Public API

prepare(nodes::Matrix{T}, kernel::RK = RK_H0()) where {T <: AbstractFloat, RK <: ReproducingKernel_0}

Prepare the spline by constructing and factoring a Gram matrix of the interpolation problem. Initialize the NormalSpline object.

Arguments

• nodes: The function value nodes. This should be an n×n_1 matrix, where n is dimension of the sampled space and n_1 is the number of function value nodes. It means that each column in the matrix defines one node.
• kernel: reproducing kernel of Bessel potential space the normal spline is constructed in. It must be a struct object of the following type: RK_H0 if the spline is constructing as a continuous function, RK_H1 if the spline is constructing as a differentiable function, RK_H2 if the spline is constructing as a twice differentiable function.

Return: the partly initialized NormalSpline object that must be passed to construct function in order to complete the spline initialization.

prepare(nodes::Matrix{T}, d_nodes::Matrix{T}, es::Matrix{T}, kernel::RK = RK_H1()) where {T <: AbstractFloat, RK <: ReproducingKernel_1}

Prepare the spline by constructing and factoring a Gram matrix of the interpolation problem. Initialize the NormalSpline object.

Arguments

• nodes: The function value nodes. This should be an n×n_1 matrix, where n is dimension of the sampled space and n_1 is the number of function value nodes. It means that each column in the matrix defines one node.
• d_nodes: The function directional derivatives nodes. This should be an n×n_2 matrix, where n is dimension of the sampled space and n_2 is the number of function directional derivative nodes.
• es: Directions of the function directional derivatives. This should be an n×n_2 matrix, where n is dimension of the sampled space and n_2 is the number of function directional derivative nodes. It means that each column in the matrix defines one direction of the function directional derivative.
• kernel: reproducing kernel of Bessel potential space the normal spline is constructed in. It must be a struct object of the following type: RK_H1 if the spline is constructing as a differentiable function, RK_H2 if the spline is constructing as a twice differentiable function.

Return: the partly initialized NormalSpline object that must be passed to construct function in order to complete the spline initialization.

prepare(nodes::Vector{T}, kernel::RK = RK_H0()) where {T <: AbstractFloat, RK <: ReproducingKernel_0}

Prepare the 1D spline by constructing and factoring a Gram matrix of the interpolation problem. Initialize the NormalSpline object.

Arguments

• nodes: function value interpolation nodes. This should be an n_1 vector where n_1 is the number of function value nodes.
• kernel: reproducing kernel of Bessel potential space the normal spline is constructed in. It must be a struct object of the following type: RK_H0 if the spline is constructing as a continuous function, RK_H1 if the spline is constructing as a differentiable function, RK_H2 if the spline is constructing as a twice differentiable function.

Return: the partly initialized NormalSpline object that must be passed to construct function in order to complete the spline initialization.

prepare(nodes::Vector{T}, d_nodes::Vector{T}, kernel::RK = RK_H1()) where {T <: AbstractFloat, RK <: ReproducingKernel_1}

Prepare the 1D interpolating normal spline by constructing and factoring a Gram matrix of the problem. Initialize the NormalSpline object.

Arguments

• nodes: function value interpolation nodes. This should be an n_1 vector where n_1 is the number of function value nodes.
• d_nodes: The function derivatives nodes. This should be an n_2 vector where n_2 is the number of function derivatives nodes.
• kernel: reproducing kernel of Bessel potential space the normal spline is constructed in. It must be a struct object of the following type: RK_H1 if the spline is constructing as a differentiable function, RK_H2 if the spline is constructing as a twice differentiable function.

Return: the partly initialized NormalSpline object that must be passed to construct function in order to complete the spline initialization.

construct(spline::NormalSpline{T, RK}, values::Vector{T}) where {T <: AbstractFloat, RK <: ReproducingKernel_0}

Construct the spline by calculating its coefficients and completely initializing the NormalSpline object.

