0, 1, 2])
      (array([ 1.,  2.,  3.]), array([ 0.]))
      >>> hermdiv([ 54.,  31.,  52.,   7.,   6.], [0, 1, 2])
      (array([ 1.,  2.,  3.]), array([ 2.,  2.]))
      >>> hermdiv([ 53.,  30.,  52.,   7.,   6.], [0, 1, 2])
      (array([ 1.,  2.,  3.]), array([ 1.,  1.]))
  
      """
      # 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 = hermmul([0]*i + [1], c2)
              q = rem[-1]/p[-1]
              rem = rem[:-1] - q*p[:-1]
              quo[i] = q
          return quo, pu.trimseq(rem)
  
  
  def hermpow(c, pow, maxpower=16) :
      """Raise a Hermite series to a power.
  
      Returns the Hermite 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  ``P_0 + 2*P_1 + 3*P_2.``
  
      Parameters
      ----------
      c : array_like
          1-D 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 unmanageable size. Default is 16
  
      Returns
      -------
      coef : ndarray
          Hermite series of power.
  
      See Also
      --------
      hermadd, hermsub, hermmul, hermdiv
  
      Examples
      --------
      >>> from numpy.polynomial.hermite import hermpow
      >>> hermpow([1, 2, 3], 2)
      array([ 81.,  52.,  82.,  12.,   9.])
  
      """
      # 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.
          prd = c
          for i in range(2, power + 1) :
              prd = hermmul(prd, c)
          return prd
  
  
  def hermder(c, m=1, scl=1, axis=0) :
      """
      Differentiate a Hermite series.
  
      Returns the Hermite 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*H_0 + 2*H_1 + 3*H_2``
      while [[1,2],[1,2]] represents ``1*H_0(x)*H_0(y) + 1*H_1(x)*H_0(y) +
      2*H_0(x)*H_1(y) + 2*H_1(x)*H_1(y)`` if axis=0 is ``x`` and axis=1 is
      ``y``.
  
      Parameters
      ----------
      c : array_like
          Array of Hermite 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
          Hermite series of the derivative.
  
      See Also
      --------
      hermint
  
      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 "unintuitive," albeit correct; see Examples section
      below.
  
      Examples
      --------
      >>> from numpy.polynomial.hermite import hermder
      >>> hermder([ 1. ,  0.5,  0.5,  0.5])
      array([ 1.,  2.,  3.])
      >>> hermder([-0.5,  1./2.,  1./8.,  1./12.,  1./16.], m=2)
      array([ 1.,  2.,  3.])
  
      """
      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, 0, -1):
                  der[j - 1] = (2*j)*c[j]
              c = der
      c = np.rollaxis(c, 0, iaxis + 1)
      return c
  
  
  def hermint(c, m=1, k=[], lbnd=0, scl=1, axis=0):
      """
      Integrate a Hermite series.
  
      Returns the Hermite 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 ``H_0 + 2*H_1 + 3*H_2`` while [[1,2],[1,2]]
      represents ``1*H_0(x)*H_0(y) + 1*H_1(x)*H_0(y) + 2*H_0(x)*H_1(y) +
      2*H_1(x)*H_1(y)`` if axis=0 is ``x`` and axis=1 is ``y``.
  
      Parameters
      ----------
      c : array_like
          Array of Hermite 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
          ``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)
      axis : int, optional
          Axis over which the integral is taken. (Default: 0).
  
          .. versionadded:: 1.7.0
  
      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
      --------
      hermder
  
      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.hermite import hermint
      >>> hermint([1,2,3]) # integrate once, value 0 at 0.
      array([ 1. ,  0.5,  0.5,  0.5])
      >>> hermint([1,2,3], m=2) # integrate twice, value & deriv 0 at 0
      array([-0.5       ,  0.5       ,  0.125     ,  0.08333333,  0.0625    ])
      >>> hermint([1,2,3], k=1) # integrate once, value 1 at 0.
      array([ 2. ,  0.5,  0.5,  0.5])
      >>> hermint([1,2,3], lbnd=-1) # integrate once, value 0 at -1
      array([-2. ,  0.5,  0.5,  0.5])
      >>> hermint([1,2,3], m=2, k=[1,2], lbnd=-1)
      array([ 1.66666667, -0.5       ,  0.125     ,  0.08333333,  0.0625    ])
  
      """
      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]/2
              for j in range(1, n):
                  tmp[j + 1] = c[j]/(2*(j + 1))
              tmp[0] += k[i] - hermval(lbnd, tmp)
              c = tmp
      c = np.rollaxis(c, 0, iaxis + 1)
      return c
  
  
  def hermval(x, c, tensor=True):
      """
      Evaluate an Hermite series at points x.
  
