TOEPLITZ_CHOLESKY
Cholesky Factor of Nonnegative Definite Toeplitz Matrix

TOEPLITZ_CHOLESKY is a C++ 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 FORTRAN90 version and a MATLAB version and a Python version.

Related Data and Programs:

ASA006, a C++ 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;

HANKEL_CHOLESKY, a C++ library which computes the upper Cholesky factor R of a nonnegative definite symmetric Hankel matrix H so that H = R' * R..

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

Reference:

1. Michael Stewart,
   Cholesky factorization of semi-definite Toeplitz matrices,
   Linear Algebra and its Applications,
   Volume 254, pages 497-525, 1997.

Source Code:

• toeplitz_cholesky.cpp, the source code.
• toeplitz_cholesky.hpp, the include file.

Examples and Tests:

• toeplitz_cholesky_test.cpp, a sample calling program.
• toeplitz_cholesky_test.txt, the output file.

List of Routines:

• I4_MAX returns the maximum of two I4's.
• I4_MIN returns the smaller of two I4's.
• R8MAT_COPY_NEW copies one R8MAT to a "new" R8MAT.
• R8MAT_MMT_NEW computes C = A * B'.
• R8MAT_MTM_NEW computes C = A' * B.
• R8MAT_PRINT prints an R8MAT.
• R8MAT_PRINT_SOME prints some of an R8MAT.
• TIMESTAMP prints the current YMDHMS date as a time stamp.
• TOEP_CHOLESKY_LOWER: lower Cholesky factor of a Toeplitz matrix.
• TOEP_CHOLESKY_UPPER: upper Cholesky factor of a Toeplitz matrix.
• TOEPLITZ_CHOLESKY_LOWER: lower Cholesky factor of a Toeplitz matrix.
• TOEPLITZ_CHOLESKY_UPPER: upper Cholesky factor of a Toeplitz matrix.

You can go up one level to the C++ source codes.