Global Bspline bases evaluations

[rayqu1126]

Hi,

@jlchan and I wonder if there is an easy way to construct a generalized Vandermonde whose columns correspond to evaluations of global Bspline bases at given points.

Right now we build BsplineVandermonde(N, K, x) = hcat(PeriodicBSplineBasis(BSplineOrder(N+1), range(-1, 1, length=K+1))(x)[2]...) BsplineVandermonde(N, K, x::AbstractVector) = vcat([BsplineVandermonde(N, K, xi) for xi in x]...) but this only returns the evaluations of local Bspline bases, as you can see from this plot x = LinRange(-1,1,500); V = BsplineVandermonde(4, 8, x); scatter(x, V[:, 1], leg=false).

Thank you for providing this useful package and your help!

[jlchan]

Hi! Agreed with @rayqu1126 - thank you for providing this package! It's very nice to have the capability to work with periodic B-splines bases.

To clarify, we were wondering if there was existing functionality to compute an array whose entries are

$b_{i} = B_i(x)$

where $B_i(x)$ is a single Bspline basis function evaluated at some single point $x$.

From what we could see from the docs, evaluate_all and PeriodicBSplineBasis return the evaluation only of B-spline bases which are supported near the evaluation point.

[jlchan]

Update: I think we have it now. evaluate is what we were looking for.

For reference, this is

function BsplineVandermonde(N, K, x) 
    bspline_basis = PeriodicBSplineBasis(BSplineOrder(N+1), range(-1, 1, length=K+1))
    return hcat([evaluate(bspline_basis, i, x) for i in 0:(N+K-1)]...)
end

which allows us to generate the following plot via

x = LinRange(-1,1,500)
V = BsplineVandermonde(3, 10, x)
scatter(x, V, legend=false)

[jlchan]

@rayqu1126 I think we can close this now