text: """
  Objects for dealing with polynomials.
  
  This module provides a number of objects (mostly functions) useful for
  dealing with polynomials, including a `Polynomial` class that
  encapsulates the usual arithmetic operations.  (General information
  on how this module represents and works with polynomial objects is in
  the docstring for its "parent" sub-package, `numpy.polynomial`).
  
  Constants
  ---------
  - `polydomain` -- Polynomial default domain, [-1,1].
  - `polyzero` -- (Coefficients of the) "zero polynomial."
  - `polyone` -- (Coefficients of the) constant polynomial 1.
  - `polyx` -- (Coefficients of the) identity map polynomial, ``f(x) = x``.
  
  Arithmetic
  ----------
  - `polyadd` -- add two polynomials.
  - `polysub` -- subtract one polynomial from another.
  - `polymul` -- multiply two polynomials.
  - `polydiv` -- divide one polynomial by another.
  - `polypow` -- raise a polynomial to an positive integer power
  - `polyval` -- evaluate a polynomial at given points.
  - `polyval2d` -- evaluate a 2D polynomial at given points.
  - `polyval3d` -- evaluate a 3D polynomial at given points.
  - `polygrid2d` -- evaluate a 2D polynomial on a Cartesian product.
  - `polygrid3d` -- evaluate a 3D polynomial on a Cartesian product.
  
  Calculus
  --------
  - `polyder` -- differentiate a polynomial.
  - `polyint` -- integrate a polynomial.
  
  Misc Functions
  --------------
  - `polyfromroots` -- create a polynomial with specified roots.
  - `polyroots` -- find the roots of a polynomial.
  - `polyvalfromroots` -- evalute a polynomial at given points from roots.
  - `polyvander` -- Vandermonde-like matrix for powers.
  - `polyvander2d` -- Vandermonde-like matrix for 2D power series.
  - `polyvander3d` -- Vandermonde-like matrix for 3D power series.
  - `polycompanion` -- companion matrix in power series form.
  - `polyfit` -- least-squares fit returning a polynomial.
  - `polytrim` -- trim leading coefficients from a polynomial.
  - `polyline` -- polynomial representing given straight line.
  
  Classes
  -------
  - `Polynomial` -- polynomial class.
  
  See Also
  --------
  `numpy.polynomial`
  
  """
  from __future__ import division, absolute_import, print_function
  
  __all__ = [
      'polyzero', 'polyone', 'polyx', 'polydomain', 'polyline', 'polyadd',
      'polysub', 'polymulx', 'polymul', 'polydiv', 'polypow', 'polyval',
      'polyvalfromroots', 'polyder', 'polyint', 'polyfromroots', 'polyvander',
      'polyfit', 'polytrim', 'polyroots', 'Polynomial', 'polyval2d', 'polyval3d',
      'polygrid2d', 'polygrid3d', 'polyvander2d', 'polyvander3d']
  
  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
  
  polytrim = pu.trimcoef
  
  #
  # These are constant arrays are of integer type so as to be compatible
  # with the widest range of other types, such as Decimal.
  #
  
  # Polynomial default domain.
  polydomain = np.array([-1, 1])
  
  # Polynomial coefficients representing zero.
  polyzero = np.array([0])
  
  # Polynomial coefficients representing one.
  polyone = np.array([1])
  
  # Polynomial coefficients representing the identity x.
  polyx = np.array([0, 1])
  
  #
  # Polynomial series functions
  #
  
  
  def polyline(off, scl):
      """
      Returns an array representing a linear polynomial.
  
      Parameters
      ----------
      off, scl : scalars
          The "y-intercept" and "slope" of the line, respectively.
  
      Returns
      -------
      y : ndarray
          This module's representation of the linear polynomial ``off +
          scl*x``.
  
      See Also
      --------
      chebline
  
      Examples
      --------
      >>> from numpy.polynomial import polynomial as P
      >>> P.polyline(1,-1)
      array([ 1, -1])
      >>> P.polyval(1, P.polyline(1,-1)) # should be 0
      0.0
  
      """
      if scl != 0:
          return np.array([off, scl])
      else:
          return np.array([off])
  
  
  def polyfromroots(roots):
      """
      Generate a monic polynomial with given roots.
  
