`array.mult`

Array multiplication

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

array.mult(a, b, x)

Arguments

Argument	Description
`a`	a numeric matrix.
`b`	a numeric matrix.
`x`	a three-dimensional array.

Details

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

\[y_{tkl} = \sum\limits_{i=1}^p\sum\limits_{j=1}^q a_{ki}x_{tij}b_{jl}\]

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.

Examples

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

array.mult