MLPY/Lib/site-packages/numpy/polynomial/polynomial.py

"""

=================================================

Power Series (:mod:`numpy.polynomial.polynomial`)

=================================================



This module provides a number of objects (mostly functions) useful for

dealing with polynomials, including a `Polynomial` class that

encapsulates the usual arithmetic operations.  (General information

on how this module represents and works with polynomial objects is in

the docstring for its "parent" sub-package, `numpy.polynomial`).



Classes

-------

.. autosummary::

   :toctree: generated/



   Polynomial



Constants

---------

.. autosummary::

   :toctree: generated/



   polydomain

   polyzero

   polyone

   polyx



Arithmetic

----------

.. autosummary::

   :toctree: generated/



   polyadd

   polysub

   polymulx

   polymul

   polydiv

   polypow

   polyval

   polyval2d

   polyval3d

   polygrid2d

   polygrid3d



Calculus

--------

.. autosummary::

   :toctree: generated/



   polyder

   polyint



Misc Functions

--------------

.. autosummary::

   :toctree: generated/



   polyfromroots

   polyroots

   polyvalfromroots

   polyvander

   polyvander2d

   polyvander3d

   polycompanion

   polyfit

   polytrim

   polyline



See Also

--------

`numpy.polynomial`



"""
__all__ = [
    'polyzero', 'polyone', 'polyx', 'polydomain', 'polyline', 'polyadd',
    'polysub', 'polymulx', 'polymul', 'polydiv', 'polypow', 'polyval',
    'polyvalfromroots', 'polyder', 'polyint', 'polyfromroots', 'polyvander',
    'polyfit', 'polytrim', 'polyroots', 'Polynomial', 'polyval2d', 'polyval3d',
    'polygrid2d', 'polygrid3d', 'polyvander2d', 'polyvander3d']

import numpy as np
import numpy.linalg as la
from numpy.core.multiarray import normalize_axis_index

from . import polyutils as pu
from ._polybase import ABCPolyBase

polytrim = pu.trimcoef

#
# These are constant arrays are of integer type so as to be compatible
# with the widest range of other types, such as Decimal.
#

# Polynomial default domain.
polydomain = np.array([-1, 1])

# Polynomial coefficients representing zero.
polyzero = np.array([0])

# Polynomial coefficients representing one.
polyone = np.array([1])

# Polynomial coefficients representing the identity x.
polyx = np.array([0, 1])

#
# Polynomial series functions
#


def polyline(off, scl):
    """

    Returns an array representing a linear polynomial.



    Parameters

    ----------

    off, scl : scalars

        The "y-intercept" and "slope" of the line, respectively.



    Returns

    -------

    y : ndarray

        This module's representation of the linear polynomial ``off +

        scl*x``.



    See Also

    --------

    numpy.polynomial.chebyshev.chebline

    numpy.polynomial.legendre.legline

    numpy.polynomial.laguerre.lagline

    numpy.polynomial.hermite.hermline

    numpy.polynomial.hermite_e.hermeline



    Examples

    --------

    >>> from numpy.polynomial import polynomial as P

    >>> P.polyline(1,-1)

    array([ 1, -1])

    >>> P.polyval(1, P.polyline(1,-1)) # should be 0

    0.0



    """
    if scl != 0:
        return np.array([off, scl])
    else:
        return np.array([off])


def polyfromroots(roots):
    """

    Generate a monic polynomial with given roots.



    Return the coefficients of the polynomial



    .. math:: p(x) = (x - r_0) * (x - r_1) * ... * (x - r_n),



    where the ``r_n`` are the roots specified in `roots`.  If a zero has

    multiplicity n, then it must appear in `roots` n times. For instance,

    if 2 is a root of multiplicity three and 3 is a root of multiplicity 2,

    then `roots` looks something like [2, 2, 2, 3, 3]. The roots can appear

    in any order.



    If the returned coefficients are `c`, then



    .. math:: p(x) = c_0 + c_1 * x + ... +  x^n



    The coefficient of the last term is 1 for monic polynomials in this

    form.



    Parameters

    ----------

    roots : array_like

        Sequence containing the roots.



    Returns

    -------

    out : ndarray

        1-D array of the polynomial's coefficients If all the roots are

        real, then `out` is also real, otherwise it is complex.  (see

        Examples below).



    See Also

    --------

    numpy.polynomial.chebyshev.chebfromroots

    numpy.polynomial.legendre.legfromroots

    numpy.polynomial.laguerre.lagfromroots

    numpy.polynomial.hermite.hermfromroots

    numpy.polynomial.hermite_e.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])



    """
    return pu._fromroots(polyline, polymul, roots)


def polyadd(c1, c2):
    """

    Add one polynomial to another.



    Returns the sum of two polynomials `c1` + `c2`.  The arguments are

    sequences of coefficients from lowest order term to highest, i.e.,

    [1,2,3] represents the polynomial ``1 + 2*x + 3*x**2``.



    Parameters

    ----------

    c1, c2 : array_like

        1-D arrays of polynomial coefficients ordered from low to high.



    Returns

    -------

    out : ndarray

        The coefficient array representing their sum.



    See Also

    --------

    polysub, polymulx, 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



    """
    return pu._add(c1, c2)


def polysub(c1, c2):
    """

    Subtract one polynomial from another.



    Returns the difference of two polynomials `c1` - `c2`.  The arguments

    are sequences of coefficients from lowest order term to highest, i.e.,

    [1,2,3] represents the polynomial ``1 + 2*x + 3*x**2``.