      Return the coefficients of the polynomial
  
      .. math:: p(x) = (x - r_0) * (x - r_1) * ... * (x - r_n),
  
      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 * x + ... +  x^n
  
      The coefficient of the last term is 1 for monic polynomials in this
      form.
  
      Parameters
      ----------
      roots : array_like
          Sequence containing the roots.
  
      Returns
      -------
      out : ndarray
          1-D array of the polynomial's coefficients If all the roots are
          real, then `out` is also real, otherwise it is complex.  (see
          Examples below).
  
      See Also
      --------
      chebfromroots, legfromroots, lagfromroots, hermfromroots
      hermefromroots
  
      Notes
      -----
      The coefficients are determined by multiplying together linear factors
      of the form `(x - r_i)`, i.e.
  
      .. math:: p(x) = (x - r_0) (x - r_1) ... (x - r_n)
  
      where ``n == len(roots) - 1``; note that this implies that `1` is always
      returned for :math:`a_n`.
  
      Examples
      --------
      >>> from numpy.polynomial import polynomial as P
      >>> P.polyfromroots((-1,0,1)) # x(x - 1)(x + 1) = x^3 - x
      array([ 0., -1.,  0.,  1.])
      >>> j = complex(0,1)
      >>> P.polyfromroots((-j,j)) # complex returned, though values are real
      array([ 1.+0.j,  0.+0.j,  1.+0.j])
  
      """
      if len(roots) == 0:
          return np.ones(1)
      else:
          [roots] = pu.as_series([roots], trim=False)
          roots.sort()
          p = [polyline(-r, 1) for r in roots]
          n = len(p)
          while n > 1:
              m, r = divmod(n, 2)
              tmp = [polymul(p[i], p[i+m]) for i in range(m)]
              if r:
                  tmp[0] = polymul(tmp[0], p[-1])
              p = tmp
              n = m
          return p[0]
  
  
  def polyadd(c1, c2):
      """
      Add one polynomial to another.
  
      Returns the sum of two polynomials `c1` + `c2`.  The arguments are
      sequences of coefficients from lowest order term to highest, i.e.,
      [1,2,3] represents the polynomial ``1 + 2*x + 3*x**2``.
  
      Parameters
      ----------
      c1, c2 : array_like
          1-D arrays of polynomial coefficients ordered from low to high.
  
      Returns
      -------
      out : ndarray
          The coefficient array representing their sum.
  
      See Also
      --------
      polysub, polymul, polydiv, polypow
  
      Examples
      --------
      >>> from numpy.polynomial import polynomial as P
      >>> c1 = (1,2,3)
      >>> c2 = (3,2,1)
      >>> sum = P.polyadd(c1,c2); sum
      array([ 4.,  4.,  4.])
      >>> P.polyval(2, sum) # 4 + 4(2) + 4(2**2)
      28.0
  
      """
      # 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 polysub(c1, c2):
      """
      Subtract one polynomial from another.
  
      Returns the difference of two polynomials `c1` - `c2`.  The arguments
      are sequences of coefficients from lowest order term to highest, i.e.,
      [1,2,3] represents the polynomial ``1 + 2*x + 3*x**2``.
  
      Parameters
      ----------
      c1, c2 : array_like
          1-D arrays of polynomial coefficients ordered from low to
          high.
  
      Returns
      -------
      out : ndarray
          Of coefficients representing their difference.
  
      See Also
      --------
      polyadd, polymul, polydiv, polypow
  
      Examples
      --------
      >>> from numpy.polynomial import polynomial as P
      >>> c1 = (1,2,3)
      >>> c2 = (3,2,1)
      >>> P.polysub(c1,c2)
      array([-2.,  0.,  2.])
      >>> P.polysub(c2,c1) # -P.polysub(c1,c2)
      array([ 2.,  0., -2.])
  
      """
      # c1, c2 are trimmed copies
      [c1, c2] = pu.as_series([c1, c2])
      if len(c1) > len(c2):
          c1[:c2.size] -= c2
          ret = c1
      else:
          c2 = -c2
          c2[:c1.size] += c1
          ret = c2
      return pu.trimseq(ret)
  
  
  def polymulx(c):
      """Multiply a polynomial by x.
  
      Multiply the polynomial `c` by x, where x is the independent
      variable.
  
