ideg = int(deg)<EOL>if ideg != deg:<EOL><INDENT>raise ValueError("<STR_LIT>")<EOL><DEDENT>if ideg < <NUM_LIT:0>:<EOL><INDENT>raise ValueError("<STR_LIT>")<EOL><DEDENT>x = np.array(x, copy=<NUM_LIT:0>, ndmin=<NUM_LIT:1>) + <NUM_LIT:0.0><EOL>dims = (ideg + <NUM_LIT:1>,) + x.shape<EOL>dtyp = x.dtype<EOL>v = np.empty(dims, dtype=dtyp)<EOL>v[<NUM_LIT:0>] = x*<NUM_LIT:0> + <NUM_LIT:1><EOL>if ideg > <NUM_LIT:0>:<EOL><INDENT>v[<NUM_LIT:1>] = x<EOL>for i in range(<NUM_LIT:2>, ideg + <NUM_LIT:1>):<EOL><INDENT>v[i] = (v[i-<NUM_LIT:1>]*x*(<NUM_LIT:2>*i - <NUM_LIT:1>) - v[i-<NUM_LIT:2>]*(i - <NUM_LIT:1>))/i<EOL><DEDENT><DEDENT>return np.rollaxis(v, <NUM_LIT:0>, v.ndim)<EOL>

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 Legendre polynomial.

    If `c` is a 1-D array of coefficients of length `n + 1` and `V` is the
    array ``V = legvander(x, n)``, then ``np.dot(V, c)`` and
    ``legval(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 Legendre 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 Legendre polynomial.  The dtype will be the same as
        the converted `x`.


ideg = [int(d) for d in deg]<EOL>is_valid = [id == d and id >= <NUM_LIT:0> for id, d in zip(ideg, deg)]<EOL>if is_valid != [<NUM_LIT:1>, <NUM_LIT:1>]:<EOL><INDENT>raise ValueError("<STR_LIT>")<EOL><DEDENT>degx, degy = ideg<EOL>x, y = np.array((x, y), copy=<NUM_LIT:0>) + <NUM_LIT:0.0><EOL>vx = legvander(x, degx)<EOL>vy = legvander(y, degy)<EOL>v = vx[..., None]*vy[..., None,:]<EOL>return v.reshape(v.shape[:-<NUM_LIT:2>] + (-<NUM_LIT:1>,))<EOL>

Pseudo-Vandermonde matrix of given degrees.

    Returns the pseudo-Vandermonde matrix of degrees `deg` and sample
    points `(x, y)`. The pseudo-Vandermonde matrix is defined by

    .. math:: V[..., deg[1]*i + j] = 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 Legendre polynomials.

    If ``V = legvander2d(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 ``legval2d(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 Legendre
    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
    --------
    legvander, legvander3d. legval2d, legval3d

    Notes
    -----

    .. versionadded::1.7.0


ideg = [int(d) for d in deg]<EOL>is_valid = [id == d and id >= <NUM_LIT:0> for id, d in zip(ideg, deg)]<EOL>if is_valid != [<NUM_LIT:1>, <NUM_LIT:1>, <NUM_LIT:1>]:<EOL><INDENT>raise ValueError("<STR_LIT>")<EOL><DEDENT>degx, degy, degz = ideg<EOL>x, y, z = np.array((x, y, z), copy=<NUM_LIT:0>) + <NUM_LIT:0.0><EOL>vx = legvander(x, degx)<EOL>vy = legvander(y, degy)<EOL>vz = legvander(z, degz)<EOL>v = vx[..., None, None]*vy[..., None,:, None]*vz[..., None, None,:]<EOL>return v.reshape(v.shape[:-<NUM_LIT:3>] + (-<NUM_LIT:1>,))<EOL>

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 Legendre polynomials.

    If ``V = legvander3d(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 ``legval3d(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 Legendre
    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
    --------
    legvander, legvander3d. legval2d, legval3d

    Notes
    -----

    .. versionadded::1.7.0


def legfit(x, y, deg, rcond=None, full=False, w=None):

order = int(deg) + <NUM_LIT:1><EOL>x = np.asarray(x) + <NUM_LIT:0.0><EOL>y = np.asarray(y) + <NUM_LIT:0.0><EOL>if deg < <NUM_LIT:0>:<EOL><INDENT>raise ValueError("<STR_LIT>")<EOL><DEDENT>if x.ndim != <NUM_LIT:1>:<EOL><INDENT>raise TypeError("<STR_LIT>")<EOL><DEDENT>if x.size == <NUM_LIT:0>:<EOL><INDENT>raise TypeError("<STR_LIT>")<EOL><DEDENT>if y.ndim < <NUM_LIT:1> or y.ndim > <NUM_LIT:2>:<EOL><INDENT>raise TypeError("<STR_LIT>")<EOL><DEDENT>if len(x) != len(y):<EOL><INDENT>raise TypeError("<STR_LIT>")<EOL><DEDENT>lhs = legvander(x, deg).T<EOL>rhs = y.T<EOL>if w is not None:<EOL><INDENT>w = np.asarray(w) + <NUM_LIT:0.0><EOL>if w.ndim != <NUM_LIT:1>:<EOL><INDENT>raise TypeError("<STR_LIT>")<EOL><DEDENT>if len(x) != len(w):<EOL><INDENT>raise TypeError("<STR_LIT>")<EOL><DEDENT>lhs = lhs * w<EOL>rhs = rhs * w<EOL><DEDENT>if rcond is None:<EOL><INDENT>rcond = len(x)*np.finfo(x.dtype).eps<EOL><DEDENT>if issubclass(lhs.dtype.type, np.complexfloating):<EOL><INDENT>scl = np.sqrt((np.square(lhs.real) + np.square(lhs.imag)).sum(<NUM_LIT:1>))<EOL><DEDENT>else:<EOL><INDENT>scl = np.sqrt(np.square(lhs).sum(<NUM_LIT:1>))<EOL><DEDENT>scl[scl == <NUM_LIT:0>] = <NUM_LIT:1><EOL>c, resids, rank, s = la.lstsq(lhs.T/scl, rhs.T, rcond)<EOL>c = (c.T/scl).T<EOL>if rank != order and not full:<EOL><INDENT>msg = "<STR_LIT>"<EOL>warnings.warn(msg, pu.RankWarning)<EOL><DEDENT>if full:<EOL><INDENT>return c, [resids, rank, s, rcond]<EOL><DEDENT>else:<EOL><INDENT>return c<EOL><DEDENT>

Least squares fit of Legendre series to data.

Return the coefficients of a Legendre series of degree `deg` that is the
least squares fit to the data values `y` given at points `x`. If `y` is
1-D the returned coefficients will also be 1-D. If `y` is 2-D multiple
fits are done, one for each column of `y`, and the resulting
coefficients are stored in the corresponding columns of a 2-D return.
The fitted polynomial(s) are in the form

.. math::  p(x) = c_0 + c_1 * 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
    Degree of the fitting polynomial
rcond : float, optional
    Relative condition number of the fit. Singular values smaller than
    this relative to the largest singular value will be ignored. The
    default value is len(x)*eps, where eps is the relative precision of
    the float type, about 2e-16 in most cases.
full : bool, optional
    Switch determining nature of return value. When it is False (the
    default) just the coefficients are returned, when True diagnostic
    information from the singular value decomposition is also returned.
w : array_like, shape (`M`,), optional
    Weights. If not None, the contribution of each point
    ``(x[i],y[i])`` to the fit is weighted by `w[i]`. Ideally the
    weights are chosen so that the errors of the products ``w[i]*y[i]``
    all have the same variance.  The default value is None.

    .. versionadded:: 1.5.0

Returns
-------
coef : ndarray, shape (M,) or (M, K)
    Legendre 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, polyfit, lagfit, hermfit, hermefit
legval : Evaluates a Legendre series.
legvander : Vandermonde matrix of Legendre series.
legweight : Legendre weight function (= 1).
linalg.lstsq : Computes a least-squares fit from the matrix.
scipy.interpolate.UnivariateSpline : Computes spline fits.

Notes
-----
The solution is the coefficients of the Legendre series `p` that
minimizes the sum of the weighted squared errors

.. math:: E = \\sum_j w_j^2 * |y_j - p(x_j)|^2,

where :math:`w_j` are the weights. This problem is solved by setting up
as the (typically) overdetermined matrix equation

.. math:: V(x) * c = w * y,

where `V` is the weighted pseudo Vandermonde matrix of `x`, `c` are the
coefficients to be solved for, `w` are the weights, and `y` are the
observed values.  This equation is then solved using the singular value
decomposition of `V`.

