Partitioned Triangular Tridiagonalization

Abstract. We present a partitioned algorithm for reducing a symmetric matrix to a tridiagonal form, with partial pivoting. That is, the algorithm computes a factorization P AP T = LTLT where P is a permutation matrix, L is lower triangular with a unit diagonal and entries’ magnitudes bounded by 1, and T is symmetric and tridiagonal. The algorithm is based on the basic (non partitioned) methods of Parlett and Reid and of Aasen. We show that our factorization algorithm is component-wise backward stable (provided that the growth factor is not too large), with a similar behavior to that of Aasen’s basic algorithm. Our implementation also computes the QR factorization of T and solves linear systems of equations using the computed factorization. The partitioning allows our algorithm 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 perfo...

• Type: Journal
• Year: 2011
• Where: TOMS
• Authors: Miroslav Rozlozník, Gil Shklarski, Sivan Toledo
• Keywords: Algorithms, Factorization, Partitioned Algorithm, TOMS 2011