  
      Parameters
      ----------
      c : array_like
          1-D array of polynomial coefficients ordered from low to
          high.
  
      Returns
      -------
      out : ndarray
          Array representing the result of the multiplication.
  
      Notes
      -----
  
      .. versionadded:: 1.5.0
  
      """
      # c is a trimmed copy
      [c] = pu.as_series([c])
      # The zero series needs special treatment
      if len(c) == 1 and c[0] == 0:
          return c
  
      prd = np.empty(len(c) + 1, dtype=c.dtype)
      prd[0] = c[0]*0
      prd[1:] = c
      return prd
  
  
  def polymul(c1, c2):
      """
      Multiply one polynomial by another.
  
      Returns the product of two polynomials `c1` * `c2`.  The arguments are
      sequences of coefficients, from lowest order term to highest, e.g.,
      [1,2,3] represents the polynomial ``1 + 2*x + 3*x**2.``
  
      Parameters
      ----------
      c1, c2 : array_like
          1-D arrays of coefficients representing a polynomial, relative to the
          "standard" basis, and ordered from lowest order term to highest.
  
      Returns
      -------
      out : ndarray
          Of the coefficients of their product.
  
      See Also
      --------
      polyadd, polysub, polydiv, polypow
  
      Examples
      --------
      >>> from numpy.polynomial import polynomial as P
      >>> c1 = (1,2,3)
      >>> c2 = (3,2,1)
      >>> P.polymul(c1,c2)
      array([  3.,   8.,  14.,   8.,   3.])
  
      """
      # c1, c2 are trimmed copies
      [c1, c2] = pu.as_series([c1, c2])
      ret = np.convolve(c1, c2)
      return pu.trimseq(ret)
  
  
  def polydiv(c1, c2):
      """
      Divide one polynomial by another.
  
      Returns the quotient-with-remainder of two polynomials `c1` / `c2`.
      The arguments are sequences of coefficients, from lowest order term
      to highest, e.g., [1,2,3] represents ``1 + 2*x + 3*x**2``.
  
      Parameters
      ----------
      c1, c2 : array_like
          1-D arrays of polynomial coefficients ordered from low to high.
  
      Returns
      -------
      [quo, rem] : ndarrays
          Of coefficient series representing the quotient and remainder.
  
      See Also
      --------
      polyadd, polysub, polymul, polypow
  
      Examples
      --------
      >>> from numpy.polynomial import polynomial as P
      >>> c1 = (1,2,3)
      >>> c2 = (3,2,1)
      >>> P.polydiv(c1,c2)
      (array([ 3.]), array([-8., -4.]))
      >>> P.polydiv(c2,c1)
      (array([ 0.33333333]), array([ 2.66666667,  1.33333333]))
  
      """
      # c1, c2 are trimmed copies
      [c1, c2] = pu.as_series([c1, c2])
      if c2[-1] == 0:
          raise ZeroDivisionError()
  
      len1 = len(c1)
      len2 = len(c2)
      if len2 == 1:
          return c1/c2[-1], c1[:1]*0
      elif len1 < len2:
          return c1[:1]*0, c1
      else:
          dlen = len1 - len2
          scl = c2[-1]
          c2 = c2[:-1]/scl
          i = dlen
          j = len1 - 1
          while i >= 0:
              c1[i:j] -= c2*c1[j]
              i -= 1
              j -= 1
          return c1[j+1:]/scl, pu.trimseq(c1[:j+1])
  
  
  def polypow(c, pow, maxpower=None):
      """Raise a polynomial to a power.
  
      Returns the polynomial `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  ``1 + 2*x + 3*x**2.``
  
      Parameters
      ----------
      c : array_like
          1-D array of array of series coefficients ordered from low to
          high degree.
      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
          Power series of power.
  
      See Also
      --------
      polyadd, polysub, polymul, polydiv
  
      Examples
      --------
  
      """
      # c is a trimmed copy
      [c] = pu.as_series([c])
      power = int(pow)
      if power != pow or power < 0:
          raise ValueError("Power must be a non-negative integer.")
      elif maxpower is not None and power > maxpower:
          raise ValueError("Power is too large")
      elif power == 0:
          return np.array([1], dtype=c.dtype)
      elif power == 1:
          return c
      else:
          # This can be made more efficient by using powers of two
          # in the usual way.
          prd = c
          for i in range(2, power + 1):
              prd = np.convolve(prd, c)
          return prd
  
  
  def polyder(c, m=1, scl=1, axis=0):
      """
      Differentiate a polynomial.
  
