Fast direct solvers for some complex symmetric block Toeplitz linear systems

We consider the solution of a class of complex symmetric block Toeplitz linear systems, arising from integral equation problems. Algorithms that exploit the Toeplitz structure provide considerable savings on the number of arithmetic operations, compared to the classical Cholesky factorization. We propose a generalized Schur algorithm adapted to the complex symmetric case. We detail blocked variants, that perform better by using BLAS 3 primitives. We also propose a solver, based on an augmented matrix approach, that allows a substantial decrease in the use of memory, by avoiding an explicit assembly of the Cholesky factor. All algorithms have been implemented and numerical results are included to illustrate the effectiveness of our approach.