sym2one {Splinets}

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)

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.

Value

A matrix that is the respective transformation of the input.

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,smorder=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