text: """
  Objects for dealing with Laguerre series.
  
  This module provides a number of objects (mostly functions) useful for
  dealing with Laguerre series, including a `Laguerre` 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
  ---------
  - `lagdomain` -- Laguerre series default domain, [-1,1].
  - `lagzero` -- Laguerre series that evaluates identically to 0.
  - `lagone` -- Laguerre series that evaluates identically to 1.
  - `lagx` -- Laguerre series for the identity map, ``f(x) = x``.
  
  Arithmetic
  ----------
  - `lagmulx` -- multiply a Laguerre series in ``P_i(x)`` by ``x``.
  - `lagadd` -- add two Laguerre series.
  - `lagsub` -- subtract one Laguerre series from another.
  - `lagmul` -- multiply two Laguerre series.
  - `lagdiv` -- divide one Laguerre series by another.
  - `lagval` -- evaluate a Laguerre series at given points.
  - `lagval2d` -- evaluate a 2D Laguerre series at given points.
  - `lagval3d` -- evaluate a 3D Laguerre series at given points.
  - `laggrid2d` -- evaluate a 2D Laguerre series on a Cartesian product.
  - `laggrid3d` -- evaluate a 3D Laguerre series on a Cartesian product.
  
  Calculus
  --------
  - `lagder` -- differentiate a Laguerre series.
  - `lagint` -- integrate a Laguerre series.
  
  Misc Functions
  --------------
  - `lagfromroots` -- create a Laguerre series with specified roots.
  - `lagroots` -- find the roots of a Laguerre series.
  - `lagvander` -- Vandermonde-like matrix for Laguerre polynomials.
  - `lagvander2d` -- Vandermonde-like matrix for 2D power series.
  - `lagvander3d` -- Vandermonde-like matrix for 3D power series.
  - `laggauss` -- Gauss-Laguerre quadrature, points and weights.
  - `lagweight` -- Laguerre weight function.
  - `lagcompanion` -- symmetrized companion matrix in Laguerre form.
  - `lagfit` -- least-squares fit returning a Laguerre series.
  - `lagtrim` -- trim leading coefficients from a Laguerre series.
  - `lagline` -- Laguerre series of given straight line.
  - `lag2poly` -- convert a Laguerre series to a polynomial.
  - `poly2lag` -- convert a polynomial to a Laguerre series.
  
  Classes
  -------
  - `Laguerre` -- A Laguerre series class.
  
  See also
  --------
  `numpy.polynomial`
  
  """
  from __future__ import division, absolute_import, print_function
  
  import warnings
  import numpy as np
  import numpy.linalg as la
  from numpy.core.multiarray import normalize_axis_index
  
  from . import polyutils as pu
  from ._polybase import ABCPolyBase
  
  __all__ = [
      'lagzero', 'lagone', 'lagx', 'lagdomain', 'lagline', 'lagadd',
      'lagsub', 'lagmulx', 'lagmul', 'lagdiv', 'lagpow', 'lagval', 'lagder',
      'lagint', 'lag2poly', 'poly2lag', 'lagfromroots', 'lagvander',
      'lagfit', 'lagtrim', 'lagroots', 'Laguerre', 'lagval2d', 'lagval3d',
      'laggrid2d', 'laggrid3d', 'lagvander2d', 'lagvander3d', 'lagcompanion',
      'laggauss', 'lagweight']
  
  lagtrim = pu.trimcoef
  
  
  def poly2lag(pol):
      """
      poly2lag(pol)
  
      Convert a polynomial to a Laguerre 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 Laguerre 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 Laguerre
          series.
  
      See Also
      --------
      lag2poly
  
      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.laguerre import poly2lag
      >>> poly2lag(np.arange(4))
      array([ 23., -63.,  58., -18.])
  
      """
      [pol] = pu.as_series([pol])
      deg = len(pol) - 1
      res = 0
      for i in range(deg, -1, -1):
          res = lagadd(lagmulx(res), pol[i])
      return res
  
  
  def lag2poly(c):
      """
      Convert a Laguerre series to a polynomial.
  
      Convert an array representing the coefficients of a Laguerre 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 Laguerre 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
      --------
      poly2lag
  
      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.laguerre import lag2poly
      >>> lag2poly([ 23., -63.,  58., -18.])
      array([ 0.,  1.,  2.,  3.])
  