Arguments

• spline: the partly initialized NormalSpline object returned by prepare function.
• values: function values at nodes nodes.

Return: the completely initialized NormalSpline object that can be passed to evaluate function to interpolate the data to required points.

construct(spline::NormalSpline{T, RK}, values::Vector{T}, d_values::Vector{T}) where {T <: AbstractFloat, RK <: ReproducingKernel_1}

Construct the spline by calculating its coefficients and completely initializing the NormalSpline object.

Arguments

• spline: the partly initialized NormalSpline object returned by prepare function.
• values: function values at nodes nodes.
• d_values: function directional derivative values at d_nodes nodes.

Return: the completely initialized NormalSpline object that can be passed to evaluate function to interpolate the data to required points.

interpolate(nodes::Matrix{T}, values::Vector{T}, kernel::RK = RK_H0()) where {T <: AbstractFloat, RK <: ReproducingKernel_0}

Prepare and construct the spline.

Arguments

• nodes: The function value nodes. This should be an n×n_1 matrix, where n is dimension of the sampled space and n_1 is the number of function value nodes. It means that each column in the matrix defines one node.
• values: function values at nodes nodes.
• kernel: reproducing kernel of Bessel potential space the normal spline is constructed in. It must be a struct object of the following type: RK_H0 if the spline is constructing as a continuous function, RK_H1 if the spline is constructing as a differentiable function, RK_H2 if the spline is constructing as a twice differentiable function.

Return: the completely initialized NormalSpline object that can be passed to evaluate function.

interpolate(nodes::Matrix{T}, values::Vector{T}, d_nodes::Matrix{T}, es::Matrix{T}, d_values::Vector{T}, kernel::RK = RK_H1()) where {T <: AbstractFloat, RK <: ReproducingKernel_1}

Prepare and construct the spline.

Arguments

• nodes: The function value nodes. This should be an n×n_1 matrix, where n is dimension of the sampled space and n_1 is the number of function value nodes. It means that each column in the matrix defines one node.
• values: function values at nodes nodes.
• d_nodes: The function directional derivative nodes. This should be an n×n_2 matrix, where n is dimension of the sampled space and n_2 is the number of function directional derivative nodes.
• es: Directions of the function directional derivatives. This should be an n×n_2 matrix, where n is dimension of the sampled space and n_2 is the number of function directional derivative nodes. It means that each column in the matrix defines one direction of the function directional derivative.
• d_values: function directional derivative values at d_nodes nodes.
• kernel: reproducing kernel of Bessel potential space the normal spline is constructed in. It must be a struct object of the following type: RK_H1 if the spline is constructing as a differentiable function, RK_H2 if the spline is constructing as a twice differentiable function.

Return: the completely initialized NormalSpline object that can be passed to evaluate function.

interpolate(nodes::Vector{T}, values::Vector{T}, kernel::RK = RK_H0()) where {T <: AbstractFloat, RK <: ReproducingKernel_0}

Prepare and construct the 1D spline.

Arguments

• nodes: function value interpolation nodes. This should be an n_1 vector where n_1 is the number of function value nodes.
• values: function values at n_1 interpolation nodes.
• kernel: reproducing kernel of Bessel potential space the normal spline is constructed in. It must be a struct object of the following type: RK_H0 if the spline is constructing as a continuous function, RK_H1 if the spline is constructing as a differentiable function, RK_H2 if the spline is constructing as a twice differentiable function.

Return: the completely initialized NormalSpline object that can be passed to evaluate function.

interpolate(nodes::Vector{T}, values::Vector{T}, d_nodes::Vector{T}, d_values::Vector{T}, kernel::RK = RK_H1()) where {T <: AbstractFloat, RK <: ReproducingKernel_1}

Prepare and construct the 1D interpolating normal spline.

