sgpykit.tools.polynomials_functions

Functions

cheb_eval(x, k, a, b)	Evaluate the k-th Chebyshev polynomial of the first kind on [a,b].
cheb_eval_multidim(X, k, a, b)	Evaluate the multidimensional Chebyshev polynomial of the first kind of order k (multi-index) on [a,b]^N.
generalized_lagu_eval(x, k, alpha, beta)	Evaluate the k-th generalized Laguerre polynomial orthonormal in [0, +inf).
generalized_lagu_eval_multidim(X, k, alpha, beta)	Evaluate the multidimensional generalized Laguerre polynomial of order k (multi-index) orthonormal on [0, +inf)^N with respect to the weight function: rho = prod_i beta_i^(alpha_i+1)/Gamma(alpha_i+1) * x^alpha_i * exp(-beta_i * x), where alpha_i > -1 and beta_i > 0.
herm_eval(x, k, mi, sigma)	Evaluate the k-th orthonormal Hermite polynomial at points x.
herm_eval_multidim(X, k, mu, sig)	Evaluate the multidimensional Hermite polynomial of order k (multi-index) orthonormal on [-inf, +inf]^N with respect to the Gaussian weight function.
jacobi_prob_eval(x, k, alpha, beta, a, b)	Evaluate the k-th Jacobi probabilistic polynomial orthonormal in [a,b].
jacobi_prob_eval_multidim(X, k, alpha, beta, ...)	Evaluate the multidimensional probabilistic Jacobi polynomial of order k (multi-index) orthonormal on [a1,b1] x [a2,b2] x [a3,b3] x .
lagr_eval(current_knot, other_knots, ...)	Evaluate the Lagrange basis polynomial at non-grid points.
lagr_eval_fast(current_knot, other_knots, ...)	Evaluate the Lagrange basis polynomial at non-grid points.
lagr_eval_multidim(current_knot, ...)	Evaluate the multidimensional Lagrange function centered at current_knot in non_grid_points.
lagu_eval(x, k, lambda_)	Evaluate the k-th Laguerre polynomial orthonormal in [0, +inf) w.r.t.
lagu_eval_multidim(X, k, lambda_)	Evaluate the multidimensional Laguerre polynomial of order k (multi-index) orthonormal on [0, +inf)^N with respect to rho=prod_i lambda_i*exp(-lambda_i*x), lambda_i>0 on the list of points X (each column is a point in R^N).
lege_eval(x, k, a, b)	Evaluate the k-th orthonormal Legendre polynomial on the interval [a, b].
lege_eval_multidim(X, k, a, b)	Evaluate the multidimensional Legendre polynomial of order k (multi-index) orthonormal on [a,b]^N on the list of points X (each column is a point in R^N).
standard_lege_eval(x, k)	Evaluate the k-th standard Legendre polynomial.
univariate_interpolant(grid_points, ...)	Interpolates values at non_grid_points using a Lagrange interpolant built from given grid_points and tabulated function_on_grid.

sgpykit.tools.polynomials_functions.cheb_eval(x, k, a, b)

Evaluate the k-th Chebyshev polynomial of the first kind on [a,b].

Parameters:

xarray_like

Points at which to evaluate the Chebyshev polynomial.

kint

Degree of the Chebyshev polynomial.

afloat

Lower bound of the interval.

bfloat

Upper bound of the interval.

Returns:

Lndarray

Values of the k-th Chebyshev polynomial evaluated at x.

sgpykit.tools.polynomials_functions.cheb_eval_multidim(X, k, a, b)

Evaluate the multidimensional Chebyshev polynomial of the first kind of order k (multi-index) on [a,b]^N.

Evaluates the multidim Chebyshev polynomial of the first kind of order k (multi-index) on [a,b]^N on the list of points X (each column is a point in R^N). a,b are scalar values a,b may differ in each direction, so that actually the domain is not [a,b]^N, but [a1,b1] x [a2,b2] x [a3,b3] x … [aN,bN] in this case, a,b are defined as vectors, each with N components

Parameters:

Xnumpy.ndarray

Array of points, shape (N, nb_pts), where each column is a point in R^N.

knumpy.ndarray

Multi-index of the polynomial order, shape (N,).

anumpy.ndarray or float

Lower bounds of the domain. If scalar, it is expanded to a vector of length N.

bnumpy.ndarray or float

Upper bounds of the domain. If scalar, it is expanded to a vector of length N.

Returns:

Cnumpy.ndarray

Evaluated polynomial values at each point, shape (nb_pts,).

sgpykit.tools.polynomials_functions.generalized_lagu_eval(x, k, alpha, beta)

Evaluate the k-th generalized Laguerre polynomial orthonormal in [0, +inf).