      """
      from .polynomial import polyadd, polysub, polymulx
  
      [c] = pu.as_series([c])
      n = len(c)
      if n == 1:
          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*(i - 1))/i)
              c1 = polyadd(tmp, polysub((2*i - 1)*c1, polymulx(c1))/i)
          return polyadd(c0, polysub(c1, 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.
  #
  
  # Laguerre
  lagdomain = np.array([0, 1])
  
  # Laguerre coefficients representing zero.
  lagzero = np.array([0])
  
  # Laguerre coefficients representing one.
  lagone = np.array([1])
  
  # Laguerre coefficients representing the identity x.
  lagx = np.array([1, -1])
  
  
  def lagline(off, scl):
      """
      Laguerre 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 Laguerre series for
          ``off + scl*x``.
  
      See Also
      --------
      polyline, chebline
  
      Examples
      --------
      >>> from numpy.polynomial.laguerre import lagline, lagval
      >>> lagval(0,lagline(3, 2))
      3.0
      >>> lagval(1,lagline(3, 2))
      5.0
  
      """
      if scl != 0:
          return np.array([off + scl, -scl])
      else:
          return np.array([off])
  
  
  def lagfromroots(roots):
      """
      Generate a Laguerre 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 Laguerre 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 * L_1(x) + ... +  c_n * L_n(x)
  
      The coefficient of the last term is not generally 1 for monic
      polynomials in Laguerre 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, chebfromroots, hermfromroots,
      hermefromroots.
  
      Examples
      --------
      >>> from numpy.polynomial.laguerre import lagfromroots, lagval
      >>> coef = lagfromroots((-1, 0, 1))
      >>> lagval((-1, 0, 1), coef)
      array([ 0.,  0.,  0.])
      >>> coef = lagfromroots((-1j, 1j))
      >>> lagval((-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 = [lagline(-r, 1) for r in roots]
          n = len(p)
          while n > 1:
              m, r = divmod(n, 2)
              tmp = [lagmul(p[i], p[i+m]) for i in range(m)]
              if r:
                  tmp[0] = lagmul(tmp[0], p[-1])
              p = tmp
              n = m
          return p[0]
  
  
  def lagadd(c1, c2):
      """
      Add one Laguerre series to another.
  
      Returns the sum of two Laguerre 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 Laguerre series coefficients ordered from low to
          high.
  
      Returns
      -------
      out : ndarray
          Array representing the Laguerre series of their sum.
  
      See Also
      --------
      lagsub, lagmul, lagdiv, lagpow
  
      Notes
      -----
      Unlike multiplication, division, etc., the sum of two Laguerre series
      is a Laguerre 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.laguerre import lagadd
      >>> lagadd([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 lagsub(c1, c2):
      """
      Subtract one Laguerre series from another.
  
      Returns the difference of two Laguerre 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 Laguerre series coefficients ordered from low to
          high.
  
      Returns
      -------
      out : ndarray
          Of Laguerre series coefficients representing their difference.
  
      See Also
      --------
      lagadd, lagmul, lagdiv, lagpow
  
      Notes
      -----
      Unlike multiplication, division, etc., the difference of two Laguerre
      series is a Laguerre 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.laguerre import lagsub
      >>> lagsub([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 lagmulx(c):
      """Multiply a Laguerre series by x.
  
      Multiply the Laguerre series `c` by x, where x is the independent
      variable.
  
  
      Parameters
      ----------
      c : array_like
          1-D array of Laguerre 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 Laguerre
      polynomials in the form
  
      .. math::
  
      xP_i(x) = (-(i + 1)*P_{i + 1}(x) + (2i + 1)P_{i}(x) - iP_{i - 1}(x))
  
      Examples
      --------
      >>> from numpy.polynomial.laguerre import lagmulx
      >>> lagmulx([1, 2, 3])
      array([ -1.,  -1.,  11.,  -9.])
  
      """
      # 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]
      prd[1] = -c[0]
      for i in range(1, len(c)):
          prd[i + 1] = -c[i]*(i + 1)
          prd[i] += c[i]*(2*i + 1)
          prd[i - 1] -= c[i]*i
      return prd
  
  
  def lagmul(c1, c2):
      """
      Multiply one Laguerre series by another.
  
      Returns the product of two Laguerre 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 Laguerre series coefficients ordered from low to
          high.
  
      Returns
      -------
      out : ndarray
          Of Laguerre series coefficients representing their product.
  