If some of the singular values of `V` are so small that they are
neglected, then a `RankWarning` will be issued. This means that the
coefficient values may be poorly determined. Using a lower order fit
will usually get rid of the warning.  The `rcond` parameter can also be
set to a value smaller than its default, but the resulting fit may be
spurious and have large contributions from roundoff error.

Fits using Legendre series are usually better conditioned than fits
using power series, but much can depend on the distribution of the
sample points and the smoothness of the data. If the quality of the fit
is inadequate splines may be a good alternative.

References
----------
.. [1] Wikipedia, "Curve fitting",
       http://en.wikipedia.org/wiki/Curve_fitting

Examples
--------


<EOL>[c] = pu.as_series([c])<EOL>if len(c) < <NUM_LIT:2>:<EOL><INDENT>raise ValueError('<STR_LIT>')<EOL><DEDENT>if len(c) == <NUM_LIT:2>:<EOL><INDENT>return np.array([[-c[<NUM_LIT:0>]/c[<NUM_LIT:1>]]])<EOL><DEDENT>n = len(c) - <NUM_LIT:1><EOL>mat = np.zeros((n, n), dtype=c.dtype)<EOL>scl = <NUM_LIT:1.>/np.sqrt(<NUM_LIT:2>*np.arange(n) + <NUM_LIT:1>)<EOL>top = mat.reshape(-<NUM_LIT:1>)[<NUM_LIT:1>::n+<NUM_LIT:1>]<EOL>bot = mat.reshape(-<NUM_LIT:1>)[n::n+<NUM_LIT:1>]<EOL>top[...] = np.arange(<NUM_LIT:1>, n)*scl[:n-<NUM_LIT:1>]*scl[<NUM_LIT:1>:n]<EOL>bot[...] = top<EOL>mat[:, -<NUM_LIT:1>] -= (c[:-<NUM_LIT:1>]/c[-<NUM_LIT:1>])*(scl/scl[-<NUM_LIT:1>])*(n/(<NUM_LIT:2>*n - <NUM_LIT:1>))<EOL>return mat<EOL>

Return the scaled companion matrix of c.

    The basis polynomials are scaled so that the companion matrix is
    symmetric when `c` is an Legendre basis polynomial. This provides
    better eigenvalue estimates than the unscaled case and for basis
    polynomials the eigenvalues are guaranteed to be real if
    `numpy.linalg.eigvalsh` is used to obtain them.

    Parameters
    ----------
    c : array_like
        1-D array of Legendre series coefficients ordered from low to high
        degree.

    Returns
    -------
    mat : ndarray
        Scaled companion matrix of dimensions (deg, deg).

    Notes
    -----

    .. versionadded::1.7.0


<EOL>[c] = pu.as_series([c])<EOL>if len(c) < <NUM_LIT:2>:<EOL><INDENT>return np.array([], dtype=c.dtype)<EOL><DEDENT>if len(c) == <NUM_LIT:2>:<EOL><INDENT>return np.array([-c[<NUM_LIT:0>]/c[<NUM_LIT:1>]])<EOL><DEDENT>m = legcompanion(c)<EOL>r = la.eigvals(m)<EOL>r.sort()<EOL>return r<EOL>

Compute the roots of a Legendre 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, chebroots, lagroots, hermroots, hermeroots

Notes
-----
The root estimates are obtained as the eigenvalues of the companion
matrix, Roots far from the origin of the complex plane may have large
errors due to the numerical instability of the series for such values.
Roots with multiplicity greater than 1 will also show larger errors as
the value of the series near such points is relatively insensitive to
errors in the roots. Isolated roots near the origin can be improved by
a few iterations of Newton's method.

The Legendre series basis polynomials aren't powers of ``x`` so the
results of this function may seem unintuitive.

Examples
--------
>>> import numpy.polynomial.legendre as leg
>>> leg.legroots((1, 2, 3, 4)) # 4L_3 + 3L_2 + 2L_1 + 1L_0, all real roots
array([-0.85099543, -0.11407192,  0.51506735])


ideg = int(deg)<EOL>if ideg != deg or ideg < <NUM_LIT:1>:<EOL><INDENT>raise ValueError("<STR_LIT>")<EOL><DEDENT>c = np.array([<NUM_LIT:0>]*deg + [<NUM_LIT:1>])<EOL>m = legcompanion(c)<EOL>x = la.eigvals(m)<EOL>x.sort()<EOL>dy = legval(x, c)<EOL>df = legval(x, legder(c))<EOL>x -= dy/df<EOL>fm = legval(x, c[<NUM_LIT:1>:])<EOL>fm /= np.abs(fm).max()<EOL>df /= np.abs(df).max()<EOL>w = <NUM_LIT:1>/(fm * df)<EOL>w = (w + w[::-<NUM_LIT:1>])/<NUM_LIT:2><EOL>x = (x - x[::-<NUM_LIT:1>])/<NUM_LIT:2><EOL>w *= <NUM_LIT> / w.sum()<EOL>return x, w<EOL>

Gauss-Legendre quadrature.

Computes the sample points and weights for Gauss-Legendre quadrature.
These sample points and weights will correctly integrate polynomials of
degree :math:`2*deg - 1` or less over the interval :math:`[-1, 1]` with
the weight function :math:`f(x) = 1`.

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.


w = x*<NUM_LIT:0.0> + <NUM_LIT:1.0><EOL>return w<EOL>

Weight function of the Legendre polynomials.

The weight function is :math:`1` and the interval of integration is
:math:`[-1, 1]`. The Legendre 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


if scl != <NUM_LIT:0>:<EOL><INDENT>return np.array([off, scl])<EOL><DEDENT>else:<EOL><INDENT>return np.array([off])<EOL><DEDENT>

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 len(roots) == <NUM_LIT:0>:<EOL><INDENT>return np.ones(<NUM_LIT:1>)<EOL><DEDENT>else:<EOL><INDENT>[roots] = pu.as_series([roots], trim=False)<EOL>roots.sort()<EOL>p = [polyline(-r, <NUM_LIT:1>) for r in roots]<EOL>n = len(p)<EOL>while n > <NUM_LIT:1>:<EOL><INDENT>m, r = divmod(n, <NUM_LIT:2>)<EOL>tmp = [polymul(p[i], p[i+m]) for i in range(m)]<EOL>if r:<EOL><INDENT>tmp[<NUM_LIT:0>] = polymul(tmp[<NUM_LIT:0>], p[-<NUM_LIT:1>])<EOL><DEDENT>p = tmp<EOL>n = m<EOL><DEDENT>return p[<NUM_LIT:0>]<EOL><DEDENT>

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])


<EOL>[c1, c2] = pu.as_series([c1, c2])<EOL>if len(c1) > len(c2):<EOL><INDENT>c1[:c2.size] += c2<EOL>ret = c1<EOL><DEDENT>else:<EOL><INDENT>c2[:c1.size] += c1<EOL>ret = c2<EOL><DEDENT>return pu.trimseq(ret)<EOL>

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


<EOL>[c1, c2] = pu.as_series([c1, c2])<EOL>if len(c1) > len(c2):<EOL><INDENT>c1[:c2.size] -= c2<EOL>ret = c1<EOL><DEDENT>else:<EOL><INDENT>c2 = -c2<EOL>c2[:c1.size] += c1<EOL>ret = c2<EOL><DEDENT>return pu.trimseq(ret)<EOL>

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.])


<EOL>[c] = pu.as_series([c])<EOL>if len(c) == <NUM_LIT:1> and c[<NUM_LIT:0>] == <NUM_LIT:0>:<EOL><INDENT>return c<EOL><DEDENT>prd = np.empty(len(c) + <NUM_LIT:1>, dtype=c.dtype)<EOL>prd[<NUM_LIT:0>] = c[<NUM_LIT:0>]*<NUM_LIT:0><EOL>prd[<NUM_LIT:1>:] = c<EOL>return prd<EOL>

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


<EOL>[c1, c2] = pu.as_series([c1, c2])<EOL>ret = np.convolve(c1, c2)<EOL>return pu.trimseq(ret)<EOL>

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.])


