Splinets (version 1.5.1)

sym2one: Switching between representations of the matrices of derivatives

Description

A technical but useful transformation of the matrix of derivatives form the one-sided to symmetric representations, or a reverse one. It allows for switching between the standard representation of the matrix of the derivatives for Splinets which is symmetric around the central knot(s) to the one-sided that yields the RHS limits at the knots, which is more convenient for computations.

Usage

sym2one(S, supp = NULL, inv = FALSE)

Value

A matrix that is the respective transformation of the input.

Arguments

S

(m+2) x (k+1) numeric matrix, the derivatives in one of the two representations;

supp

(Nsupp x 2) or NULL matrix, row-wise the endpoint indices of the support intervals; If it is equal to NULL (which is also the default), then the full support is assumed.

inv

logical; If FALSE (default), then the function assumes that the input is in the symmetric format and transforms it to the left-to-right format. If TRUE, then the inverse transformation is applied.

Details

The transformation essentially changes only the last column in S, i.e. the highest (discontinuous) derivatives so that the one-sided representation yields the right-hand-side limit. It is expected that the number of rows in S is the same as the total size of the support as indicated by supp, i.e. if supp!=NULL, then sum(supp[,2]-supp[,1]+1)=m+2. If the latter is true, than all derivative submatrices of the components in S will be reversed. However, this condition formally is not checked in the code, which may lead to switch of the representations only for parts of the matrix S.

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>

See Also

Splinets-class for the description of the Splinets-class; is.splinets for diagnostic of Splinets-objects;

Examples

#-----------------------------------------------------#
#-------Representations of derivatives at knots-------#
#-----------------------------------------------------#
n=10; k=3; xi=seq(0,1,by=1/(n+1)) #the even number of equally spaced knots 
set.seed(5)
S=matrix(rnorm((n+2)*(k+1)),ncol=(k+1))
spl=construct(xi,k,S) #construction of a spline
a=spl@der[[1]]
b=sym2one(a)
aa=sym2one(b,inv=TRUE) # matrix 'aa' is the same as 'a'

n=11; xi2=seq(0,1,by=1/(n+1)) #the odd number of knots case
S2=matrix(rnorm((n+2)*(k+1)),ncol=(k+1))
spl2=construct(xi2,k,S2) #construction of a spline
a2=spl2@der[[1]]
b2=sym2one(a2)
aa2=sym2one(b2, inv=TRUE) # matrix 'aa2' is the same as 'a2'

#-----------------------------------------------------#
#--------------More complex support sets--------------#
#-----------------------------------------------------#
#Zero order splines, non-equidistant case, support with three components
n=43; xi=seq(0,1,by=1/(n+1)); k=3; xi=sort(runif(n+2)); xi[1]=0; xi[n+2]=1;
support=list(matrix(c(2,14,17,30,32,43),ncol=2,byrow = TRUE))
#Third order splines
ssp=new("Splinets",knots=xi,supp=support,degree=k) #with partial support

m=sum(ssp@supp[[1]][,2]-ssp@supp[[1]][,1]+1) #the total number of knots in the support
ssp@der=list(matrix(rnorm(m*(k+1)),ncol=(k+1)))  #the derivative matrix at random
IS=is.splinets(ssp) 
IS$robject@der
IS$robject@supp
b=sym2one(IS$robject@der[[1]],IS$robject@supp[[1]]) #the RHS limits at the knots
a=sym2one(b,IS$robject@supp[[1]],inv=TRUE) #is the same as the SLOT supp in IS@robject