Computing the Minimum Eigenvalue of a Symmetric Positive Definite Toeplitz Matrix with Spectral Transformation Lanczos Methods

Abstract

A matrix T is Toeplitz if the elements on each diagonal are all equal. Thus, for the real symmetric case T is of the form

$$ T = T({{t}_{0}},{{t}_{1}}, \ldots ,{{t}_{{n - 1}}}): = \left( {\begin{array}{*{20}{c}} {{{t}_{0}}} & {{{t}_{1}}} & \cdots & \cdots & {{{t}_{{n - 1}}}} \\ {{{t}_{1}}} & {{{t}_{0}}} & {{{t}_{1}}} & {} & \vdots \\ \vdots & {{{t}_{1}}} & \ddots & \ddots & \vdots \\ \vdots & {} & \ddots & \ddots & {{{t}_{1}}} \\ {{{t}_{{n - 1}}}} & \cdots & \cdots & {{{t}_{1}}} & {{{t}_{0}}} \\ \end{array} } \right). $$