TOEPLITZ_CHOLESKY
Cholesky Factor of Nonnegative Definite Toeplitz Matrix

TOEPLITZ_CHOLESKY is a MATLAB library which computes the Cholesky factorization of a nonnegative definite symmetric Toeplitz matrix.

A Toeplitz matrix is a matrix which is constant along all diagonals. A schematic of a 3x4 Toeplitz matrix would be

        a  b  c  d
        e  a  b  c
        f  e  a  b
      

A symmetric matrix is a matrix with N rows and N columns, such that A(I,J) = A(J,I) for all indices I and J. All the eigenvalues of a symmetric matrix are real.

A symmetric Toeplitz matrix is a matrix which is symmetric and Toeplitz. A schematic of a 4x4 symmetric Toeplitz matrix would be

        a  b  c  d
        b  a  b  c
        c  b  a  b
        d  c  b  a
      

A nonnegative definite symmetric matrix A is a symmetric matrix whose eigenvalues are all nonnegative.

Given a nonnegative definite symmetric matrix A, the upper Cholesky factor R is an upper triangular matrix such that A = R' * R; the lower Cholesky factor L is a lower triangular matrix such that A = L L'. Obviously, L = R'.

A Toeplitz matrix can be represented in a compressed format that stores the first row and the first column (omitting the first entry). One convenient format would be to create the 2xN array G as follows:

       G(1,1:N) = A(1,1:N)
       G(2,1)   = 0.0
       G(2,2:N) = A(2:N,1)
      

A symmetric Toeplitz matrix can be represented in a compressed format that stores just the first row.

Licensing:

The computer code and data files described and made available on this web page are distributed under the GNU LGPL license.

Languages:

TOEPLITZ_CHOLESKY is available in a C version and a C++ version and a FORTRAN77 version and a FORTRAN90 version and a MATLAB version.

Related Data and Programs:

ASA006, a MATLAB library which computes the Cholesky factorization of a symmetric positive definite matrix, by Michael Healy. This is a MATLAB version of Applied Statistics Algorithm 6;

LINPACK_D, a MATLAB library which factors and solves linear systems using double precision real arithmetic, by Jack Dongarra, Jim Bunch, Cleve Moler, Pete Stewart.

LINPLUS, a MATLAB library which carries out simple manipulations of matrices in a variety of formats, including matrices stored in banded, border-banded, circulant, lower triangular, pentadiagonal, sparse, symmetric, toeplitz, tridiagonal, upper triangular and vandermonde formats.

Reference:

1. Michael Stewart,
   Cholesky factorization of semi-definite Toeplitz matrices.

Source Code:

• r8mat.m, prints an R8MAT.
• r8mat_print_some.m, prints some of an R8MAT.
• timestamp.m, prints the current YMDHMS date as a time stamp.
• toep_cholesky_lower.m, given a compressed Toeplitz matrix, computes the lower Cholesky factor.
• toep_cholesky_upper.m, given a compressed Toeplitz matrix, computes the upper Cholesky factor.
• toeplitz_cholesky_lower.m, given a Toeplitz matrix, computes the lower Cholesky factor.
• toeplitz_cholesky_upper.m, given a Toeplitz matrix, computes the upper Cholesky factor.

Examples and Tests:

• toeplitz_cholesky_test.m, calls all the tests;
• toeplitz_cholesky_test021.m, calls all the codes for a simple 3x3 case.
• toeplitz_cholesky_test_output.txt, the output file.

You can go up one level to the MATLAB source codes.