    Parameters

    ----------

    c1, c2 : array_like

        1-D arrays of polynomial coefficients ordered from low to

        high.



    Returns

    -------

    out : ndarray

        Of coefficients representing their difference.



    See Also

    --------

    polyadd, polymulx, 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.])



    """
    return pu._sub(c1, c2)


def polymulx(c):
    """Multiply a polynomial by x.



    Multiply the polynomial `c` by x, where x is the independent

    variable.





    Parameters

    ----------

    c : array_like

        1-D array of polynomial coefficients ordered from low to

        high.



    Returns

    -------

    out : ndarray

        Array representing the result of the multiplication.



    See Also

    --------

    polyadd, polysub, polymul, polydiv, polypow



    Notes

    -----



    .. versionadded:: 1.5.0



    """
    # c is a trimmed copy
    [c] = pu.as_series([c])
    # The zero series needs special treatment
    if len(c) == 1 and c[0] == 0:
        return c

    prd = np.empty(len(c) + 1, dtype=c.dtype)
    prd[0] = c[0]*0
    prd[1:] = c
    return prd


def polymul(c1, c2):
    """

    Multiply one polynomial by another.



    Returns the product of two polynomials `c1` * `c2`.  The arguments are

    sequences of coefficients, from lowest order term to highest, e.g.,

    [1,2,3] represents the polynomial ``1 + 2*x + 3*x**2.``



    Parameters

    ----------

    c1, c2 : array_like

        1-D arrays of coefficients representing a polynomial, relative to the

        "standard" basis, and ordered from lowest order term to highest.



    Returns

    -------

    out : ndarray

        Of the coefficients of their product.



    See Also

    --------

    polyadd, polysub, polymulx, polydiv, polypow



    Examples

    --------

    >>> from numpy.polynomial import polynomial as P

    >>> c1 = (1,2,3)

    >>> c2 = (3,2,1)

    >>> P.polymul(c1,c2)

    array([  3.,   8.,  14.,   8.,   3.])



    """
    # c1, c2 are trimmed copies
    [c1, c2] = pu.as_series([c1, c2])
    ret = np.convolve(c1, c2)
    return pu.trimseq(ret)


def polydiv(c1, c2):
    """

    Divide one polynomial by another.



    Returns the quotient-with-remainder of two polynomials `c1` / `c2`.

    The arguments are sequences of coefficients, from lowest order term

    to highest, e.g., [1,2,3] represents ``1 + 2*x + 3*x**2``.



    Parameters

    ----------

    c1, c2 : array_like

        1-D arrays of polynomial coefficients ordered from low to high.



    Returns

    -------

    [quo, rem] : ndarrays

        Of coefficient series representing the quotient and remainder.



    See Also

    --------

    polyadd, polysub, polymulx, 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])) # may vary



    """
    # c1, c2 are trimmed copies
    [c1, c2] = pu.as_series([c1, c2])
    if c2[-1] == 0:
        raise ZeroDivisionError()

    # note: this is more efficient than `pu._div(polymul, c1, c2)`
    lc1 = len(c1)
    lc2 = len(c2)
    if lc1 < lc2:
        return c1[:1]*0, c1
    elif lc2 == 1:
        return c1/c2[-1], c1[:1]*0
    else:
        dlen = lc1 - lc2
        scl = c2[-1]
        c2 = c2[:-1]/scl
        i = dlen
        j = lc1 - 1
        while i >= 0:
            c1[i:j] -= c2*c1[j]
            i -= 1
            j -= 1
        return c1[j+1:]/scl, pu.trimseq(c1[:j+1])


def polypow(c, pow, maxpower=None):
    """Raise a polynomial to a power.



    Returns the polynomial `c` raised to the power `pow`. The argument

    `c` is a sequence of coefficients ordered from low to high. i.e.,

    [1,2,3] is the series  ``1 + 2*x + 3*x**2.``



    Parameters

    ----------

    c : array_like

        1-D array of array of series coefficients ordered from low to

        high degree.

    pow : integer

        Power to which the series will be raised

    maxpower : integer, optional

        Maximum power allowed. This is mainly to limit growth of the series

        to unmanageable size. Default is 16



    Returns

    -------

    coef : ndarray

        Power series of power.



    See Also

    --------

    polyadd, polysub, polymulx, polymul, polydiv



    Examples

    --------

    >>> from numpy.polynomial import polynomial as P

    >>> P.polypow([1,2,3], 2)

    array([ 1., 4., 10., 12., 9.])



    """
    # note: this is more efficient than `pu._pow(polymul, c1, c2)`, as it
    # avoids calling `as_series` repeatedly
    return pu._pow(np.convolve, c, pow, maxpower)


def polyder(c, m=1, scl=1, axis=0):
    """

    Differentiate a polynomial.



    Returns the polynomial coefficients `c` differentiated `m` times along

    `axis`.  At each iteration the result is multiplied by `scl` (the

    scaling factor is for use in a linear change of variable).  The

    argument `c` is an array of coefficients from low to high degree along

    each axis, e.g., [1,2,3] represents the polynomial ``1 + 2*x + 3*x**2``

    while [[1,2],[1,2]] represents ``1 + 1*x + 2*y + 2*x*y`` if axis=0 is

    ``x`` and axis=1 is ``y``.