This function returns the values of the k-th generalized Laguerre polynomial orthonormal in [0, +inf) with respect to the weight function rho = beta^(alpha+1)/Gamma(alpha+1) * x^alpha * exp(-beta*x) at the points x. The polynomials start from k=0: L_0(x) = 1, L_1(x) = -x*beta + alpha + 1. This function expresses L as a function of the standard generalized Laguerre “probabilistic” polynomial (i.e., orthogonal w.r.t. rho = x^alpha * exp(-x)/Gamma(alpha+1)), which are recursively calculated through the function standard_generalized_lagu_eval.

Parameters:

xarray_like

Points at which to evaluate the polynomial. Can be a matrix.

kint

Degree of the polynomial.

alphafloat

Parameter of the generalized Laguerre polynomial.

betafloat

Parameter of the generalized Laguerre polynomial.

Returns:

Lndarray

Values of the k-th generalized Laguerre polynomial at the points x.

sgpykit.tools.polynomials_functions.generalized_lagu_eval_multidim(X, k, alpha, beta)

Evaluate the multidimensional generalized Laguerre polynomial of order k (multi-index) orthonormal on [0, +inf)^N with respect to the weight function: rho = prod_i beta_i^(alpha_i+1)/Gamma(alpha_i+1) * x^alpha_i * exp(-beta_i * x), where alpha_i > -1 and beta_i > 0.

Parameters:

Xnumpy.ndarray

Array of points where the polynomial is evaluated. Each column is a point in R^N.

knumpy.ndarray

Multi-index defining the polynomial order.

alphanumpy.ndarray or scalar

Parameter alpha for the generalized Laguerre polynomial. Can be scalar or array.

betanumpy.ndarray or scalar

Parameter beta for the generalized Laguerre polynomial. Can be scalar or array.

Returns:

Lnumpy.ndarray

Array of polynomial evaluations at the given points.

sgpykit.tools.polynomials_functions.herm_eval(x, k, mi, sigma)

Evaluate the k-th orthonormal Hermite polynomial at points x.

This function returns the values of the k-th Hermite polynomial orthonormal in (-inf, +inf) with respect to the weight function rho = 1/sqrt(2 pi sigma^2) * exp(-(x-mi)^2/(2*sigma^2)) at the points x (x can be a matrix as well).

The polynomials start from k=0: H_0(x) = 1, H_1(x) = (x - mi)/sigma. This function expresses H as a function of the standard Hermite “probabilistic” polynomial (i.e., orthogonal with respect to rho=1/sqrt(2 pi) * exp(-x^2/2)), which are recursively calculated through the function standard_herm_eval.

Parameters:

xarray_like

Points at which to evaluate the polynomial.

kint

Degree of the Hermite polynomial.

mifloat

Mean of the Gaussian distribution.

sigmafloat

Standard deviation of the Gaussian distribution.

Returns:

Hndarray

Values of the k-th orthonormal Hermite polynomial at points x.

sgpykit.tools.polynomials_functions.herm_eval_multidim(X, k, mu, sig)

Evaluate the multidimensional Hermite polynomial of order k (multi-index) orthonormal on [-inf, +inf]^N with respect to the Gaussian weight function.

Parameters:

Xnumpy.ndarray

Array of points where the polynomial is evaluated. Each column is a point in R^N.

knumpy.ndarray

Multi-index defining the order of the polynomial in each dimension.

mufloat or numpy.ndarray

Mean of the Gaussian weight function. Can be a scalar or a vector of length N.

sigfloat or numpy.ndarray

Standard deviation of the Gaussian weight function. Can be a scalar or a vector of length N.

Returns:

Hnumpy.ndarray

Array of polynomial evaluations at the points in X.

sgpykit.tools.polynomials_functions.jacobi_prob_eval(x, k, alpha, beta, a, b)

Evaluate the k-th Jacobi probabilistic polynomial orthonormal in [a,b].

This function returns the values of the k-th Jacobi “probabilistic” polynomial orthonormal in [a,b] with respect to the weight function: rho(x) = Gamma(alpha+beta+2)/(Gamma(alpha+1)*Gamma(beta+1)*(b-a)^(alpha+beta+1)) * (x-a)^alpha * (b-x)^beta in the points x (x can be a matrix as well).

Note that these are NOT the classical Jacobi polynomials (which are orthonormal in [a,b] w.r.t. (b-x)^alpha*(x-a)^beta).

Parameters:

xarray_like

Points at which to evaluate the polynomial.

kint

Degree of the polynomial.

alphafloat

Parameter of the Jacobi polynomial.

betafloat

Parameter of the Jacobi polynomial.

afloat

Lower bound of the interval.

bfloat

Upper bound of the interval.

Returns:

Lndarray

Values of the k-th Jacobi probabilistic polynomial at points x.