      If `c` is of length `n + 1`, this function returns the value:
  
      .. math:: p(x) = c_0 * H_0(x) + c_1 * H_1(x) + ... + c_n * H_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
      --------
      hermval2d, hermgrid2d, hermval3d, hermgrid3d
  
      Notes
      -----
      The evaluation uses Clenshaw recursion, aka synthetic division.
  
      Examples
      --------
      >>> from numpy.polynomial.hermite import hermval
      >>> coef = [1,2,3]
      >>> hermval(1, coef)
      11.0
      >>> hermval([[1,2],[3,4]], coef)
      array([[  11.,   51.],
             [ 115.,  203.]])
  
      """
      c = np.array(c, ndmin=1, copy=0)
      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)
  
      x2 = x*2
      if len(c) == 1 :
          c0 = c[0]
          c1 = 0
      elif len(c) == 2 :
          c0 = c[0]
          c1 = c[1]
      else :
          nd = len(c)
          c0 = c[-2]
          c1 = c[-1]
          for i in range(3, len(c) + 1) :
              tmp = c0
              nd =  nd - 1
              c0 = c[-i] - c1*(2*(nd - 1))
              c1 = tmp + c1*x2
      return c0 + c1*x2
  
  
  def hermval2d(x, y, c):
      """
      Evaluate a 2-D Hermite series at points (x, y).
  
      This function returns the values:
  
      .. math:: p(x,y) = \\sum_{i,j} c_{i,j} * H_i(x) * H_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 two the remaining indices enumerate multiple
          sets of coefficients.
  
      Returns
      -------
      values : ndarray, compatible object
          The values of the two dimensional polynomial at points formed with
          pairs of corresponding values from `x` and `y`.
  
      See Also
      --------
      hermval, hermgrid2d, hermval3d, hermgrid3d
  
      Notes
      -----
  
      .. versionadded::1.7.0
  
      """
      try:
          x, y = np.array((x, y), copy=0)
      except:
          raise ValueError('x, y are incompatible')
  
      c = hermval(x, c)
      c = hermval(y, c, tensor=False)
      return c
  
  
  def hermgrid2d(x, y, c):
      """
      Evaluate a 2-D Hermite series on the Cartesian product of x and y.
  
      This function returns the values:
  
      .. math:: p(a,b) = \sum_{i,j} c_{i,j} * H_i(a) * H_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.
  
      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 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
      --------
      hermval, hermval2d, hermval3d, hermgrid3d
  
      Notes
      -----
  
      .. versionadded::1.7.0
  
      """
      c = hermval(x, c)
      c = hermval(y, c)
      return c
  
  
  def hermval3d(x, y, z, c):
      """
      Evaluate a 3-D Hermite series at points (x, y, z).
  
      This function returns the values:
  
      .. math:: p(x,y,z) = \\sum_{i,j,k} c_{i,j,k} * H_i(x) * H_j(y) * H_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
      --------
      hermval, hermval2d, hermgrid2d, hermgrid3d
  
      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 = hermval(x, c)
      c = hermval(y, c, tensor=False)
      c = hermval(z, c, tensor=False)
      return c
  
  
  def hermgrid3d(x, y, z, c):
      """
      Evaluate a 3-D Hermite 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} * H_i(a) * H_j(b) * H_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
      --------
      hermval, hermval2d, hermgrid2d, hermval3d
  
      Notes
      -----
  
      .. versionadded::1.7.0
  
      """
      c = hermval(x, c)
      c = hermval(y, c)
      c = hermval(z, c)
      return c
  
  
  def hermvander(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] = H_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 Hermite polynomial.
  
      If `c` is a 1-D array of coefficients of length `n + 1` and `V` is the
      array ``V = hermvander(x, n)``, then ``np.dot(V, c)`` and
      ``hermval(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 Hermite 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 Hermite polynomial.  The dtype will be the same as
          the converted `x`.
  
      Examples
      --------
      >>> from numpy.polynomial.hermite import hermvander
      >>> x = np.array([-1, 0, 1])
      >>> hermvander(x, 3)
      array([[ 1., -2.,  2.,  4.],
             [ 1.,  0., -2., -0.],
             [ 1.,  2.,  2., -4.]])
  
      """
      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)
      v[0] = x*0 + 1
      if ideg > 0 :
          x2 = x*2
          v[1] = x2
          for i in range(2, ideg + 1) :
              v[i] = (v[i-1]*x2 - v[i-2]*(2*(i - 1)))
      return np.rollaxis(v, 0, v.ndim)
  
  
  def hermvander2d(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] = H_i(x) * H_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 Hermite polynomials.
  