    Parameters

    ----------

    c : array_like

        Array of polynomial coefficients. If c is multidimensional the

        different axis correspond to different variables with the degree

        in each axis given by the corresponding index.

    m : int, optional

        Number of derivatives taken, must be non-negative. (Default: 1)

    scl : scalar, optional

        Each differentiation is multiplied by `scl`.  The end result is

        multiplication by ``scl**m``.  This is for use in a linear change

        of variable. (Default: 1)

    axis : int, optional

        Axis over which the derivative is taken. (Default: 0).



        .. versionadded:: 1.7.0



    Returns

    -------

    der : ndarray

        Polynomial coefficients of the derivative.



    See Also

    --------

    polyint



    Examples

    --------

    >>> from numpy.polynomial import polynomial as P

    >>> c = (1,2,3,4) # 1 + 2x + 3x**2 + 4x**3

    >>> P.polyder(c) # (d/dx)(c) = 2 + 6x + 12x**2

    array([  2.,   6.,  12.])

    >>> P.polyder(c,3) # (d**3/dx**3)(c) = 24

    array([24.])

    >>> P.polyder(c,scl=-1) # (d/d(-x))(c) = -2 - 6x - 12x**2

    array([ -2.,  -6., -12.])

    >>> P.polyder(c,2,-1) # (d**2/d(-x)**2)(c) = 6 + 24x

    array([  6.,  24.])



    """
    c = np.array(c, ndmin=1, copy=True)
    if c.dtype.char in '?bBhHiIlLqQpP':
        # astype fails with NA
        c = c + 0.0
    cdt = c.dtype
    cnt = pu._deprecate_as_int(m, "the order of derivation")
    iaxis = pu._deprecate_as_int(axis, "the axis")
    if cnt < 0:
        raise ValueError("The order of derivation must be non-negative")
    iaxis = normalize_axis_index(iaxis, c.ndim)

    if cnt == 0:
        return c

    c = np.moveaxis(c, iaxis, 0)
    n = len(c)
    if cnt >= n:
        c = c[:1]*0
    else:
        for i in range(cnt):
            n = n - 1
            c *= scl
            der = np.empty((n,) + c.shape[1:], dtype=cdt)
            for j in range(n, 0, -1):
                der[j - 1] = j*c[j]
            c = der
    c = np.moveaxis(c, 0, iaxis)
    return c


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

    Integrate a polynomial.



    Returns the polynomial coefficients `c` integrated `m` times from

    `lbnd` along `axis`.  At each iteration the resulting series is

    **multiplied** by `scl` and an integration constant, `k`, is added.

    The scaling factor is for use in a linear change of variable.  ("Buyer

    beware": note that, depending on what one is doing, one may want `scl`

    to be the reciprocal of what one might expect; for more information,

    see the Notes section below.) The argument `c` is an array of

    coefficients, from low to high degree along each axis, e.g., [1,2,3]

    represents the polynomial ``1 + 2*x + 3*x**2`` while [[1,2],[1,2]]

    represents ``1 + 1*x + 2*y + 2*x*y`` if axis=0 is ``x`` and axis=1 is

    ``y``.



    Parameters

    ----------

    c : array_like

        1-D array of polynomial coefficients, ordered from low to high.

    m : int, optional

        Order of integration, must be positive. (Default: 1)

    k : {[], list, scalar}, optional

        Integration constant(s).  The value of the first integral at zero

        is the first value in the list, the value of the second integral

        at zero is the second value, etc.  If ``k == []`` (the default),

        all constants are set to zero.  If ``m == 1``, a single scalar can

        be given instead of a list.

    lbnd : scalar, optional

        The lower bound of the integral. (Default: 0)

    scl : scalar, optional

        Following each integration the result is *multiplied* by `scl`

        before the integration constant is added. (Default: 1)

    axis : int, optional

        Axis over which the integral is taken. (Default: 0).



        .. versionadded:: 1.7.0



    Returns

    -------

    S : ndarray

        Coefficient array of the integral.



    Raises

    ------

    ValueError

        If ``m < 1``, ``len(k) > m``, ``np.ndim(lbnd) != 0``, or

        ``np.ndim(scl) != 0``.



    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, # may vary

             0.05      ])

    >>> P.polyint(c,k=3) # should return array([3, 1, 1, 1])

    array([3.,  1.,  1.,  1.])

    >>> P.polyint(c,lbnd=-2) # should return array([6, 1, 1, 1])

    array([6.,  1.,  1.,  1.])

    >>> P.polyint(c,scl=-2) # should return array([0, -2, -2, -2])

    array([ 0., -2., -2., -2.])



    """
    c = np.array(c, ndmin=1, copy=True)
    if c.dtype.char in '?bBhHiIlLqQpP':
        # astype doesn't preserve mask attribute.
        c = c + 0.0
    cdt = c.dtype
    if not np.iterable(k):
        k = [k]
    cnt = pu._deprecate_as_int(m, "the order of integration")
    iaxis = pu._deprecate_as_int(axis, "the axis")
    if cnt < 0:
        raise ValueError("The order of integration must be non-negative")
    if len(k) > cnt:
        raise ValueError("Too many integration constants")
    if np.ndim(lbnd) != 0:
        raise ValueError("lbnd must be a scalar.")
    if np.ndim(scl) != 0:
        raise ValueError("scl must be a scalar.")
    iaxis = normalize_axis_index(iaxis, c.ndim)

    if cnt == 0:
        return c

    k = list(k) + [0]*(cnt - len(k))
    c = np.moveaxis(c, iaxis, 0)
    for i in range(cnt):
        n = len(c)
        c *= scl
        if n == 1 and np.all(c[0] == 0):
            c[0] += k[i]
        else:
            tmp = np.empty((n + 1,) + c.shape[1:], dtype=cdt)
            tmp[0] = c[0]*0
            tmp[1] = c[0]
            for j in range(1, n):
                tmp[j + 1] = c[j]/(j + 1)
            tmp[0] += k[i] - polyval(lbnd, tmp)
            c = tmp
    c = np.moveaxis(c, 0, iaxis)
    return c


def polyval(x, c, tensor=True):
    """

    Evaluate a polynomial at points x.