Arguments

• nodes: function value interpolation nodes. This should be an n_1 vector where n_1 is the number of function value nodes.
• values: function values at nodes nodes.
• d_nodes: The function derivatives nodes. This should be an n_2 vector where n_2 is the number of function derivatives nodes.
• d_values: function derivative values at d_nodes nodes.
• kernel: reproducing kernel of Bessel potential space the normal spline is constructed in. It must be a struct object of the following type: RK_H1 if the spline is constructing as a differentiable function, RK_H2 if the spline is constructing as a twice differentiable function.

Return: the completely initialized NormalSpline object that can be passed to evaluate function.

evaluate(spline::NormalSpline{T, RK}, points::Matrix{T}) where {T <: AbstractFloat, RK <: ReproducingKernel_0}

Evaluate the spline values at the locations defined in points.

Arguments

• spline: theNormalSplineobject returned byinterpolateorconstruct` function.
• points: locations at which spline values are evaluating. This should be an n×m matrix, where n is dimension of the sampled space and m is the number of locations where spline values are evaluating. It means that each column in the matrix defines one location.

Return: Vector{T} of the spline values at the locations defined in points.

evaluate(spline::NormalSpline{T, RK}, points::Vector{T}) where {T <: AbstractFloat, RK <: ReproducingKernel_0}

Evaluate the spline values/value at the points locations.

Arguments

• spline: the NormalSpline object returned by interpolate or construct function.
• points: locations at which spline values are evaluating. This should be a vector of size m where m is the number of evaluating points.

Return: spline value at the point location.

evaluate_one(spline::NormalSpline{T, RK}, point::Vector{T}) where {T <: AbstractFloat, RK <: ReproducingKernel_0}

Evaluate the spline value at the point location.

Arguments

• spline: the NormalSpline object returned by interpolate or construct function.
• point: location at which spline value is evaluating. This should be a vector of size n, where n is dimension of the sampled space.

Return: the spline value at the location defined in point.

evaluate_one(spline::NormalSpline{T, RK}, point::T) where {T <: AbstractFloat, RK <: ReproducingKernel_0}

Evaluate the 1D spline value at the point location.

Arguments

• spline: the NormalSpline object returned by interpolate or construct function.
• point: location at which spline value is evaluating.

Return: spline value at the point location.

evaluate_gradient(spline::NormalSpline{T, RK}, point::Vector{T}) where {T <: AbstractFloat, RK <: ReproducingKernel_0}

Evaluate gradient of the spline at the location defined in point.

Arguments

• spline: the NormalSpline object returned by interpolate or construct function.
• point: location at which gradient value is evaluating. This should be a vector of size n, where n is dimension of the sampled space.

Note: Gradient of spline built with reproducing kernel RK_H0 does not exist at the spline nodes.

Return: Vector{T} - gradient of the spline at the location defined in point.

evaluate_derivative(spline::NormalSpline{T, RK}, point::T) where {T <: AbstractFloat, RK <: ReproducingKernel_0}

Evaluate the 1D spline derivative at the point location.

Arguments

• spline: the NormalSpline object returned by interpolate or construct function.
• point: location at which spline derivative is evaluating.

Note: Derivative of spline built with reproducing kernel RK_H0 does not exist at the spline nodes.

Return: the spline derivative value at the point location.

estimate_accuracy(spline::NormalSpline{T, RK}) where {T <: AbstractFloat, RK <: ReproducingKernel_0}

Assess accuracy of interpolation results by analyzing residuals.

Arguments

• spline: the NormalSpline object returned by construct or interpolate function.

Return: an estimation of the number of significant digits in the interpolation result.

estimate_cond(spline::NormalSpline{T, RK}) where {T <: AbstractFloat, RK <: ReproducingKernel_0}

Get an estimation of the Gram matrix condition number. It needs the spline object is prepared and requires O(N^2) operations. (C. Brás, W. Hager, J. Júdice, An investigation of feasible descent algorithms for estimating the condition number of a matrix. TOP Vol.20, No.3, 2012.)

Arguments

• spline: the NormalSpline object returned by prepare, construct or interpolate function.

