Splines

Spline{T} <: Function

Represents a spline function.

Spline(B::AbstractBSplineBasis, coefs::AbstractVector)

Construct a spline from a B-spline basis and a vector of B-spline coefficients.

Examples

julia> B = BSplineBasis(BSplineOrder(4), -1:0.2:1);

julia> coefs = rand(length(B));

julia> S = Spline(B, coefs)
13-element Spline{Float64}:
 basis: 13-element BSplineBasis of order 4, domain [-1.0, 1.0]
 order: 4
 knots: [-1.0, -1.0, -1.0, -1.0, -0.8, -0.6, -0.4, -0.2, 0.0, 0.2, 0.4, 0.6, 0.8, 1.0, 1.0, 1.0, 1.0]
 coefficients: [0.173575, 0.321662, 0.258585, 0.166439, 0.527015, 0.483022, 0.390663, 0.802763, 0.721983, 0.372347, 0.0301856, 0.0793339, 0.663758]

Spline{T = Float64}(undef, B::AbstractBSplineBasis)

Construct a spline with uninitialised vector of coefficients.

(S::Spline)(x)

Evaluate spline at coordinate x.

coefficients(S::Spline)

Get B-spline coefficients of the spline.

eltype(::Type{<:Spline})
eltype(S::Spline)

Returns type of element returned when evaluating the Spline.

length(S::Spline)

Returns the number of coefficients in the spline.

Note that this is equal to the number of basis functions, length(basis(S)).

basis(S::Spline) -> AbstractBSplineBasis

Returns the associated B-spline basis.

PeriodicVector{T} <: AbstractVector{T}

Describes a periodic (or "circular") vector wrapping a regular vector.

Used to store the coefficients of periodic splines.

The vector has an effective length N associated to a single period, but it is possible to index it outside of this "main" interval.

This is similar to BSplines.PeriodicKnots. It is simpler though, since here there is no notion of coordinates or of a period L. Periodicity is only manifest in the indexation of the vector, e.g. a PeriodicVector vs satisfies vs[i + N] == vs[i].

PeriodicVector(cs::AbstractVector)

Wraps coefficient vector cs such that it can be indexed in a periodic manner.

*(op::Derivative, S::Spline) -> Spline

Returns N-th derivative of spline S as a new spline.

See also diff.

Examples

julia> B = BSplineBasis(BSplineOrder(4), -1:0.2:1);

julia> S = Spline(B, rand(length(B)))
13-element Spline{Float64}:
 basis: 13-element BSplineBasis of order 4, domain [-1.0, 1.0]
 order: 4
 knots: [-1.0, -1.0, -1.0, -1.0, -0.8, -0.6, -0.4, -0.2, 0.0, 0.2, 0.4, 0.6, 0.8, 1.0, 1.0, 1.0, 1.0]
 coefficients: [0.173575, 0.321662, 0.258585, 0.166439, 0.527015, 0.483022, 0.390663, 0.802763, 0.721983, 0.372347, 0.0301856, 0.0793339, 0.663758]

julia> Derivative(0) * S === S
true

julia> Derivative(1) * S
12-element Spline{Float64}:
 basis: 12-element BSplineBasis of order 3, domain [-1.0, 1.0]
 order: 3
 knots: [-1.0, -1.0, -1.0, -0.8, -0.6, -0.4, -0.2, 0.0, 0.2, 0.4, 0.6, 0.8, 1.0, 1.0, 1.0]
 coefficients: [2.22131, -0.473071, -0.460734, 1.80288, -0.219964, -0.461794, 2.0605, -0.403899, -1.74818, -1.71081, 0.368613, 8.76636]

julia> Derivative(2) * S
11-element Spline{Float64}:
 basis: 11-element BSplineBasis of order 2, domain [-1.0, 1.0]
 order: 2
 knots: [-1.0, -1.0, -0.8, -0.6, -0.4, -0.2, 0.0, 0.2, 0.4, 0.6, 0.8, 1.0, 1.0]
 coefficients: [-26.9438, 0.0616849, 11.3181, -10.1142, -1.20915, 12.6114, -12.322, -6.72141, 0.186876, 10.3971, 83.9775]

diff(S::Spline, [op::Derivative = Derivative(1)]) -> Spline

Same as op * S.

Returns N-th derivative of spline S as a new spline.

integral(S::Spline)

Returns an antiderivative of the given spline as a new spline.

The algorithm is described in de Boor 2001, p. 127.

Note that the integral spline I returned by this function is defined up to a constant. By convention, here the returned spline I is zero at the left boundary of the domain. One usually cares about the integral of S from point a to point b, which can be obtained as I(b) - I(a).

SplineWrapper{S <: Spline} <: Function

Abstract type representing a type that wraps a Spline.

Such a type implements all common operations on splines, including evaluation, differentiation, etc…

spline(w::SplineWrapper) -> Spline

Returns the Spline wrapped by the object.