Stabilizing Trench's algorithm to invert symmetric Toeplitz matrices (Q1316134)

The original Trench algorithm for the inversion of Toeplitz matrices may be applied only to matrices with non-vanishing leading minors. When leading minors of the Toeplitz matrix \(T\) are vanishing or nearly vanishing, the authors propose to apply \textit{S. Zohar's} formulation of Trench's algorithm [J. Assoc. Comput. Machin. 16, 592-601 (1969; Zbl 0194.181) and ibid. 21, 272-276 (1974; Zbl 0276.65014)] to the positive definite \(T+\alpha I\) and to use persymmetric diagonal modifications of this matrix to evaluate \(T^{-1}\). The efficiency of the algorithm is achieved by careful exploitation of the symmetry and persymmetry properties of the matrices involved in computations.