chol.tensor {tensorA} | R Documentation |
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
X |
The tensor to be decomposed |
i |
The image dimensions of the linear mapping |
j |
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 |
... |
for generic use only |
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.tensor
Computes for a tensor
a_{i_1 \ldots i_dj_1 \ldots j_d}
representing a positive definit mapping formR_j
toR_i
with equal dimension structure ini
andj
its "Cholesky" decompositionL_{i_1 \ldots i_d \lambda{}}
such thata_{i_1...i_dj_1...j_d}=\sum_{\lambda{}} L_{i_1...i_d \lambda{}}L_{j_1...j_d \lambda{}}
Value
a tensor
Note
A by
argument is not necessary, since both processing
dimensions have to be given.
Author(s)
K. Gerald van den Boogaart
See Also
Examples
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