      If ``V = hermvander2d(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 ``hermval2d(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 Hermite
      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
      --------
      hermvander, hermvander3d. hermval2d, hermval3d
  
      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 = hermvander(x, degx)
      vy = hermvander(y, degy)
      v = vx[..., None]*vy[..., None, :]
      return v.reshape(v.shape[:-2] + (-1,))
  
  
  def hermvander3d(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] = H_i(x)*H_j(y)*H_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 Hermite polynomials.
  
      If ``V = hermvander3d(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 ``hermval3d(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 Hermite
      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
      --------
      hermvander, hermvander3d. hermval2d, hermval3d
  
      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 = hermvander(x, degx)
      vy = hermvander(y, degy)
      vz = hermvander(z, degz)
      v = vx[..., None, None]*vy[..., None, :, None]*vz[..., None, None, :]
      return v.reshape(v.shape[:-3] + (-1,))
  
  
  def hermfit(x, y, deg, rcond=None, full=False, w=None):
      """
      Least squares fit of Hermite series to data.
  
      Return the coefficients of a Hermite 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 * H_1(x) + ... + c_n * H_n(x),
  
      where `n` is `deg`.
  
      Since numpy version 1.7.0, hermfit 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 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, lagfit, polyfit, hermefit
      hermval : Evaluates a Hermite series.
      hermvander : Vandermonde matrix of Hermite series.
      hermweight : Hermite 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 Hermite 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 weighted pseudo Vandermonde matrix of `x`, `c` are the
      coefficients to be solved for, `w` are the weights, `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 Hermite series are probably most useful when the data can be
      approximated by ``sqrt(w(x)) * p(x)``, where `w(x)` is the Hermite
      weight. In that case the weight ``sqrt(w(x[i])`` should be used
      together with data values ``y[i]/sqrt(w(x[i])``. The weight function is
      available as `hermweight`.
  
      References
      ----------
      .. [1] Wikipedia, "Curve fitting",
             http://en.wikipedia.org/wiki/Curve_fitting
  
      Examples
      --------
      >>> from numpy.polynomial.hermite import hermfit, hermval
      >>> x = np.linspace(-10, 10)
      >>> err = np.random.randn(len(x))/10
      >>> y = hermval(x, [1, 2, 3]) + err
      >>> hermfit(x, y, 2)
      array([ 0.97902637,  1.99849131,  3.00006   ])
  
      """
      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 = hermvander(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 hermcompanion(c):
      """Return the scaled companion matrix of c.
  
      The basis polynomials are scaled so that the companion matrix is
      symmetric when `c` is an Hermite 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 Hermite series coefficients ordered from low to high
          degree.
  
      Returns
      -------
      mat : ndarray
          Scaled companion matrix of dimensions (deg, deg).
  
      Notes
      -----
  
      .. versionadded::1.7.0
  
      """
      accprod = np.multiply.accumulate
      # 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(-.5*c[0]/c[1])
  
      n = len(c) - 1
      mat = np.zeros((n, n), dtype=c.dtype)
      scl = np.hstack((1., np.sqrt(2.*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(.5*np.arange(1,n))
      bot[...] = top
      mat[:,-1] -= (c[:-1]/c[-1])*(scl/scl[-1])*.5
      return mat
  
  
  def hermroots(c):
      """
      Compute the roots of a Hermite series.
  
      Return the roots (a.k.a. "zeros") of the polynomial
  
      .. math:: p(x) = \\sum_i c[i] * H_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, chebroots, 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 Hermite series basis polynomials aren't powers of `x` so the
      results of this function may seem unintuitive.
  
      Examples
      --------
      >>> from numpy.polynomial.hermite import hermroots, hermfromroots
      >>> coef = hermfromroots([-1, 0, 1])
      >>> coef
      array([ 0.   ,  0.25 ,  0.   ,  0.125])
      >>> hermroots(coef)
      array([ -1.00000000e+00,  -1.38777878e-17,   1.00000000e+00])
  
      """
      # c is a trimmed copy
      [c] = pu.as_series([c])
      if len(c) <= 1 :
          return np.array([], dtype=c.dtype)
      if len(c) == 2 :
          return np.array([-.5*c[0]/c[1]])
  
      m = hermcompanion(c)
      r = la.eigvals(m)
      r.sort()
      return r
  
  
  def hermgauss(deg):
      """
      Gauss-Hermite quadrature.
  
