Fast matrix multiplication and division for Toeplitz and Hankel matrices in Julia
A Toeplitz matrix has constant diagonals. It can be constructed using
Toeplitz(vc,vr)where vc are the entries in the first column and vr are the entries in the first row, where vc[1] must equal vr[1]. For example.
Toeplitz([1.,2.,3.],[1.,4.,5.])is a sparse representation of the matrix
[ 1.0 4.0 5.0
2.0 1.0 4.0
3.0 2.0 1.0 ]A triangular Toeplitz matrix can be constructed using
TriangularToeplitz(ve,uplo)where uplo is either :L or :U and ve are the rows or columns, respectively. For example,
TriangularToeplitz([1.,2.,3.],:L)is a sparse representation of the matrix
[ 1.0 0.0 0.0
2.0 1.0 0.0
3.0 2.0 1.0 ]A Hankel matrix has constant anti-diagonals. It can be constructed using
Hankel(vc,vr)where vc are the entries in the first column and vr are the entries in the last row, where vc[end] must equal vr[1]. For example.
Hankel([1.,2.,3.],[3.,4.,5.])is a sparse representation of the matrix
[ 1.0 2.0 3.0
2.0 3.0 4.0
3.0 4.0 5.0 ]A circulant matrix is a special case of a Toeplitz matrix with periodic end conditions. It can be constructed using
Circulant(vc)where vc is a vector with the entries for the first column.
For example:
Circulant(1:3)is a sparse representation of the matrix
[ 1.0 3.0 2.0
2.0 1.0 3.0
3.0 2.0 1.0 ]