      See Also
      --------
      lagadd, lagsub, lagdiv, lagpow
  
      Notes
      -----
      In general, the (polynomial) product of two C-series results in terms
      that are not in the Laguerre polynomial basis set.  Thus, to express
      the product as a Laguerre 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.laguerre import lagmul
      >>> lagmul([1, 2, 3], [0, 1, 2])
      array([  8., -13.,  38., -51.,  36.])
  
      """
      # 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 = lagsub(c[-i]*xs, (c1*(nd - 1))/nd)
              c1 = lagadd(tmp, lagsub((2*nd - 1)*c1, lagmulx(c1))/nd)
      return lagadd(c0, lagsub(c1, lagmulx(c1)))
  
  
  def lagdiv(c1, c2):
      """
      Divide one Laguerre series by another.
  
      Returns the quotient-with-remainder of two Laguerre 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 Laguerre series coefficients ordered from low to
          high.
  
      Returns
      -------
      [quo, rem] : ndarrays
          Of Laguerre series coefficients representing the quotient and
          remainder.
  
      See Also
      --------
      lagadd, lagsub, lagmul, lagpow
  
      Notes
      -----
      In general, the (polynomial) division of one Laguerre series by another
      results in quotient and remainder terms that are not in the Laguerre
      polynomial basis set.  Thus, to express these results as a Laguerre
      series, it is necessary to "reproject" the results onto the Laguerre
      basis set, which may produce "unintuitive" (but correct) results; see
      Examples section below.
  
      Examples
      --------
      >>> from numpy.polynomial.laguerre import lagdiv
      >>> lagdiv([  8., -13.,  38., -51.,  36.], [0, 1, 2])
      (array([ 1.,  2.,  3.]), array([ 0.]))
      >>> lagdiv([  9., -12.,  38., -51.,  36.], [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 = lagmul([0]*i + [1], c2)
              q = rem[-1]/p[-1]
              rem = rem[:-1] - q*p[:-1]
              quo[i] = q
          return quo, pu.trimseq(rem)
  
  
  def lagpow(c, pow, maxpower=16):
      """Raise a Laguerre series to a power.
  
      Returns the Laguerre 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 Laguerre 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
          Laguerre series of power.
  
      See Also
      --------
      lagadd, lagsub, lagmul, lagdiv
  
      Examples
      --------
      >>> from numpy.polynomial.laguerre import lagpow
      >>> lagpow([1, 2, 3], 2)
      array([ 14., -16.,  56., -72.,  54.])
  
      """
      # 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 = lagmul(prd, c)
          return prd
  
  
  def lagder(c, m=1, scl=1, axis=0):
      """
      Differentiate a Laguerre series.
  
      Returns the Laguerre 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*L_0 + 2*L_1 + 3*L_2``
      while [[1,2],[1,2]] represents ``1*L_0(x)*L_0(y) + 1*L_1(x)*L_0(y) +
      2*L_0(x)*L_1(y) + 2*L_1(x)*L_1(y)`` if axis=0 is ``x`` and axis=1 is
      ``y``.
  
      Parameters
      ----------
      c : array_like
          Array of Laguerre 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
          Laguerre series of the derivative.
  
      See Also
      --------
      lagint
  
      Notes
      -----
      In general, the result of differentiating a Laguerre 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.laguerre import lagder
      >>> lagder([ 1.,  1.,  1., -3.])
      array([ 1.,  2.,  3.])
      >>> lagder([ 1.,  0.,  0., -4.,  3.], 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")
      iaxis = normalize_axis_index(iaxis, c.ndim)
  
      if cnt == 0:
          return c
  
      c = np.moveaxis(c, iaxis, 0)
      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, 1, -1):
                  der[j - 1] = -c[j]
                  c[j - 1] += c[j]
              der[0] = -c[1]
              c = der
      c = np.moveaxis(c, 0, iaxis)
      return c
  
  
  def lagint(c, m=1, k=[], lbnd=0, scl=1, axis=0):
      """
      Integrate a Laguerre series.
  
      Returns the Laguerre 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 ``L_0 + 2*L_1 + 3*L_2`` while [[1,2],[1,2]]
      represents ``1*L_0(x)*L_0(y) + 1*L_1(x)*L_0(y) + 2*L_0(x)*L_1(y) +
      2*L_1(x)*L_1(y)`` if axis=0 is ``x`` and axis=1 is ``y``.
  