      Returns the polynomial 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 polynomial ``1 + 2*x + 3*x**2``
      while [[1,2],[1,2]] represents ``1 + 1*x + 2*y + 2*x*y`` if axis=0 is
      ``x`` and axis=1 is ``y``.
  
      Parameters
      ----------
      c : array_like
          Array of polynomial 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
          Polynomial coefficients of the derivative.
  
      See Also
      --------
      polyint
  
      Examples
      --------
      >>> from numpy.polynomial import polynomial as P
      >>> c = (1,2,3,4) # 1 + 2x + 3x**2 + 4x**3
      >>> P.polyder(c) # (d/dx)(c) = 2 + 6x + 12x**2
      array([  2.,   6.,  12.])
      >>> P.polyder(c,3) # (d**3/dx**3)(c) = 24
      array([ 24.])
      >>> P.polyder(c,scl=-1) # (d/d(-x))(c) = -2 - 6x - 12x**2
      array([ -2.,  -6., -12.])
      >>> P.polyder(c,2,-1) # (d**2/d(-x)**2)(c) = 6 + 24x
      array([  6.,  24.])
  
      """
      c = np.array(c, ndmin=1, copy=1)
      if c.dtype.char in '?bBhHiIlLqQpP':
          # astype fails with NA
          c = c + 0.0
      cdt = c.dtype
      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=cdt)
              for j in range(n, 0, -1):
                  der[j - 1] = j*c[j]
              c = der
      c = np.moveaxis(c, 0, iaxis)
      return c
  
  
  def polyint(c, m=1, k=[], lbnd=0, scl=1, axis=0):
      """
      Integrate a polynomial.
  
      Returns the polynomial 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 polynomial ``1 + 2*x + 3*x**2`` while [[1,2],[1,2]]
      represents ``1 + 1*x + 2*y + 2*x*y`` if axis=0 is ``x`` and axis=1 is
      ``y``.
  
      Parameters
      ----------
      c : array_like
          1-D array of polynomial coefficients, ordered from low to high.
      m : int, optional
          Order of integration, must be positive. (Default: 1)
      k : {[], list, scalar}, optional
          Integration constant(s).  The value of the first integral at zero
          is the first value in the list, the value of the second integral
          at zero is the second value, etc.  If ``k == []`` (the default),
          all constants are set to zero.  If ``m == 1``, a single scalar can
          be given instead of a list.
      lbnd : scalar, optional
          The lower bound of the integral. (Default: 0)
      scl : scalar, optional
          Following each integration the result is *multiplied* by `scl`
          before the integration constant is added. (Default: 1)
      axis : int, optional
          Axis over which the integral is taken. (Default: 0).
  
          .. versionadded:: 1.7.0
  
      Returns
      -------
      S : ndarray
          Coefficient array of the integral.
  
      Raises
      ------
      ValueError
          If ``m < 1``, ``len(k) > m``.
  
      See Also
      --------
      polyder
  
      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.
  
      Examples
      --------
      >>> from numpy.polynomial import polynomial as P
      >>> c = (1,2,3)
      >>> P.polyint(c) # should return array([0, 1, 1, 1])
      array([ 0.,  1.,  1.,  1.])
      >>> P.polyint(c,3) # should return array([0, 0, 0, 1/6, 1/12, 1/20])
      array([ 0.        ,  0.        ,  0.        ,  0.16666667,  0.08333333,
              0.05      ])
      >>> P.polyint(c,k=3) # should return array([3, 1, 1, 1])
      array([ 3.,  1.,  1.,  1.])
      >>> P.polyint(c,lbnd=-2) # should return array([6, 1, 1, 1])
      array([ 6.,  1.,  1.,  1.])
      >>> P.polyint(c,scl=-2) # should return array([0, -2, -2, -2])
      array([ 0., -2., -2., -2.])
  
