Vector Space bases

AbstractBasis

Super type for all bases. A basis must implement the following methods: nVariates, length, getType, isDifferentiable, value. If the basis contains only differentiable functions, in which case isDifferentiable returns true, it must also implement derivative.

nVariates(b::AbstractBasis)

Return the number of variates of the functions inside the basis.

length(b::AbstractBasis)

Return the number of elements in the basis.

size(b::AbstractBasis)

Return the tuple (nVariates, length).

getType(b::AbstractBasis)

Return the internal basis type.

isDifferentiable(b::AbstractBasis)

Return true if the functions in the basis are differentiable. In this case, a specific method derivative must be implemented.

value(b::AbstractBasis, x::AbstractVector{<:Real}, index::Integer)

Compute the value of the index-th basis function at point x.

derivative(b::AbstractBasis, x::AbstractVector{<:Real}, index::Integer, derivativeIndex::Integer)

Compute the value of the first derivative of the index-th basis function w.r.t to the derivativeIndex variate at point x.

PolynomialBasis

Represent a multivariate polynomial

• degree::Int64: maximum total degree
• nVariates::Int6: number of variates
• dim::Int6: dimension of the generated vector space
• type::PolynomialType: a value from PolynomialType
• tensor::SparseMatrixCSC{Int64, Int64}: the sparse tensor representation of the polynomial. Polynomials are stored by column.

PolynomialType

List the families of polynomials available through the PolynomialBasis type

• Canonic
• Hermite
• Tchebychev

PolynomialBasis(type::PolynomialType, nVariates::Integer, degree::Integer)

Create a polynomial basis with type, nVariates variables and total maximum degree

PiecewiseConstantBasis

Represent a basis of local functions defined on $[0,1]^d$

• nVariates is the dimension d of the space.
• nIntervals is a vector of size d defining the number of sub-intervals to use along each dimension

PiecewiseConstantBasis(nVariates::Integer, nIntervals::Integer)

Create a PiecewiseConstantBasis with nIntervals along each direction.

PiecewiseConstantBasis(nVariates::Integer, nIntervals::Vector{<:Integer})

Create a PiecewiseConstantBasis by specifying the number of intervals per direction (possibly different to allow non squared grids).