ChebyshevApprox.chebyshev_derivativeMethod

Computes the first derivative of a complete Chebyshev polynomial with respect to the i'th variable, 'pos', evaluated at 'x', given the polynomial weights, 'weights', the polynomial's order, 'order', and the domain, 'domain'.

weights –- an N-dimensional array containinbg the polynomial weights.

x –- a vector containing the point at which to evaluate the derivative.

pos –- an integer specifying the variable that the function is being differentiated with respect to.

order –- an integer specifying the polynomial's maximal order along each spacial dimension.

domain –- a matrix containing the upper and lower bounds on the domain for each spacial dimension.

ChebyshevApprox.chebyshev_derivativeMethod

Computes the first derivative of a tensor-product Chebyshev polynomial with respect to the i'th variable, 'pos', evaluated at 'x', given the polynomial weights, 'weights', the polynomial's order, 'order', and the domain, 'domain'.

weights –- an N-dimensional array containinbg the polynomial weights.

x –- a vector containing the point at which to evaluate the derivative.

pos –- an integer specifying the variable that the function is being differentiated with respect to.

order –- a tuple or vector specifying the polynomial's order along each spacial dimension.

domain –- a matrix containing the upper and lower bounds on the domain for each spacial dimension.

ChebyshevApprox.chebyshev_evaluateMethod

Evaluate a complete Chebyshev polynomial at point, 'x', given the polynomial weights, 'weights', the order of the polynomial, 'order', and the domain, 'domain'.

weights –- an N-dimensional array containing the polynomial weights.

x –- a vector of Numbers representing the evaluation point.

order –- an integer specifying the polynomial's maximal order along each spacial dimension.

domain –- a matrix containing the upper and lower bounds on the domain for each spacial dimension.

ChebyshevApprox.chebyshev_evaluateMethod

Evaluate a tensor-product Chebyshev polynomial at point, 'x', given the polynomial weights, 'weights', the order of the polynomial, 'order', and the domain, 'domain'.

weights –- an N-dimensional array containing the polynomial weights.

x –- a vector of Numbers representing the evaluation point.

order –- a tuple or vector specifying the polynomial's order along each spacial dimension.

domain –- a matrix containing the upper and lower bounds on the domain for each spacial dimension.

ChebyshevApprox.chebyshev_extendedMethod

Compute 'N' extended Chebyshev points and scale the points to the interval given in 'domain'.

N –- an integer specifying the number of extended Chebyshev points.

domain –- (with with default [1.0,-1.0]) is a 2-element vector specifying the upper and lower bounds on the domain.

ChebyshevApprox.chebyshev_extremaMethod

Compute 'N' extrema of the Chebyshev polynomial and scale the roots to the interval given in 'domain'.

N –- an integer specifying the number of Chebyshev extrema.

domain –- (with with default [1.0,-1.0]) is a 2-element vector specifying the upper and lower bounds on the domain.

ChebyshevApprox.chebyshev_gradientMethod

Creates a function that evaluates the gradient of a Chebyshev polynomial evaluated at 'x', given an approximation plan, 'plan'.

y –- an N-dimensional array containing function evaluated on the approximation grid.

plan –- a structure specifying how the approximation should be undertaken.

ChebyshevApprox.chebyshev_gradientMethod

Computes the gradient of a complete Chebyshev polynomial evaluated at 'x', given the polynomial weights, 'weights', the polynomial's order, 'order', and the domain, 'domain'.

weights –- an N-dimensional array containinbg the polynomial weights.

x –- a vector containing the point at which to evaluate the derivative.

order –- an integer specifying the polynomial's maximal order along each spacial dimension.

domain –- a matrix containing the upper and lower bounds on the domain for each spacial dimension.

ChebyshevApprox.chebyshev_gradientMethod

Computes the gradient of a tensor-product Chebyshev polynomial evaluated at 'x', given the polynomial weights, 'weights', the polynomial's order, 'order', and the domain, 'domain'.

weights –- an N-dimensional array containinbg the polynomial weights.

x –- a vector containing the point at which to evaluate the derivative.

order –- a tuple or vector specifying the polynomial's order along each spacial dimension.

domain –- a matrix containing the upper and lower bounds on the domain for each spacial dimension.

