package np

Get an attribute of this module as a Py.Object.t. This is useful to pass a Python function to another function.

Convert a Legendre series to a polynomial.

Convert an array representing the coefficients of a Legendre 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 Legendre 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 -------- poly2leg

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

Examples -------- >>> from numpy import polynomial as P >>> c = P.Legendre(range(4)) >>> c Legendre(0., 1., 2., 3., domain=-1, 1, window=-1, 1) >>> p = c.convert(kind=P.Polynomial) >>> p Polynomial(-1. , -3.5, 3. , 7.5, domain=-1., 1., window=-1., 1.) >>> P.leg2poly(range(4)) array(-1. , -3.5, 3. , 7.5)

Add one Legendre series to another.

Returns the sum of two Legendre 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 Legendre series coefficients ordered from low to high.

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

See Also -------- legsub, legmulx, legmul, legdiv, legpow

Notes ----- Unlike multiplication, division, etc., the sum of two Legendre series is a Legendre 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 import legendre as L >>> c1 = (1,2,3) >>> c2 = (3,2,1) >>> L.legadd(c1,c2) array(4., 4., 4.)

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

Differentiate a Legendre series.

Returns the Legendre series coefficients `c` differentiated `m` times along `axis`. At each iteration the result is multiplied by `scl` (the scaling factor is for use in a linear change of variable). The argument `c` is an array of coefficients from low to high degree along each axis, e.g., 1,2,3 represents the series ``1*L_0 + 2*L_1 + 3*L_2`` while [1,2],[1,2] represents ``1*L_0(x)*L_0(y) + 1*L_1(x)*L_0(y) + 2*L_0(x)*L_1(y) + 2*L_1(x)*L_1(y)`` if axis=0 is ``x`` and axis=1 is ``y``.

Parameters ---------- c : array_like Array of Legendre 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 Legendre series of the derivative.

See Also -------- legint

Notes ----- In general, the result of differentiating a Legendre 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 import legendre as L >>> c = (1,2,3,4) >>> L.legder(c) array( 6., 9., 20.) >>> L.legder(c, 3) array(60.) >>> L.legder(c, scl=-1) array( -6., -9., -20.) >>> L.legder(c, 2,-1) array( 9., 60.)

Divide one Legendre series by another.

Returns the quotient-with-remainder of two Legendre 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 Legendre series coefficients ordered from low to high.

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

See Also -------- legadd, legsub, legmulx, legmul, legpow

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

Examples -------- >>> from numpy.polynomial import legendre as L >>> c1 = (1,2,3) >>> c2 = (3,2,1) >>> L.legdiv(c1,c2) # quotient 'intuitive,' remainder not (array(3.), array(-8., -4.)) >>> c2 = (0,1,2,3) >>> L.legdiv(c2,c1) # neither 'intuitive' (array(-0.07407407, 1.66666667), array(-1.03703704, -2.51851852)) # may vary

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 ``(xi, yi)``. y : array_like, shape (M,) or (M, K) y-coordinates of the sample points. Several data sets of sample points sharing the same x-coordinates can be fitted at once by passing in a 2D-array that contains one dataset per column. deg : int or 1-D array_like Degree(s) of the fitting polynomials. If `deg` is a single integer all terms up to and including the `deg`'th term are included in the fit. For NumPy versions >= 1.11.0 a list of integers specifying the degrees of the terms to include may be used instead. rcond : float, optional Relative condition number of the fit. Singular values smaller than this relative to the largest singular value will be ignored. The default value is len(x)*eps, where eps is the relative precision of the float type, about 2e-16 in most cases. full : bool, optional Switch determining nature of return value. When it is False (the default) just the coefficients are returned, when True diagnostic information from the singular value decomposition is also returned. w : array_like, shape (`M`,), optional Weights. If not None, the contribution of each point ``(xi,yi)`` to the fit is weighted by `wi`. Ideally the weights are chosen so that the errors of the products ``wi*yi`` 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`. If `deg` is specified as a list, coefficients for terms not included in the fit are set equal to zero in the returned `coef`.

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', np.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', https://en.wikipedia.org/wiki/Curve_fitting

Examples --------

Generate a Legendre 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 Legendre form, where the `r_n` are the roots specified in `roots`. If a zero has multiplicity n, then it must appear in `roots` n times. For instance, if 2 is a root of multiplicity three and 3 is a root of multiplicity 2, then `roots` looks something like 2, 2, 2, 3, 3. The roots can appear in any order.

If the returned coefficients are `c`, then

.. math:: p(x) = c_0 + c_1 * L_1(x) + ... + c_n * L_n(x)

The coefficient of the last term is not generally 1 for monic polynomials in Legendre 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, chebfromroots, lagfromroots, hermfromroots, hermefromroots

Examples -------- >>> import numpy.polynomial.legendre as L >>> L.legfromroots((-1,0,1)) # x^3 - x relative to the standard basis array( 0. , -0.4, 0. , 0.4) >>> j = complex(0,1) >>> L.legfromroots((-j,j)) # x^2 + 1 relative to the standard basis array( 1.33333333+0.j, 0.00000000+0.j, 0.66666667+0.j) # may vary

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.

