stokes (version 1.2-1)

tensorprod: Tensor products of \(k\)-tensors

Description

Tensor products of \(k\)-tensors

Usage

tensorprod(U, ...)
tensorprod2(U1,U2)

Value

The functions documented here all return a spray object.

Arguments

U,U1,U2

Object of class ktensor

...

Further arguments, currently ignored

Author

Robin K. S. Hankin

Details

Given a \(k\)-tensor \(S\) and an \(l\)-tensor \(T\), we can form the tensor product \(S\otimes T\), defined as

$$S\otimes T\left(v_1,\ldots,v_k,v_{k+1},\ldots, v_{k+l}\right)= S\left(v_1,\ldots v_k\right)\cdot T\left(v_{k+1},\ldots v_{k+l}\right).$$

Package idiom for this includes tensorprod(S,T) and S %X% T; note that the tensor product is not commutative. Function tensorprod() can take any number of arguments (the result is well-defined because the tensor product is associative); it uses tensorprod2() as a low-level helper function.

References

Spivak 1961

See Also

ktensor

Examples

(A <- ktensor(spray(matrix(c(1,1,2,2,3,3),2,3,byrow=TRUE),1:2)))
(B <- ktensor(spray(10+matrix(4:9,3,2),5:7)))
tensorprod(A,B)

A %X% B - B %X% A


Va <- matrix(rnorm(9),3,3)
Vb <- matrix(rnorm(38),19,2)

LHS <- as.function(A %X% B)(cbind(rbind(Va,matrix(0,19-3,3)),Vb))
RHS <-  as.function(A)(Va) * as.function(B)(Vb)

c(LHS=LHS,RHS=RHS,diff=LHS-RHS)