tensorA (version 0.36.1)

solve.tensor: Solving linear equations with tensors

Description

We can formulate linear equation systems with tensors. This functions solves these systems or gives a least squares fit of minimal norm.

Usage

solve.tensor(a,b,i,j=i,...,allowSingular=FALSE,eps=1E-10,by=NULL)

Arguments

a

The a of ax=b

b

The a of ax=b

i

The dimensions of the equation in a

j

The dimensions of the equation in b

allowSingular

A boolean, indicating the that a least squares fit should be generated with singular equations systems.

…

further arguments for generic use

eps

The limit for the smallest singular value in inversion

by

the operation is done in parallel for these dimensions

Value

a tensor such that ax=b as good as possible for each combination of by values.

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 \(i_1...i_d\).

• solve.tensorSolves the equation for \(a_{i_1...i_dk_1...k_p}\), \(b_{j_1...j_dl_1...l_q}\) and \(x_{k_1...k_pl_1...l_q}\) the equation $$ \sum_{k_1,...,k_p} a_{i_1...i_dk_1...k_p}x_{k_1...k_pl_1...l_q}= b_{j_1...j_dl_1...l_q}$$.

See Also

to.tensor, svd.tensor, inv.tensor, chol.tensor, power.tensor

Examples

# NOT RUN {
R1  <- matrix(rnorm(9),nrow=3)
R1i <- solve(R1)
R2 <- to.tensor(R1,c(a=3,b=3),what=1:2)
R2i <- to.tensor(R1i,c(b=3,a=3),what=1:2)

inv.tensor(R2,"a","b") - R2i
inv.tensor(R2,"a","b",allowSingular=TRUE) - R2i

inv.tensor(rep(R2,4,1,"K"),"a","b",by="K") - rep(R2i,4,1,"K")
inv.tensor(rep(R2,4,1,"K"),"a","b",by="K",allowSingular=TRUE) - rep(R2i,4,3,"K")

R3 <- to.tensor(rnorm(15),c(a=3,z=5))

mul.tensor(R2i,"b",mul.tensor(R2,"a",R3)) # R3

solve.tensor(R2i,R3[[z=1]],"a")
mul.tensor(R2,"a",R3[[z=1]])

solve.tensor(R2i,R3,"a")
mul.tensor(R2,"a",R3)

solve.tensor(R2i,R3[[z=1]],"a",allowSingular=TRUE)
mul.tensor(R2,"a",R3[[z=1]])

solve.tensor(R2i,R3,"a",allowSingular=TRUE)
mul.tensor(R2,"a",R3)

solve.tensor(rep(R2i,4,1,"K"),R3[[z=1]],"a",by="K")
rep(mul.tensor(R2,"a",R3[[z=1]]),4,1,"K")

solve.tensor(rep(R2i,4,1,"K"),rep(R3[[z=1]],4,1,"K"),"a",by="K")
rep(mul.tensor(R2,"a",R3[[z=1]]),4,1,"K")

# }