hankel_spd

hankel_spd, an Octave code which can compute a lower triangular matrix L which is the Cholesky factor of a symmetric positive definite (SPD) Hankel matrix H, that is, H = L * L'.

A Hankel matrix is a matrix which is constant along all antidiagonals. A schematic of a 5x5 Hankel matrix would be:

        a  b  c  d  e
        b  c  d  e  f
        c  d  e  f  g
        d  e  f  g  h
        e  f  g  h  i
      

Let J represent the exchange matrix, formed by reversing the order of the columns of the identity matrix. If H is a Hankel matrix, then J*H and H*J are Toeplitz matrices, and similarly in the other direction. Hence many algorithms that apply to one class can be easily adapted to the other.

Licensing:

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

Languages:

hankel_spd is available in a C version and a C++ version and a Fortran90 version and a MATLAB version and an Octave version and a Python version.

Related Data and Programs:

hankel_spd_test

asa006, an Octave code which computes the Cholesky factorization of a symmetric positive definite (SPD) matrix, by Michael Healy. This is a MATLAB version of Applied Statistics Algorithm 6;

hankel_cholesky, an Octave code which computes the upper Cholesky factor R of a symmetric positive definite (SPD) Hankel matrix H so that H = R' * R..

hankel_inverse, an Octave code which computes the inverse of a Hankel matrix.

Reference:

1. S Al-Homidan, M Alshahrani,
   Positive Definite Hankel Matrices Using Cholesky Factorization,
   Computational Methods in Applied Mathematics,
   Volume 9, Number 3, 2009, pages 221-225.

Source Code:

• hankel_spd_cholesky_lower.m, given data for the main diagonal and first subdiagonal, returns a lower triangular matrix L which is the lower Cholesky factor of a symmetric positive definite (SPD) Hankel matrix.
• r8mat_print.m, prints an R8MAT.
• r8mat_print_some.m, prints some of an R8MAT.