Description Usage Arguments Details Value Note Author(s) See Also Examples
A tensor can be seen as a linear mapping of a tensor to a tensor. This function computes its Cholesky decomposition.
1 | chol.tensor(X,i,j,...,name="lambda")
|
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 e_i |
... |
for generic use only |
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...i_dj_1...j_d} representing a positive definit mapping form R_j to R_j with equal dimension structure in i and j its "Cholesky" decomposition L_{i_1...i_d lambda} such that
a_{i_1...i_dj_1...j_d}=∑_{λ{}} L_{i_1...i_d λ{}}L_{j_1...j_d λ{}}
a tensor
A by
argument is not necessary, since both processing
dimensions have to be given.
K. Gerald van den Boogaart
1 2 3 4 5 6 7 8 9 10 11 12 | 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
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