  
      Parameters
      ----------
      c : array_like
          Array of Laguerre 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
          Laguerre series coefficients of the integral.
  
      Raises
      ------
      ValueError
          If ``m < 0``, ``len(k) > m``, ``np.ndim(lbnd) != 0``, or
          ``np.ndim(scl) != 0``.
  
      See Also
      --------
      lagder
  
      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.laguerre import lagint
      >>> lagint([1,2,3])
      array([ 1.,  1.,  1., -3.])
      >>> lagint([1,2,3], m=2)
      array([ 1.,  0.,  0., -4.,  3.])
      >>> lagint([1,2,3], k=1)
      array([ 2.,  1.,  1., -3.])
      >>> lagint([1,2,3], lbnd=-1)
      array([ 11.5,   1. ,   1. ,  -3. ])
      >>> lagint([1,2], m=2, k=[1,2], lbnd=-1)
      array([ 11.16666667,  -5.        ,  -3.        ,   2.        ])
  
      """
      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 np.ndim(lbnd) != 0:
          raise ValueError("lbnd must be a scalar.")
      if np.ndim(scl) != 0:
          raise ValueError("scl must be a scalar.")
      if iaxis != axis:
          raise ValueError("The axis must be integer")
      iaxis = normalize_axis_index(iaxis, c.ndim)
  
      if cnt == 0:
          return c
  
      c = np.moveaxis(c, iaxis, 0)
      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]
              tmp[1] = -c[0]
              for j in range(1, n):
                  tmp[j] += c[j]
                  tmp[j + 1] = -c[j]
              tmp[0] += k[i] - lagval(lbnd, tmp)
              c = tmp
      c = np.moveaxis(c, 0, iaxis)
      return c
  
  
  def lagval(x, c, tensor=True):
      """
      Evaluate a Laguerre series at points x.
  
      If `c` is of length `n + 1`, this function returns the value:
  
      .. math:: p(x) = c_0 * L_0(x) + c_1 * L_1(x) + ... + c_n * L_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
      --------
      lagval2d, laggrid2d, lagval3d, laggrid3d
  
      Notes
      -----
      The evaluation uses Clenshaw recursion, aka synthetic division.
  
      Examples
      --------
      >>> from numpy.polynomial.laguerre import lagval
      >>> coef = [1,2,3]
      >>> lagval(1, coef)
      -0.5
      >>> lagval([[1,2],[3,4]], coef)
      array([[-0.5, -4. ],
             [-4.5, -2. ]])
  
      """
      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)
  
      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*(nd - 1))/nd
              c1 = tmp + (c1*((2*nd - 1) - x))/nd
      return c0 + c1*(1 - x)
  
  
  def lagval2d(x, y, c):
      """
      Evaluate a 2-D Laguerre series at points (x, y).
  
      This function returns the values:
  
      .. math:: p(x,y) = \\sum_{i,j} c_{i,j} * L_i(x) * L_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
      --------
      lagval, laggrid2d, lagval3d, laggrid3d
  
      Notes
      -----
  
      .. versionadded:: 1.7.0
  
      """
      try:
          x, y = np.array((x, y), copy=0)
      except Exception:
          raise ValueError('x, y are incompatible')
  
      c = lagval(x, c)
      c = lagval(y, c, tensor=False)
      return c
  
  
  def laggrid2d(x, y, c):
      """
      Evaluate a 2-D Laguerre series on the Cartesian product of x and y.
  
      This function returns the values:
  
      .. math:: p(a,b) = \\sum_{i,j} c_{i,j} * L_i(a) * L_j(b)
  
      where the points `(a, b)` consist of all pairs formed by taking
      `a` from `x` and `b` from `y`. The resulting points form a grid with
      `x` in the first dimension and `y` in the second.
  
      The parameters `x` and `y` are converted to arrays only if they are
      tuples or a lists, otherwise they are treated as a scalars. In either
      case, either `x` and `y` or their elements must support multiplication
      and addition both with themselves and with the elements of `c`.
  
      If `c` has fewer than two dimensions, ones are implicitly appended to
      its shape to make it 2-D. The shape of the result will be c.shape[2:] +
      x.shape + y.shape.
  