Return: an estimation of the Gram matrix condition number.

get_epsilon(spline::NormalSpline{T, RK}) where {T <: AbstractFloat, RK <: ReproducingKernel_0}

Get the 'scaling parameter' of Bessel Potential space the spline was built in.

Arguments

• spline: the NormalSpline object returned by prepare, construct or interpolate function.

Return: ε - the 'scaling parameter'.

estimate_epsilon(nodes::Matrix{T}, kernel::RK = RK_H0()) where {T <: AbstractFloat, RK <: ReproducingKernel_0}

Get the estimation of the 'scaling parameter' of Bessel Potential space the spline being built in. It coincides with the result returned by get_epsilon function.

Arguments

• nodes: The function value nodes. This should be an n×n_1 matrix, where n is dimension of the sampled space and n_1 is the number of function value nodes. It means that each column in the matrix defines one node.
• kernel: reproducing kernel of Bessel potential space the normal spline will be constructed in. It must be a struct object of the following type: RK_H0 if the spline is constructing as a continuous function, RK_H1 if the spline is constructing as a differentiable function, RK_H2 if the spline is constructing as a twice differentiable function.

Return: estimation of ε.

estimate_epsilon(nodes::Matrix{T}, d_nodes::Matrix{T}, kernel::RK = RK_H1()) where {T <: AbstractFloat, RK <: ReproducingKernel_1}

Get an the estimation of the 'scaling parameter' of Bessel Potential space the spline being built in. It coincides with the result returned by get_epsilon function.

Arguments

• nodes: The function value nodes. This should be an n×n_1 matrix, where n is dimension of the sampled space and n_1 is the number of function value nodes. It means that each column in the matrix defines one node.
• d_nodes: The function directional derivative nodes. This should be an n×n_2 matrix, where n is dimension of the sampled space and n_2 is the number of function directional derivative nodes.
• kernel: reproducing kernel of Bessel potential space the normal spline will be constructed in. It must be a struct object of the following type: RK_H1 if the spline is constructing as a differentiable function, RK_H2 if the spline is constructing as a twice differentiable function.

Return: estimation of ε.

estimate_epsilon(nodes::Vector{T}, kernel::RK = RK_H0()) where {T <: AbstractFloat, RK <: ReproducingKernel_0}

Get an the estimation of the 'scaling parameter' of Bessel Potential space the spline being built in. It coincides with the result returned by get_epsilon function.

Arguments

• nodes: The function value nodes.
• kernel: reproducing kernel of Bessel potential space the normal spline is constructed in. It must be a struct object of the following type: RK_H0 if the spline is constructing as a continuous function, RK_H1 if the spline is constructing as a differentiable function, RK_H2 if the spline is constructing as a twice differentiable function.

Return: estimation of ε.

estimate_epsilon(nodes::Vector{T}, d_nodes::Vector{T}, kernel::RK = RK_H1()) where {T <: AbstractFloat, RK <: ReproducingKernel_1}

Get an the estimation of the 'scaling parameter' of Bessel Potential space the spline being built in. It coincides with the result returned by get_epsilon function.

Arguments

• nodes: The function value nodes.
• d_nodes: The function derivative nodes.
• kernel: reproducing kernel of Bessel potential space the normal spline is constructed in. It must be a struct object of the following type: RK_H1 if the spline is constructing as a differentiable function, RK_H2 if the spline is constructing as a twice differentiable function.

Return: estimation of ε.

get_cond(nodes::Matrix{T}, kernel::RK = RK_H0()) where {T <: AbstractFloat, RK <: ReproducingKernel_0}

Get a value of the Gram matrix spectral condition number. It is obtained by means of the matrix SVD decomposition and requires $O(N^3)$ operations.

