# array.mult: Array multiplication In fastmatrix: Fast Computation of some Matrices Useful in Statistics

## Description

Multiplication of 3-dimensional arrays was first introduced by Bates and Watts (1980). More extensions and technical details can be found in Wei (1998).

## Usage

 1 array.mult(a, b, x) 

## Arguments

 a a numeric matrix. b a numeric matrix. x a three-dimensional array.

## Details

Let \bold{X} = (x_{tij}) be a 3-dimensional n\times p\times q where indices t, i and j indicate face, row and column, respectively. The product \bold{Y} = \bold{AXB} is an n\times r\times s array, with \bold{A} and \bold{B} are r\times p and q\times s matrices respectively. The elements of \bold{Y} are defined as:

y_{tkl} = ā\limits_{i=1}^pā\limits_{j=1}^q a_{ki}x_{tij}b_{jl}

## Value

array.mult returns a 3-dimensional array of dimension n\times r\times s.

## References

Bates, D.M., Watts, D.G. (1980). Relative curvature measures of nonlinearity. Journal of the Royal Statistical Society, Series B 42, 1-25.

Wei, B.C. (1998). Exponential Family Nonlinear Models. Springer, New York.

array, matrix, bracket.prod.
  1 2 3 4 5 6 7 8 9 10 x <- array(0, dim = c(2,3,3)) # 2 x 3 x 3 array x[,,1] <- c(1,2,2,4,3,6) x[,,2] <- c(2,4,4,8,6,12) x[,,3] <- c(3,6,6,12,9,18) a <- matrix(1, nrow = 2, ncol = 3) b <- matrix(1, nrow = 3, ncol = 2) y <- array.mult(a, b, x) # a 2 x 2 x 2 array y