circulant {magic} | R Documentation |
Creates and tests for circulant matrices of any order
circulant(vec) is.circulant(m,dir=rep(1,length(dim(m))))
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 |
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.
Robin K. S. Hankin
Arthur T. Benjamin and K. Yasuda. Magic “Squares” Indeed!, American Mathematical Monthly, vol 106(2), pp152-156, Feb 1999
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))