      """
      c = np.array(c, ndmin=1, copy=1)
      if c.dtype.char in '?bBhHiIlLqQpP':
          # astype doesn't preserve mask attribute.
          c = c + 0.0
      cdt = c.dtype
      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")
      iaxis = normalize_axis_index(iaxis, c.ndim)
  
      if cnt == 0:
          return c
  
      k = list(k) + [0]*(cnt - len(k))
      c = np.moveaxis(c, iaxis, 0)
      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=cdt)
              tmp[0] = c[0]*0
              tmp[1] = c[0]
              for j in range(1, n):
                  tmp[j + 1] = c[j]/(j + 1)
              tmp[0] += k[i] - polyval(lbnd, tmp)
              c = tmp
      c = np.moveaxis(c, 0, iaxis)
      return c
  
  
  def polyval(x, c, tensor=True):
      """
      Evaluate a polynomial at points x.
  
      If `c` is of length `n + 1`, this function returns the value
  
      .. math:: p(x) = c_0 + c_1 * x + ... + c_n * x^n
  
      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, compatible object
          The shape of the returned array is described above.
  
      See Also
      --------
      polyval2d, polygrid2d, polyval3d, polygrid3d
  
      Notes
      -----
      The evaluation uses Horner's method.
  
      Examples
      --------
      >>> from numpy.polynomial.polynomial import polyval
      >>> polyval(1, [1,2,3])
      6.0
      >>> a = np.arange(4).reshape(2,2)
      >>> a
      array([[0, 1],
             [2, 3]])
      >>> polyval(a, [1,2,3])
      array([[  1.,   6.],
             [ 17.,  34.]])
      >>> coef = np.arange(4).reshape(2,2) # multidimensional coefficients
      >>> coef
      array([[0, 1],
             [2, 3]])
      >>> polyval([1,2], coef, tensor=True)
      array([[ 2.,  4.],
             [ 4.,  7.]])
      >>> polyval([1,2], coef, tensor=False)
      array([ 2.,  7.])
  
      """
      c = np.array(c, ndmin=1, copy=0)
      if c.dtype.char in '?bBhHiIlLqQpP':
          # astype fails with NA
          c = c + 0.0
      if isinstance(x, (tuple, list)):
          x = np.asarray(x)
      if isinstance(x, np.ndarray) and tensor:
          c = c.reshape(c.shape + (1,)*x.ndim)
  
      c0 = c[-1] + x*0
      for i in range(2, len(c) + 1):
          c0 = c[-i] + c0*x
      return c0
  
  
  def polyvalfromroots(x, r, tensor=True):
      """
      Evaluate a polynomial specified by its roots at points x.
  
      If `r` is of length `N`, this function returns the value
  
      .. math:: p(x) = \\prod_{n=1}^{N} (x - r_n)
  
      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 `r`.
  
      If `r` is a 1-D array, then `p(x)` will have the same shape as `x`.  If `r`
      is multidimensional, then the shape of the result depends on the value of
      `tensor`. If `tensor is ``True`` the shape will be r.shape[1:] + x.shape;
      that is, each polynomial is evaluated at every value of `x`. If `tensor` is
      ``False``, the shape will be r.shape[1:]; that is, each polynomial is
      evaluated only for the corresponding broadcast value of `x`. Note that
      scalars have shape (,).
  
      .. versionadded:: 1.12
  
      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 `r`.
      r : array_like
          Array of roots. If `r` is multidimensional the first index is the
          root index, while the remaining indices enumerate multiple
          polynomials. For instance, in the two dimensional case the roots
          of each polynomial may be thought of as stored in the columns of `r`.
      tensor : boolean, optional
          If True, the shape of the roots 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 `r` is
          evaluated for every element of `x`. If False, `x` is broadcast over the
          columns of `r` for the evaluation.  This keyword is useful when `r` is
          multidimensional. The default value is True.
  
      Returns
      -------
      values : ndarray, compatible object
          The shape of the returned array is described above.
  
