integra {Splinets}

Indefinite integrals of splines

Description

The function generates the indefinite integrals for given input splines. The integral is a function of the upper limit of the definite integral and is a spline of the higher order that does not satisfy the zero boundary conditions at the RHS endpoint, unless the definite integral over the whole range is equal to zero. Moreover, the support of the function is extended in the RHS up to the RHS end point unless the definite integral of the input is zero, in which the case the support is extracted from the obtained spline.

Usage

integra(object, epsilon = 1e-07)

Arguments

object

a Splinets object of the smoothness order k;

epsilon

non-negative number indicating accuracy when close to zero value are detected; This accuracy is used in when the boundary conditions of the integral are checked.

Details

The value on the RHS is not zero, so the zero boundary condition typically is not satisfied and the support is is extended to the RHS end of the whole domain of splines. However, the function returns proper support if the original spline is a derivative of a spline that satisfies the boundary conditons.

Value

A Splinets-object with order k+1 that contains the indefinite integrals of the input object.

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

deriva for computing derivatives of splines; dintegra for the definite integral;

Examples

#------------------------------------#
#--- Generate indefinite integral ---#
#------------------------------------#
n=18; k=3; xi=sort(runif(n+2)); xi[1]=0; xi[n+2]=1
# generate a random matrix S
set.seed(5)
S=matrix(rnorm((n+2)*(k+1)),ncol=(k+1))
spl=construct(xi,k,S) #constructing a spline
plot(spl)

dspl = deriva(spl) #derivative
plot(dspl)
is.splinets(dspl)
dintegra(dspl) #the definite integral is 0 (the boundary conditions for 'spl')

ispl = integra(spl) #the integral of a spline
plot(ispl) #the boundary condition on the rhs not satisfied (non-zero value)
ispl@smorder
is.splinets(ispl) #the object does not satisfy the boundary condition for the spline

spll = integra(dspl)
plot(spll)
is.splinets(spll) #the boundary conditions of the integral of the derivative satisfied.


#----------------------------------------------#
#--- Examples with different support ranges ---#
#----------------------------------------------#
n=25; k=2; 
set.seed(5)
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 proper spline
nspl=construct(xi,k,SS2,supp[2,],'CRFC')
spl=gather(spl,nspl) #the second and the first together
nspl=construct(xi,k,SS3,supp[3,],'CRLC')
spl=gather(spl,nspl) #the third is added

plot(spl)
spl@supp


dspl = deriva(spl) #derivative of the splines
plot(dspl)

dintegra(dspl) #the definite integral over the entire range of knots is zero

idspl = integra(dspl) #integral of the derivative returns the original splines
plot(idspl)
is.splinets(idspl)  #and confirms that the object is a spline with boundary conditions 
                    #satified


idspl@supp #Since integral is taken over a function that integrates to zero over 
spl@supp  #each of the support interval, the support of all three objects are the same. 
dspl@supp

ispl=integra(spl)
plot(ispl) #the zero boundary condition at the RHS-end for the splines are not satisfied.
is.splinets(ispl) #thus the object is reported as a non-spline

plot(deriva(ispl))
displ=deriva(ispl) 
displ@supp #Comparison of the supports
spl@supp   
           #Here the integrals have extended support as it is taken from a function
ispl@supp  #that does not integrate to zero.



#---------------------------------------#
#---Example with complicated supports---#
#---------------------------------------#
n=40; xi=seq(0,1,by=1/(n+1)); k=3; 
support=list(matrix(c(2,12,15,27,30,40),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]] #Comparison of the corrected and the original 'der' matrices

support=list(matrix(c(2,13,17,30),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]] 
sp = gather(sp1,sp2) #a group of two splines
plot(sp)

dsp = deriva(sp) #derivative
plot(dsp)

spl = integra(dsp)
plot(spl) #the spline retrieved
spl@supp  #the supports are retrieved as well
sp@supp
is.splinets(spl) #the proper splinet object that satisfies the boundaries

ispl = integra(sp)
plot(ispl) 
ispl@supp #full support shown by empty list in SLOT 'supp'
is.splinets(ispl) #diagnostic confirms no zeros at the boundaries

spll = deriva(ispl)
plot(spll)
spll@supp