Notes

The polynomials start from k=0: L_0(x) = 1, L_1(x) = 0.5*(alpha-beta) + 0.5*(alpha+beta+2)*(-2*x+a+b)/(b-a)

This function expresses L as a function of the standard Jacobi polynomial (i.e. orthoGONAL w.r.t. rho(x)=(1-x)^alpha*(1+x)^beta, which are recursively calculated through the function standard_jac_eval.

sgpykit.tools.polynomials_functions.jacobi_prob_eval_multidim(X, k, alpha, beta, a, b)

Evaluate the multidimensional probabilistic Jacobi polynomial of order k (multi-index) orthonormal on [a1,b1] x [a2,b2] x [a3,b3] x … [aN,bN] with respect to rho=prod_i Gamma(alpha_i+beta_i+2)/(Gamma(alpha_i+1)*Gamma(beta_i+1)*(b-a)^(alpha_i+beta_i+1))*(x-a)^alpha_i*(b-x)^beta_i, alpha_i,beta_i>-1 on the list of points X (each column is a point in R^N).

Parameters:

Xnumpy.ndarray

Array of points, shape (N, nb_pts), where N is the dimension and nb_pts is the number of points.

knumpy.ndarray

Multi-index of the polynomial, shape (N,).

alphanumpy.ndarray or float

Parameter alpha for the Jacobi polynomial, can be scalar or array of shape (N,).

betanumpy.ndarray or float

Parameter beta for the Jacobi polynomial, can be scalar or array of shape (N,).

anumpy.ndarray or float

Lower bound of the interval, can be scalar or array of shape (N,).

bnumpy.ndarray or float

Upper bound of the interval, can be scalar or array of shape (N,).

Returns:

Lnumpy.ndarray

Evaluated polynomial values, shape (nb_pts,).

sgpykit.tools.polynomials_functions.lagr_eval(current_knot, other_knots, non_grid_points)

Evaluate the Lagrange basis polynomial at non-grid points.

Builds the Lagrange function L(x) such that: L(current_knot) = 1 L(other_knots) = 0

and returns L = L(non_grid_points)

Parameters:

current_knotfloat

The knot where the Lagrange function should be 1.

other_knotsarray_like

A list or array of knots where the Lagrange function should be 0.

non_grid_pointsarray_like

The points at which to evaluate the Lagrange function.

Returns:

Lndarray

The evaluated Lagrange function at non_grid_points.

sgpykit.tools.polynomials_functions.lagr_eval_fast(current_knot, other_knots, ok_len, non_grid_points, ng_size)

Evaluate the Lagrange basis polynomial at non-grid points.

This function builds the Lagrange function L(x) such that L(current_knot)=1 and L(other_knots)=0, and returns L evaluated at non_grid_points.

Parameters:

current_knotfloat

The knot where the Lagrange polynomial equals 1.

other_knotsarray_like

The knots where the Lagrange polynomial equals 0.

ok_lenint

The length of other_knots.

non_grid_pointsarray_like

The points at which to evaluate the Lagrange polynomial.

ng_sizeint or tuple

The size of non_grid_points. If scalar, it is treated as the size of a square matrix. If tuple, it is treated as the shape of non_grid_points.

Returns:

Lndarray

The evaluated Lagrange polynomial at non_grid_points.

sgpykit.tools.polynomials_functions.lagr_eval_multidim(current_knot, knots_per_dim, non_grid_points)

Evaluate the multidimensional Lagrange function centered at current_knot in non_grid_points.

Parameters:

current_knotarray_like

The knot where the Lagrange function is centered.

knots_per_dimlist or array_like

List of arrays, where each array contains all the knots in each dimension.

non_grid_pointsarray_like

Matrix where each row is a point to be evaluated.

Returns:

Lndarray

The evaluated multidimensional Lagrange function at non_grid_points.

sgpykit.tools.polynomials_functions.lagu_eval(x, k, lambda_)

Evaluate the k-th Laguerre polynomial orthonormal in [0, +inf) w.r.t. rho=lambda*exp(-lambda*x).

This function computes the values of the k-th Laguerre polynomial, which is orthonormal with respect to the weight function rho = lambda * exp(-lambda * x) on the interval [0, +inf). The polynomials are defined starting from k=0, with L_0(x) = 1 and L_1(x) = -x*lambda + 1.

Parameters:

xarray_like

Points at which to evaluate the polynomial (can be a matrix).

kint

Degree of the Laguerre polynomial.

lambda_float

Parameter of the weight function (must be positive).

Returns:

Lndarray

Values of the k-th Laguerre polynomial at the points x.

sgpykit.tools.polynomials_functions.lagu_eval_multidim(X, k, lambda_)

Evaluate the multidimensional Laguerre polynomial of order k (multi-index) orthonormal on [0, +inf)^N with respect to rho=prod_i lambda_i*exp(-lambda_i*x), lambda_i>0 on the list of points X (each column is a point in R^N). Lambda can be a scalar value.