      See Also
      --------
      polyroots, polyfromroots, polyval
  
      Examples
      --------
      >>> from numpy.polynomial.polynomial import polyvalfromroots
      >>> polyvalfromroots(1, [1,2,3])
      0.0
      >>> a = np.arange(4).reshape(2,2)
      >>> a
      array([[0, 1],
             [2, 3]])
      >>> polyvalfromroots(a, [-1, 0, 1])
      array([[ -0.,   0.],
             [  6.,  24.]])
      >>> r = np.arange(-2, 2).reshape(2,2) # multidimensional coefficients
      >>> r # each column of r defines one polynomial
      array([[-2, -1],
             [ 0,  1]])
      >>> b = [-2, 1]
      >>> polyvalfromroots(b, r, tensor=True)
      array([[-0.,  3.],
             [ 3., 0.]])
      >>> polyvalfromroots(b, r, tensor=False)
      array([-0.,  0.])
      """
      r = np.array(r, ndmin=1, copy=0)
      if r.dtype.char in '?bBhHiIlLqQpP':
          r = r.astype(np.double)
      if isinstance(x, (tuple, list)):
          x = np.asarray(x)
      if isinstance(x, np.ndarray):
          if tensor:
              r = r.reshape(r.shape + (1,)*x.ndim)
          elif x.ndim >= r.ndim:
              raise ValueError("x.ndim must be < r.ndim when tensor == False")
      return np.prod(x - r, axis=0)
  
  
  def polyval2d(x, y, c):
      """
      Evaluate a 2-D polynomial at points (x, y).
  
      This function returns the value
  
      .. math:: p(x,y) = \\sum_{i,j} c_{i,j} * x^i * y^j
  
      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` 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 `(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
      --------
      polyval, polygrid2d, polyval3d, polygrid3d
  
      Notes
      -----
  
      .. versionadded:: 1.7.0
  
      """
      try:
          x, y = np.array((x, y), copy=0)
      except Exception:
          raise ValueError('x, y are incompatible')
  
      c = polyval(x, c)
      c = polyval(y, c, tensor=False)
      return c
  
  
  def polygrid2d(x, y, c):
      """
      Evaluate a 2-D polynomial on the Cartesian product of x and y.
  
      This function returns the values:
  
      .. math:: p(a,b) = \\sum_{i,j} c_{i,j} * a^i * b^j
  
      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 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
      --------
      polyval, polyval2d, polyval3d, polygrid3d
  
      Notes
      -----
  
      .. versionadded:: 1.7.0
  
      """
      c = polyval(x, c)
      c = polyval(y, c)
      return c
  
  
  def polyval3d(x, y, z, c):
      """
      Evaluate a 3-D polynomial at points (x, y, z).
  
      This function returns the values:
  
      .. math:: p(x,y,z) = \\sum_{i,j,k} c_{i,j,k} * x^i * y^j * z^k
  
      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
      --------
      polyval, polyval2d, polygrid2d, polygrid3d
  
      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 = polyval(x, c)
      c = polyval(y, c, tensor=False)
      c = polyval(z, c, tensor=False)
      return c
  
  
  def polygrid3d(x, y, z, c):
      """
      Evaluate a 3-D polynomial 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} * a^i * b^j * c^k
  
      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
      --------
      polyval, polyval2d, polygrid2d, polyval3d
  
      Notes
      -----
  
      .. versionadded:: 1.7.0
  
      """
      c = polyval(x, c)
      c = polyval(y, c)
      c = polyval(z, c)
      return c
  
  
  def polyvander(x, deg):
      """Vandermonde matrix of given degree.
  
      Returns the Vandermonde matrix of degree `deg` and sample points
      `x`. The Vandermonde matrix is defined by
  
      .. math:: V[..., i] = x^i,
  
      where `0 <= i <= deg`. The leading indices of `V` index the elements of
      `x` and the last index is the power of `x`.
  
      If `c` is a 1-D array of coefficients of length `n + 1` and `V` is the
      matrix ``V = polyvander(x, n)``, then ``np.dot(V, c)`` and
      ``polyval(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 polynomials 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 Vandermonde matrix. The shape of the returned matrix is
          ``x.shape + (deg + 1,)``, where the last index is the power of `x`.
          The dtype will be the same as the converted `x`.
  
      See Also
      --------
      polyvander2d, polyvander3d
  
      """
      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] = x
          for i in range(2, ideg + 1):
              v[i] = v[i-1]*x
      return np.moveaxis(v, 0, -1)
  
  
  def polyvander2d(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] = x^i * y^j,
  
      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 powers of
      `x` and `y`.
  