      Computes the sample points and weights for Gauss-Hermite quadrature.
      These sample points and weights will correctly integrate polynomials of
      degree :math:`2*deg - 1` or less over the interval :math:`[-\inf, \inf]`
      with the weight function :math:`f(x) = \exp(-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. The weights are determined by using the fact that
  
      .. math:: w_k = c / (H'_n(x_k) * H_{n-1}(x_k))
  
      where :math:`c` is a constant independent of :math:`k` and :math:`x_k`
      is the k'th root of :math:`H_n`, and then scaling the results to get
      the right value when integrating 1.
  
      """
      ideg = int(deg)
      if ideg != deg or ideg < 1:
          raise ValueError("deg must be a non-negative integer")
  
      # first approximation of roots. We use the fact that the companion
      # matrix is symmetric in this case in order to obtain better zeros.
      c = np.array([0]*deg + [1])
      m = hermcompanion(c)
      x = la.eigvals(m)
      x.sort()
  
      # improve roots by one application of Newton
      dy = hermval(x, c)
      df = hermval(x, hermder(c))
      x -= dy/df
  
      # compute the weights. We scale the factor to avoid possible numerical
      # overflow.
      fm = hermval(x, c[1:])
      fm /= np.abs(fm).max()
      df /= np.abs(df).max()
      w = 1/(fm * df)
  
      # for Hermite we can also symmetrize
      w = (w + w[::-1])/2
      x = (x - x[::-1])/2
  
      # scale w to get the right value
      w *= np.sqrt(np.pi) / w.sum()
  
      return x, w
  
  
  def hermweight(x):
      """
      Weight function of the Hermite polynomials.
  
      The weight function is :math:`\exp(-x^2)` and the interval of
      integration is :math:`[-\inf, \inf]`. the Hermite 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 = np.exp(-x**2)
      return w
  
  
  #
  # Hermite series class
  #
  
  exec(polytemplate.substitute(name='Hermite', nick='herm', domain='[-1,1]'))
   """Tests for hermite_e module.
  
  """
  from __future__ import division, absolute_import
  
  import numpy as np
  import numpy.polynomial.hermite_e as herme
  from numpy.polynomial.polynomial import polyval
  from numpy.testing import *
  
  He0 = np.array([ 1 ])
  He1 = np.array([ 0 , 1 ])
  He2 = np.array([ -1 ,0 , 1 ])
  He3 = np.array([ 0 , -3 ,0 , 1 ])
  He4 = np.array([ 3 ,0 , -6 ,0 , 1 ])
  He5 = np.array([ 0 , 15 ,0 , -10 ,0 , 1 ])
  He6 = np.array([ -15 ,0 , 45 ,0 , -15 ,0 , 1 ])
  He7 = np.array([ 0 , -105 ,0 , 105 ,0 , -21 ,0 , 1 ])
  He8 = np.array([ 105 ,0 , -420 ,0 , 210 ,0 , -28 ,0 , 1 ])
  He9 = np.array([ 0 , 945 ,0 , -1260 ,0 , 378 ,0 , -36 ,0 , 1 ])
  
  Helist = [He0, He1, He2, He3, He4, He5, He6, He7, He8, He9]
  
  def trim(x) :
      return herme.hermetrim(x, tol=1e-6)
  
  
  class TestConstants(TestCase) :
  
      def test_hermedomain(self) :
          assert_equal(herme.hermedomain, [-1, 1])
  
      def test_hermezero(self) :
          assert_equal(herme.hermezero, [0])
  
      def test_hermeone(self) :
          assert_equal(herme.hermeone, [1])
  
      def test_hermex(self) :
          assert_equal(herme.hermex, [0, 1])
  
  
  class TestArithmetic(TestCase) :
      x = np.linspace(-3, 3, 100)
  
      def test_hermeadd(self) :
          for i in range(5) :
              for j in range(5) :
                  msg = "At i=%d, j=%d" % (i,j)
                  tgt = np.zeros(max(i,j) + 1)
                  tgt[i] += 1
                  tgt[j] += 1
                  res = herme.hermeadd([0]*i + [1], [0]*j + [1])
                  assert_equal(trim(res), trim(tgt), err_msg=msg)
  
      def test_hermesub(self) :
          for i in range(5) :
              for j in range(5) :
                  msg = "At i=%d, j=%d" % (i,j)
                  tgt = np.zeros(max(i,j) + 1)
                  tgt[i] += 1
                  tgt[j] -= 1
                  res = herme.hermesub([0]*i + [1], [0]*j + [1])
                  assert_equal(trim(res), trim(tgt), err_msg=msg)
  
      def test_hermemulx(self):
          assert_equal(herme.hermemulx([0]), [0])
          assert_equal(herme.hermemulx([1]), [0,1])
          for i in range(1, 5):
              ser = [0]*i + [1]
              tgt = [0]*(i - 1) + [i, 0, 1]
              assert_equal(herme.hermemulx(ser), tgt)
  
