BibSonomy publications for /user/ejr/siamTue Oct 09 07:04:05 CEST 2007SIAM Conference on Computational Science and EngineeringFebruaryParallel Bipartite Matching for Sparse Matrix Computations2003combinatorialoptimization parallelalgorithms siam sparsematrix Tue Oct 09 07:04:05 CEST 2007SIAM Annual MeetingJunePractical Alternatives for Parallel Pivoting2003linearalgebra siam sparsematrix Tue Oct 09 07:04:05 CEST 2007SIAM Parallel Processing for Scientific ComputingfebParallel Weighted Bipartite Matching and Applications2004combinatorialoptimization parallelalgorithms siam sparsematrix Tue Oct 09 07:04:05 CEST 2007SIAM Workshop on Combinatorial Scientific ComputingfebSparse Data Structures for Weighted Bipartite Matching2004combinatorialoptimization siam sparsematrix Tue Oct 09 07:04:05 CEST 2007SIAM Conference on Computational Science and EngineeringfebParallel Combinatorial Computing and Sparse Matrices2005combinatorialoptimization parallelalgorithms siam sparsematrix Tue Oct 09 06:57:00 CEST 2007SIAM Journal on Scientific Computing51613--1633Benefits of {IEEE-754} Features in Modern Symmetric Tridiagonal Eigensolvers282006eigenvalue floatingpoint ieee754 siam sisc Bisection is one of the most common methods used to compute the eigenvalues of symmetric tridiagonal matrices. Bisection relies on the Sturm count: For a given shift sigma, the number of negative pivots in the factorization $T - \sigma I = LDL^T$ equals the number of eigenvalues of T that are smaller than sigma. In IEEE-754 arithmetic, the value $\infty$ permits the computation to continue past a zero pivot, producing a correct Sturm count when $T$ is unreduced. Demmel and Li showed [IEEE Trans. Comput., 43 (1994), pp. 983–992] that using $\infty$ rather than testing for zero pivots within the loop could significantly improve performance on certain architectures. When eigenvalues are to be computed to high relative accuracy, it is often preferable to work with $LDL^T$ factorizations instead of the original tridiagonal $T$. One important example is the MRRR algorithm. When bisection is applied to the factored matrix, the Sturm count is computed from $LDL^T$ which makes differential stationary and progressive qds algorithms the methods of choice. While it seems trivial to replace $T$ by $LDL^T$, in reality these algorithms are more complicated: In IEEE-754 arithmetic, a zero pivot produces an overflow followed by an invalid exception (NaN, or "Not a Number") that renders the Sturm count incorrect. We present alternative, safe formulations that are guaranteed to produce the correct result. Benchmarking these algorithms on a variety of platforms shows that the original formulation without tests is always faster provided that no exception occurs. The transforms see speed-ups of up to 2.6x over the careful formulations. Tests on industrial matrices show that encountering exceptions in practice is rare. This leads to the following design: First, compute the Sturm count by the fast but unsafe algorithm. Then, if an exception occurs, recompute the count by a safe, slower alternative. The new Sturm count algorithms improve the speed of bisection by up to 2x on our test matrices. Furthermore, unlike the traditional tiny-pivot substitution, proper use of IEEE-754 features provides a careful formulation that imposes no input range restrictions.