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



    .. math:: p(x) = c_0 + c_1 * x + ... + c_n * x^n



    The parameter `x` is converted to an array only if it is a tuple or a

    list, otherwise it is treated as a scalar. In either case, either `x`

    or its elements must support multiplication and addition both with

    themselves and with the elements of `c`.



    If `c` is a 1-D array, then `p(x)` will have the same shape as `x`.  If

    `c` is multidimensional, then the shape of the result depends on the

    value of `tensor`. If `tensor` is true the shape will be c.shape[1:] +

    x.shape. If `tensor` is false the shape will be c.shape[1:]. Note that

    scalars have shape (,).



    Trailing zeros in the coefficients will be used in the evaluation, so

    they should be avoided if efficiency is a concern.



    Parameters

    ----------

    x : array_like, compatible object

        If `x` is a list or tuple, it is converted to an ndarray, otherwise

        it is left unchanged and treated as a scalar. In either case, `x`

        or its elements must support addition and multiplication with

        with themselves and with the elements of `c`.

    c : array_like

        Array of coefficients ordered so that the coefficients for terms of

        degree n are contained in c[n]. If `c` is multidimensional the

        remaining indices enumerate multiple polynomials. In the two

        dimensional case the coefficients may be thought of as stored in

        the columns of `c`.

    tensor : boolean, optional

        If True, the shape of the coefficient array is extended with ones

        on the right, one for each dimension of `x`. Scalars have dimension 0

        for this action. The result is that every column of coefficients in

        `c` is evaluated for every element of `x`. If False, `x` is broadcast

        over the columns of `c` for the evaluation.  This keyword is useful

        when `c` is multidimensional. The default value is True.



        .. versionadded:: 1.7.0



    Returns

    -------

    values : ndarray, compatible object

        The shape of the returned array is described above.



    See Also

    --------

    polyval2d, polygrid2d, polyval3d, polygrid3d



    Notes

    -----

    The evaluation uses Horner's method.



    Examples

    --------

    >>> from numpy.polynomial.polynomial import polyval

    >>> polyval(1, [1,2,3])

    6.0

    >>> a = np.arange(4).reshape(2,2)

    >>> a

    array([[0, 1],

           [2, 3]])

    >>> polyval(a, [1,2,3])

    array([[ 1.,   6.],

           [17.,  34.]])

    >>> coef = np.arange(4).reshape(2,2) # multidimensional coefficients

    >>> coef

    array([[0, 1],

           [2, 3]])

    >>> polyval([1,2], coef, tensor=True)

    array([[2.,  4.],

           [4.,  7.]])

    >>> polyval([1,2], coef, tensor=False)

    array([2.,  7.])



    """
    c = np.array(c, ndmin=1, copy=False)
    if c.dtype.char in '?bBhHiIlLqQpP':
        # astype fails with NA
        c = c + 0.0
    if isinstance(x, (tuple, list)):
        x = np.asarray(x)
    if isinstance(x, np.ndarray) and tensor:
        c = c.reshape(c.shape + (1,)*x.ndim)

    c0 = c[-1] + x*0
    for i in range(2, len(c) + 1):
        c0 = c[-i] + c0*x
    return c0


def polyvalfromroots(x, r, tensor=True):
    """

    Evaluate a polynomial specified by its roots at points x.



    If `r` is of length `N`, this function returns the value



    .. math:: p(x) = \\prod_{n=1}^{N} (x - r_n)



    The parameter `x` is converted to an array only if it is a tuple or a

    list, otherwise it is treated as a scalar. In either case, either `x`

    or its elements must support multiplication and addition both with

    themselves and with the elements of `r`.



    If `r` is a 1-D array, then `p(x)` will have the same shape as `x`.  If `r`

    is multidimensional, then the shape of the result depends on the value of

    `tensor`. If `tensor is ``True`` the shape will be r.shape[1:] + x.shape;

    that is, each polynomial is evaluated at every value of `x`. If `tensor` is

    ``False``, the shape will be r.shape[1:]; that is, each polynomial is

    evaluated only for the corresponding broadcast value of `x`. Note that

    scalars have shape (,).



    .. versionadded:: 1.12



    Parameters

    ----------

    x : array_like, compatible object

        If `x` is a list or tuple, it is converted to an ndarray, otherwise

        it is left unchanged and treated as a scalar. In either case, `x`

        or its elements must support addition and multiplication with

        with themselves and with the elements of `r`.

    r : array_like

        Array of roots. If `r` is multidimensional the first index is the

        root index, while the remaining indices enumerate multiple

        polynomials. For instance, in the two dimensional case the roots

        of each polynomial may be thought of as stored in the columns of `r`.

    tensor : boolean, optional

        If True, the shape of the roots array is extended with ones on the

        right, one for each dimension of `x`. Scalars have dimension 0 for this

        action. The result is that every column of coefficients in `r` is

        evaluated for every element of `x`. If False, `x` is broadcast over the

        columns of `r` for the evaluation.  This keyword is useful when `r` is

        multidimensional. The default value is True.