<EOL>[c1, c2] = pu.as_series([c1, c2])<EOL>if c2[-<NUM_LIT:1>] == <NUM_LIT:0>:<EOL><INDENT>raise ZeroDivisionError()<EOL><DEDENT>len1 = len(c1)<EOL>len2 = len(c2)<EOL>if len2 == <NUM_LIT:1>:<EOL><INDENT>return c1/c2[-<NUM_LIT:1>], c1[:<NUM_LIT:1>]*<NUM_LIT:0><EOL><DEDENT>elif len1 < len2:<EOL><INDENT>return c1[:<NUM_LIT:1>]*<NUM_LIT:0>, c1<EOL><DEDENT>else:<EOL><INDENT>dlen = len1 - len2<EOL>scl = c2[-<NUM_LIT:1>]<EOL>c2 = c2[:-<NUM_LIT:1>]/scl<EOL>i = dlen<EOL>j = len1 - <NUM_LIT:1><EOL>while i >= <NUM_LIT:0>:<EOL><INDENT>c1[i:j] -= c2*c1[j]<EOL>i -= <NUM_LIT:1><EOL>j -= <NUM_LIT:1><EOL><DEDENT>return c1[j+<NUM_LIT:1>:]/scl, pu.trimseq(c1[:j+<NUM_LIT:1>])<EOL><DEDENT>

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]))


<EOL>[c] = pu.as_series([c])<EOL>power = int(pow)<EOL>if power != pow or power < <NUM_LIT:0>:<EOL><INDENT>raise ValueError("<STR_LIT>")<EOL><DEDENT>elif maxpower is not None and power > maxpower:<EOL><INDENT>raise ValueError("<STR_LIT>")<EOL><DEDENT>elif power == <NUM_LIT:0>:<EOL><INDENT>return np.array([<NUM_LIT:1>], dtype=c.dtype)<EOL><DEDENT>elif power == <NUM_LIT:1>:<EOL><INDENT>return c<EOL><DEDENT>else:<EOL><INDENT>prd = c<EOL>for i in range(<NUM_LIT:2>, power + <NUM_LIT:1>):<EOL><INDENT>prd = np.convolve(prd, c)<EOL><DEDENT>return prd<EOL><DEDENT>

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
    --------


def polyder(c, m=<NUM_LIT:1>, scl=<NUM_LIT:1>, axis=<NUM_LIT:0>):

c = np.array(c, ndmin=<NUM_LIT:1>, copy=<NUM_LIT:1>)<EOL>if c.dtype.char in '<STR_LIT>':<EOL><INDENT>c = c + <NUM_LIT:0.0><EOL><DEDENT>cdt = c.dtype<EOL>cnt, iaxis = [int(t) for t in [m, axis]]<EOL>if cnt != m:<EOL><INDENT>raise ValueError("<STR_LIT>")<EOL><DEDENT>if cnt < <NUM_LIT:0>:<EOL><INDENT>raise ValueError("<STR_LIT>")<EOL><DEDENT>if iaxis != axis:<EOL><INDENT>raise ValueError("<STR_LIT>")<EOL><DEDENT>if not -c.ndim <= iaxis < c.ndim:<EOL><INDENT>raise ValueError("<STR_LIT>")<EOL><DEDENT>if iaxis < <NUM_LIT:0>:<EOL><INDENT>iaxis += c.ndim<EOL><DEDENT>if cnt == <NUM_LIT:0>:<EOL><INDENT>return c<EOL><DEDENT>c = np.rollaxis(c, iaxis)<EOL>n = len(c)<EOL>if cnt >= n:<EOL><INDENT>c = c[:<NUM_LIT:1>]*<NUM_LIT:0><EOL><DEDENT>else:<EOL><INDENT>for i in range(cnt):<EOL><INDENT>n = n - <NUM_LIT:1><EOL>c *= scl<EOL>der = np.empty((n,) + c.shape[<NUM_LIT:1>:], dtype=cdt)<EOL>for j in range(n, <NUM_LIT:0>, -<NUM_LIT:1>):<EOL><INDENT>der[j - <NUM_LIT:1>] = j*c[j]<EOL><DEDENT>c = der<EOL><DEDENT><DEDENT>c = np.rollaxis(c, <NUM_LIT:0>, iaxis + <NUM_LIT:1>)<EOL>return c<EOL>

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.])


def polyint(c, m=<NUM_LIT:1>, k=[], lbnd=<NUM_LIT:0>, scl=<NUM_LIT:1>, axis=<NUM_LIT:0>):

c = np.array(c, ndmin=<NUM_LIT:1>, copy=<NUM_LIT:1>)<EOL>if c.dtype.char in '<STR_LIT>':<EOL><INDENT>c = c + <NUM_LIT:0.0><EOL><DEDENT>cdt = c.dtype<EOL>if not np.iterable(k):<EOL><INDENT>k = [k]<EOL><DEDENT>cnt, iaxis = [int(t) for t in [m, axis]]<EOL>if cnt != m:<EOL><INDENT>raise ValueError("<STR_LIT>")<EOL><DEDENT>if cnt < <NUM_LIT:0>:<EOL><INDENT>raise ValueError("<STR_LIT>")<EOL><DEDENT>if len(k) > cnt:<EOL><INDENT>raise ValueError("<STR_LIT>")<EOL><DEDENT>if iaxis != axis:<EOL><INDENT>raise ValueError("<STR_LIT>")<EOL><DEDENT>if not -c.ndim <= iaxis < c.ndim:<EOL><INDENT>raise ValueError("<STR_LIT>")<EOL><DEDENT>if iaxis < <NUM_LIT:0>:<EOL><INDENT>iaxis += c.ndim<EOL><DEDENT>if cnt == <NUM_LIT:0>:<EOL><INDENT>return c<EOL><DEDENT>k = list(k) + [<NUM_LIT:0>]*(cnt - len(k))<EOL>c = np.rollaxis(c, iaxis)<EOL>for i in range(cnt):<EOL><INDENT>n = len(c)<EOL>c *= scl<EOL>if n == <NUM_LIT:1> and np.all(c[<NUM_LIT:0>] == <NUM_LIT:0>):<EOL><INDENT>c[<NUM_LIT:0>] += k[i]<EOL><DEDENT>else:<EOL><INDENT>tmp = np.empty((n + <NUM_LIT:1>,) + c.shape[<NUM_LIT:1>:], dtype=cdt)<EOL>tmp[<NUM_LIT:0>] = c[<NUM_LIT:0>]*<NUM_LIT:0><EOL>tmp[<NUM_LIT:1>] = c[<NUM_LIT:0>]<EOL>for j in range(<NUM_LIT:1>, n):<EOL><INDENT>tmp[j + <NUM_LIT:1>] = c[j]/(j + <NUM_LIT:1>)<EOL><DEDENT>tmp[<NUM_LIT:0>] += k[i] - polyval(lbnd, tmp)<EOL>c = tmp<EOL><DEDENT><DEDENT>c = np.rollaxis(c, <NUM_LIT:0>, iaxis + <NUM_LIT:1>)<EOL>return c<EOL>

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=<NUM_LIT:1>, copy=<NUM_LIT:0>)<EOL>if c.dtype.char in '<STR_LIT>':<EOL><INDENT>c = c + <NUM_LIT:0.0><EOL><DEDENT>if isinstance(x, (tuple, list)):<EOL><INDENT>x = np.asarray(x)<EOL><DEDENT>if isinstance(x, np.ndarray) and tensor:<EOL><INDENT>c = c.reshape(c.shape + (<NUM_LIT:1>,)*x.ndim)<EOL><DEDENT>c0 = c[-<NUM_LIT:1>] + x*<NUM_LIT:0><EOL>for i in range(<NUM_LIT:2>, len(c) + <NUM_LIT:1>):<EOL><INDENT>c0 = c[-i] + c0*x<EOL><DEDENT>return c0<EOL>

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.])


try:<EOL><INDENT>x, y = np.array((x, y), copy=<NUM_LIT:0>)<EOL><DEDENT>except:<EOL><INDENT>raise ValueError('<STR_LIT>')<EOL><DEDENT>c = polyval(x, c)<EOL>c = polyval(y, c, tensor=False)<EOL>return c<EOL>

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


c = polyval(x, c)<EOL>c = polyval(y, c)<EOL>return c<EOL>

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


try:<EOL><INDENT>x, y, z = np.array((x, y, z), copy=<NUM_LIT:0>)<EOL><DEDENT>except:<EOL><INDENT>raise ValueError('<STR_LIT>')<EOL><DEDENT>c = polyval(x, c)<EOL>c = polyval(y, c, tensor=False)<EOL>c = polyval(z, c, tensor=False)<EOL>return c<EOL>

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


c = polyval(x, c)<EOL>c = polyval(y, c)<EOL>c = polyval(z, c)<EOL>return c<EOL>

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


ideg = int(deg)<EOL>if ideg != deg:<EOL><INDENT>raise ValueError("<STR_LIT>")<EOL><DEDENT>if ideg < <NUM_LIT:0>:<EOL><INDENT>raise ValueError("<STR_LIT>")<EOL><DEDENT>x = np.array(x, copy=<NUM_LIT:0>, ndmin=<NUM_LIT:1>) + <NUM_LIT:0.0><EOL>dims = (ideg + <NUM_LIT:1>,) + x.shape<EOL>dtyp = x.dtype<EOL>v = np.empty(dims, dtype=dtyp)<EOL>v[<NUM_LIT:0>] = x*<NUM_LIT:0> + <NUM_LIT:1><EOL>if ideg > <NUM_LIT:0>:<EOL><INDENT>v[<NUM_LIT:1>] = x<EOL>for i in range(<NUM_LIT:2>, ideg + <NUM_LIT:1>):<EOL><INDENT>v[i] = v[i-<NUM_LIT:1>]*x<EOL><DEDENT><DEDENT>return np.rollaxis(v, <NUM_LIT:0>, v.ndim)<EOL>

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(d) for d in deg]<EOL>is_valid = [id == d and id >= <NUM_LIT:0> for id, d in zip(ideg, deg)]<EOL>if is_valid != [<NUM_LIT:1>, <NUM_LIT:1>]:<EOL><INDENT>raise ValueError("<STR_LIT>")<EOL><DEDENT>degx, degy = ideg<EOL>x, y = np.array((x, y), copy=<NUM_LIT:0>) + <NUM_LIT:0.0><EOL>vx = polyvander(x, degx)<EOL>vy = polyvander(y, degy)<EOL>v = vx[..., None]*vy[..., None,:]<EOL>return v.reshape(v.shape[:-<NUM_LIT:2>] + (-<NUM_LIT:1>,))<EOL>

Pseudo-Vandermonde matrix of given degrees.