ChebyshevApprox.chebyshev_gradient_threadedMethod

Creates a function that evaluates using multi-threading the gradient of a Chebyshev polynomial evaluated at 'x', given an approximation plan, 'plan'.

y –- an N-dimensional array containing function evaluated on the approximation grid.

plan –- a structure specifying how the approximation should be undertaken.

ChebyshevApprox.chebyshev_hessianMethod

Creates a function that evaluates the hessian of a Chebyshev polynomial evaluated at 'x', given an approximation plan, 'plan'.

y –- an N-dimensional array containing function evaluated on the approximation grid.

plan –- a structure specifying how the approximation should be undertaken.

ChebyshevApprox.chebyshev_hessianMethod

Computes the hessian of a complete Chebyshev polynomial evaluated at 'x', given the polynomial weights, 'weights', the polynomial's order, 'order', and the domain, 'domain'.

weights –- an N-dimensional array containinbg the polynomial weights.

x –- a vector containing the point at which to evaluate the derivative.

order –- a tuple or vector specifying the polynomial's order along each spacial dimension.

domain –- a matrix containing the upper and lower bounds on the domain for each spacial dimension.

ChebyshevApprox.chebyshev_hessianMethod

Computes the hessian of a tensor-product Chebyshev polynomial evaluated at 'x', given the polynomial weights, 'weights', the polynomial's order, 'order', and the domain, 'domain'.

weights –- an N-dimensional array containinbg the polynomial weights.

x –- a vector containing the point at which to evaluate the derivative.

order –- a tuple or vector specifying the polynomial's order along each spacial dimension.

domain –- a matrix containing the upper and lower bounds on the domain for each spacial dimension.

ChebyshevApprox.chebyshev_hessian_threadedMethod

Creates a function that uses multi-threading to evaluate the hessian of a Chebyshev polynomial evaluated at 'x', given an approximation plan, 'plan'.

y –- an N-dimensional array containing function evaluated on the approximation grid.

plan –- a structure specifying how the approximation should be undertaken.

ChebyshevApprox.chebyshev_interpMethod

Creates an interpolating function given an approximation plan, 'plan', that can evaluate the Chebyshev polynomial at any point in the state-space.

y –- an N-dimensional array containing the function evaluated at each point of the approximation grid.

plan –- a structure specifying how the approximation should be undertaken.

ChebyshevApprox.chebyshev_interp_threadedMethod

Creates an interpolating function given an approximation plan, 'plan', that can evaluate using multi-threading the Chebyshev polynomial at any point in the state-space.

y –- an N-dimensional array containing the function evaluated at each point of the approximation grid.

plan –- a structure specifying how the approximation should be undertaken.

ChebyshevApprox.chebyshev_nodesMethod

Compute 'N' roots of the Chebyshev polynomial and scale the roots to the interval given in 'domain'.

N –- an integer specifying the number of Chebyshev roots.

domain –- (with with default [1.0,-1.0]) is a 2-element vector specifying the upper and lower bounds on the domain.

ChebyshevApprox.chebyshev_weightsMethod

Computes the Chebyshev weights given the approximation sample, 'y', and the approximation plan, 'plan'.

y –- a N-dimensional array of data evaluated at the approximation points.

plan –- a structure specifying how the approximation should be undertaken.

ChebyshevApprox.chebyshev_weightsMethod

Computes the Chebyshev weights in a complete polynomial given the data sample, 'f', the Chebyshev polynomials evaluated at the Chebyshev roots, 'poly', and the order of the polynomial, 'order'.

f –- an array containing the function evaluated on the approximation grid.

poly –- a tuple of matrices containing the Chebyshev polynomials evaluated at the Chebyshev roots.

order –- an integer specifying the polynomial's maximal order along each spacial dimension.

ChebyshevApprox.chebyshev_weightsMethod

Computes the Chebyshev weights in a tensor-product polynomial given the data sample, 'f', the Chebyshev polynomials evaluated at the Chebyshev roots, 'poly', and the order of the polynomial, 'order'.

f –- an array containing the function evaluated on the approximation grid.

poly –- a tuple of matrices containing the Chebyshev polynomials evaluated at the Chebyshev roots.

order –- a tuple of a vector specifying the polynomial's order for each spacial dimension.