      def test_hermemul(self) :
          # check values of result
          for i in range(5) :
              pol1 = [0]*i + [1]
              val1 = herme.hermeval(self.x, pol1)
              for j in range(5) :
                  msg = "At i=%d, j=%d" % (i,j)
                  pol2 = [0]*j + [1]
                  val2 = herme.hermeval(self.x, pol2)
                  pol3 = herme.hermemul(pol1, pol2)
                  val3 = herme.hermeval(self.x, pol3)
                  assert_(len(pol3) == i + j + 1, msg)
                  assert_almost_equal(val3, val1*val2, err_msg=msg)
  
      def test_hermediv(self) :
          for i in range(5) :
              for j in range(5) :
                  msg = "At i=%d, j=%d" % (i,j)
                  ci = [0]*i + [1]
                  cj = [0]*j + [1]
                  tgt = herme.hermeadd(ci, cj)
                  quo, rem = herme.hermediv(tgt, ci)
                  res = herme.hermeadd(herme.hermemul(quo, ci), rem)
                  assert_equal(trim(res), trim(tgt), err_msg=msg)
  
  
  class TestEvaluation(TestCase) :
      # coefficients of 1 + 2*x + 3*x**2
      c1d = np.array([4., 2., 3.])
      c2d = np.einsum('i,j->ij', c1d, c1d)
      c3d = np.einsum('i,j,k->ijk', c1d, c1d, c1d)
  
      # some random values in [-1, 1)
      x = np.random.random((3, 5))*2 - 1
      y = polyval(x, [1., 2., 3.])
  
  
      def test_hermeval(self) :
          #check empty input
          assert_equal(herme.hermeval([], [1]).size, 0)
  
          #check normal input)
          x = np.linspace(-1,1)
          y = [polyval(x, c) for c in Helist]
          for i in range(10) :
              msg = "At i=%d" % i
              ser = np.zeros
              tgt = y[i]
              res = herme.hermeval(x, [0]*i + [1])
              assert_almost_equal(res, tgt, err_msg=msg)
  
          #check that shape is preserved
          for i in range(3) :
              dims = [2]*i
              x = np.zeros(dims)
              assert_equal(herme.hermeval(x, [1]).shape, dims)
              assert_equal(herme.hermeval(x, [1,0]).shape, dims)
              assert_equal(herme.hermeval(x, [1,0,0]).shape, dims)
  
      def test_hermeval2d(self):
          x1, x2, x3 = self.x
          y1, y2, y3 = self.y
  
          #test exceptions
          assert_raises(ValueError, herme.hermeval2d, x1, x2[:2], self.c2d)
  
          #test values
          tgt = y1*y2
          res = herme.hermeval2d(x1, x2, self.c2d)
          assert_almost_equal(res, tgt)
  
          #test shape
          z = np.ones((2,3))
          res = herme.hermeval2d(z, z, self.c2d)
          assert_(res.shape == (2,3))
  
      def test_hermeval3d(self):
          x1, x2, x3 = self.x
          y1, y2, y3 = self.y
  
          #test exceptions
          assert_raises(ValueError, herme.hermeval3d, x1, x2, x3[:2], self.c3d)
  
          #test values
          tgt = y1*y2*y3
          res = herme.hermeval3d(x1, x2, x3, self.c3d)
          assert_almost_equal(res, tgt)
  
          #test shape
          z = np.ones((2,3))
          res = herme.hermeval3d(z, z, z, self.c3d)
          assert_(res.shape == (2,3))
  
      def test_hermegrid2d(self):
          x1, x2, x3 = self.x
          y1, y2, y3 = self.y
  
          #test values
          tgt = np.einsum('i,j->ij', y1, y2)
          res = herme.hermegrid2d(x1, x2, self.c2d)
          assert_almost_equal(res, tgt)
  
          #test shape
          z = np.ones((2,3))
          res = herme.hermegrid2d(z, z, self.c2d)
          assert_(res.shape == (2, 3)*2)
  
      def test_hermegrid3d(self):
          x1, x2, x3 = self.x
          y1, y2, y3 = self.y
  
          #test values
          tgt = np.einsum('i,j,k->ijk', y1, y2, y3)
          res = herme.hermegrid3d(x1, x2, x3, self.c3d)
          assert_almost_equal(res, tgt)
  
