R: Circulant matrices of any order

circulant {magic} | R Documentation

## 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[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 function `is.circulant()`, for arbitrary dimensional arrays, 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.

### 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(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))

[Package magic version 1.5-8 Index]