    Returns

    -------

    values : ndarray, compatible object

        The shape of the returned array is described above.



    See Also

    --------

    polyroots, polyfromroots, polyval



    Examples

    --------

    >>> from numpy.polynomial.polynomial import polyvalfromroots

    >>> polyvalfromroots(1, [1,2,3])

    0.0

    >>> a = np.arange(4).reshape(2,2)

    >>> a

    array([[0, 1],

           [2, 3]])

    >>> polyvalfromroots(a, [-1, 0, 1])

    array([[-0.,   0.],

           [ 6.,  24.]])

    >>> r = np.arange(-2, 2).reshape(2,2) # multidimensional coefficients

    >>> r # each column of r defines one polynomial

    array([[-2, -1],

           [ 0,  1]])

    >>> b = [-2, 1]

    >>> polyvalfromroots(b, r, tensor=True)

    array([[-0.,  3.],

           [ 3., 0.]])

    >>> polyvalfromroots(b, r, tensor=False)

    array([-0.,  0.])

    """
    r = np.array(r, ndmin=1, copy=False)
    if r.dtype.char in '?bBhHiIlLqQpP':
        r = r.astype(np.double)
    if isinstance(x, (tuple, list)):
        x = np.asarray(x)
    if isinstance(x, np.ndarray):
        if tensor:
            r = r.reshape(r.shape + (1,)*x.ndim)
        elif x.ndim >= r.ndim:
            raise ValueError("x.ndim must be < r.ndim when tensor == False")
    return np.prod(x - r, axis=0)


def polyval2d(x, y, c):
    """

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



    This function returns the value



    .. math:: p(x,y) = \\sum_{i,j} c_{i,j} * x^i * y^j



    The parameters `x` and `y` are converted to arrays only if they are

    tuples or a lists, otherwise they are treated as a scalars and they

    must have the same shape after conversion. In either case, either `x`

    and `y` or their elements must support multiplication and addition both

    with themselves and with the elements of `c`.



    If `c` has fewer than two dimensions, ones are implicitly appended to

    its shape to make it 2-D. The shape of the result will be c.shape[2:] +

    x.shape.



    Parameters

    ----------

    x, y : array_like, compatible objects

        The two dimensional series is evaluated at the points `(x, y)`,

        where `x` and `y` must have the same shape. If `x` or `y` is a list

        or tuple, it is first converted to an ndarray, otherwise it is left

        unchanged and, if it isn't an ndarray, it is treated as a scalar.

    c : array_like

        Array of coefficients ordered so that the coefficient of the term

        of multi-degree i,j is contained in `c[i,j]`. If `c` has

        dimension greater than two the remaining indices enumerate multiple

        sets of coefficients.



    Returns

    -------

    values : ndarray, compatible object

        The values of the two dimensional polynomial at points formed with

        pairs of corresponding values from `x` and `y`.



    See Also

    --------

    polyval, polygrid2d, polyval3d, polygrid3d



    Notes

    -----



    .. versionadded:: 1.7.0



    """
    return pu._valnd(polyval, c, x, y)


def polygrid2d(x, y, c):
    """

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



    This function returns the values:



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



    where the points `(a, b)` consist of all pairs formed by taking

    `a` from `x` and `b` from `y`. The resulting points form a grid with

    `x` in the first dimension and `y` in the second.



    The parameters `x` and `y` are converted to arrays only if they are

    tuples or a lists, otherwise they are treated as a scalars. In either

    case, either `x` and `y` or their elements must support multiplication

    and addition both with themselves and with the elements of `c`.



    If `c` has fewer than two dimensions, ones are implicitly appended to

    its shape to make it 2-D. The shape of the result will be c.shape[2:] +

    x.shape + y.shape.



    Parameters

    ----------

    x, y : array_like, compatible objects

        The two dimensional series is evaluated at the points in the

        Cartesian product of `x` and `y`.  If `x` or `y` is a list or

        tuple, it is first converted to an ndarray, otherwise it is left

        unchanged and, if it isn't an ndarray, it is treated as a scalar.

    c : array_like

        Array of coefficients ordered so that the coefficients for terms of

        degree i,j are contained in ``c[i,j]``. If `c` has dimension

        greater than two the remaining indices enumerate multiple sets of

        coefficients.



    Returns

    -------

    values : ndarray, compatible object

        The values of the two dimensional polynomial at points in the Cartesian

        product of `x` and `y`.



    See Also

    --------

    polyval, polyval2d, polyval3d, polygrid3d



    Notes

    -----



    .. versionadded:: 1.7.0



    """
    return pu._gridnd(polyval, c, x, y)


def polyval3d(x, y, z, c):
    """

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



    This function returns the values:



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



    The parameters `x`, `y`, and `z` are converted to arrays only if

    they are tuples or a lists, otherwise they are treated as a scalars and

    they must have the same shape after conversion. In either case, either

    `x`, `y`, and `z` or their elements must support multiplication and

    addition both with themselves and with the elements of `c`.



    If `c` has fewer than 3 dimensions, ones are implicitly appended to its

    shape to make it 3-D. The shape of the result will be c.shape[3:] +

    x.shape.



