gramian {Splinets}

Gramian matrix, norms, and inner products of splines

Description

The function performs evaluation of the matrix of the inner products \int S(t) \cdot T(t) dt of all the pairs of splines S, T from the input object. The program utilizes the Taylor expansion of splines, see the reference for details.

Usage

gramian(Sp, norm_only = FALSE, sID = NULL, Sp2 = NULL, s2ID = NULL)

Arguments

Sp

Splinets object;

norm_only

logical, indicates if only the square norm of the elements in the input object is calculated; The default is norm_only=FALSE;

sID

vector of integers, the indicies specifying splines in the Splinets list Sp to be evaluated; If sID=NULL (default), then the inner products for all the pairs taken from the object are evaluated.

Sp2

Splinets object, the optional second Splinets-object; The inner products between splines in Sp and in Sp2 are evaluated, i.e. the cross-gramian matrix.

s2ID

vector of integers, the indicies specifying splines in the Sp2 to be considered in the cross-gramian;

Details

If there is only one input Splinet-object, then the non-negative symmetrix matrix of the splines in this object is returned. If there are two input Splinet-objects, then the m \times r matrix of the cross-inner product is returned, where m is the number of splines in the first object and r is their number in the second one. If only the norms are evaluated (norm_only= TRUE) it is always evaluating the norms of the first object. In the case of two input Splinets-objects, they should be over the same set of knots and of the same smoothness order.

Value

• norm_only=FALSE – the Gram matrix of inner products of the splines within the input Splinets-objects is returned,

• Sp2 = NULL – the non-negative definite matrix of the inner products of splines in Sp is returned,

• both Sp and Sp2 are non-NULL and contain splines S_i's and T_j's, respectively == the cross-gramian matris of the inner products for the pairs of splines (S_i,T_j) is returned,

• norm_only=FALSE– the vector of the norms of Sp is returned.

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

lincomb for evaluation of a linear combination of splines; project for projections to the spaces of Splines;

Examples

#---------------------------------------#
#---- Simple three splines example -----# 
#---------------------------------------#
n=25; k=3
xi=sort(runif(n+2)); xi[1]=0; xi[n+2]=1
#Defining support ranges for three splines
supp=matrix(c(2,12,4,20,6,25),byrow=TRUE,ncol=2)
#Initial random matrices of the derivative for each spline
SS1=matrix(rnorm((supp[1,2]-supp[1,1]+1)*(k+1)),ncol=(k+1)) 
SS2=matrix(rnorm((supp[2,2]-supp[2,1]+1)*(k+1)),ncol=(k+1)) 
SS3=matrix(rnorm((supp[3,2]-supp[3,1]+1)*(k+1)),ncol=(k+1)) 
spl=construct(xi,k,SS1,supp[1,]) #constructing the first correct spline
nspl=construct(xi,k,SS2,supp[2,])
spl=gather(spl,nspl) #the second and the first ones
nspl=construct(xi,k,SS3,supp[3,])
spl=gather(spl,nspl) #the third is added

plot(spl)
gramian(spl)
gramian(spl, norm_only = TRUE)
gramian(spl, sID = c(1,3))
gramian(spl,sID=c(2,3),Sp2=spl,s2ID=c(1)) #the cross-Gramian matrix  

#-----------------------------------------#
#--- Example with varying support sets ---#
#-----------------------------------------#
n=40; xi=seq(0,1,by=1/(n+1)); k=2; 
support=list(matrix(c(2,9,15,24,30,37),ncol=2,byrow = TRUE))
sp=new("Splinets",knots=xi,smorder=k,supp=support) 
m=sum(sp@supp[[1]][,2]-sp@supp[[1]][,1]+1) #the number of knots in the support
sp@der=list(matrix(rnorm(m*(k+1)),ncol=(k+1))) #the derivative matrix at random
sp1 = is.splinets(sp)[[2]] #the correction of 'der' matrices

support=list(matrix(c(5,12,17,29),ncol=2,byrow = TRUE))
sp=new("Splinets",knots=xi,smorder=k,supp=support) 
m=sum(sp@supp[[1]][,2]-sp@supp[[1]][,1]+1) #the number of knots in the support
sp@der=list(matrix(rnorm(m*(k+1)),ncol=(k+1))) #the derivative matrix at random
sp2 = is.splinets(sp)[[2]] 

spp = gather(sp1,sp2)

support=list(matrix(c(3,10,14,21,27,34),ncol=2,byrow = TRUE))
sp=new("Splinets",knots=xi,smorder=k,supp=support) 
m=sum(sp@supp[[1]][,2]-sp@supp[[1]][,1]+1) #the number of knots in the support
sp@der=list(matrix(rnorm(m*(k+1)),ncol=(k+1))) #the derivative matrix at random
sp3 = is.splinets(sp)[[2]] 

spp = gather(spp, sp3)

plot(spp)
gramian(spp) #the regular gramian matrix
spp2=subsample(spp,sample(1:3,size=3,rep=TRUE))
gramian(Sp=spp,Sp2=spp2) #cross-Gramian matrix


#-----------------------------------------#
#--------- Grammian for B-splines --------#
#-----------------------------------------#

n=25; xi=seq(0,1,by=1/(n+1)); k=2; 
Sp=splinet(xi) #B-splines and corresponding splinet

gramian(Sp$bs) #band grammian matrix for B-splines
gramian(Sp$os) #diagonal gramian matrix for the splinet
A=gramian(Sp=Sp$bs,Sp2=Sp$os) #cross-Gramian matrix, the coefficients of
                              #the decomposition of the B-splines

plot(Sp$bs)
plot(lincomb(Sp$os,A))