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

This function returns the values:

.. math:: p(a,b) = \sum_,j c_,j * L_i(a) * L_j(b)

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

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

If `c` has fewer than two dimensions, ones are implicitly appended to its shape to make it 2-D. The shape of the result will be c.shape2: + x.shape + y.shape.

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

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

See Also -------- legval, legval2d, legval3d, leggrid3d

Notes -----

.. versionadded:: 1.7.0

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

This function returns the values:

.. math:: p(a,b,c) = \sum_,j,k c_,j,k * L_i(a) * L_j(b) * L_k(c)

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

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

If `c` has fewer than three dimensions, ones are implicitly appended to its shape to make it 3-D. The shape of the result will be c.shape3: + 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 ``ci,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 -------- legval, legval2d, leggrid2d, legval3d

Notes -----

.. versionadded:: 1.7.0

Integrate a Legendre series.

Returns the Legendre series coefficients `c` integrated `m` times from `lbnd` along `axis`. At each iteration the resulting series is **multiplied** by `scl` and an integration constant, `k`, is added. The scaling factor is for use in a linear change of variable. ('Buyer beware': note that, depending on what one is doing, one may want `scl` to be the reciprocal of what one might expect; for more information, see the Notes section below.) The argument `c` is an array of coefficients from low to high degree along each axis, e.g., 1,2,3 represents the series ``L_0 + 2*L_1 + 3*L_2`` while [1,2],[1,2] represents ``1*L_0(x)*L_0(y) + 1*L_1(x)*L_0(y) + 2*L_0(x)*L_1(y) + 2*L_1(x)*L_1(y)`` if axis=0 is ``x`` and axis=1 is ``y``.

Parameters ---------- c : array_like Array of Legendre 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
    Legendre series coefficient array of the integral.

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

See Also
--------
legder

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 import legendre as L
>>> c = (1,2,3)
>>> L.legint(c)
array([ 0.33333333,  0.4       ,  0.66666667,  0.6       ]) # may vary
>>> L.legint(c, 3)
array([  1.66666667e-02,  -1.78571429e-02,   4.76190476e-02, # may vary
         -1.73472348e-18,   1.90476190e-02,   9.52380952e-03])
>>> L.legint(c, k=3)
 array([ 3.33333333,  0.4       ,  0.66666667,  0.6       ]) # may vary
>>> L.legint(c, lbnd=-2)
array([ 7.33333333,  0.4       ,  0.66666667,  0.6       ]) # may vary
>>> L.legint(c, scl=2)
array([ 0.66666667,  0.8       ,  1.33333333,  1.2       ]) # may vary

Legendre 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 Legendre series for ``off + scl*x``.

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

Examples -------- >>> import numpy.polynomial.legendre as L >>> L.legline(3,2) array(3, 2) >>> L.legval(-3, L.legline(3,2)) # should be -3 -3.0

Multiply one Legendre series by another.

Returns the product of two Legendre 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 Legendre series coefficients ordered from low to high.

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

See Also -------- legadd, legsub, legmulx, legdiv, legpow

Notes ----- In general, the (polynomial) product of two C-series results in terms that are not in the Legendre polynomial basis set. Thus, to express the product as a Legendre 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 import legendre as L >>> c1 = (1,2,3) >>> c2 = (3,2) >>> L.legmul(c1,c2) # multiplication requires 'reprojection' array( 4.33333333, 10.4 , 11.66666667, 3.6 ) # may vary

Multiply a Legendre series by x.

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

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

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

See Also -------- legadd, legmul, legmul, legdiv, legpow

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

.. math::

xP_i(x) = ((i + 1)*P_+ 1(x) + i*P_- 1(x))/(2i + 1)

Examples -------- >>> from numpy.polynomial import legendre as L >>> L.legmulx(1,2,3) array( 0.66666667, 2.2, 1.33333333, 1.8) # may vary

Raise a Legendre series to a power.

Returns the Legendre 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 Legendre 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 Legendre series of power.

See Also -------- legadd, legsub, legmulx, legmul, legdiv

Examples --------

Compute the roots of a Legendre series.

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

.. math:: p(x) = \sum_i ci * 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) # may vary

Subtract one Legendre series from another.

Returns the difference of two Legendre 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 Legendre series coefficients ordered from low to high.

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

See Also -------- legadd, legmulx, legmul, legdiv, legpow

Notes ----- Unlike multiplication, division, etc., the difference of two Legendre series is a Legendre 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 import legendre as L >>> c1 = (1,2,3) >>> c2 = (3,2,1) >>> L.legsub(c1,c2) array(-2., 0., 2.) >>> L.legsub(c2,c1) # -C.legsub(c1,c2) array( 2., 0., -2.)

Remove 'small' 'trailing' coefficients from a polynomial.