      If ``V = polyvander2d(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 ``polyval2d(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 polynomials
      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
      --------
      polyvander, polyvander3d. polyval2d, polyval3d
  
      """
      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 = polyvander(x, degx)
      vy = polyvander(y, degy)
      v = vx[..., None]*vy[..., None,:]
      # einsum bug
      #v = np.einsum("...i,...j->...ij", vx, vy)
      return v.reshape(v.shape[:-2] + (-1,))
  
  
  def polyvander3d(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] = x^i * y^j * z^k,
  
      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 powers of `x`, `y`, and `z`.
  
      If ``V = polyvander3d(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 ``polyval3d(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 polynomials
      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
      --------
      polyvander, polyvander3d. polyval2d, polyval3d
  
      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 = polyvander(x, degx)
      vy = polyvander(y, degy)
      vz = polyvander(z, degz)
      v = vx[..., None, None]*vy[..., None,:, None]*vz[..., None, None,:]
      # einsum bug
      #v = np.einsum("...i, ...j, ...k->...ijk", vx, vy, vz)
      return v.reshape(v.shape[:-3] + (-1,))
  
  
  def polyfit(x, y, deg, rcond=None, full=False, w=None):
      """
      Least-squares fit of a polynomial to data.
  
      Return the coefficients of a polynomial 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 * x + ... + c_n * x^n,
  
      where `n` is `deg`.
  
      Parameters
      ----------
      x : array_like, shape (`M`,)
          x-coordinates of the `M` sample (data) points ``(x[i], y[i])``.
      y : array_like, shape (`M`,) or (`M`, `K`)
          y-coordinates of the sample points.  Several sets of sample points
          sharing the same x-coordinates can be (independently) fit with one
          call to `polyfit` by passing in for `y` a 2-D array that contains
          one data set 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 `rcond`, relative to the largest singular value, will be
          ignored.  The default value is ``len(x)*eps``, where `eps` is the
          relative precision of the platform's float type, about 2e-16 in
          most cases.
      full : bool, optional
          Switch determining the nature of the return value.  When ``False``
          (the default) just the coefficients are returned; when ``True``,
          diagnostic information from the singular value decomposition (used
          to solve the fit's matrix equation) is also returned.
      w : array_like, shape (`M`,), optional
          Weights. If not None, the contribution of each point
          ``(x[i],y[i])`` to the fit is weighted by `w[i]`. Ideally the
          weights are chosen so that the errors of the products ``w[i]*y[i]``
          all have the same variance.  The default value is None.
  
          .. versionadded:: 1.5.0
  
      Returns
      -------
      coef : ndarray, shape (`deg` + 1,) or (`deg` + 1, `K`)
          Polynomial coefficients ordered from low to high.  If `y` was 2-D,
          the coefficients in column `k` of `coef` represent the polynomial
          fit to the data in `y`'s `k`-th column.
  
      [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`.
  
      Raises
      ------
      RankWarning
          Raised if the matrix in the least-squares fit is rank 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, hermfit, hermefit
      polyval : Evaluates a polynomial.
      polyvander : Vandermonde matrix for powers.
      linalg.lstsq : Computes a least-squares fit from the matrix.
      scipy.interpolate.UnivariateSpline : Computes spline fits.
  
      Notes
      -----
      The solution is the coefficients of the polynomial `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) over-determined 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 (and `full` == ``False``), a `RankWarning` will be raised.
      This means that the coefficient values may be poorly determined.
      Fitting to a lower order polynomial will usually get rid of the warning
      (but may not be what you want, of course; if you have independent
      reason(s) for choosing the degree which isn't working, you may have to:
      a) reconsider those reasons, and/or b) reconsider the quality of your
      data).  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.
  
      Polynomial fits using double precision tend to "fail" at about
      (polynomial) degree 20. Fits using Chebyshev or Legendre series are
      generally better conditioned, but much can still depend on the
      distribution of the sample points and the smoothness of the data.  If
      the quality of the fit is inadequate, splines may be a good
      alternative.
  
