Data interpolation

interpolate(x, y, BSplineOrder(k), [bc = nothing])

Interpolate values y at locations x using B-splines of order k.

Grid points x must be real-valued and are assumed to be in increasing order.

Returns a SplineInterpolation which can be evaluated at any intermediate point.

Optionally, one may pass one of the boundary conditions listed in the Boundary conditions section. Currently, the Natural and Periodic boundary conditions are available.

See also interpolate!.

Examples

julia> xs = -1:0.1:1;

julia> ys = cospi.(xs);

julia> S = interpolate(xs, ys, BSplineOrder(4))
SplineInterpolation containing the 21-element Spline{Float64}:
 basis: 21-element BSplineBasis of order 4, domain [-1.0, 1.0]
 order: 4
 knots: [-1.0, -1.0, -1.0, -1.0, -0.8, -0.7, -0.6, -0.5, -0.4, -0.3  …  0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 1.0, 1.0, 1.0, 1.0]
 coefficients: [-1.0, -1.00111, -0.8975, -0.597515, -0.314147, 1.3265e-6, 0.314142, 0.597534, 0.822435, 0.96683  …  0.96683, 0.822435, 0.597534, 0.314142, 1.3265e-6, -0.314147, -0.597515, -0.8975, -1.00111, -1.0]
 interpolation points: -1.0:0.1:1.0

julia> S(-1)
-1.0

julia> (Derivative(1) * S)(-1)
-0.01663433622896893

julia> (Derivative(2) * S)(-1)
10.52727328755495

julia> Snat = interpolate(xs, ys, BSplineOrder(4), Natural())
SplineInterpolation containing the 21-element Spline{Float64}:
 basis: 21-element RecombinedBSplineBasis of order 4, domain [-1.0, 1.0], BCs {left => (D{2},), right => (D{2},)}
 order: 4
 knots: [-1.0, -1.0, -1.0, -1.0, -0.9, -0.8, -0.7, -0.6, -0.5, -0.4  …  0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1.0, 1.0, 1.0, 1.0]
 coefficients: [-0.833333, -0.647516, -0.821244, -0.597853, -0.314057, -2.29076e-5, 0.314148, 0.597532, 0.822435, 0.96683  …  0.96683, 0.822435, 0.597532, 0.314148, -2.29076e-5, -0.314057, -0.597853, -0.821244, -0.647516, -0.833333]
 interpolation points: -1.0:0.1:1.0

julia> Snat(-1)
-1.0

julia> (Derivative(1) * Snat)(-1)
0.28726186708894824

julia> (Derivative(2) * Snat)(-1)
0.0

Periodic boundary conditions

Interpolate $f(x) = \cos(πx)$ for $x ∈ [-1, 1)$. Note that the period is $L = 2$ and that the endpoint ($x = 1$) must not be included in the data points.

julia> xp = -1:0.1:0.9;

julia> yp = cospi.(xp);

julia> Sper = interpolate(xp, yp, BSplineOrder(4), Periodic(2))
SplineInterpolation containing the 20-element Spline{Float64}:
 basis: 20-element PeriodicBSplineBasis of order 4, domain [-1.0, 1.0), period 2.0
 order: 4
 knots: [..., -1.2, -1.1, -1.0, -0.9, -0.8, -0.7, -0.6, -0.5, -0.4, -0.3  …  0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1.0, 1.1, ...]
 coefficients: [..., -1.01659, -0.96683, -0.822435, -0.597534, -0.314142, 0.0, 0.314142, 0.597534, 0.822435, 0.96683, 1.01659, 0.96683, 0.822435, 0.597534, 0.314142, -3.34668e-17, -0.314142, -0.597534, -0.822435, -0.96683, ...]
 interpolation points: -1.0:0.1:0.9

As expected, the periodic spline does a better job at approximating the periodic function $f(x) = \cos(πx)$ near the boundaries than the other interpolations:

julia> x = -0.99; cospi(x), Sper(x), Snat(x), S(x)
(-0.9995065603657316, -0.9995032595823043, -0.9971071640321146, -0.9996420091470221)

julia> x = 0.998; cospi(x), Sper(x), Snat(x), S(x)
(-0.9999802608561371, -0.9999801044078943, -0.9994253145274461, -1.0000122303614758)

interpolate!(I::SplineInterpolation, y::AbstractVector)

Update spline interpolation with new data.

This function allows to reuse a SplineInterpolation returned by a previous call to interpolate, using new data on the same locations x.

See interpolate for details.

SplineInterpolation

Spline interpolation.

This is the type returned by interpolate.

A SplineInterpolation I can be evaluated at any point x using the I(x) syntax.

It can also be updated with new data on the same data points using interpolate!.

SplineInterpolation(undef, B::AbstractBSplineBasis, x::AbstractVector, [T = eltype(x)])

Initialise a SplineInterpolation from B-spline basis and a set of interpolation (or collocation) points x.

Note that the length of x must be equal to the number of B-splines.

Use interpolate! to actually interpolate data known on the x locations.