Solving Unsymmetric Sparse Systems of Linear Equations with PARDISO
Supernode pivoting for unsymmetric matrices coupled with supernode partitioning and asynchronous computation can achieve high gigaflop rates for parallel sparse LU factorization on shared memory parallel computers. The progress in weighted graph matching algorithms helps to extend these concepts further and prepermutation of rows is used to place large matrix entries on the diagonal. Supernode pivoting allows dynamical interchanges of columns and rows during the factorization process. The BLAS-3 level efficiency is retained. An enhanced left—right looking scheduling scheme is uneffected and results in good speedup on SMP machines without increasing the operation count. These algorithms have been integrated into the recent unsymmetric version of the PARDISO solver. Experiments demonstrate that a wide set of unsymmetric linear systems can be solved and high performance is consistently achieved for large sparse unsymmetric matrices from real world applications.
KeywordsSparse Matrice Sparse Linear System Operation Count Partial Pivoting Nest Dissection
- 1.P. R. Amestoy, I. S. Duff, J.-Y. L’Excellent, and X. S. Li. Analysis and comparison of two general sparse solvers for distributed memory computers. Technical Report TR/PA/00/90, CERFACS, Toulouse, France, December 2000. Submitted to ACM Trans. Math. Softw. Google Scholar
- 6.Timothy A. Davis. UMFPACK. Software for unsymmetric multifrontal method. In NA Digest, 01(11), March 18, 2001., http://www.cise.ufl.edu/research/sparse/umfpack.
- 9.I. S. Duff and J. Koster. The design and use of algorithms for permuting large entries to the diagonal of sparse matrices. Technical Report TR/PA/97/45, CERFACS, Toulouse, France, 1997. Also appeared as Report RAL-TR-97-059, Rutherford Appleton Laboratories, Oxfordshire.Google Scholar
- 11.A. Gupta. Fast and effective algorithms for solving graph partitioning and sparse matrix ordering. IBM Journal of Research and Development, 41(1/2):171–183, January/March 1997.Google Scholar
- 12.A. Gupta. Improved symbolic and numerical factorization algorithms for unsymmetric sparse matrices. Technical Report RC 22137 (99131), IBM T. J. Watson Research Center, Yorktown Heights, NY, August 1, 2001.Google Scholar
- 13.A. Gupta. Recent advances in direct methods for solving unsymmetric sparse systems of linear equations. Technical Report RC 22039 (98933), IBM T. J. Watson Research Center, Yorktown Heights, NY, April 20, 2001.Google Scholar
- 15.X.S. Li and J.W Demmel. A scalable sparse direct solver using static pivoting. In Proceeding of the 9th SIAM conference on Parallel Processing for Scientic Computing, San Antonio, Texas, March 22-34, 1999.Google Scholar
- 17.O. Schenk. Scalable Parallel Sparse LU Factorization Methods on Shared Memory Multiprocessors. PhD thesis, ETH Zürich, 2000.Google Scholar
- 18.O. Schenk and K. Gärtner. Two-level scheduling in PARDISO: Improved scalability on shared memory multiprocessing systems. Accepted for publication in Parallel Computing.Google Scholar
- 19.O. Schenk and K. Gärtner. PARDISO: a high performance serial and parallel sparse linear solver in semiconductor device simulation. Future Generation Computer Systems, 789(1):1–9, 2001.Google Scholar