seq2dyad: Organizing indices in a spline basis in the net form

Description

This auxiliary function generates the map between the sequential order and the dyadic net structure of a spline basis. It works only with indices so it can be utilized to any basis in the space of splines with the zero-boundary conditions. The function is useful for creating the dyadic structure of the graphs and whenever a reference to the k-tuples and the levels of support is needed.

Usage

seq2dyad(n_sp, k)

Value

The double indexed list of single row matrices of positive integers in the range 1:n_sp. Each vector has typically the length k and some of them may correspond to incomplete tuplets and thus can be shorter. The first index in the list points to the level in the dyadic structure, the second one to the the number of the tuplet at the given level. The integers in the vector pointed by the list correspond to the sequential index of the element belonging to this tuplet.

Arguments

n_sp: positive integer, the number of splines to be organized into the dyadic net; The dyadic net does not need to be fully dyadic, i.e. n_sp does not need to be equal to \(k2^n-1\), where \(n\) is the number of the internal knots. See the references for more details.
k: the size of a tuple in the dyadic net; It naturally corresponds to the degree of splines for which the net is build.

References

Liu, X., Nassar, H., Podg\(\mbox{\'o}\)rski, K. "Dyadic diagonalization of positive definite band matrices and efficient B-spline orthogonalization." Journal of Computational and Applied Mathematics (2022) <https://doi.org/10.1016/j.cam.2022.114444>.

Podg\(\mbox{\'o}\)rski, K. (2021) "Splinets -- splines through the Taylor expansion, their support sets and orthogonal bases." <arXiv:2102.00733>.

Nassar, H., Podg\(\mbox{\'o}\)rski, K. (2023) "Splinets 1.5.0 -- Periodic Splinets." <arXiv:2302.07552>

Examples

Run this code

#-------------------------------------------------------#
#--The support layers of the dyadic structure of bases--#
#-------------------------------------------------------#
k=4 # order
n = 36 # number of the internal knots (excluding the endpoints)
xi = seq(0, 1, length.out = n+2)
spnt=splinet(xi,k)

plot(spnt$os)                #standard plotting 
plot(spnt$bs,type='dyadic')  #dyadic format of plots

net=seq2dyad(n-k+1,k)  #retrieving dyadic structure
ind1=c(net[[4]][[1]],net[[4]][[2]])

plot(subsample(spnt$os,ind1))

ind2=c(net[[4]][[3]],net[[4]][[4]]) #the lowest support in the dyadic net

lines(subsample(spnt$bs,ind2))