          #test shape
          z = np.ones((2,3))
          res = herme.hermegrid3d(z, z, z, self.c3d)
          assert_(res.shape == (2, 3)*3)
  
  
  class TestIntegral(TestCase):
  
      def test_hermeint(self) :
          # check exceptions
          assert_raises(ValueError, herme.hermeint, [0], .5)
          assert_raises(ValueError, herme.hermeint, [0], -1)
          assert_raises(ValueError, herme.hermeint, [0], 1, [0,0])
  
          # test integration of zero polynomial
          for i in range(2, 5):
              k = [0]*(i - 2) + [1]
              res = herme.hermeint([0], m=i, k=k)
              assert_almost_equal(res, [0, 1])
  
          # check single integration with integration constant
          for i in range(5) :
              scl = i + 1
              pol = [0]*i + [1]
              tgt = [i] + [0]*i + [1/scl]
              hermepol = herme.poly2herme(pol)
              hermeint = herme.hermeint(hermepol, m=1, k=[i])
              res = herme.herme2poly(hermeint)
              assert_almost_equal(trim(res), trim(tgt))
  
          # check single integration with integration constant and lbnd
          for i in range(5) :
              scl = i + 1
              pol = [0]*i + [1]
              hermepol = herme.poly2herme(pol)
              hermeint = herme.hermeint(hermepol, m=1, k=[i], lbnd=-1)
              assert_almost_equal(herme.hermeval(-1, hermeint), i)
  
          # check single integration with integration constant and scaling
          for i in range(5) :
              scl = i + 1
              pol = [0]*i + [1]
              tgt = [i] + [0]*i + [2/scl]
              hermepol = herme.poly2herme(pol)
              hermeint = herme.hermeint(hermepol, m=1, k=[i], scl=2)
              res = herme.herme2poly(hermeint)
              assert_almost_equal(trim(res), trim(tgt))
  
          # check multiple integrations with default k
          for i in range(5) :
              for j in range(2,5) :
                  pol = [0]*i + [1]
                  tgt = pol[:]
                  for k in range(j) :
                      tgt = herme.hermeint(tgt, m=1)
                  res = herme.hermeint(pol, m=j)
                  assert_almost_equal(trim(res), trim(tgt))
  
          # check multiple integrations with defined k
          for i in range(5) :
              for j in range(2,5) :
                  pol = [0]*i + [1]
                  tgt = pol[:]
                  for k in range(j) :
                      tgt = herme.hermeint(tgt, m=1, k=[k])
                  res = herme.hermeint(pol, m=j, k=list(range(j)))
                  assert_almost_equal(trim(res), trim(tgt))
  
          # check multiple integrations with lbnd
          for i in range(5) :
              for j in range(2,5) :
                  pol = [0]*i + [1]
                  tgt = pol[:]
                  for k in range(j) :
                      tgt = herme.hermeint(tgt, m=1, k=[k], lbnd=-1)
                  res = herme.hermeint(pol, m=j, k=list(range(j)), lbnd=-1)
                  assert_almost_equal(trim(res), trim(tgt))
  
          # check multiple integrations with scaling
          for i in range(5) :
              for j in range(2,5) :
                  pol = [0]*i + [1]
                  tgt = pol[:]
                  for k in range(j) :
                      tgt = herme.hermeint(tgt, m=1, k=[k], scl=2)
                  res = herme.hermeint(pol, m=j, k=list(range(j)), scl=2)
                  assert_almost_equal(trim(res), trim(tgt))
  
      def test_hermeint_axis(self):
          # check that axis keyword works
          c2d = np.random.random((3, 4))
  
          tgt = np.vstack([herme.hermeint(c) for c in c2d.T]).T
          res = herme.hermeint(c2d, axis=0)
          assert_almost_equal(res, tgt)
  
          tgt = np.vstack([herme.hermeint(c) for c in c2d])
          res = herme.hermeint(c2d, axis=1)
          assert_almost_equal(res, tgt)
  
          tgt = np.vstack([herme.hermeint(c, k=3) for c in c2d])
          res = herme.hermeint(c2d, k=3, axis=1)
          assert_almost_equal(res, tgt)
  
  
  class TestDerivative(TestCase) :
  
      def test_hermeder(self) :
          # check exceptions
          assert_raises(ValueError, herme.hermeder, [0], .5)
          assert_raises(ValueError, herme.hermeder, [0], -1)
  
          # check that zeroth deriviative does nothing
          for i in range(5) :
              tgt = [0]*i + [1]
              res = herme.hermeder(tgt, m=0)
              assert_equal(trim(res), trim(tgt))
  
          # check that derivation is the inverse of integration
          for i in range(5) :
              for j in range(2,5) :
                  tgt = [0]*i + [1]
                  res = herme.hermeder(herme.hermeint(tgt, m=j), m=j)
                  assert_almost_equal(trim(res), trim(tgt))
  