    Returns the pseudo-Vandermonde matrix of degrees `deg` and sample
    points `(x, y)`. The pseudo-Vandermonde matrix is defined by

    .. math:: V[..., deg[1]*i + j] = 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]<EOL>is_valid = [id == d and id >= <NUM_LIT:0> for id, d in zip(ideg, deg)]<EOL>if is_valid != [<NUM_LIT:1>, <NUM_LIT:1>, <NUM_LIT:1>]:<EOL><INDENT>raise ValueError("<STR_LIT>")<EOL><DEDENT>degx, degy, degz = ideg<EOL>x, y, z = np.array((x, y, z), copy=<NUM_LIT:0>) + <NUM_LIT:0.0><EOL>vx = polyvander(x, degx)<EOL>vy = polyvander(y, degy)<EOL>vz = polyvander(z, degz)<EOL>v = vx[..., None, None]*vy[..., None,:, None]*vz[..., None, None,:]<EOL>return v.reshape(v.shape[:-<NUM_LIT:3>] + (-<NUM_LIT:1>,))<EOL>

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


def polyfit(x, y, deg, rcond=None, full=False, w=None):

order = int(deg) + <NUM_LIT:1><EOL>x = np.asarray(x) + <NUM_LIT:0.0><EOL>y = np.asarray(y) + <NUM_LIT:0.0><EOL>if deg < <NUM_LIT:0>:<EOL><INDENT>raise ValueError("<STR_LIT>")<EOL><DEDENT>if x.ndim != <NUM_LIT:1>:<EOL><INDENT>raise TypeError("<STR_LIT>")<EOL><DEDENT>if x.size == <NUM_LIT:0>:<EOL><INDENT>raise TypeError("<STR_LIT>")<EOL><DEDENT>if y.ndim < <NUM_LIT:1> or y.ndim > <NUM_LIT:2>:<EOL><INDENT>raise TypeError("<STR_LIT>")<EOL><DEDENT>if len(x) != len(y):<EOL><INDENT>raise TypeError("<STR_LIT>")<EOL><DEDENT>lhs = polyvander(x, deg).T<EOL>rhs = y.T<EOL>if w is not None:<EOL><INDENT>w = np.asarray(w) + <NUM_LIT:0.0><EOL>if w.ndim != <NUM_LIT:1>:<EOL><INDENT>raise TypeError("<STR_LIT>")<EOL><DEDENT>if len(x) != len(w):<EOL><INDENT>raise TypeError("<STR_LIT>")<EOL><DEDENT>lhs = lhs * w<EOL>rhs = rhs * w<EOL><DEDENT>if rcond is None:<EOL><INDENT>rcond = len(x)*np.finfo(x.dtype).eps<EOL><DEDENT>if issubclass(lhs.dtype.type, np.complexfloating):<EOL><INDENT>scl = np.sqrt((np.square(lhs.real) + np.square(lhs.imag)).sum(<NUM_LIT:1>))<EOL><DEDENT>else:<EOL><INDENT>scl = np.sqrt(np.square(lhs).sum(<NUM_LIT:1>))<EOL><DEDENT>scl[scl == <NUM_LIT:0>] = <NUM_LIT:1><EOL>c, resids, rank, s = la.lstsq(lhs.T/scl, rhs.T, rcond)<EOL>c = (c.T/scl).T<EOL>if rank != order and not full:<EOL><INDENT>msg = "<STR_LIT>"<EOL>warnings.warn(msg, pu.RankWarning)<EOL><DEDENT>if full:<EOL><INDENT>return c, [resids, rank, s, rcond]<EOL><DEDENT>else:<EOL><INDENT>return c<EOL><DEDENT>

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
    Degree of the polynomial(s) to be fit.
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]


<EOL>[c] = pu.as_series([c])<EOL>if len(c) < <NUM_LIT:2>:<EOL><INDENT>raise ValueError('<STR_LIT>')<EOL><DEDENT>if len(c) == <NUM_LIT:2>:<EOL><INDENT>return np.array([[-c[<NUM_LIT:0>]/c[<NUM_LIT:1>]]])<EOL><DEDENT>n = len(c) - <NUM_LIT:1><EOL>mat = np.zeros((n, n), dtype=c.dtype)<EOL>bot = mat.reshape(-<NUM_LIT:1>)[n::n+<NUM_LIT:1>]<EOL>bot[...] = <NUM_LIT:1><EOL>mat[:, -<NUM_LIT:1>] -= c[:-<NUM_LIT:1>]/c[-<NUM_LIT:1>]<EOL>return mat<EOL>

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


<EOL>[c] = pu.as_series([c])<EOL>if len(c) < <NUM_LIT:2>:<EOL><INDENT>return np.array([], dtype=c.dtype)<EOL><DEDENT>if len(c) == <NUM_LIT:2>:<EOL><INDENT>return np.array([-c[<NUM_LIT:0>]/c[<NUM_LIT:1>]])<EOL><DEDENT>m = polycompanion(c)<EOL>r = la.eigvals(m)<EOL>r.sort()<EOL>return r<EOL>

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])


[pol] = pu.as_series([pol])<EOL>deg = len(pol) - <NUM_LIT:1><EOL>res = <NUM_LIT:0><EOL>for i in range(deg, -<NUM_LIT:1>, -<NUM_LIT:1>):<EOL><INDENT>res = hermeadd(hermemulx(res), pol[i])<EOL><DEDENT>return res<EOL>

poly2herme(pol)

Convert a polynomial to a Hermite series.

Convert an array representing the coefficients of a polynomial (relative
to the "standard" basis) ordered from lowest degree to highest, to an
array of the coefficients of the equivalent Hermite series, ordered
from lowest to highest degree.

Parameters
----------
pol : array_like
    1-D array containing the polynomial coefficients

Returns
-------
c : ndarray
    1-D array containing the coefficients of the equivalent Hermite
    series.

See Also
--------
herme2poly

Notes
-----
The easy way to do conversions between polynomial basis sets
is to use the convert method of a class instance.

Examples
--------
>>> from numpy.polynomial.hermite_e import poly2herme
>>> poly2herme(np.arange(4))
array([  2.,  10.,   2.,   3.])


from .polynomial import polyadd, polysub, polymulx<EOL>[c] = pu.as_series([c])<EOL>n = len(c)<EOL>if n == <NUM_LIT:1>:<EOL><INDENT>return c<EOL><DEDENT>if n == <NUM_LIT:2>:<EOL><INDENT>return c<EOL><DEDENT>else:<EOL><INDENT>c0 = c[-<NUM_LIT:2>]<EOL>c1 = c[-<NUM_LIT:1>]<EOL>for i in range(n - <NUM_LIT:1>, <NUM_LIT:1>, -<NUM_LIT:1>):<EOL><INDENT>tmp = c0<EOL>c0 = polysub(c[i - <NUM_LIT:2>], c1*(i - <NUM_LIT:1>))<EOL>c1 = polyadd(tmp, polymulx(c1))<EOL><DEDENT>return polyadd(c0, polymulx(c1))<EOL><DEDENT>

Convert a Hermite series to a polynomial.

Convert an array representing the coefficients of a Hermite series,
ordered from lowest degree to highest, to an array of the coefficients
of the equivalent polynomial (relative to the "standard" basis) ordered
from lowest to highest degree.

Parameters
----------
c : array_like
    1-D array containing the Hermite series coefficients, ordered
    from lowest order term to highest.