      Examples
      --------
      >>> from numpy.polynomial import polynomial as P
      >>> x = np.linspace(-1,1,51) # x "data": [-1, -0.96, ..., 0.96, 1]
      >>> y = x**3 - x + np.random.randn(len(x)) # x^3 - x + N(0,1) "noise"
      >>> c, stats = P.polyfit(x,y,3,full=True)
      >>> c # c[0], c[2] should be approx. 0, c[1] approx. -1, c[3] approx. 1
      array([ 0.01909725, -1.30598256, -0.00577963,  1.02644286])
      >>> stats # note the large SSR, explaining the rather poor results
      [array([ 38.06116253]), 4, array([ 1.38446749,  1.32119158,  0.50443316,
      0.28853036]), 1.1324274851176597e-014]
  
      Same thing without the added noise
  
      >>> y = x**3 - x
      >>> c, stats = P.polyfit(x,y,3,full=True)
      >>> c # c[0], c[2] should be "very close to 0", c[1] ~= -1, c[3] ~= 1
      array([ -1.73362882e-17,  -1.00000000e+00,  -2.67471909e-16,
               1.00000000e+00])
      >>> stats # note the minuscule SSR
      [array([  7.46346754e-31]), 4, array([ 1.38446749,  1.32119158,
      0.50443316,  0.28853036]), 1.1324274851176597e-014]
  
      """
      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 = polyvander(x, lmax)
      else:
          deg = np.sort(deg)
          lmax = deg[-1]
          order = len(deg)
          van = polyvander(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 == 1:
          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 polycompanion(c):
      """
      Return the companion matrix of c.
  
      The companion matrix for power series cannot be made symmetric by
      scaling the basis, so this function differs from those for the
      orthogonal polynomials.
  
      Parameters
      ----------
      c : array_like
          1-D array of polynomial 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([[-c[0]/c[1]]])
  
      n = len(c) - 1
      mat = np.zeros((n, n), dtype=c.dtype)
      bot = mat.reshape(-1)[n::n+1]
      bot[...] = 1
      mat[:, -1] -= c[:-1]/c[-1]
      return mat
  
  
  def polyroots(c):
      """
      Compute the roots of a polynomial.
  
      Return the roots (a.k.a. "zeros") of the polynomial
  
      .. math:: p(x) = \\sum_i c[i] * x^i.
  
      Parameters
      ----------
      c : 1-D array_like
          1-D array of polynomial coefficients.
  
      Returns
      -------
      out : ndarray
          Array of the roots of the polynomial. If all the roots are real,
          then `out` is also real, otherwise it is complex.
  
      See Also
      --------
      chebroots
  
      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 power 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.
  
      Examples
      --------
      >>> import numpy.polynomial.polynomial as poly
      >>> poly.polyroots(poly.polyfromroots((-1,0,1)))
      array([-1.,  0.,  1.])
      >>> poly.polyroots(poly.polyfromroots((-1,0,1))).dtype
      dtype('float64')
      >>> j = complex(0,1)
      >>> poly.polyroots(poly.polyfromroots((-j,0,j)))
      array([  0.00000000e+00+0.j,   0.00000000e+00+1.j,   2.77555756e-17-1.j])
  
      """
      # c is a trimmed copy
      [c] = pu.as_series([c])
      if len(c) < 2:
          return np.array([], dtype=c.dtype)
      if len(c) == 2:
          return np.array([-c[0]/c[1]])
  
      m = polycompanion(c)
      r = la.eigvals(m)
      r.sort()
      return r
  
  
  #
  # polynomial class
  #
  
  class Polynomial(ABCPolyBase):
      """A power series class.
  
      The Polynomial 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
          Polynomial coefficients in order of increasing degree, i.e.,
          ``(1, 2, 3)`` give ``1 + 2*x + 3*x**2``.
      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 [-1, 1].
      window : (2,) array_like, optional
          Window, see `domain` for its use. The default value is [-1, 1].
  
          .. versionadded:: 1.6.0
  
      """
      # Virtual Functions
      _add = staticmethod(polyadd)
      _sub = staticmethod(polysub)
      _mul = staticmethod(polymul)
      _div = staticmethod(polydiv)
      _pow = staticmethod(polypow)
      _val = staticmethod(polyval)
      _int = staticmethod(polyint)
      _der = staticmethod(polyder)
      _fit = staticmethod(polyfit)
      _line = staticmethod(polyline)
      _roots = staticmethod(polyroots)
      _fromroots = staticmethod(polyfromroots)
  
      # Virtual properties
      nickname = 'poly'
      domain = np.array(polydomain)
      window = np.array(polydomain)
  