          # check derivation with scaling
          for i in range(5) :
              for j in range(2,5) :
                  tgt = [0]*i + [1]
                  res = herme.hermeder(herme.hermeint(tgt, m=j, scl=2), m=j, scl=.5)
                  assert_almost_equal(trim(res), trim(tgt))
  
      def test_hermeder_axis(self):
          # check that axis keyword works
          c2d = np.random.random((3, 4))
  
          tgt = np.vstack([herme.hermeder(c) for c in c2d.T]).T
          res = herme.hermeder(c2d, axis=0)
          assert_almost_equal(res, tgt)
  
          tgt = np.vstack([herme.hermeder(c) for c in c2d])
          res = herme.hermeder(c2d, axis=1)
          assert_almost_equal(res, tgt)
  
  
  class TestVander(TestCase):
      # some random values in [-1, 1)
      x = np.random.random((3, 5))*2 - 1
  
  
      def test_hermevander(self) :
          # check for 1d x
          x = np.arange(3)
          v = herme.hermevander(x, 3)
          assert_(v.shape == (3, 4))
          for i in range(4) :
              coef = [0]*i + [1]
              assert_almost_equal(v[..., i], herme.hermeval(x, coef))
  
          # check for 2d x
          x = np.array([[1, 2], [3, 4], [5, 6]])
          v = herme.hermevander(x, 3)
          assert_(v.shape == (3, 2, 4))
          for i in range(4) :
              coef = [0]*i + [1]
              assert_almost_equal(v[..., i], herme.hermeval(x, coef))
  
      def test_hermevander2d(self) :
          # also tests hermeval2d for non-square coefficient array
          x1, x2, x3 = self.x
          c = np.random.random((2, 3))
          van = herme.hermevander2d(x1, x2, [1, 2])
          tgt = herme.hermeval2d(x1, x2, c)
          res = np.dot(van, c.flat)
          assert_almost_equal(res, tgt)
  
          # check shape
          van = herme.hermevander2d([x1], [x2], [1, 2])
          assert_(van.shape == (1, 5, 6))
  
  
      def test_hermevander3d(self) :
          # also tests hermeval3d for non-square coefficient array
          x1, x2, x3 = self.x
          c = np.random.random((2, 3, 4))
          van = herme.hermevander3d(x1, x2, x3, [1, 2, 3])
          tgt = herme.hermeval3d(x1, x2, x3, c)
          res = np.dot(van, c.flat)
          assert_almost_equal(res, tgt)
  
          # check shape
          van = herme.hermevander3d([x1], [x2], [x3], [1, 2, 3])
          assert_(van.shape == (1, 5, 24))
  
  
  class TestFitting(TestCase):
  
      def test_hermefit(self) :
          def f(x) :
              return x*(x - 1)*(x - 2)
  
          # Test exceptions
          assert_raises(ValueError, herme.hermefit, [1],    [1],     -1)
          assert_raises(TypeError,  herme.hermefit, [[1]],  [1],      0)
          assert_raises(TypeError,  herme.hermefit, [],     [1],      0)
          assert_raises(TypeError,  herme.hermefit, [1],    [[[1]]],  0)
          assert_raises(TypeError,  herme.hermefit, [1, 2], [1],      0)
          assert_raises(TypeError,  herme.hermefit, [1],    [1, 2],   0)
          assert_raises(TypeError,  herme.hermefit, [1],    [1],   0, w=[[1]])
          assert_raises(TypeError,  herme.hermefit, [1],    [1],   0, w=[1,1])
  
          # Test fit
          x = np.linspace(0,2)
          y = f(x)
          #
          coef3 = herme.hermefit(x, y, 3)
          assert_equal(len(coef3), 4)
          assert_almost_equal(herme.hermeval(x, coef3), y)
          #
          coef4 = herme.hermefit(x, y, 4)
          assert_equal(len(coef4), 5)
          assert_almost_equal(herme.hermeval(x, coef4), y)
          #
          coef2d = herme.hermefit(x, np.array([y,y]).T, 3)
          assert_almost_equal(coef2d, np.array([coef3,coef3]).T)
          # test weighting
          w = np.zeros_like(x)
          yw = y.copy()
          w[1::2] = 1
          y[0::2] = 0
          wcoef3 = herme.hermefit(x, yw, 3, w=w)
          assert_almost_equal(wcoef3, coef3)
          #
          wcoef2d = herme.hermefit(x, np.array([yw,yw]).T, 3, w=w)
          assert_almost_equal(wcoef2d, np.array([coef3,coef3]).T)
          # test scaling with complex values x points whose square
          # is zero when summed.
          x = [1, 1j, -1, -1j]
          assert_almost_equal(herme.hermefit(x, x, 1), [0, 1])
  
  class TestGauss(TestCase):
  
      def test_100(self):
          x, w = herme.hermegauss(100)
  