Returns
-------
pol : ndarray
    1-D array containing the coefficients of the equivalent polynomial
    (relative to the "standard" basis) ordered from lowest order term
    to highest.

See Also
--------
poly2herme

Notes
-----
The easy way to do conversions between polynomial basis sets
is to use the convert method of a class instance.

Examples
--------
>>> from numpy.polynomial.hermite_e import herme2poly
>>> herme2poly([  2.,  10.,   2.,   3.])
array([ 0.,  1.,  2.,  3.])


if scl != <NUM_LIT:0>:<EOL><INDENT>return np.array([off, scl])<EOL><DEDENT>else:<EOL><INDENT>return np.array([off])<EOL><DEDENT>

Hermite series whose graph is a straight line.

Parameters
----------
off, scl : scalars
    The specified line is given by ``off + scl*x``.

Returns
-------
y : ndarray
    This module's representation of the Hermite series for
    ``off + scl*x``.

See Also
--------
polyline, chebline

Examples
--------
>>> from numpy.polynomial.hermite_e import hermeline
>>> from numpy.polynomial.hermite_e import hermeline, hermeval
>>> hermeval(0,hermeline(3, 2))
3.0
>>> hermeval(1,hermeline(3, 2))
5.0


if len(roots) == <NUM_LIT:0>:<EOL><INDENT>return np.ones(<NUM_LIT:1>)<EOL><DEDENT>else:<EOL><INDENT>[roots] = pu.as_series([roots], trim=False)<EOL>roots.sort()<EOL>p = [hermeline(-r, <NUM_LIT:1>) for r in roots]<EOL>n = len(p)<EOL>while n > <NUM_LIT:1>:<EOL><INDENT>m, r = divmod(n, <NUM_LIT:2>)<EOL>tmp = [hermemul(p[i], p[i+m]) for i in range(m)]<EOL>if r:<EOL><INDENT>tmp[<NUM_LIT:0>] = hermemul(tmp[<NUM_LIT:0>], p[-<NUM_LIT:1>])<EOL><DEDENT>p = tmp<EOL>n = m<EOL><DEDENT>return p[<NUM_LIT:0>]<EOL><DEDENT>

Generate a HermiteE 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 HermiteE 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 * He_1(x) + ... +  c_n * He_n(x)

The coefficient of the last term is not generally 1 for monic
polynomials in HermiteE form.

Parameters
----------
roots : array_like
    Sequence containing the roots.

Returns
-------
out : ndarray
    1-D array of coefficients.  If all roots are real then `out` is a
    real array, if some of the roots are complex, then `out` is complex
    even if all the coefficients in the result are real (see Examples
    below).

See Also
--------
polyfromroots, legfromroots, lagfromroots, hermfromroots,
chebfromroots.

Examples
--------
>>> from numpy.polynomial.hermite_e import hermefromroots, hermeval
>>> coef = hermefromroots((-1, 0, 1))
>>> hermeval((-1, 0, 1), coef)
array([ 0.,  0.,  0.])
>>> coef = hermefromroots((-1j, 1j))
>>> hermeval((-1j, 1j), coef)
array([ 0.+0.j,  0.+0.j])


<EOL>[c1, c2] = pu.as_series([c1, c2])<EOL>if len(c1) > len(c2):<EOL><INDENT>c1[:c2.size] += c2<EOL>ret = c1<EOL><DEDENT>else:<EOL><INDENT>c2[:c1.size] += c1<EOL>ret = c2<EOL><DEDENT>return pu.trimseq(ret)<EOL>

Add one Hermite series to another.

Returns the sum of two Hermite series `c1` + `c2`.  The arguments
are sequences of coefficients ordered from lowest order term to
highest, i.e., [1,2,3] represents the series ``P_0 + 2*P_1 + 3*P_2``.

Parameters
----------
c1, c2 : array_like
    1-D arrays of Hermite series coefficients ordered from low to
    high.

Returns
-------
out : ndarray
    Array representing the Hermite series of their sum.

See Also
--------
hermesub, hermemul, hermediv, hermepow

Notes
-----
Unlike multiplication, division, etc., the sum of two Hermite series
is a Hermite series (without having to "reproject" the result onto
the basis set) so addition, just like that of "standard" polynomials,
is simply "component-wise."

Examples
--------
>>> from numpy.polynomial.hermite_e import hermeadd
>>> hermeadd([1, 2, 3], [1, 2, 3, 4])
array([ 2.,  4.,  6.,  4.])


<EOL>[c1, c2] = pu.as_series([c1, c2])<EOL>if len(c1) > len(c2):<EOL><INDENT>c1[:c2.size] -= c2<EOL>ret = c1<EOL><DEDENT>else:<EOL><INDENT>c2 = -c2<EOL>c2[:c1.size] += c1<EOL>ret = c2<EOL><DEDENT>return pu.trimseq(ret)<EOL>

Subtract one Hermite series from another.

Returns the difference of two Hermite series `c1` - `c2`.  The
sequences of coefficients are from lowest order term to highest, i.e.,
[1,2,3] represents the series ``P_0 + 2*P_1 + 3*P_2``.

Parameters
----------
c1, c2 : array_like
    1-D arrays of Hermite series coefficients ordered from low to
    high.

Returns
-------
out : ndarray
    Of Hermite series coefficients representing their difference.

See Also
--------
hermeadd, hermemul, hermediv, hermepow

Notes
-----
Unlike multiplication, division, etc., the difference of two Hermite
series is a Hermite series (without having to "reproject" the result
onto the basis set) so subtraction, just like that of "standard"
polynomials, is simply "component-wise."

Examples
--------
>>> from numpy.polynomial.hermite_e import hermesub
>>> hermesub([1, 2, 3, 4], [1, 2, 3])
array([ 0.,  0.,  0.,  4.])


<EOL>[c] = pu.as_series([c])<EOL>if len(c) == <NUM_LIT:1> and c[<NUM_LIT:0>] == <NUM_LIT:0>:<EOL><INDENT>return c<EOL><DEDENT>prd = np.empty(len(c) + <NUM_LIT:1>, dtype=c.dtype)<EOL>prd[<NUM_LIT:0>] = c[<NUM_LIT:0>]*<NUM_LIT:0><EOL>prd[<NUM_LIT:1>] = c[<NUM_LIT:0>]<EOL>for i in range(<NUM_LIT:1>, len(c)):<EOL><INDENT>prd[i + <NUM_LIT:1>] = c[i]<EOL>prd[i - <NUM_LIT:1>] += c[i]*i<EOL><DEDENT>return prd<EOL>

Multiply a Hermite series by x.

    Multiply the Hermite series `c` by x, where x is the independent
    variable.

    Parameters
    ----------
    c : array_like
        1-D array of Hermite series coefficients ordered from low to
        high.

    Returns
    -------
    out : ndarray
        Array representing the result of the multiplication.

    Notes
    -----
    The multiplication uses the recursion relationship for Hermite
    polynomials in the form

    .. math::

    xP_i(x) = (P_{i + 1}(x) + iP_{i - 1}(x)))

    Examples
    --------
    >>> from numpy.polynomial.hermite_e import hermemulx
    >>> hermemulx([1, 2, 3])
    array([ 2.,  7.,  2.,  3.])


<EOL>[c1, c2] = pu.as_series([c1, c2])<EOL>if len(c1) > len(c2):<EOL><INDENT>c = c2<EOL>xs = c1<EOL><DEDENT>else:<EOL><INDENT>c = c1<EOL>xs = c2<EOL><DEDENT>if len(c) == <NUM_LIT:1>:<EOL><INDENT>c0 = c[<NUM_LIT:0>]*xs<EOL>c1 = <NUM_LIT:0><EOL><DEDENT>elif len(c) == <NUM_LIT:2>:<EOL><INDENT>c0 = c[<NUM_LIT:0>]*xs<EOL>c1 = c[<NUM_LIT:1>]*xs<EOL><DEDENT>else:<EOL><INDENT>nd = len(c)<EOL>c0 = c[-<NUM_LIT:2>]*xs<EOL>c1 = c[-<NUM_LIT:1>]*xs<EOL>for i in range(<NUM_LIT:3>, len(c) + <NUM_LIT:1>):<EOL><INDENT>tmp = c0<EOL>nd = nd - <NUM_LIT:1><EOL>c0 = hermesub(c[-i]*xs, c1*(nd - <NUM_LIT:1>))<EOL>c1 = hermeadd(tmp, hermemulx(c1))<EOL><DEDENT><DEDENT>return hermeadd(c0, hermemulx(c1))<EOL>

Multiply one Hermite series by another.

Returns the product of two Hermite series `c1` * `c2`.  The arguments
are sequences of coefficients, from lowest order "term" to highest,
e.g., [1,2,3] represents the series ``P_0 + 2*P_1 + 3*P_2``.