      Parameters
      ----------
      x, y : array_like, compatible objects
          The two dimensional series is evaluated at the points in the
          Cartesian product of `x` and `y`.  If `x` or `y` is a list or
          tuple, it is first converted to an ndarray, otherwise it is left
          unchanged and, if it isn't an ndarray, it is treated as a scalar.
      c : array_like
          Array of coefficients ordered so that the coefficient of the term of
          multi-degree i,j is contained in `c[i,j]`. If `c` has dimension
          greater than two the remaining indices enumerate multiple sets of
          coefficients.
  
      Returns
      -------
      values : ndarray, compatible object
          The values of the two dimensional Chebyshev series at points in the
          Cartesian product of `x` and `y`.
  
      See Also
      --------
      lagval, lagval2d, lagval3d, laggrid3d
  
      Notes
      -----
  
      .. versionadded:: 1.7.0
  
      """
      c = lagval(x, c)
      c = lagval(y, c)
      return c
  
  
  def lagval3d(x, y, z, c):
      """
      Evaluate a 3-D Laguerre series at points (x, y, z).
  
      This function returns the values:
  
      .. math:: p(x,y,z) = \\sum_{i,j,k} c_{i,j,k} * L_i(x) * L_j(y) * L_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 multidimension polynomial on points formed with
          triples of corresponding values from `x`, `y`, and `z`.
  
      See Also
      --------
      lagval, lagval2d, laggrid2d, laggrid3d
  
      Notes
      -----
  
      .. versionadded:: 1.7.0
  
      """
      try:
          x, y, z = np.array((x, y, z), copy=0)
      except Exception:
          raise ValueError('x, y, z are incompatible')
  
      c = lagval(x, c)
      c = lagval(y, c, tensor=False)
      c = lagval(z, c, tensor=False)
      return c
  
  
  def laggrid3d(x, y, z, c):
      """
      Evaluate a 3-D Laguerre 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} * L_i(a) * L_j(b) * L_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
      --------
      lagval, lagval2d, laggrid2d, lagval3d
  
      Notes
      -----
  
      .. versionadded:: 1.7.0
  
      """
      c = lagval(x, c)
      c = lagval(y, c)
      c = lagval(z, c)
      return c
  
  
  def lagvander(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] = L_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 Laguerre polynomial.
  
      If `c` is a 1-D array of coefficients of length `n + 1` and `V` is the
      array ``V = lagvander(x, n)``, then ``np.dot(V, c)`` and
      ``lagval(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 Laguerre 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 Laguerre polynomial.  The dtype will be the same as
          the converted `x`.
  
      Examples
      --------
      >>> from numpy.polynomial.laguerre import lagvander
      >>> x = np.array([0, 1, 2])
      >>> lagvander(x, 3)
      array([[ 1.        ,  1.        ,  1.        ,  1.        ],
             [ 1.        ,  0.        , -0.5       , -0.66666667],
             [ 1.        , -1.        , -1.        , -0.33333333]])
  
      """
      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:
          v[1] = 1 - x
          for i in range(2, ideg + 1):
              v[i] = (v[i-1]*(2*i - 1 - x) - v[i-2]*(i - 1))/i
      return np.moveaxis(v, 0, -1)
  
  
  def lagvander2d(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] + 1)*i + j] = L_i(x) * L_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 Laguerre polynomials.
  
      If ``V = lagvander2d(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 ``lagval2d(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 Laguerre
      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
      --------
      lagvander, lagvander3d. lagval2d, lagval3d
  
      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 = lagvander(x, degx)
      vy = lagvander(y, degy)
      v = vx[..., None]*vy[..., None,:]
      return v.reshape(v.shape[:-2] + (-1,))
  
  
  def lagvander3d(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] = L_i(x)*L_j(y)*L_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 Laguerre polynomials.
  
      If ``V = lagvander3d(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 ``lagval3d(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 Laguerre
      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
      --------
      lagvander, lagvander3d. lagval2d, lagval3d
  
      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 = lagvander(x, degx)
      vy = lagvander(y, degy)
      vz = lagvander(z, degz)
      v = vx[..., None, None]*vy[..., None,:, None]*vz[..., None, None,:]
      return v.reshape(v.shape[:-3] + (-1,))
  
  
  def lagfit(x, y, deg, rcond=None, full=False, w=None):
      """
      Least squares fit of Laguerre series to data.
  