    Parameters

    ----------

    x, y, z : array_like, compatible object

        The three dimensional series is evaluated at the points

        `(x, y, z)`, where `x`, `y`, and `z` must have the same shape.  If

        any of `x`, `y`, or `z` is a list or tuple, it is first converted

        to an ndarray, otherwise it is left unchanged and if it isn't an

        ndarray it is  treated as a scalar.

    c : array_like

        Array of coefficients ordered so that the coefficient of the term of

        multi-degree i,j,k is contained in ``c[i,j,k]``. If `c` has dimension

        greater than 3 the remaining indices enumerate multiple sets of

        coefficients.



    Returns

    -------

    values : ndarray, compatible object

        The values of the multidimensional polynomial on points formed with

        triples of corresponding values from `x`, `y`, and `z`.



    See Also

    --------

    polyval, polyval2d, polygrid2d, polygrid3d



    Notes

    -----



    .. versionadded:: 1.7.0



    """
    return pu._valnd(polyval, c, x, y, z)


def polygrid3d(x, y, z, c):
    """

    Evaluate a 3-D polynomial on the Cartesian product of x, y and z.



    This function returns the values:



    .. math:: p(a,b,c) = \\sum_{i,j,k} c_{i,j,k} * a^i * b^j * c^k



    where the points `(a, b, c)` consist of all triples formed by taking

    `a` from `x`, `b` from `y`, and `c` from `z`. The resulting points form

    a grid with `x` in the first dimension, `y` in the second, and `z` in

    the third.



    The parameters `x`, `y`, and `z` are converted to arrays only if they

    are tuples or a lists, otherwise they are treated as a scalars. In

    either case, either `x`, `y`, and `z` or their elements must support

    multiplication and addition both with themselves and with the elements

    of `c`.



    If `c` has fewer than three dimensions, ones are implicitly appended to

    its shape to make it 3-D. The shape of the result will be c.shape[3:] +

    x.shape + y.shape + z.shape.



    Parameters

    ----------

    x, y, z : array_like, compatible objects

        The three dimensional series is evaluated at the points in the

        Cartesian product of `x`, `y`, and `z`.  If `x`,`y`, or `z` is a

        list or tuple, it is first converted to an ndarray, otherwise it is

        left unchanged and, if it isn't an ndarray, it is treated as a

        scalar.

    c : array_like

        Array of coefficients ordered so that the coefficients for terms of

        degree i,j are contained in ``c[i,j]``. If `c` has dimension

        greater than two the remaining indices enumerate multiple sets of

        coefficients.



    Returns

    -------

    values : ndarray, compatible object

        The values of the two dimensional polynomial at points in the Cartesian

        product of `x` and `y`.



    See Also

    --------

    polyval, polyval2d, polygrid2d, polyval3d



    Notes

    -----



    .. versionadded:: 1.7.0



    """
    return pu._gridnd(polyval, c, x, y, z)


def polyvander(x, deg):
    """Vandermonde matrix of given degree.



    Returns the Vandermonde matrix of degree `deg` and sample points

    `x`. The Vandermonde matrix is defined by



    .. math:: V[..., i] = x^i,



    where `0 <= i <= deg`. The leading indices of `V` index the elements of

    `x` and the last index is the power of `x`.



    If `c` is a 1-D array of coefficients of length `n + 1` and `V` is the

    matrix ``V = polyvander(x, n)``, then ``np.dot(V, c)`` and

    ``polyval(x, c)`` are the same up to roundoff. This equivalence is

    useful both for least squares fitting and for the evaluation of a large

    number of polynomials of the same degree and sample points.



    Parameters

    ----------

    x : array_like

        Array of points. The dtype is converted to float64 or complex128

        depending on whether any of the elements are complex. If `x` is

        scalar it is converted to a 1-D array.

    deg : int

        Degree of the resulting matrix.



    Returns

    -------

    vander : ndarray.

        The Vandermonde matrix. The shape of the returned matrix is

        ``x.shape + (deg + 1,)``, where the last index is the power of `x`.

        The dtype will be the same as the converted `x`.



    See Also

    --------

    polyvander2d, polyvander3d



    """
    ideg = pu._deprecate_as_int(deg, "deg")
    if ideg < 0:
        raise ValueError("deg must be non-negative")

    x = np.array(x, copy=False, ndmin=1) + 0.0
    dims = (ideg + 1,) + x.shape
    dtyp = x.dtype
    v = np.empty(dims, dtype=dtyp)
    v[0] = x*0 + 1
    if ideg > 0:
        v[1] = x
        for i in range(2, ideg + 1):
            v[i] = v[i-1]*x
    return np.moveaxis(v, 0, -1)


def polyvander2d(x, y, deg):
    """Pseudo-Vandermonde matrix of given degrees.



    Returns the pseudo-Vandermonde matrix of degrees `deg` and sample

    points `(x, y)`. The pseudo-Vandermonde matrix is defined by



    .. math:: V[..., (deg[1] + 1)*i + j] = x^i * y^j,



    where `0 <= i <= deg[0]` and `0 <= j <= deg[1]`. The leading indices of

    `V` index the points `(x, y)` and the last index encodes the powers of

    `x` and `y`.



    If ``V = polyvander2d(x, y, [xdeg, ydeg])``, then the columns of `V`

    correspond to the elements of a 2-D coefficient array `c` of shape

    (xdeg + 1, ydeg + 1) in the order



    .. math:: c_{00}, c_{01}, c_{02} ... , c_{10}, c_{11}, c_{12} ...



    and ``np.dot(V, c.flat)`` and ``polyval2d(x, y, c)`` will be the same

    up to roundoff. This equivalence is useful both for least squares

    fitting and for the evaluation of a large number of 2-D polynomials

    of the same degrees and sample points.



