This program is written by Egor Maximenko and Gabino Sánchez Arzate.

The formulas used in this program are based on the article by William F. Trench (1985):
Explicit inversion formulas for Toeplitz band matrices.
DOI: 10.1137/0606054

On this page we construct and visualize the inverse matrix *T*^{−1},
where *T* is the tridiagonal real symmetric Toeplitz matrix of order *n*,
with generating symbol *g*(*x*) = 4 (sin(*x*/2))^{2} + α.

We suppose that α > 0. In this case the matrix *T* is positive definite.

The first column of *T* is 2 + α, −1, 0, …, 0.

Choose the value of the parameter *α* and the order *n* of the matrix:

*α* =

*n* =

In this example *T*^{−1} = *S* − *H*,
where *S* is a Toeplitz matrix and *H* is a Hankel matrix.

Next picture shows*T*^{−1}, *S*, and *H*.
All their entries are positive. Black color corresponds to the maximum values.

Next picture shows

= | − |

All these matrices are symmetric and persymmetric.
Next picture shows the first column of *S*:

Code in MATLAB language (tested in GNU Octave)