      Return the coefficients of a Laguerre 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 * L_1(x) + ... + c_n * L_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 or 1-D array_like
          Degree(s) of the fitting polynomials. If `deg` is a single integer
          all terms up to and including the `deg`'th term are included in the
          fit. For NumPy versions >= 1.11.0 a list of integers specifying the
          degrees of the terms to include may be used instead.
      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)
          Laguerre 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] : list
          These values are only returned if `full` = True
  
          resid -- sum of squared residuals of the least squares fit
          rank -- the numerical rank of the scaled Vandermonde matrix
          sv -- singular values of the scaled Vandermonde matrix
          rcond -- 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, hermefit
      lagval : Evaluates a Laguerre series.
      lagvander : pseudo Vandermonde matrix of Laguerre series.
      lagweight : Laguerre 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 Laguerre 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 as the (typically) overdetermined matrix equation
  
      .. math:: V(x) * c = w * y,
  
      where `V` is the weighted pseudo Vandermonde matrix of `x`, `c` are the
      coefficients to be solved for, `w` are the weights, and `y` are the
      observed values.  This equation is then solved using the singular value
      decomposition of `V`.
  
      If some of the singular values of `V` are so small that they are
      neglected, then a `RankWarning` will be issued. This means that the
      coefficient values may be poorly determined. Using a lower order fit
      will usually get rid of the warning.  The `rcond` parameter can also be
      set to a value smaller than its default, but the resulting fit may be
      spurious and have large contributions from roundoff error.
  
      Fits using Laguerre series are probably most useful when the data can
      be approximated by ``sqrt(w(x)) * p(x)``, where `w(x)` is the Laguerre
      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 `lagweight`.
  
      References
      ----------
      .. [1] Wikipedia, "Curve fitting",
             http://en.wikipedia.org/wiki/Curve_fitting
  
      Examples
      --------
      >>> from numpy.polynomial.laguerre import lagfit, lagval
      >>> x = np.linspace(0, 10)
      >>> err = np.random.randn(len(x))/10
      >>> y = lagval(x, [1, 2, 3]) + err
      >>> lagfit(x, y, 2)
      array([ 0.96971004,  2.00193749,  3.00288744])
  
      """
      x = np.asarray(x) + 0.0
      y = np.asarray(y) + 0.0
      deg = np.asarray(deg)
  
      # check arguments.
      if deg.ndim > 1 or deg.dtype.kind not in 'iu' or deg.size == 0:
          raise TypeError("deg must be an int or non-empty 1-D array of int")
      if deg.min() < 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")
  
      if deg.ndim == 0:
          lmax = deg
          order = lmax + 1
          van = lagvander(x, lmax)
      else:
          deg = np.sort(deg)
          lmax = deg[-1]
          order = len(deg)
          van = lagvander(x, lmax)[:, deg]
  
      # set up the least squares matrices in transposed form
      lhs = van.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
  
      # Expand c to include non-fitted coefficients which are set to zero
      if deg.ndim > 0:
          if c.ndim == 2:
              cc = np.zeros((lmax+1, c.shape[1]), dtype=c.dtype)
          else:
              cc = np.zeros(lmax+1, dtype=c.dtype)
          cc[deg] = c
          c = cc
  
      # warn on rank reduction
      if rank != order and not full:
          msg = "The fit may be poorly conditioned"
          warnings.warn(msg, pu.RankWarning, stacklevel=2)
  
      if full:
          return c, [resids, rank, s, rcond]
      else:
          return c
  
  
  def lagcompanion(c):
      """
      Return the companion matrix of c.
  
      The usual companion matrix of the Laguerre polynomials is already
      symmetric when `c` is a basis Laguerre polynomial, so no scaling is
      applied.
  
      Parameters
      ----------
      c : array_like
          1-D array of Laguerre series coefficients ordered from low to high
          degree.
  
      Returns
      -------
      mat : ndarray
          Companion matrix of dimensions (deg, deg).
  
      Notes
      -----
  
      .. versionadded:: 1.7.0
  
      """
      # c is a trimmed copy
      [c] = pu.as_series([c])
      if len(c) < 2:
          raise ValueError('Series must have maximum degree of at least 1.')
      if len(c) == 2:
          return np.array([[1 + c[0]/c[1]]])
  
      n = len(c) - 1
      mat = np.zeros((n, n), dtype=c.dtype)
      top = mat.reshape(-1)[1::n+1]
      mid = mat.reshape(-1)[0::n+1]
      bot = mat.reshape(-1)[n::n+1]
      top[...] = -np.arange(1, n)
      mid[...] = 2.*np.arange(n) + 1.
      bot[...] = top
      mat[:, -1] += (c[:-1]/c[-1])*n
      return mat
  
  
  def lagroots(c):
      """
      Compute the roots of a Laguerre series.
  
