riemann.tensor {tensorA}

Tensor multiplication with Riemann's convention

Description

Multiplies tensors by multiplying over all pairs with one covariate and one contravariate variable with the same name according to Riemann's summing convention.

Usage

riemann.tensor(...,only=NULL,by=NULL)
## Methods for class tensor
# x %r% y
## Default method
# x %r% y

Arguments

...

some tensors, or a renaming code

only

an optional list of the dimension names to be recognized for duplication to allow parallel processing on lists of tensors

x

a tensor

y

a tensor

by

Riemannian summing is done in parallel in these dimensions.

Details

see mul.tensor on details on tensor multiplication. In einstein.tensor complex operations can be performed by command and renaming code: The arguments are processed from left to right and multiplied. Unnamed attributes are regarded as tensors or scalars and multiplied with the current result by the Riemann summing convention, which means an inner product over all pairs of covariate and contravariate indices with the same name. Named attributes can either have the name diag, which performs a diagmul according to the same-name convention or be of the form A="B" or "A"="B", for which we have two cases. Typically both are given covariate. The first specifies the covariate to be used in the multiplication and the second the contravariate. If both names are present in the current result, an inner multiplication (trace) of on these two dimensions is performed. If only the covariate or the contravariate is present up to this point, the specific dimension is renamed to the second name, but keeps its type. This renaming might be visible in the result or inducing a multiplication according to the Riemann convention later if the other shows up.

Value

the tensor product of all the tensors along all duplicate dimensions.

Author(s)

K. Gerald van den Boogaart

See Also

mul.tensor, to.tensor, riemann.tensor

Examples

A <- to.tensor(1:20,c(U=2,"^V"=2,W=5))
B <- to.tensor(1:20,c("^U"=2,V=2,Q=5))
riemann.tensor(A,B)
A %r% B