    Parameters

    ----------

    x, y : array_like

        Arrays of point coordinates, all of the same shape. The dtypes

        will be converted to either float64 or complex128 depending on

        whether any of the elements are complex. Scalars are converted to

        1-D arrays.

    deg : list of ints

        List of maximum degrees of the form [x_deg, y_deg].



    Returns

    -------

    vander2d : ndarray

        The shape of the returned matrix is ``x.shape + (order,)``, where

        :math:`order = (deg[0]+1)*(deg([1]+1)`.  The dtype will be the same

        as the converted `x` and `y`.



    See Also

    --------

    polyvander, polyvander3d, polyval2d, polyval3d



    """
    return pu._vander_nd_flat((polyvander, polyvander), (x, y), deg)


def polyvander3d(x, y, z, deg):
    """Pseudo-Vandermonde matrix of given degrees.



    Returns the pseudo-Vandermonde matrix of degrees `deg` and sample

    points `(x, y, z)`. If `l, m, n` are the given degrees in `x, y, z`,

    then The pseudo-Vandermonde matrix is defined by



    .. math:: V[..., (m+1)(n+1)i + (n+1)j + k] = x^i * y^j * z^k,



    where `0 <= i <= l`, `0 <= j <= m`, and `0 <= j <= n`.  The leading

    indices of `V` index the points `(x, y, z)` and the last index encodes

    the powers of `x`, `y`, and `z`.



    If ``V = polyvander3d(x, y, z, [xdeg, ydeg, zdeg])``, then the columns

    of `V` correspond to the elements of a 3-D coefficient array `c` of

    shape (xdeg + 1, ydeg + 1, zdeg + 1) in the order



    .. math:: c_{000}, c_{001}, c_{002},... , c_{010}, c_{011}, c_{012},...



    and  ``np.dot(V, c.flat)`` and ``polyval3d(x, y, z, c)`` will be the

    same up to roundoff. This equivalence is useful both for least squares

    fitting and for the evaluation of a large number of 3-D polynomials

    of the same degrees and sample points.



    Parameters

    ----------

    x, y, z : array_like

        Arrays of point coordinates, all of the same shape. The dtypes will

        be converted to either float64 or complex128 depending on whether

        any of the elements are complex. Scalars are converted to 1-D

        arrays.

    deg : list of ints

        List of maximum degrees of the form [x_deg, y_deg, z_deg].



    Returns

    -------

    vander3d : ndarray

        The shape of the returned matrix is ``x.shape + (order,)``, where

        :math:`order = (deg[0]+1)*(deg([1]+1)*(deg[2]+1)`.  The dtype will

        be the same as the converted `x`, `y`, and `z`.



    See Also

    --------

    polyvander, polyvander3d, polyval2d, polyval3d



    Notes

    -----



    .. versionadded:: 1.7.0



    """
    return pu._vander_nd_flat((polyvander, polyvander, polyvander), (x, y, z), deg)


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

    Least-squares fit of a polynomial to data.



    Return the coefficients of a polynomial of degree `deg` that is the

    least squares fit to the data values `y` given at points `x`. If `y` is

    1-D the returned coefficients will also be 1-D. If `y` is 2-D multiple

    fits are done, one for each column of `y`, and the resulting

    coefficients are stored in the corresponding columns of a 2-D return.

    The fitted polynomial(s) are in the form



    .. math::  p(x) = c_0 + c_1 * x + ... + c_n * x^n,



    where `n` is `deg`.



    Parameters

    ----------

    x : array_like, shape (`M`,)

        x-coordinates of the `M` sample (data) points ``(x[i], y[i])``.

    y : array_like, shape (`M`,) or (`M`, `K`)

        y-coordinates of the sample points.  Several sets of sample points

        sharing the same x-coordinates can be (independently) fit with one

        call to `polyfit` by passing in for `y` a 2-D array that contains

        one data set per column.

    deg : int or 1-D array_like

        Degree(s) of the fitting polynomials. If `deg` is a single integer

        all terms up to and including the `deg`'th term are included in the

        fit. For NumPy versions >= 1.11.0 a list of integers specifying the

        degrees of the terms to include may be used instead.

    rcond : float, optional

        Relative condition number of the fit.  Singular values smaller

        than `rcond`, relative to the largest singular value, will be

        ignored.  The default value is ``len(x)*eps``, where `eps` is the

        relative precision of the platform's float type, about 2e-16 in

        most cases.

    full : bool, optional

        Switch determining the nature of the return value.  When ``False``

        (the default) just the coefficients are returned; when ``True``,

        diagnostic information from the singular value decomposition (used

        to solve the fit's matrix equation) is also returned.

    w : array_like, shape (`M`,), optional

        Weights. If not None, the contribution of each point

        ``(x[i],y[i])`` to the fit is weighted by ``w[i]``. Ideally the

        weights are chosen so that the errors of the products ``w[i]*y[i]``

        all have the same variance.  The default value is None.



        .. versionadded:: 1.5.0



    Returns

    -------

    coef : ndarray, shape (`deg` + 1,) or (`deg` + 1, `K`)

        Polynomial coefficients ordered from low to high.  If `y` was 2-D,

        the coefficients in column `k` of `coef` represent the polynomial

        fit to the data in `y`'s `k`-th column.