ChebyshevApprox.chebyshev_weightsMethod

Computes the Chebyshev weights in a complete polynomial given the data sample, 'f', the Chebyshev roots, 'nodes', the order of the polynomial, 'order', and the domain for the sampling points, 'domain'.

f –- an array containing the function evaluated on the approximation grid.

nodes –- a tuple of vectors containing the Chebyshev nodes used for approximation along each spacial dimension.

order –- an integer specifying the polynomial's maximal order along each spacial dimension.

domain –- a matrix containing the upper and lower bounds on the domain for each spacial dimension.

ChebyshevApprox.chebyshev_weightsMethod

Computes the Chebyshev weights in a tensor-product polynomial given the data sample, 'f', the Chebyshev roots, 'nodes', the order of the polynomial, 'order', and the domain for the sampling points, 'domain'.

f –- an array containing the function evaluated on the approximation grid.

nodes –- a tuple of vectors containing the Chebyshev nodes used for approximation along each spacial dimension.

order –- a tuple of a vector specifying the polynomial's order for each spacial dimension.

domain –- a matrix containing the upper and lower bounds on the domain for each spacial dimension.

ChebyshevApprox.chebyshev_weights_extendedMethod

Computes the Chebyshev weights in a complete polynomial given the data sample, 'f', the Chebyshev polynomials evaluated at the extended Chebyshev points, 'poly', and the order of the polynomial, 'order'.

f –- an array containing the function evaluated on the approximation grid.

poly –- a tuple of matrices containing the Chebyshev polynomials evaluated at the extended Chebyshev points.

order –- an integer specifying the polynomial's maximal order along each spacial dimension.

ChebyshevApprox.chebyshev_weights_extendedMethod

Computes the Chebyshev weights in a tensor-product polynomial given the data sample, 'f', the Chebyshev polynomials evaluated at the extended Chebyshev points, 'poly', and the order of the polynomial, 'order'.

f –- an array containing the function evaluated on the approximation grid.

poly –- a tuple of matrices containing the Chebyshev polynomials evaluated at the extended Chebyshev points.

order –- a tuple of a vector specifying the polynomial's order for each spacial dimension.

ChebyshevApprox.chebyshev_weights_extendedMethod

Computes the Chebyshev weights in a complete polynomial given the data sample, 'f', the extended Chebyshev points, 'nodes', the order of the polynomial, 'order', and the domain for the sampling points, 'domain'.

f –- an array containing the function evaluated on the approximation grid.

nodes –- a tuple of vectors containing the extended Chebyshev points used for approximation along each spacial dimension.

order –- an integer specifying the polynomial's maximal order along each spacial dimension.

domain –- a matrix containing the upper and lower bounds on the domain for each spacial dimension.

ChebyshevApprox.chebyshev_weights_extendedMethod

Computes the Chebyshev weights in a tensor-product polynomial given the data sample, 'f', the extended Chebyshev points, 'nodes', the order of the polynomial, 'order', and the domain for the sampling points, 'domain'.

f –- an array containing the function evaluated on the approximation grid.

nodes –- a tuple of vectors containing the extended Chebyshev points used for approximation along each spacial dimension.

order –- a tuple of a vector specifying the polynomial's order for each spacial dimension.

domain –- a matrix containing the upper and lower bounds on the domain for each spacial dimension.

ChebyshevApprox.chebyshev_weights_extended_threadedMethod

Computes the Chebyshev weights using multi-threading in a complete polynomial given the data sample, 'f', the Chebyshev polynomials evaluated at the extended Chebyshev points, 'poly', and the order of the polynomial, 'order'.

f –- an array containing the function evaluated on the approximation grid.

poly –- a tuple of matrices containing the Chebyshev polynomials evaluated at the extended Chebyshev points.

order –- an integer specifying the polynomial's maximal order along each spacial dimension.

ChebyshevApprox.chebyshev_weights_extended_threadedMethod

Computes the Chebyshev weights using multi-threading in a tensor-product polynomial given the data sample, 'f', the Chebyshev polynomials evaluated at the extended Chebyshev points, 'poly', and the order of the polynomial, 'order'.

f –- an array containing the function evaluated on the approximation grid.

poly –- a tuple of matrices containing the Chebyshev polynomials evaluated at the Chebyshev roots.

order –- an integer specifying the polynomial's maximal order along each spacial dimension.