'Small' means 'small in absolute value' and is controlled by the parameter `tol`; 'trailing' means highest order coefficient(s), e.g., in ``0, 1, 1, 0, 0`` (which represents ``0 + x + x**2 + 0*x**3 + 0*x**4``) both the 3-rd and 4-th order coefficients would be 'trimmed.'

Parameters ---------- c : array_like 1-d array of coefficients, ordered from lowest order to highest. tol : number, optional Trailing (i.e., highest order) elements with absolute value less than or equal to `tol` (default value is zero) are removed.

Returns ------- trimmed : ndarray 1-d array with trailing zeros removed. If the resulting series would be empty, a series containing a single zero is returned.

Raises ------ ValueError If `tol` < 0

See Also -------- trimseq

Examples -------- >>> from numpy.polynomial import polyutils as pu >>> pu.trimcoef((0,0,3,0,5,0,0)) array(0., 0., 3., 0., 5.) >>> pu.trimcoef((0,0,1e-3,0,1e-5,0,0),1e-3) # item == tol is trimmed array(0.) >>> i = complex(0,1) # works for complex >>> pu.trimcoef((3e-4,1e-3*(1-i),5e-4,2e-5*(1+i)), 1e-3) array(0.0003+0.j , 0.001 -0.001j)

Evaluate a Legendre series at points x.

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

.. math:: p(x) = c_0 * L_0(x) + c_1 * L_1(x) + ... + c_n * L_n(x)

The parameter `x` is converted to an array only if it is a tuple or a list, otherwise it is treated as a scalar. In either case, either `x` or its elements must support multiplication and addition both with themselves and with the elements of `c`.

If `c` is a 1-D array, then `p(x)` will have the same shape as `x`. If `c` is multidimensional, then the shape of the result depends on the value of `tensor`. If `tensor` is true the shape will be c.shape1: + x.shape. If `tensor` is false the shape will be c.shape1:. 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 cn. 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 -------- legval2d, leggrid2d, legval3d, leggrid3d

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

Examples --------

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

This function returns the values:

.. math:: p(x,y) = \sum_,j c_,j * L_i(x) * L_j(y)

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

If `c` is a 1-D array a one is implicitly appended to its shape to make it 2-D. The shape of the result will be c.shape2: + 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 ``ci,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 Legendre series at points formed from pairs of corresponding values from `x` and `y`.

See Also -------- legval, leggrid2d, legval3d, leggrid3d

Notes -----

.. versionadded:: 1.7.0

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

This function returns the values:

.. math:: p(x,y,z) = \sum_,j,k c_,j,k * L_i(x) * L_j(y) * L_k(z)

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

If `c` has fewer than 3 dimensions, ones are implicitly appended to its shape to make it 3-D. The shape of the result will be c.shape3: + 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 ``ci,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 -------- legval, legval2d, leggrid2d, leggrid3d

Notes -----

.. versionadded:: 1.7.0

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`.

Pseudo-Vandermonde matrix of given degrees.

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

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

where `0 <= i <= deg0` and `0 <= j <= deg1`. 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_

, c_

, c_

... , c_

, c_

, c_

...

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 = (deg0+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

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_

, c_

, c_

,... , c_

, c_

, c_

,...

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 = (deg0+1)*(deg(1+1)*(deg2+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

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

normalize_axis_index(axis, ndim, msg_prefix=None)

Normalizes an axis index, `axis`, such that is a valid positive index into the shape of array with `ndim` dimensions. Raises an AxisError with an appropriate message if this is not possible.

Used internally by all axis-checking logic.

.. versionadded:: 1.13.0

Parameters ---------- axis : int The un-normalized index of the axis. Can be negative ndim : int The number of dimensions of the array that `axis` should be normalized against msg_prefix : str A prefix to put before the message, typically the name of the argument

Returns ------- normalized_axis : int The normalized axis index, such that `0 <= normalized_axis < ndim`

Raises ------ AxisError If the axis index is invalid, when `-ndim <= axis < ndim` is false.

Examples -------- >>> normalize_axis_index(0, ndim=3) 0 >>> normalize_axis_index(1, ndim=3) 1 >>> normalize_axis_index(-1, ndim=3) 2

>>> normalize_axis_index(3, ndim=3) Traceback (most recent call last): ... AxisError: axis 3 is out of bounds for array of dimension 3 >>> normalize_axis_index(-4, ndim=3, msg_prefix='axes_arg') Traceback (most recent call last): ... AxisError: axes_arg: axis -4 is out of bounds for array of dimension 3

Convert a polynomial to a Legendre 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 Legendre 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 Legendre series.

See Also -------- leg2poly

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

Examples -------- >>> from numpy import polynomial as P >>> p = P.Polynomial(np.arange(4)) >>> p Polynomial(0., 1., 2., 3., domain=-1, 1, window=-1, 1) >>> c = P.Legendre(P.legendre.poly2leg(p.coef)) >>> c Legendre( 1. , 3.25, 1. , 0.75, domain=-1, 1, window=-1, 1) # may vary