Towards stable mixed pivoting strategies for the sequential and parallel solution of sparse symmetric indefinite systems

SIAM Journal on Matrix Analysis and Applications

Volumn 29, Issue 3, 2007, Pages 1007-1024

  Duff, Iain S ^a,b   Pralet, Stéphane ^c  

^a CERFACS   (France)

^b RUTHERFORD APPLETON LABORATORY   (United Kingdom)

^c IRIT   (France)

Author keywords

Direct solver for sparse symmetric indefinite matrices; Gaussian elimination; Matrix perturbation; Parallel multifrontal methods; Static pivoting

Indexed keywords

ESTIMATION; FACTORIZATION; PERTURBATION TECHNIQUES;

APPLIED MATHEMATICS; APPROXIMATE FACTORIZATION (AF); DIRECT SOLUTION; GROWTH FACTORS; INDEFINITE SYSTEMS; ITERATIVE REFINEMENT; PARALLEL IMPLEMENTATIONS;

ITERATIVE METHODS;

EID: 48249141435     PISSN: 08954798     EISSN: 10957162     Source Type: Journal    
DOI: 10.1137/050629598     Document Type: Article

References (28)

1. P. R. AMESTOY, I. S. DUFF, J.-Y. L'EXCELLENT, AND J. KOSTER, A fully asynchronous multifrontal solver using distributed dynamic scheduling, SIAM J. Matrix Anal. Appl., 23 (2001), pp. 15-41.
2. P. R. AMESTOY, I. S. DUFF, AND J.-Y. L'EXCELLENT, Multifrontal parallel distributed symmetric and unsymmetric solvers, Comput. Methods Appl. Mech. Engrg., 184 (2000), pp. 501-520.
3. M. ARIOLI, J. W. DEMMEL, AND I. S. DUFF, Solving sparse linear systems with sparse backward error, SIAM J. Matrix Anal. Appl., 10 (1989), pp. 165-190.
4. C. ASHCRAFT, R. G. GRIMES, AND J. G. LEWIS, Accurate symmetric indefinite linear equation solvers, SIAM J. Matrix Anal. Appl., 20 (1998), pp. 513-561.
5. J. R. BUNCH AND L. KAUFMAN, Some stable methods for calculating inertia and solving symmetric linear systems, Math. Comput., 31 (1977), pp. 162-179.
6. J. R. BUNCH AND B. N. PARLETT, Direct methods for solving symmetric indefinite systems of linear equations, SIAM J. Numer. Anal., 8 (1971), pp. 639-655.
7. S. H. CHENG AND N. J. HIGHAM, A modified Cholesky algorithm based on a symmetric indefinite factorization, SIAM J. Matrix Anal. Appl., 19 (1998), pp. 1097-1110.
8. T. A. DAVIS, University of Florida Sparse Matrix Collection, 2002, http://www.cise.ufl.edu/research/sparse/matrices.
9. J. J. DONGARRA, I. S. DUFF, D. C. SORENSEN, AND H. A. VAN DER VORST, Numerical Linear Algebra on High-Performance Computers, Software Environ. Tools 7, SIAM, Philadelphia, 1998.
10. I. S. DUFF, MA57-A code for the solution of sparse symmetric indefinite systems, ACM Trans. Math. Software, 30 (2004), pp. 118-144.
11. I. S. DUFF AND J. KOSTER, On algorithms for permuting large entries to the diagonal of a sparse matrix, SIAM J. Matrix Anal. Appl., 22 (2001), pp. 973-996.
12. I. S. DUFF AND S. PRALET, Strategies for scaling and pivoting for sparse symmetric indefinite problems, SIAM J. Matrix Anal. Appl., 27 (2005), pp. 313-340.
13. I. S. DUFF AND J. K. REID, The multifrontal solution of indefinite sparse symmetric linear systems, ACM Trans. Math. Software, 9 (1983), pp. 302-325.
14. I. S. DUFF AND J. K. REID, Exploiting zeros on the diagonal in the direct solution of indefinite sparse symmetric linear systems, ACM Trans. Math. Software, 22 (1996), pp. 227-257.
15. E. ESKOW AND R. B. SCHNABEL, Algorithm 695: Software for a new modified Cholesky factorization, ACM Trans. Math. Software, 17 (1991), pp. 306-312.
16. P. E. GILL AND W. MURRAY, Newton-type methods for unconstrained and linearly constrained optimization, Math. Programming, 28 (1974), pp. 311-350.
17. N. I. M. GOULD, J. A. SCOTT, AND Y. HU, A numerical evaluation of sparse direct solvers for the solution of large sparse symmetric linear systems of equations, ACM Trans. Math. Softw., 33 (2007).
18. N. I. M. GOULD, D. ORBAN, AND P. L. TOINT, CUTEr (and SifDec), A Constrained and Unconstrained Testing Environment, Revisited, Technical Report 2002-009, Rutherford Appleton Laboratory, Oxon, UK, 2002.
19. N. I. M. GOULD AND J. A. SCOTT, A numerical evaluation of HSL packages for the direct solution of large sparse, symmetric linear systems of equations, ACM Trans. Math. Software, 30 (2004), pp. 300-325.
20. N. J. HIGHAM, Accuracy and Stability of Numerical Algorithms, 2nd ed., SIAM, Philadelphia, 2002.
21. X. S. LI AND J. W. DEMMEL, Making sparse Gaussian elimination scalable by static pivoting, in Proceedings of Supercomputing, San Jose, CA, 1998, IEEE Computer Society Press, Washington, DC, 1998, pp. 1-17.
22. X. S. LI AND J. W. DEMMEL, SuperLU_DIST: A scalable distributed-memory sparse direct solver for unsymmetric linear systems, ACM Trans. Math. Software, 29 (2003).
23. J. W. H. LIU, The role of elimination trees in sparse factorization, SIAM J. Matrix Anal. Appl., 11 (1990), pp. 134-172.
24. J. W. H. LIU, E. G. NG, AND B. W. PEYTON, On finding super-nodes for sparse matrix computations, SIAM J. Matrix Anal. Appl., 14 (1993), pp. 242-252.
25. I. MAROS AND C. MESZAROS, A repository of convex quadratic programming problems, Optim. Methods Softw., 11-12 (1999), pp. 671-681.
26. S. PRALET, Constrained Orderings and Scheduling for Parallel Sparse Linear Algebra, Ph.D thesis, Institut National Polytechnique de Toulouse, Toulouse, France, 2004; also available as CERFACS Technical Report TH/PA/04/105.
27. O. SCHENK AND K. GÄRTNER, On Fast Factorization Pivoting Methods for Sparse Symmetric Indefinite Systems, Technical Report CS-2004-004, Computer Science Department, University of Basel, Basel, Switzerland, 2004.
28. G. W. STEWART, Modifying pivot elements in Gaussian elimination, Math. Comput., 28 (1974), pp. 537-542.