Parameters
----------
c1, c2 : array_like
    1-D arrays of Hermite series coefficients ordered from low to
    high.

Returns
-------
out : ndarray
    Of Hermite series coefficients representing their product.

See Also
--------
hermeadd, hermesub, hermediv, hermepow

Notes
-----
In general, the (polynomial) product of two C-series results in terms
that are not in the Hermite polynomial basis set.  Thus, to express
the product as a Hermite series, it is necessary to "reproject" the
product onto said basis set, which may produce "unintuitive" (but
correct) results; see Examples section below.

Examples
--------
>>> from numpy.polynomial.hermite_e import hermemul
>>> hermemul([1, 2, 3], [0, 1, 2])
array([ 14.,  15.,  28.,   7.,   6.])


<EOL>[c1, c2] = pu.as_series([c1, c2])<EOL>if c2[-<NUM_LIT:1>] == <NUM_LIT:0>:<EOL><INDENT>raise ZeroDivisionError()<EOL><DEDENT>lc1 = len(c1)<EOL>lc2 = len(c2)<EOL>if lc1 < lc2:<EOL><INDENT>return c1[:<NUM_LIT:1>]*<NUM_LIT:0>, c1<EOL><DEDENT>elif lc2 == <NUM_LIT:1>:<EOL><INDENT>return c1/c2[-<NUM_LIT:1>], c1[:<NUM_LIT:1>]*<NUM_LIT:0><EOL><DEDENT>else:<EOL><INDENT>quo = np.empty(lc1 - lc2 + <NUM_LIT:1>, dtype=c1.dtype)<EOL>rem = c1<EOL>for i in range(lc1 - lc2, - <NUM_LIT:1>, -<NUM_LIT:1>):<EOL><INDENT>p = hermemul([<NUM_LIT:0>]*i + [<NUM_LIT:1>], c2)<EOL>q = rem[-<NUM_LIT:1>]/p[-<NUM_LIT:1>]<EOL>rem = rem[:-<NUM_LIT:1>] - q*p[:-<NUM_LIT:1>]<EOL>quo[i] = q<EOL><DEDENT>return quo, pu.trimseq(rem)<EOL><DEDENT>

Divide one Hermite series by another.

Returns the quotient-with-remainder of two Hermite series
`c1` / `c2`.  The arguments are sequences of coefficients from lowest
order "term" to highest, e.g., [1,2,3] represents the series
``P_0 + 2*P_1 + 3*P_2``.

Parameters
----------
c1, c2 : array_like
    1-D arrays of Hermite series coefficients ordered from low to
    high.

Returns
-------
[quo, rem] : ndarrays
    Of Hermite series coefficients representing the quotient and
    remainder.

See Also
--------
hermeadd, hermesub, hermemul, hermepow

Notes
-----
In general, the (polynomial) division of one Hermite series by another
results in quotient and remainder terms that are not in the Hermite
polynomial basis set.  Thus, to express these results as a Hermite
series, it is necessary to "reproject" the results onto the Hermite
basis set, which may produce "unintuitive" (but correct) results; see
Examples section below.

Examples
--------
>>> from numpy.polynomial.hermite_e import hermediv
>>> hermediv([ 14.,  15.,  28.,   7.,   6.], [0, 1, 2])
(array([ 1.,  2.,  3.]), array([ 0.]))
>>> hermediv([ 15.,  17.,  28.,   7.,   6.], [0, 1, 2])
(array([ 1.,  2.,  3.]), array([ 1.,  2.]))


def hermepow(c, pow, maxpower=<NUM_LIT:16>):

<EOL>[c] = pu.as_series([c])<EOL>power = int(pow)<EOL>if power != pow or power < <NUM_LIT:0>:<EOL><INDENT>raise ValueError("<STR_LIT>")<EOL><DEDENT>elif maxpower is not None and power > maxpower:<EOL><INDENT>raise ValueError("<STR_LIT>")<EOL><DEDENT>elif power == <NUM_LIT:0>:<EOL><INDENT>return np.array([<NUM_LIT:1>], dtype=c.dtype)<EOL><DEDENT>elif power == <NUM_LIT:1>:<EOL><INDENT>return c<EOL><DEDENT>else:<EOL><INDENT>prd = c<EOL>for i in range(<NUM_LIT:2>, power + <NUM_LIT:1>):<EOL><INDENT>prd = hermemul(prd, c)<EOL><DEDENT>return prd<EOL><DEDENT>

Raise a Hermite series to a power.

    Returns the Hermite series `c` raised to the power `pow`. The
    argument `c` is a sequence of coefficients ordered from low to high.
    i.e., [1,2,3] is the series  ``P_0 + 2*P_1 + 3*P_2.``

    Parameters
    ----------
    c : array_like
        1-D array of Hermite series coefficients ordered from low to
        high.
    pow : integer
        Power to which the series will be raised
    maxpower : integer, optional
        Maximum power allowed. This is mainly to limit growth of the series
        to unmanageable size. Default is 16

    Returns
    -------
    coef : ndarray
        Hermite series of power.

    See Also
    --------
    hermeadd, hermesub, hermemul, hermediv

    Examples
    --------
    >>> from numpy.polynomial.hermite_e import hermepow
    >>> hermepow([1, 2, 3], 2)
    array([ 23.,  28.,  46.,  12.,   9.])


def hermeder(c, m=<NUM_LIT:1>, scl=<NUM_LIT:1>, axis=<NUM_LIT:0>):

c = np.array(c, ndmin=<NUM_LIT:1>, copy=<NUM_LIT:1>)<EOL>if c.dtype.char in '<STR_LIT>':<EOL><INDENT>c = c.astype(np.double)<EOL><DEDENT>cnt, iaxis = [int(t) for t in [m, axis]]<EOL>if cnt != m:<EOL><INDENT>raise ValueError("<STR_LIT>")<EOL><DEDENT>if cnt < <NUM_LIT:0>:<EOL><INDENT>raise ValueError("<STR_LIT>")<EOL><DEDENT>if iaxis != axis:<EOL><INDENT>raise ValueError("<STR_LIT>")<EOL><DEDENT>if not -c.ndim <= iaxis < c.ndim:<EOL><INDENT>raise ValueError("<STR_LIT>")<EOL><DEDENT>if iaxis < <NUM_LIT:0>:<EOL><INDENT>iaxis += c.ndim<EOL><DEDENT>if cnt == <NUM_LIT:0>:<EOL><INDENT>return c<EOL><DEDENT>c = np.rollaxis(c, iaxis)<EOL>n = len(c)<EOL>if cnt >= n:<EOL><INDENT>return c[:<NUM_LIT:1>]*<NUM_LIT:0><EOL><DEDENT>else:<EOL><INDENT>for i in range(cnt):<EOL><INDENT>n = n - <NUM_LIT:1><EOL>c *= scl<EOL>der = np.empty((n,) + c.shape[<NUM_LIT:1>:], dtype=c.dtype)<EOL>for j in range(n, <NUM_LIT:0>, -<NUM_LIT:1>):<EOL><INDENT>der[j - <NUM_LIT:1>] = j*c[j]<EOL><DEDENT>c = der<EOL><DEDENT><DEDENT>c = np.rollaxis(c, <NUM_LIT:0>, iaxis + <NUM_LIT:1>)<EOL>return c<EOL>

Differentiate a Hermite_e series.

Returns the 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*He_0 + 2*He_1 + 3*He_2``
while [[1,2],[1,2]] represents ``1*He_0(x)*He_0(y) + 1*He_1(x)*He_0(y)
+ 2*He_0(x)*He_1(y) + 2*He_1(x)*He_1(y)`` if axis=0 is ``x`` and axis=1
is ``y``.

Parameters
----------
c : array_like
    Array of Hermite_e series coefficients. If `c` is multidimensional
    the different axis correspond to different variables with the
    degree in each axis given by the corresponding index.
m : int, optional
    Number of derivatives taken, must be non-negative. (Default: 1)
scl : scalar, optional
    Each differentiation is multiplied by `scl`.  The end result is
    multiplication by ``scl**m``.  This is for use in a linear change of
    variable. (Default: 1)
axis : int, optional
    Axis over which the derivative is taken. (Default: 0).

    .. versionadded:: 1.7.0

Returns
-------
der : ndarray
    Hermite series of the derivative.

See Also
--------
hermeint

Notes
-----
In general, the result of differentiating a Hermite series does not
resemble the same operation on a power series. Thus the result of this
function may be "unintuitive," albeit correct; see Examples section
below.

