numpy.polynomial.hermite

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.

>>> from numpy.polynomial.hermite import herm2poly
>>> herm2poly([ 1.   ,  2.75 ,  0.5  ,  0.375])
array([0., 1., 2., 3.])

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.

>>> from numpy.polynomial.hermite import hermadd
>>> hermadd([1, 2, 3], [1, 2, 3, 4])
array([2., 4., 6., 4.])

Return the scaled companion matrix of c.

The basis polynomials are scaled so that the companion matrix is symmetric when c is an Hermite 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.

Differentiate a Hermite series.

Returns the Hermite 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*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.

>>> from numpy.polynomial.hermite import hermder
>>> hermder([ 1. ,  0.5,  0.5,  0.5])
array([1., 2., 3.])
>>> hermder([-0.5,  1./2.,  1./8.,  1./12.,  1./16.], m=2)
array([1., 2., 3.])

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.

>>> from numpy.polynomial.hermite import hermdiv
>>> hermdiv([ 52.,  29.,  52.,   7.,   6.], [0, 1, 2])
(array([1., 2., 3.]), array([0.]))
>>> hermdiv([ 54.,  31.,  52.,   7.,   6.], [0, 1, 2])
(array([1., 2., 3.]), array([2., 2.]))
>>> hermdiv([ 53.,  30.,  52.,   7.,   6.], [0, 1, 2])
(array([1., 2., 3.]), array([1., 1.]))

Least squares fit of Hermite series to data.

Return the coefficients of a Hermite 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

where n is deg.

>>> from numpy.polynomial.hermite import hermfit, hermval
>>> x = np.linspace(-10, 10)
>>> err = np.random.randn(len(x))/10
>>> y = hermval(x, [1, 2, 3]) + err
>>> hermfit(x, y, 2)
array([1.0218, 1.9986, 2.9999]) # may vary

>>> import warnings
>>> warnings.simplefilter('ignore', np.RankWarning)

Generate a Hermite series with given roots.

The function returns the coefficients of the polynomial

in Hermite 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

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

>>> from numpy.polynomial.hermite import hermfromroots, hermval
>>> coef = hermfromroots((-1, 0, 1))
>>> hermval((-1, 0, 1), coef)
array([0.,  0.,  0.])
>>> coef = hermfromroots((-1j, 1j))
>>> hermval((-1j, 1j), coef)
array([0.+0.j, 0.+0.j])

Gauss-Hermite quadrature.

Computes the sample points and weights for Gauss-Hermite quadrature. These sample points and weights will correctly integrate polynomials of degree 2*deg − 1 or less over the interval [ − inf, inf] with the weight function f(x) = exp( − x²).

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

This function returns the values:

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.

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

This function returns the values:

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.

Integrate a Hermite series.

Returns the Hermite 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.

>>> from numpy.polynomial.hermite import hermint
>>> hermint([1,2,3]) # integrate once, value 0 at 0.
array([1. , 0.5, 0.5, 0.5])
>>> hermint([1,2,3], m=2) # integrate twice, value & deriv 0 at 0
array([-0.5       ,  0.5       ,  0.125     ,  0.08333333,  0.0625    ]) # may vary
>>> hermint([1,2,3], k=1) # integrate once, value 1 at 0.
array([2. , 0.5, 0.5, 0.5])
>>> hermint([1,2,3], lbnd=-1) # integrate once, value 0 at -1
array([-2. ,  0.5,  0.5,  0.5])
>>> hermint([1,2,3], m=2, k=[1,2], lbnd=-1)
array([ 1.66666667, -0.5       ,  0.125     ,  0.08333333,  0.0625    ]) # may vary

Hermite series whose graph is a straight line.

>>> from numpy.polynomial.hermite import hermline, hermval
>>> hermval(0,hermline(3, 2))
3.0
>>> hermval(1,hermline(3, 2))
5.0

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.

>>> from numpy.polynomial.hermite import hermmul
>>> hermmul([1, 2, 3], [0, 1, 2])
array([52.,  29.,  52.,   7.,   6.])

Multiply a Hermite series by x.

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

>>> from numpy.polynomial.hermite import hermmulx
>>> hermmulx([1, 2, 3])
array([2. , 6.5, 1. , 1.5])

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.

>>> from numpy.polynomial.hermite import hermpow
>>> hermpow([1, 2, 3], 2)
array([81.,  52.,  82.,  12.,   9.])

Compute the roots of a Hermite series.

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

>>> from numpy.polynomial.hermite import hermroots, hermfromroots
>>> coef = hermfromroots([-1, 0, 1])
>>> coef
array([0.   ,  0.25 ,  0.   ,  0.125])
>>> hermroots(coef)
array([-1.00000000e+00, -1.38777878e-17,  1.00000000e+00])

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.

>>> from numpy.polynomial.hermite import hermsub
>>> hermsub([1, 2, 3, 4], [1, 2, 3])
array([0.,  0.,  0.,  4.])

Evaluate an Hermite series at points x.

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

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.

>>> from numpy.polynomial.hermite import hermval
>>> coef = [1,2,3]
>>> hermval(1, coef)
11.0
>>> hermval([[1,2],[3,4]], coef)
array([[ 11.,   51.],
       [115.,  203.]])

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

This function returns the values:

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.

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

This function returns the values:

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.

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

where 0 <= i <= deg. The leading indices of V index the elements of x and the last index is the degree of the Hermite polynomial.

If c is a 1-D array of coefficients of length n + 1 and V is the array V = hermvander(x, n), then np.dot(V, c) and hermval(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 Hermite series of the same degree and sample points.

>>> from numpy.polynomial.hermite import hermvander
>>> x = np.array([-1, 0, 1])
>>> hermvander(x, 3)
array([[ 1., -2.,  2.,  4.],
       [ 1.,  0., -2., -0.],
       [ 1.,  2.,  2., -4.]])

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

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 Hermite polynomials.

If V = hermvander2d(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

and np.dot(V, c.flat) and hermval2d(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 Hermite series of the same degrees and sample points.

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

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 Hermite polynomials.

If V = hermvander3d(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

and np.dot(V, c.flat) and hermval3d(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 Hermite series of the same degrees and sample points.

Weight function of the Hermite polynomials.

The weight function is exp( − x²) and the interval of integration is [ − inf, inf]. the Hermite polynomials are orthogonal, but not normalized, with respect to this weight function.

poly2herm(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.

>>> from numpy.polynomial.hermite import poly2herm
>>> poly2herm(np.arange(4))
array([1.   ,  2.75 ,  0.5  ,  0.375])