magic (version 1.3-20)

circulant: Circulant matrices of any order

Description

Creates and tests for circulant matrices of any order

Usage

circulant(vec)
is.circulant(m,dir=rep(1,length(dim(m))))

Arguments

vec

In circulant(), vector of elements of the first row. If of length one, interpret as the order of the matrix and use 1:vec.

m

In is.circulant(), matrix to be tested for circulantism

dir

In is.circulant(), the direction of the diagonal. In a matrix, the default value (c(1,1)) traces the major diagonals.

Details

A matrix $a$ is circulant if all major diagonals, including broken diagonals, are uniform; ie if $a_{ij}=a_{kl}$ when $i-j=k-l$ (modulo $n$). The standard values to use give 1:n for the top row.

In the case of arbitrary dimensional arrays, giving the default value for dir checks that a[v]==a[v+rep(1,d)]==...=a[v+rep((n-1),d)] for all v (that is, following lines parallel to the major diagonal); indices are passed through process().

For general dir, function is.circulant() checks that a[v]==a[v+dir]==a[v+2*dir]==...==a[v+(n-1)*d] for all v.

A Toeplitz matrix is one in which a[i,j]=a[i',j'] whenever |i-j|=|i'-j'|. See function toeplitz() of the stats package for details.

References

Arthur T. Benjamin and K. Yasuda. Magic Squares Indeed!, American Mathematical Monthly, vol 106(2), pp152-156, Feb 1999

Examples

circulant(5)
circulant(2^(0:4))
is.circulant(circulant(5))

 a <- outer(1:3,1:3,"+")%%3
 is.circulant(a)
 is.circulant(a,c(1,2))

 is.circulant(array(c(1:4,4:1),rep(2,3)))

 is.circulant(magic(5)%%5,c(1,-2))