Arguments

• nodes: The function value nodes. This should be an n×n_1 matrix, where n is dimension of the sampled space and n_1 is the number of function value nodes. It means that each column in the matrix defines one node.
• kernel: reproducing kernel of Bessel potential space the normal spline is constructed in. It must be a struct object of the following type: RK_H0 if the spline is constructing as a continuous function, RK_H1 if the spline is constructing as a differentiable function, RK_H2 if the spline is constructing as a twice differentiable function.

Return: a value of the Gram matrix spectral condition number.

get_cond(nodes::Matrix{T}, d_nodes::Matrix{T}, es::Matrix{T}, kernel::RK = RK_H1()) where {T <: AbstractFloat, RK <: ReproducingKernel_1}

Get a value of the Gram matrix spectral condition number. It is obtained by means of the matrix SVD decomposition and requires $O(N^3)$ operations.

Arguments

• nodes: The function value nodes. This should be an n×n_1 matrix, where n is dimension of the sampled space and n_1 is the number of function value nodes. It means that each column in the matrix defines one node.
• d_nodes: The function directional derivatives nodes. This should be an n×n_2 matrix, where n is dimension of the sampled space and n_2 is the number of function directional derivative nodes.
• es: Directions of the function directional derivatives. This should be an n×n_2 matrix, where n is dimension of the sampled space and n_2 is the number of function directional derivative nodes. It means that each column in the matrix defines one direction of the function directional derivative.
• kernel: reproducing kernel of Bessel potential space the normal spline is constructed in. It must be a struct object of the following type: RK_H1 if the spline is constructing as a differentiable function, RK_H2 if the spline is constructing as a twice differentiable function.

Return: a value of the Gram matrix spectral condition number.

struct RK_H0{T <: AbstractFloat} <: ReproducingKernel_0

Defines a type of reproducing kernel of Bessel Potential space $H^{n/2 + 1/2}_ε (R^n)$ ('Basic Matérn kernel'):

\[V(\eta , \xi, \varepsilon) = \exp (-\varepsilon |\xi - \eta|) \, .\]

Fields

• ε::T: 'scaling parameter' from the Bessel Potential space definition, it may be omitted in the struct constructor otherwise it must be greater than zero

struct RK_H1{T <: AbstractFloat} <: ReproducingKernel_1

Defines a type of reproducing kernel of Bessel Potential space $H^{n/2 + 3/2}_ε (R^n)$ ('Linear Matérn kernel'):

\[V(\eta , \xi, \varepsilon) = \exp (-\varepsilon |\xi - \eta|) (1 + \varepsilon |\xi - \eta|) \, .\]

Fields

• ε::T: 'scaling parameter' from the Bessel Potential space definition, it may be omitted in the struct constructor otherwise it must be greater than zero

struct RK_H2{T <: AbstractFloat} <: ReproducingKernel_2

Defines a type of reproducing kernel of Bessel Potential space $H^{n/2 + 5/2}_ε (R^n)$ ('Quadratic Matérn kernel'):

\[V(\eta , \xi, \varepsilon) = \exp (-\varepsilon |\xi - \eta|) (3 + 3\varepsilon |\xi - \eta| + \varepsilon ^2 |\xi - \eta| ^2 ) \, .\]

Fields

• ε::T: 'scaling parameter' from the Bessel Potential space definition, it may be omitted in the struct constructor otherwise it must be greater than zero

struct NormalSpline{T, RK} <: AbstractSpline where {T <: AbstractFloat, RK <: ReproducingKernel_0}

Define a structure containing full information of a normal spline

Fields

• _kernel: a reproducing kernel spline was built with
• _compression: factor of transforming the original node locations into unit hypercube
• _nodes: transformed function value nodes
• _values: function values at interpolation nodes
• _d_nodes: transformed function directional derivative nodes
• _es: normalized derivative directions
• _d_values: function directional derivative values
• _min_bound: minimal bounds of the original node locations area
• _gram: Gram matrix of the problem
• _chol: Cholesky factorization of the Gram matrix
• _mu: spline coefficients
• _cond: estimation of the Gram matrix condition number