ChebyshevApprox.chebyshev_weights_extended_threadedMethod

Computes the Chebyshev weights using multi-threading in a complete polynomial given the data sample, 'f', the Chebyshev polynomials evaluated at the extended Chebyshev points, 'poly', and the order of the polynomial, 'order'.

f –- an array containing the function evaluated on the approximation grid.

poly –- a tuple of matrices containing the Chebyshev polynomials evaluated at the extended Chebyshev points.

order –- an integer specifying the polynomial's maximal order along each spacial dimension.

ChebyshevApprox.chebyshev_weights_extended_threadedMethod

Computes the Chebyshev weights using mutli-threading in a complete polynomial given the data sample, 'f', the extended Chebyshev points, 'nodes', the order of the polynomial, 'order', and the domain for the sampling points, 'domain'.

f –- an array containing the function evaluated on the approximation grid.

nodes –- a tuple of vectors containing the extended Chebyshev points used for approximation along each spacial dimension.

order –- a vector of integers specifying the polynomial's order along each spacial dimension.

domain –- a matrix containing the upper and lower bounds on the domain for each spacial dimension.

ChebyshevApprox.chebyshev_weights_extremaMethod

Computes the Chebyshev weights in a complete polynomial given the data sample, 'f', the Chebyshev polynomials evaluated at the Chebyshev extrema, 'poly', and the order of the polynomial, 'order'.

f –- an array containing the function evaluated on the approximation grid.

poly –- a tuple of matrices containing the Chebyshev polynomials evaluated at the Chebyshev extrema.

order –- an integer specifying the polynomial's maximal order along each spacial dimension.

ChebyshevApprox.chebyshev_weights_extremaMethod

Computes the Chebyshev weights in a tensor-product polynomial given the data sample, 'f', the Chebyshev polynomials evaluated at the Chebyshev extrema, 'poly', and the order of the polynomial, 'order'.

f –- an array containing the function evaluated on the approximation grid.

poly –- a tuple of matrices containing the Chebyshev polynomials evaluated at the Chebyshev extrema.

order –- a tuple of a vector specifying the polynomial's order for each spacial dimension.

ChebyshevApprox.chebyshev_weights_extremaMethod

Computes the Chebyshev weights in a complete polynomial given the data sample, 'f', the Chebyshev extrema, 'nodes', the order of the polynomial, 'order', and the domain for the sampling points, 'domain'.

f –- an array containing the function evaluated on the approximation grid.

nodes –- a tuple of vectors containing the Chebyshev extrema used for approximation along each spacial dimension.

order –- an integer specifying the polynomial's maximal order along each spacial dimension.

domain –- a matrix containing the upper and lower bounds on the domain for each spacial dimension.

ChebyshevApprox.chebyshev_weights_extremaMethod

Computes the Chebyshev weights in a tensor-product polynomial given the data sample, 'f', the Chebyshev extrema, 'nodes', the order of the polynomial, 'order', and the domain for the sampling points, 'domain'.

f –- an array containing the function evaluated on the approximation grid.

nodes –- a tuple of vectors containing the Chebyshev extrema used for approximation along each spacial dimension.

order –- a tuple of a vector specifying the polynomial's order for each spacial dimension.

domain –- a matrix containing the upper and lower bounds on the domain for each spacial dimension.

ChebyshevApprox.chebyshev_weights_extrema_threadedMethod

Computes the Chebyshev weights using multi-threading in a complete polynomial given the data sample, 'f', the Chebyshev polynomials evaluated at the Chebyshev extrema, 'poly', and the order of the polynomial, 'order'.

f –- an array containing the function evaluated on the approximation grid.

poly –- a tuple of matrices containing the Chebyshev polynomials evaluated at the Chebyshev extrema.

order –- an integer specifying the polynomial's maximal order along each spacial dimension.

ChebyshevApprox.chebyshev_weights_extrema_threadedMethod

Computes the Chebyshev weights using multi-threading in a tensor-product polynomial given the data sample, 'f', the Chebyshev polynomials evaluated at the Chebyshev extrema, 'poly', and the order of the polynomial, 'order'.

f –- an array containing the function evaluated on the approximation grid.

poly –- a tuple of matrices containing the Chebyshev polynomials evaluated at the Chebyshev extrema.

order –- an integer specifying the polynomial's maximal order along each spacial dimension.

