PTAk (version 2.0.0)

CONTRACTION: Contraction of two tensors

Description

Computes the contraction product of two tensors as a generalisation of matrix product.

Usage

CONTRACTION(X,z, Xwiz=NULL,zwiX=NULL,rezwiX=FALSE,usetensor=TRUE)
 CONTRACTION.list(X,zlist,moins=1,zwiX=NULL,usetensor=TRUE,withapply=FALSE)

Value

A tensor of dimension c(dim(X)[-Xwiz],dim(z)[-zwiX])

if X has got a bigger order than z.

Arguments

X

a tensor(as an array) of any order

z

another tensor (with at least one space in common)

zlist

a list of lists like a solution.PTAk at least with v for every list(here v can be any array)

Xwiz

Xwiz is to specify the entries of X to contract with entries of z specified by zwiX, if Xwiz NULL dim(z)[zwiX] matching dim(X) will do without ambiguity (taking all z dimensions if zwiX is NULL). In CONTRACTION.list it is not set as one supposes the contractions in the list to operate follow the dimensions of X

zwiX

idem as Xwiz. If both Xwiz and zwiX are NULL zwiXis replaced by full possibilities (1:length(dimz)) then Xwiz is looked for. In CONTRACTION.list it is the vector for dimensions in the v to contract with X. Only 1-way dimension for each v.

moins

the elements in zlist to skip (see also TENSELE)

rezwiX

logical if TRUE (and zwiX is NULL) rematches the dimensions in for zwiX: useful only if the dimensions of z were not following the Xwiz order and are not equals.

usetensor

if TRUE uses tensor (add-on package)

withapply

if TRUE (only for vectors in zlist uses apply

Author

Didier G. Leibovici

Details

Like two matrices contract according to the appropriate dimensions (columns matching rows) when one performs a matrix product, this operation does pretty much the same thing for tensors(array) and specified contraction dimensions given by Xwiz and zwiX which should match. The function is actually written like: transforms both tensors as matrices with the ``matching tensor product" of their contraction dimensions in columns (for higher order tensor) and rows (the other one), performs the matrix product and rebuild the result as a tensor(array). Without using tensor, if Xwiz and/or zwiX are not specified the functions tries to match all z dimensions onto the dimensions of X where X is the higher order tensor (if it is not the case in the arguments the function swaps them).

References

Leibovici D and Sabatier R (1998) A Singular Value Decomposition of a k-ways array for a Principal Component Analysis of multi-way data, the PTA-k. Linear Algebra and its Applications, 269:307-329.

Schwartz L (1975) Les Tenseurs. Herman, Paris.

See Also

PTAk, APSOLUk

Examples

 library(tensor)
   z <-  array(1:12,c(2,3,2))
   X <- array(1:48,c(3,4,2,2))
   Xcz <- CONTRACTION(X,z,Xwiz=c(1,3,4),zwiX=c(2,3,1))
   dim(Xcz)   # 4
   Xcz1 <- CONTRACTION(X,z,Xwiz=c(3,4),zwiX=c(1,3))
   dim(Xcz1) # 3,4,3
   Xcz2 <- CONTRACTION(X,z,Xwiz=c(3,4),zwiX=c(3,1))
   Xcz1[,,1]
   Xcz2[,,1]
   #######
   sval0 <- list(list(v=c(1,2,3,4)),list(v=rep(1,3)),list(v=c(1,3)))
   tew <- array(1:24,c(4,3,2))
    CONTRACTION.list(tew,sval0,moins=1)
       #this is equivalent to the following which may be too expensive for big datasets
    CONTRACTION(tew,TENSELE(sval0,moins=1),Xwiz=c(2,3))
   ##
     CONTRACTION.list(tew,sval0,moins=c(1,2)) #must be equal to
     CONTRACTION(tew,sval0[[3]]$v,Xwiz=3)