Parameters:

Xnumpy.ndarray

Array of points, shape (N, nb_pts), where N is the dimension and nb_pts is the number of points.

knumpy.ndarray

Multi-index of the polynomial, shape (N,).

lambda_float or numpy.ndarray

Scale parameter(s) for the Laguerre polynomial. If scalar, it is expanded to a vector of length N.

Returns:

Lnumpy.ndarray

Evaluated polynomial values, shape (nb_pts,).

sgpykit.tools.polynomials_functions.lege_eval(x, k, a, b)

Evaluate the k-th orthonormal Legendre polynomial on the interval [a, b].

This function computes the values of the k-th Legendre polynomial, which is orthonormal with respect to the uniform weight function rho = 1/(b-a), at the points specified by x. The polynomials are defined starting from k=0: L_0(x) = 1, L_1(x) = (2*x - a - b)/(b - a).

The computation is performed by transforming the input points from the interval [a, b] to the standard interval [-1, 1], evaluating the standard Legendre polynomial (orthogonal with respect to rho=1) at these transformed points, and then applying the necessary normalizations to account for the interval [a, b].

Parameters:

xarray_like

The points at which to evaluate the polynomial. Can be a scalar, list, or ndarray.

kint

The degree of the Legendre polynomial.

afloat

The lower bound of the interval.

bfloat

The upper bound of the interval.

Returns:

Lndarray

The values of the k-th orthonormal Legendre polynomial evaluated at x.

sgpykit.tools.polynomials_functions.lege_eval_multidim(X, k, a, b)

Evaluate the multidimensional Legendre polynomial of order k (multi-index) orthonormal on [a,b]^N on the list of points X (each column is a point in R^N). a,b are scalar values.

Parameters:

Xnumpy.ndarray

Array of points, shape (N, nb_pts), where N is the dimension and nb_pts is the number of points.

knumpy.ndarray

Multi-index of the polynomial, shape (N,).

aint or numpy.ndarray

Lower bound(s) of the domain. If scalar, it is broadcasted to all dimensions.

bint or numpy.ndarray

Upper bound(s) of the domain. If scalar, it is broadcasted to all dimensions.

Returns:

Lnumpy.ndarray

Evaluated polynomial values, shape (nb_pts,).

Notes

a,b may differ in each direction, so that actually the domain is not [a,b]^N, but [a1,b1] x [a2,b2] x [a3,b3] x … [aN,bN] in this case, a,b are defined as vectors, each with N components.

sgpykit.tools.polynomials_functions.standard_lege_eval(x, k)

Evaluate the k-th standard Legendre polynomial.

This function computes the values of the k-th standard Legendre polynomial, which is orthogonal (but not orthonormal) with respect to the uniform weight function rho=1 on the interval [-1, 1], at the points specified by x. The polynomials are defined starting from k=0: L_0(x) = 1, L_1(x) = x.

Parameters:

xarray_like

The points at which to evaluate the polynomial. Can be a scalar, list, or ndarray.

kint

The degree of the Legendre polynomial.

Returns:

Lndarray

The values of the k-th standard Legendre polynomial evaluated at x.

sgpykit.tools.polynomials_functions.univariate_interpolant(grid_points, function_on_grid, non_grid_points)

Interpolates values at non_grid_points using a Lagrange interpolant built from given grid_points and tabulated function_on_grid.

Parameters:

grid_points(M,) array_like

Interpolation nodes.

function_on_grid(V, M) array_like

Values of V different functions sampled at the M grid points.

non_grid_points(N,) array_like

Points where we want to evaluate the interpolation.

Returns:

f_values(V, N) ndarray

Interpolated values at non_grid_points.

Notes

UNIVARIATE_INTERPOLANT interpolates a univariate function on grid points, i.e. evaluates the Lagrange polynomial approximation on a generic point of the parameter space.

F_VALUES = UNIVARIATE_INTERPOLANT(GRID_POINTS,FUNCTION_ON_GRID,NON_GRID_POINTS) evaluates the

lagrangian interpolant of a vector function F: R -> R^V based on the points contained in the vector GRID_POINTS.

GRID POINTS is a row vector with the nodes of interpolation. FUNCTION_ON_GRID is a matrix containing the evaluation of F on the points GRID_POINTS.

Its dimensions are V x length(GRID_POINTS).

NON_GRID_POINTS is a row vector of points where one wants to evaluate the polynomial approximation. F_VALUES is a matrix containing the evaluation of the function F in each of the NON_GRID_POINTS.

Its dimensions are V X length(NON_GRID_POINTS)