Towards a Parallel Data Mining Toolbox
IPDPS '01 Proceedings of the 15th International Parallel & Distributed Processing Symposium
Object-Oriented Design for Sparse Direct Solvers
ISCOPE '98 Proceedings of the Second International Symposium on Computing in Object-Oriented Parallel Environments
An iterative working-set method for large-scale nonconvex quadratic programming
Applied Numerical Mathematics
An out-of-core sparse symmetric-indefinite factorization method
ACM Transactions on Mathematical Software (TOMS)
ACM Transactions on Mathematical Software (TOMS)
Experiences of sparse direct symmetric solvers
ACM Transactions on Mathematical Software (TOMS)
Relaxed forms of BBK algorithm and FBP algorithm for symmetric indefinite linear systems
Computers & Mathematics with Applications
On solving sparse symmetric linear systems whose definiteness is unknown
Applied Numerical Mathematics
FGMRES preconditioning by symmetric/skew-symmetric decomposition of generalized stokes problems
Mathematics and Computers in Simulation
Computational Optimization and Applications
Issues in the design of scalable out-of-core dense symmetric indefinite factorization algorithms
ICCS'03 Proceedings of the 2003 international conference on Computational science: PartIII
Prospectus for the next LAPACK and ScaLAPACK libraries
PARA'06 Proceedings of the 8th international conference on Applied parallel computing: state of the art in scientific computing
Partitioned Triangular Tridiagonalization
ACM Transactions on Mathematical Software (TOMS)
SelInv---An Algorithm for Selected Inversion of a Sparse Symmetric Matrix
ACM Transactions on Mathematical Software (TOMS)
Partial factorization of a dense symmetric indefinite matrix
ACM Transactions on Mathematical Software (TOMS)
Novel modifications of parallel Jacobi algorithms
Numerical Algorithms
PARA'04 Proceedings of the 7th international conference on Applied Parallel Computing: state of the Art in Scientific Computing
Reducing the amount of pivoting in symmetric indefinite systems
PPAM'11 Proceedings of the 9th international conference on Parallel Processing and Applied Mathematics - Volume Part I
Pivoting strategies for tough sparse indefinite systems
ACM Transactions on Mathematical Software (TOMS)
Hi-index | 0.02 |
The Bunch-Kaufman factorization is widely accepted as the algorithm of choice for the direct solution of symmetric indefinite linear equations; it is the algorithm employed in both LINPACK and LAPACK. It has also been adapted to sparse symmetric indefinite linear systems.While the Bunch--Kaufman factorization is normwise backward stable, its factors can have unusual scaling, with entries bounded by terms depending both on |A| and on $\kappa(A)$. This scaling, combined with the block nature of the algorithm, may degrade the accuracy of computed solutions unnecessarily. Overlooking the lack of a triangular factor bound leads to a further complication in LAPACK such that the LAPACK Bunch--Kaufman factorization can be unstable. We present two alternative algorithms, close cousins of the Bunch-Kaufman factorization, for solving dense symmetric indefinite systems. Both share the positive attributes of the Bunch-Kaufman algorithm but provide better accuracy by bounding the triangular factors. The price of higher accuracy can be kept low by choosing between our two algorithms. One is appropriate as the replacement for the blocked LAPACK Bunch-Kaufman factorization; the other would replace the LINPACK-like unblocked factorization in LAPACK.Solving sparse symmetric indefinite systems is more problematic. We conclude that the Bunch-Kaufman algorithm cannot be rescued effectively in the sparse case. Imposing the constraint of bounding the triangular factors leads naturally to one particular version of the Duff-Reid algorithm, which we show gives better accuracy than Liu's sparse variant of the Bunch-Kaufman algorithm. We extend the work of Duff and Reid in two respects that often provide higher efficiency: a more effective procedure for finding pivot blocks and a stable extension to pivot blocks of size larger than two.