circulant {magic}R Documentation

Circulant matrices of any order

Description

Creates and tests for circulant matrices of any order

Usage

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

Arguments

n Order of circulant matrix in circulant()
vec In circulant(), vector of elements of the first row, defaulting to 1:n.
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 are uniform, ie if a[i,j]==a[k,j] 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, the function checks that a[v]==a[v+dir]==a[v+2*dir]==...==a[v+(n-1)*d] for all v.

Author(s)

Robin K. S. Hankin

References

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

Examples

circulant(5)
circulant(5,vec=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(2,-1))


[Package magic version 1.3-15 Index]