          # test orthogonality. Note that the results need to be normalized,
          # otherwise the huge values that can arise from fast growing
          # functions like Laguerre can be very confusing.
          v = herme.hermevander(x, 99)
          vv = np.dot(v.T * w, v)
          vd = 1/np.sqrt(vv.diagonal())
          vv = vd[:,None] * vv * vd
          assert_almost_equal(vv, np.eye(100))
  
          # check that the integral of 1 is correct
          tgt = np.sqrt(2*np.pi)
          assert_almost_equal(w.sum(), tgt)
  
  
  class TestMisc(TestCase) :
  
      def test_hermefromroots(self) :
          res = herme.hermefromroots([])
          assert_almost_equal(trim(res), [1])
          for i in range(1,5) :
              roots = np.cos(np.linspace(-np.pi, 0, 2*i + 1)[1::2])
              pol = herme.hermefromroots(roots)
              res = herme.hermeval(roots, pol)
              tgt = 0
              assert_(len(pol) == i + 1)
              assert_almost_equal(herme.herme2poly(pol)[-1], 1)
              assert_almost_equal(res, tgt)
  
      def test_hermeroots(self) :
          assert_almost_equal(herme.hermeroots([1]), [])
          assert_almost_equal(herme.hermeroots([1, 1]), [-1])
          for i in range(2,5) :
              tgt = np.linspace(-1, 1, i)
              res = herme.hermeroots(herme.hermefromroots(tgt))
              assert_almost_equal(trim(res), trim(tgt))
  
      def test_hermetrim(self) :
          coef = [2, -1, 1, 0]
  
          # Test exceptions
          assert_raises(ValueError, herme.hermetrim, coef, -1)
  
          # Test results
          assert_equal(herme.hermetrim(coef), coef[:-1])
          assert_equal(herme.hermetrim(coef, 1), coef[:-3])
          assert_equal(herme.hermetrim(coef, 2), [0])
  
      def test_hermeline(self) :
          assert_equal(herme.hermeline(3,4), [3, 4])
  
      def test_herme2poly(self) :
          for i in range(10) :
              assert_almost_equal(herme.herme2poly([0]*i + [1]), Helist[i])
  
      def test_poly2herme(self) :
          for i in range(10) :
              assert_almost_equal(herme.poly2herme(Helist[i]), [0]*i + [1])
  
      def test_weight(self):
          x = np.linspace(-5, 5, 11)
          tgt = np.exp(-.5*x**2)
          res = herme.hermeweight(x)
          assert_almost_equal(res, tgt)
  
  
  if __name__ == "__main__":
      run_module_suite()
  
apis: numpy.sqrt numpy.ones numpy.empty numpy.rollaxis numpy.zeros numpy.iterable numpy.abs numpy.asarray numpy.exp numpy.arange numpy.linalg.eigvals numpy.linalg.lstsq numpy.all numpy.multiply.accumulate numpy.array numpy.finfo numpy.square
apis: numpy.ones numpy.polynomial.hermite_e.hermeline numpy.polynomial.hermite_e.hermevander3d numpy.polynomial.hermite_e.hermeval numpy.polynomial.hermite_e.hermemulx numpy.polynomial.hermite_e.hermetrim numpy.polynomial.hermite_e.hermesub numpy.polynomial.hermite_e.hermeint numpy.polynomial.hermite_e.hermeval3d numpy.polynomial.hermite_e.hermegauss numpy.polynomial.hermite_e.hermeder numpy.linspace numpy.polynomial.hermite_e.hermeadd numpy.sqrt numpy.eye numpy.polynomial.hermite_e.hermegrid2d numpy.polynomial.hermite_e.hermeroots numpy.polynomial.hermite_e.hermevander numpy.zeros numpy.polynomial.hermite_e.hermegrid3d numpy.arange numpy.polynomial.hermite_e.herme2poly numpy.einsum numpy.polynomial.hermite_e.hermeweight numpy.polynomial.hermite_e.hermediv numpy.polynomial.hermite_e.hermefit numpy.polynomial.hermite_e.hermeval2d numpy.polynomial.hermite_e.hermevander2d numpy.zeros_like numpy.polynomial.hermite_e.hermemul numpy.polynomial.hermite_e.poly2herme numpy.exp numpy.polynomial.polynomial.polyval numpy.random.random numpy.array numpy.polynomial.hermite_e.hermefromroots numpy.dot