Examples
--------
>>> from numpy.polynomial.hermite_e import hermeder
>>> hermeder([ 1.,  1.,  1.,  1.])
array([ 1.,  2.,  3.])
>>> hermeder([-0.25,  1.,  1./2.,  1./3.,  1./4 ], m=2)
array([ 1.,  2.,  3.])


def hermeint(c, m=<NUM_LIT:1>, k=[], lbnd=<NUM_LIT:0>, scl=<NUM_LIT:1>, axis=<NUM_LIT:0>):

c = np.array(c, ndmin=<NUM_LIT:1>, copy=<NUM_LIT:1>)<EOL>if c.dtype.char in '<STR_LIT>':<EOL><INDENT>c = c.astype(np.double)<EOL><DEDENT>if not np.iterable(k):<EOL><INDENT>k = [k]<EOL><DEDENT>cnt, iaxis = [int(t) for t in [m, axis]]<EOL>if cnt != m:<EOL><INDENT>raise ValueError("<STR_LIT>")<EOL><DEDENT>if cnt < <NUM_LIT:0>:<EOL><INDENT>raise ValueError("<STR_LIT>")<EOL><DEDENT>if len(k) > cnt:<EOL><INDENT>raise ValueError("<STR_LIT>")<EOL><DEDENT>if iaxis != axis:<EOL><INDENT>raise ValueError("<STR_LIT>")<EOL><DEDENT>if not -c.ndim <= iaxis < c.ndim:<EOL><INDENT>raise ValueError("<STR_LIT>")<EOL><DEDENT>if iaxis < <NUM_LIT:0>:<EOL><INDENT>iaxis += c.ndim<EOL><DEDENT>if cnt == <NUM_LIT:0>:<EOL><INDENT>return c<EOL><DEDENT>c = np.rollaxis(c, iaxis)<EOL>k = list(k) + [<NUM_LIT:0>]*(cnt - len(k))<EOL>for i in range(cnt):<EOL><INDENT>n = len(c)<EOL>c *= scl<EOL>if n == <NUM_LIT:1> and np.all(c[<NUM_LIT:0>] == <NUM_LIT:0>):<EOL><INDENT>c[<NUM_LIT:0>] += k[i]<EOL><DEDENT>else:<EOL><INDENT>tmp = np.empty((n + <NUM_LIT:1>,) + c.shape[<NUM_LIT:1>:], dtype=c.dtype)<EOL>tmp[<NUM_LIT:0>] = c[<NUM_LIT:0>]*<NUM_LIT:0><EOL>tmp[<NUM_LIT:1>] = c[<NUM_LIT:0>]<EOL>for j in range(<NUM_LIT:1>, n):<EOL><INDENT>tmp[j + <NUM_LIT:1>] = c[j]/(j + <NUM_LIT:1>)<EOL><DEDENT>tmp[<NUM_LIT:0>] += k[i] - hermeval(lbnd, tmp)<EOL>c = tmp<EOL><DEDENT><DEDENT>c = np.rollaxis(c, <NUM_LIT:0>, iaxis + <NUM_LIT:1>)<EOL>return c<EOL>

Integrate a Hermite_e series.

Returns the Hermite_e series coefficients `c` integrated `m` times from
`lbnd` along `axis`. At each iteration the resulting series is
**multiplied** by `scl` and an integration constant, `k`, is added.
The scaling factor is for use in a linear change of variable.  ("Buyer
beware": note that, depending on what one is doing, one may want `scl`
to be the reciprocal of what one might expect; for more information,
see the Notes section below.)  The argument `c` is an array of
coefficients from low to high degree along each axis, e.g., [1,2,3]
represents the series ``H_0 + 2*H_1 + 3*H_2`` while [[1,2],[1,2]]
represents ``1*H_0(x)*H_0(y) + 1*H_1(x)*H_0(y) + 2*H_0(x)*H_1(y) +
2*H_1(x)*H_1(y)`` if axis=0 is ``x`` and axis=1 is ``y``.

Parameters
----------
c : array_like
    Array of Hermite_e series coefficients. If c is multidimensional
    the different axis correspond to different variables with the
    degree in each axis given by the corresponding index.
m : int, optional
    Order of integration, must be positive. (Default: 1)
k : {[], list, scalar}, optional
    Integration constant(s).  The value of the first integral at
    ``lbnd`` is the first value in the list, the value of the second
    integral at ``lbnd`` is the second value, etc.  If ``k == []`` (the
    default), all constants are set to zero.  If ``m == 1``, a single
    scalar can be given instead of a list.
lbnd : scalar, optional
    The lower bound of the integral. (Default: 0)
scl : scalar, optional
    Following each integration the result is *multiplied* by `scl`
    before the integration constant is added. (Default: 1)
axis : int, optional
    Axis over which the integral is taken. (Default: 0).

    .. versionadded:: 1.7.0

Returns
-------
S : ndarray
    Hermite_e series coefficients of the integral.

Raises
------
ValueError
    If ``m < 0``, ``len(k) > m``, ``np.isscalar(lbnd) == False``, or
    ``np.isscalar(scl) == False``.

See Also
--------
hermeder

Notes
-----
Note that the result of each integration is *multiplied* by `scl`.
Why is this important to note?  Say one is making a linear change of
variable :math:`u = ax + b` in an integral relative to `x`.  Then
.. math::`dx = du/a`, so one will need to set `scl` equal to
:math:`1/a` - perhaps not what one would have first thought.

Also note that, in general, the result of integrating a C-series needs
to be "reprojected" onto the C-series basis set.  Thus, typically,
the result of this function is "unintuitive," albeit correct; see
Examples section below.

Examples
--------
>>> from numpy.polynomial.hermite_e import hermeint
>>> hermeint([1, 2, 3]) # integrate once, value 0 at 0.
array([ 1.,  1.,  1.,  1.])
>>> hermeint([1, 2, 3], m=2) # integrate twice, value & deriv 0 at 0
array([-0.25      ,  1.        ,  0.5       ,  0.33333333,  0.25      ])
>>> hermeint([1, 2, 3], k=1) # integrate once, value 1 at 0.
array([ 2.,  1.,  1.,  1.])
>>> hermeint([1, 2, 3], lbnd=-1) # integrate once, value 0 at -1
array([-1.,  1.,  1.,  1.])
>>> hermeint([1, 2, 3], m=2, k=[1, 2], lbnd=-1)
array([ 1.83333333,  0.        ,  0.5       ,  0.33333333,  0.25      ])


c = np.array(c, ndmin=<NUM_LIT:1>, copy=<NUM_LIT:0>)<EOL>if c.dtype.char in '<STR_LIT>':<EOL><INDENT>c = c.astype(np.double)<EOL><DEDENT>if isinstance(x, (tuple, list)):<EOL><INDENT>x = np.asarray(x)<EOL><DEDENT>if isinstance(x, np.ndarray) and tensor:<EOL><INDENT>c = c.reshape(c.shape + (<NUM_LIT:1>,)*x.ndim)<EOL><DEDENT>if len(c) == <NUM_LIT:1>:<EOL><INDENT>c0 = c[<NUM_LIT:0>]<EOL>c1 = <NUM_LIT:0><EOL><DEDENT>elif len(c) == <NUM_LIT:2>:<EOL><INDENT>c0 = c[<NUM_LIT:0>]<EOL>c1 = c[<NUM_LIT:1>]<EOL><DEDENT>else:<EOL><INDENT>nd = len(c)<EOL>c0 = c[-<NUM_LIT:2>]<EOL>c1 = c[-<NUM_LIT:1>]<EOL>for i in range(<NUM_LIT:3>, len(c) + <NUM_LIT:1>):<EOL><INDENT>tmp = c0<EOL>nd = nd - <NUM_LIT:1><EOL>c0 = c[-i] - c1*(nd - <NUM_LIT:1>)<EOL>c1 = tmp + c1*x<EOL><DEDENT><DEDENT>return c0 + c1*x<EOL>

Evaluate an HermiteE series at points x.

If `c` is of length `n + 1`, this function returns the value:

.. math:: p(x) = c_0 * He_0(x) + c_1 * He_1(x) + ... + c_n * He_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
--------
hermeval2d, hermegrid2d, hermeval3d, hermegrid3d

Notes
-----
The evaluation uses Clenshaw recursion, aka synthetic division.

Examples
--------
>>> from numpy.polynomial.hermite_e import hermeval
>>> coef = [1,2,3]
>>> hermeval(1, coef)
3.0
>>> hermeval([[1,2],[3,4]], coef)
array([[  3.,  14.],
       [ 31.,  54.]])


try:<EOL><INDENT>x, y = np.array((x, y), copy=<NUM_LIT:0>)<EOL><DEDENT>except:<EOL><INDENT>raise ValueError('<STR_LIT>')<EOL><DEDENT>c = hermeval(x, c)<EOL>c = hermeval(y, c, tensor=False)<EOL>return c<EOL>

Evaluate a 2-D HermiteE series at points (x, y).

