chol.tensor: Cholesky decomposition of a tensor

Description

A tensor can be seen as a linear mapping of a tensor to a tensor. This function computes its Cholesky decomposition.

Usage

chol.tensor(X,i,j,...,name="lambda")

Arguments

The tensor to be decomposed

The image dimensions of the linear mapping

The coimage dimensions of the linear mapping

name

The name of the eigenspace dimension. This is the dimension created by the decompositions, in which the eigenvectors are $e_i$

…

for generic use only

Value

a tensor

Details

A tensor can be seen as a linear mapping of a tensor to a tensor. Let denote $R_i$ the space of real tensors with dimensions $i_1...i_d$.

chol.tensorComputes for a tensor $ a_{i_1 \ldots i_dj_1 \ldots j_d} $ representing a positive definit mapping form $R_j$ to $R_i$ with equal dimension structure in $i$ and $j$ its "Cholesky" decomposition $L_{i_1 \ldots i_d \lambda{}}$ such that $$ a_{i_1...i_dj_1...j_d}=\sum_{\lambda{}} L_{i_1...i_d \lambda{}}L_{j_1...j_d \lambda{}} $$

Examples

Run this code

# NOT RUN {

A <- to.tensor(rnorm(15),c(a=3,b=5))
AAt <- einstein.tensor(A,mark(A,i="a"))
ch <- chol.tensor(AAt,"a","a'",name="lambda")
#names(ch)[1]<-"lambda"
einstein.tensor(ch,mark(ch,i="a")) # AAt

A <- to.tensor(rnorm(30),c(a=3,b=5,c=2))
AAt <- einstein.tensor(A,mark(A,i="a"),by="c")
ch <- chol.tensor(AAt,"a","a'",name="lambda")
einstein.tensor(ch,mark(ch,i="a"),by="c") #AAt

	     

# }