Blocked and Recursive Algorithms for Triangular Tridiagonalization

We present partitioned (blocked) algorithms for reducing a symmetric matrix to a tridiagonal form, with partial pivoting. That is, the algorithms compute a factorization PAP = LTL where P is a permutation matrix, L is lower triangular with a unit diagonal, and T is symmetric and tridiagonal. The algorithms are based on the column-by-column methods of Parlett and Reid and of Aasen. Our implementations also compute the QR factorization of T and solve linear systems of equations using the computed factorization. The partitioning allow our algorithms to exploit modern computer architectures (in particular, cache memories and high-performance blas libraries). Experimental results demonstrate that our algorithms achieve approximately the same level of performance as the partitioned Bunch-Kaufman factorization and solve in lapack.