ChebyshevApprox.chebyshev_weights_extrema_threadedMethod

Computes the Chebyshev weights using multi-threading in a complete polynomial given the data sample, 'f', the Chebyshev polynomials evaluated at the Chebyshev extrema, 'poly', and the order of the polynomial, 'order'.

f –- an array containing the function evaluated on the approximation grid.

poly –- a tuple of matrices containing the Chebyshev polynomials evaluated at the Chebyshev extrema.

order –- an integer specifying the polynomial's maximal order along each spacial dimension.

ChebyshevApprox.chebyshev_weights_extrema_threadedMethod

Computes the Chebyshev weights using mutli-threading in a complete polynomial given the data sample, 'f', the Chebyshev extrema, 'nodes', the order of the polynomial, 'order', and the domain for the sampling points, 'domain'.

f –- an array containing the function evaluated on the approximation grid.

nodes –- a tuple of vectors containing the Chebyshev extrema used for approximation along each spacial dimension.

order –- a vector of integers specifying the polynomial's order along each spacial dimension.

domain –- a matrix containing the upper and lower bounds on the domain for each spacial dimension.

ChebyshevApprox.chebyshev_weights_threadedMethod

Computes the Chebyshev weights using multi-threading given the approximation sample, 'y', and the approximation plan, 'plan'.

y –- a N-dimensional array of data evaluated at the approximation points.

plan –- a structure specifying how the approximation should be undertaken.

ChebyshevApprox.chebyshev_weights_threadedMethod

Computes the Chebyshev weights using multi-threading in a complete polynomial given the data sample, 'f', the Chebyshev polynomials evaluated at the Chebyshev roots, 'poly', and the order of the polynomial, 'order'.

f –- an array containing the function evaluated on the approximation grid.

poly –- a tuple of matrices containing the Chebyshev polynomials evaluated at the Chebyshev roots.

order –- an integer specifying the polynomial's maximal order along each spacial dimension.

ChebyshevApprox.chebyshev_weights_threadedMethod

Computes the Chebyshev weights using multi-threading in a tensor-product polynomial given the data sample, 'f', the Chebyshev polynomials evaluated at the Chebyshev roots, 'poly', and the order of the polynomial, 'order'.

f –- an array containing the function evaluated on the approximation grid.

poly –- a tuple of matrices containing the Chebyshev polynomials evaluated at the Chebyshev roots.

order –- an integer specifying the polynomial's maximal order along each spacial dimension.

ChebyshevApprox.chebyshev_weights_threadedMethod

Computes the Chebyshev weights using multi-threading in a complete polynomial given the data sample, 'f', the Chebyshev polynomials evaluated at the Chebyshev roots, 'poly', and the order of the polynomial, 'order'.

f –- an array containing the function evaluated on the approximation grid.

poly –- a tuple of matrices containing the Chebyshev polynomials evaluated at the Chebyshev roots.

order –- an integer specifying the polynomial's maximal order along each spacial dimension.

ChebyshevApprox.chebyshev_weights_threadedMethod

Computes the Chebyshev weights using mutli-threading in a complete polynomial given the data sample, 'f', the Chebyshev roots, 'nodes', the order of the polynomial, 'order', and the domain for the sampling points, 'domain'.

f –- an array containing the function evaluated on the approximation grid.

nodes –- a tuple of vectors containing the Chebyshev roots used for approximation along each spacial dimension.

order –- a vector of integers specifying the polynomial's order along each spacial dimension.

domain –- a matrix containing the upper and lower bounds on the domain for each spacial dimension.

ChebyshevApprox.nodesMethod

Computes 'N' points according to 'node_generator' and scales those points to the interval given in 'domain'.

N –- an integer specifying the number of points.

node_generator –- a function that computes the desired points.

domain –- (with with default [1.0,-1.0]) is a 2-element vector specifying the upper and lower bounds on the domain.

ChebyshevApprox.normalize_nodeMethod

Normalizes a point, 'node', in 'domain' to the [1.0,-1.0] interval.

node –- a number.

domain –- (with with default [1.0,-1.0]) is a 2-element vector specifying the upper and lower bounds on the domain.

ChebyshevApprox.normalize_nodeMethod

Normalizes a vector of points, 'node', with in element in 'domain' to the [1.0,-1.0] interval.

node –- a vector of numbers.

domain –- (with with default [1.0,-1.0]) is a 2-element vector specifying the upper and lower bounds on the domain.