    [residuals, rank, singular_values, rcond] : list

        These values are only returned if `full` = True



        resid -- sum of squared residuals of the least squares fit

        rank -- the numerical rank of the scaled Vandermonde matrix

        sv -- singular values of the scaled Vandermonde matrix

        rcond -- value of `rcond`.



        For more details, see `numpy.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', np.RankWarning)



    See Also

    --------

    numpy.polynomial.chebyshev.chebfit

    numpy.polynomial.legendre.legfit

    numpy.polynomial.laguerre.lagfit

    numpy.polynomial.hermite.hermfit

    numpy.polynomial.hermite_e.hermefit

    polyval : Evaluates a polynomial.

    polyvander : Vandermonde matrix for powers.

    numpy.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

    --------

    >>> np.random.seed(123)

    >>> 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)

    >>> np.random.seed(123)

    >>> 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]) # may vary

    >>> stats # note the large SSR, explaining the rather poor results

     [array([ 38.06116253]), 4, array([ 1.38446749,  1.32119158,  0.50443316, # may vary

              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([-6.36925336e-18, -1.00000000e+00, -4.08053781e-16,  1.00000000e+00])

    >>> stats # note the minuscule SSR

    [array([  7.46346754e-31]), 4, array([ 1.38446749,  1.32119158, # may vary

               0.50443316,  0.28853036]), 1.1324274851176597e-014]



    """
    return pu._fit(polyvander, x, y, deg, rcond, full, w)


def polycompanion(c):
    """

    Return the companion matrix of c.



    The companion matrix for power series cannot be made symmetric by

    scaling the basis, so this function differs from those for the

    orthogonal polynomials.



    Parameters

    ----------

    c : array_like

        1-D array of polynomial coefficients ordered from low to high

        degree.



    Returns

    -------

    mat : ndarray

        Companion matrix of dimensions (deg, deg).



    Notes

    -----



    .. versionadded:: 1.7.0



    """
    # c is a trimmed copy
    [c] = pu.as_series([c])
    if len(c) < 2:
        raise ValueError('Series must have maximum degree of at least 1.')
    if len(c) == 2:
        return np.array([[-c[0]/c[1]]])

    n = len(c) - 1
    mat = np.zeros((n, n), dtype=c.dtype)
    bot = mat.reshape(-1)[n::n+1]
    bot[...] = 1
    mat[:, -1] -= c[:-1]/c[-1]
    return mat


def polyroots(c):
    """

    Compute the roots of a polynomial.



    Return the roots (a.k.a. "zeros") of the polynomial



    .. math:: p(x) = \\sum_i c[i] * x^i.



    Parameters

    ----------

    c : 1-D array_like

        1-D array of polynomial coefficients.



    Returns

    -------

    out : ndarray

        Array of the roots of the polynomial. If all the roots are real,

        then `out` is also real, otherwise it is complex.



    See Also

    --------

    numpy.polynomial.chebyshev.chebroots

    numpy.polynomial.legendre.legroots

    numpy.polynomial.laguerre.lagroots

    numpy.polynomial.hermite.hermroots

    numpy.polynomial.hermite_e.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 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]) # may vary



    """
    # c is a trimmed copy
    [c] = pu.as_series([c])
    if len(c) < 2:
        return np.array([], dtype=c.dtype)
    if len(c) == 2:
        return np.array([-c[0]/c[1]])

    # rotated companion matrix reduces error
    m = polycompanion(c)[::-1,::-1]
    r = la.eigvals(m)
    r.sort()
    return r


#
# polynomial class
#

class Polynomial(ABCPolyBase):
    """A power series class.



    The Polynomial class provides the standard Python numerical methods

    '+', '-', '*', '//', '%', 'divmod', '**', and '()' as well as the

    attributes and methods listed in the `ABCPolyBase` documentation.



    Parameters

    ----------

    coef : array_like

        Polynomial coefficients in order of increasing degree, i.e.,

        ``(1, 2, 3)`` give ``1 + 2*x + 3*x**2``.

    domain : (2,) array_like, optional

        Domain to use. The interval ``[domain[0], domain[1]]`` is mapped

        to the interval ``[window[0], window[1]]`` by shifting and scaling.

        The default value is [-1, 1].

    window : (2,) array_like, optional

        Window, see `domain` for its use. The default value is [-1, 1].



        .. versionadded:: 1.6.0



    """
    # Virtual Functions
    _add = staticmethod(polyadd)
    _sub = staticmethod(polysub)
    _mul = staticmethod(polymul)
    _div = staticmethod(polydiv)
    _pow = staticmethod(polypow)
    _val = staticmethod(polyval)
    _int = staticmethod(polyint)
    _der = staticmethod(polyder)
    _fit = staticmethod(polyfit)
    _line = staticmethod(polyline)
    _roots = staticmethod(polyroots)
    _fromroots = staticmethod(polyfromroots)

    # Virtual properties
    domain = np.array(polydomain)
    window = np.array(polydomain)
    basis_name = None

    @classmethod
    def _str_term_unicode(cls, i, arg_str):
        return f"·{arg_str}{i.translate(cls._superscript_mapping)}"

    @staticmethod
    def _str_term_ascii(i, arg_str):
        return f" {arg_str}**{i}"

    @staticmethod
    def _repr_latex_term(i, arg_str, needs_parens):
        if needs_parens:
            arg_str = rf"\left({arg_str}\right)"
        if i == 0:
            return '1'
        elif i == 1:
            return arg_str
        else:
            return f"{arg_str}^{{{i}}}"