This function returns the values:

.. math:: p(x,y) = \\sum_{i,j} c_{i,j} * He_i(x) * He_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
--------
hermeval, hermegrid2d, hermeval3d, hermegrid3d

Notes
-----

.. versionadded::1.7.0


c = hermeval(x, c)<EOL>c = hermeval(y, c)<EOL>return c<EOL>

Evaluate a 2-D HermiteE series on the Cartesian product of x and y.

This function returns the values:

.. math:: p(a,b) = \sum_{i,j} c_{i,j} * H_i(a) * H_j(b)

where the points `(a, b)` consist of all pairs formed by taking
`a` from `x` and `b` from `y`. The resulting points form a grid with
`x` in the first dimension and `y` in the second.

The parameters `x` and `y` are converted to arrays only if they are
tuples or a lists, otherwise they are treated as a scalars. In either
case, either `x` and `y` or their elements must support multiplication
and addition both with themselves and with the elements of `c`.

If `c` has fewer than two dimensions, ones are implicitly appended to
its shape to make it 2-D. The shape of the result will be c.shape[2:] +
x.shape.

Parameters
----------
x, y : array_like, compatible objects
    The two dimensional series is evaluated at the points in the
    Cartesian product of `x` and `y`.  If `x` or `y` is a list or
    tuple, it is first converted to an ndarray, otherwise it is left
    unchanged and, if it isn't an ndarray, it is treated as a scalar.
c : array_like
    Array of coefficients ordered so that the coefficients for terms of
    degree i,j are contained in ``c[i,j]``. If `c` has dimension
    greater than two the remaining indices enumerate multiple sets of
    coefficients.

Returns
-------
values : ndarray, compatible object
    The values of the two dimensional polynomial at points in the Cartesian
    product of `x` and `y`.

See Also
--------
hermeval, hermeval2d, hermeval3d, hermegrid3d

Notes
-----

.. versionadded::1.7.0


try:<EOL><INDENT>x, y, z = np.array((x, y, z), copy=<NUM_LIT:0>)<EOL><DEDENT>except:<EOL><INDENT>raise ValueError('<STR_LIT>')<EOL><DEDENT>c = hermeval(x, c)<EOL>c = hermeval(y, c, tensor=False)<EOL>c = hermeval(z, c, tensor=False)<EOL>return c<EOL>

Evaluate a 3-D Hermite_e series at points (x, y, z).

This function returns the values:

.. math:: p(x,y,z) = \\sum_{i,j,k} c_{i,j,k} * He_i(x) * He_j(y) * He_k(z)

The parameters `x`, `y`, and `z` are converted to arrays only if
they are tuples or a lists, otherwise they are treated as a scalars and
they must have the same shape after conversion. In either case, either
`x`, `y`, and `z` or their elements must support multiplication and
addition both with themselves and with the elements of `c`.

If `c` has fewer than 3 dimensions, ones are implicitly appended to its
shape to make it 3-D. The shape of the result will be c.shape[3:] +
x.shape.

Parameters
----------
x, y, z : array_like, compatible object
    The three dimensional series is evaluated at the points
    `(x, y, z)`, where `x`, `y`, and `z` must have the same shape.  If
    any of `x`, `y`, or `z` is a list or tuple, it is first converted
    to an ndarray, otherwise it is left unchanged and if it isn't an
    ndarray it is  treated as a scalar.
c : array_like
    Array of coefficients ordered so that the coefficient of the term of
    multi-degree i,j,k is contained in ``c[i,j,k]``. If `c` has dimension
    greater than 3 the remaining indices enumerate multiple sets of
    coefficients.

Returns
-------
values : ndarray, compatible object
    The values of the multidimensional polynomial on points formed with
    triples of corresponding values from `x`, `y`, and `z`.

See Also
--------
hermeval, hermeval2d, hermegrid2d, hermegrid3d

Notes
-----

.. versionadded::1.7.0


c = hermeval(x, c)<EOL>c = hermeval(y, c)<EOL>c = hermeval(z, c)<EOL>return c<EOL>

Evaluate a 3-D HermiteE 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} * He_i(a) * He_j(b) * He_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
--------
hermeval, hermeval2d, hermegrid2d, hermeval3d

Notes
-----

.. versionadded::1.7.0


ideg = int(deg)<EOL>if ideg != deg:<EOL><INDENT>raise ValueError("<STR_LIT>")<EOL><DEDENT>if ideg < <NUM_LIT:0>:<EOL><INDENT>raise ValueError("<STR_LIT>")<EOL><DEDENT>x = np.array(x, copy=<NUM_LIT:0>, ndmin=<NUM_LIT:1>) + <NUM_LIT:0.0><EOL>dims = (ideg + <NUM_LIT:1>,) + x.shape<EOL>dtyp = x.dtype<EOL>v = np.empty(dims, dtype=dtyp)<EOL>v[<NUM_LIT:0>] = x*<NUM_LIT:0> + <NUM_LIT:1><EOL>if ideg > <NUM_LIT:0>:<EOL><INDENT>v[<NUM_LIT:1>] = x<EOL>for i in range(<NUM_LIT:2>, ideg + <NUM_LIT:1>):<EOL><INDENT>v[i] = (v[i-<NUM_LIT:1>]*x - v[i-<NUM_LIT:2>]*(i - <NUM_LIT:1>))<EOL><DEDENT><DEDENT>return np.rollaxis(v, <NUM_LIT:0>, v.ndim)<EOL>

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] = He_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 HermiteE polynomial.

    If `c` is a 1-D array of coefficients of length `n + 1` and `V` is the
    array ``V = hermevander(x, n)``, then ``np.dot(V, c)`` and
    ``hermeval(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 HermiteE 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 HermiteE polynomial.  The dtype will be the same as
        the converted `x`.

    Examples
    --------
    >>> from numpy.polynomial.hermite_e import hermevander
    >>> x = np.array([-1, 0, 1])
    >>> hermevander(x, 3)
    array([[ 1., -1.,  0.,  2.],
           [ 1.,  0., -1., -0.],
           [ 1.,  1.,  0., -2.]])


ideg = [int(d) for d in deg]<EOL>is_valid = [id == d and id >= <NUM_LIT:0> for id, d in zip(ideg, deg)]<EOL>if is_valid != [<NUM_LIT:1>, <NUM_LIT:1>]:<EOL><INDENT>raise ValueError("<STR_LIT>")<EOL><DEDENT>degx, degy = ideg<EOL>x, y = np.array((x, y), copy=<NUM_LIT:0>) + <NUM_LIT:0.0><EOL>vx = hermevander(x, degx)<EOL>vy = hermevander(y, degy)<EOL>v = vx[..., None]*vy[..., None,:]<EOL>return v.reshape(v.shape[:-<NUM_LIT:2>] + (-<NUM_LIT:1>,))<EOL>

Pseudo-Vandermonde matrix of given degrees.

    Returns the pseudo-Vandermonde matrix of degrees `deg` and sample
    points `(x, y)`. The pseudo-Vandermonde matrix is defined by

    .. math:: V[..., deg[1]*i + j] = He_i(x) * He_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 HermiteE polynomials.

    If ``V = hermevander2d(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 ``hermeval2d(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 HermiteE
    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
    --------
    hermevander, hermevander3d. hermeval2d, hermeval3d

    Notes
    -----

    .. versionadded::1.7.0


ideg = [int(d) for d in deg]<EOL>is_valid = [id == d and id >= <NUM_LIT:0> for id, d in zip(ideg, deg)]<EOL>if is_valid != [<NUM_LIT:1>, <NUM_LIT:1>, <NUM_LIT:1>]:<EOL><INDENT>raise ValueError("<STR_LIT>")<EOL><DEDENT>degx, degy, degz = ideg<EOL>x, y, z = np.array((x, y, z), copy=<NUM_LIT:0>) + <NUM_LIT:0.0><EOL>vx = hermevander(x, degx)<EOL>vy = hermevander(y, degy)<EOL>vz = hermevander(z, degz)<EOL>v = vx[..., None, None]*vy[..., None,:, None]*vz[..., None, None,:]<EOL>return v.reshape(v.shape[:-<NUM_LIT:3>] + (-<NUM_LIT:1>,))<EOL>

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 Hehe pseudo-Vandermonde matrix is defined by

    .. math:: V[..., (m+1)(n+1)i + (n+1)j + k] = He_i(x)*He_j(y)*He_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 HermiteE polynomials.

    If ``V = hermevander3d(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 ``hermeval3d(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 HermiteE
    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
    --------
    hermevander, hermevander3d. hermeval2d, hermeval3d

    Notes
    -----

    .. versionadded::1.7.0