      Return the roots (a.k.a. "zeros") of the polynomial
  
      .. math:: p(x) = \\sum_i c[i] * L_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, chebroots, hermroots, hermeroots
  
      Notes
      -----
      The root estimates are obtained as the eigenvalues of the companion
      matrix, Roots far from the origin of the complex plane may have large
      errors due to the numerical instability of the series for such
      values. Roots with multiplicity greater than 1 will also show larger
      errors as the value of the series near such points is relatively
      insensitive to errors in the roots. Isolated roots near the origin can
      be improved by a few iterations of Newton's method.
  
      The Laguerre series basis polynomials aren't powers of `x` so the
      results of this function may seem unintuitive.
  
      Examples
      --------
      >>> from numpy.polynomial.laguerre import lagroots, lagfromroots
      >>> coef = lagfromroots([0, 1, 2])
      >>> coef
      array([  2.,  -8.,  12.,  -6.])
      >>> lagroots(coef)
      array([ -4.44089210e-16,   1.00000000e+00,   2.00000000e+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([1 + c[0]/c[1]])
  
      m = lagcompanion(c)
      r = la.eigvals(m)
      r.sort()
      return r
  
  
  def laggauss(deg):
      """
      Gauss-Laguerre quadrature.
  
      Computes the sample points and weights for Gauss-Laguerre quadrature.
      These sample points and weights will correctly integrate polynomials of
      degree :math:`2*deg - 1` or less over the interval :math:`[0, \\inf]`
      with the weight function :math:`f(x) = \\exp(-x)`.
  
      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 / (L'_n(x_k) * L_{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:`L_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 = lagcompanion(c)
      x = la.eigvalsh(m)
  
      # improve roots by one application of Newton
      dy = lagval(x, c)
      df = lagval(x, lagder(c))
      x -= dy/df
  
      # compute the weights. We scale the factor to avoid possible numerical
      # overflow.
      fm = lagval(x, c[1:])
      fm /= np.abs(fm).max()
      df /= np.abs(df).max()
      w = 1/(fm * df)
  
      # scale w to get the right value, 1 in this case
      w /= w.sum()
  
      return x, w
  
  
  def lagweight(x):
      """Weight function of the Laguerre polynomials.
  
      The weight function is :math:`exp(-x)` and the interval of integration
      is :math:`[0, \\inf]`. The Laguerre 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)
      return w
  
  #
  # Laguerre series class
  #
  
  class Laguerre(ABCPolyBase):
      """A Laguerre series class.
  
      The Laguerre class provides the standard Python numerical methods
      '+', '-', '*', '//', '%', 'divmod', '**', and '()' as well as the
      attributes and methods listed in the `ABCPolyBase` documentation.
  
      Parameters
      ----------
      coef : array_like
          Laguerre coefficients in order of increasing degree, i.e,
          ``(1, 2, 3)`` gives ``1*L_0(x) + 2*L_1(X) + 3*L_2(x)``.
      domain : (2,) array_like, optional
          Domain to use. The interval ``[domain[0], domain[1]]`` is mapped
          to the interval ``[window[0], window[1]]`` by shifting and scaling.
          The default value is [0, 1].
      window : (2,) array_like, optional
          Window, see `domain` for its use. The default value is [0, 1].
  
          .. versionadded:: 1.6.0
  
      """
      # Virtual Functions
      _add = staticmethod(lagadd)
      _sub = staticmethod(lagsub)
      _mul = staticmethod(lagmul)
      _div = staticmethod(lagdiv)
      _pow = staticmethod(lagpow)
      _val = staticmethod(lagval)
      _int = staticmethod(lagint)
      _der = staticmethod(lagder)
      _fit = staticmethod(lagfit)
      _line = staticmethod(lagline)
      _roots = staticmethod(lagroots)
      _fromroots = staticmethod(lagfromroots)
  
      # Virtual properties
      nickname = 'lag'
      domain = np.array(lagdomain)
      window = np.array(lagdomain)
  