Using the Sherman–Morrison–Woodbury inversion formula for a fast solution of tridiagonal block Toeplitz systems

A fast numerical algorithm for solving systems of linear equations with tridiagonal block Toeplitz matrices is presented. The algorithm is based on a preliminary factorization of the generating quadratic matrix polynomial associated with the Toeplitz matrix, followed by the Sherman–Morrison–Woodbury inversion formula and solution of two bidiagonal and one diagonal block Toeplitz systems. Tight estimates of the condition numbers are provided for the matrix system and the main matrix systems generated during the preliminary factorization. The emphasis is put on rigorous stability analysis to rounding errors of the Sherman–Morrison–Woodbury inversion. Numerical experiments are provided to illustrate the theory.