article

Free access

A compact row storage scheme for Cholesky factors using elimination trees

Editor: John R. Rice Author:

Joseph W. LiuAuthors Info & Claims

ACM Transactions on Mathematical Software (TOMS), Volume 12, Issue 2

Pages 127 - 148

https://doi.org/10.1145/6497.6499

Published: 01 June 1986 Publication History

PDF eReader

Abstract

For a given sparse symmetric positive definite matrix, a compact row-oriented storage scheme for its Cholesky factor is introduced. The scheme is based on the structure of an elimination tree defined for the given matrix. This new storage scheme has the distinct advantage of having the amount of overhead storage required for indexing always bounded by the number of nonzeros in the original matrix. The structural representation may be viewed as storing the minimal structure of the given matrix that will preserve the symbolic Cholesky factor. Experimental results on practical problems indicate that the amount of savings in overhead storage can be substantial when compared with Sherman's compressed column storage scheme.

References

[1]

3~o, ~. v., t-tu~t;rtur"r, o. m., ANU ULLMAN, O. t). lJata orructures arm ~tgort~nms. ~uumon- Wesley, Reading, Mass., 1983.

Google Scholar

[2]

DUFF, I.S. Full matrix techniques in sparse Gaussian elimination. In Proceedings of the Dundee Conference on Numerical Analysis. Lecture Notes in Mathematics, 912, G. A. Watson, Ed., Springer-Verlag, New York, N.Y. 1982, pp. 71-84.

Google Scholar

[3]

DUFF, I.S. Parallel implementation of multifrontal methods. Technical Memorandum No. 49. Mathematics and Computer Science Division, Argonne National Laboratory, Argonne, IL, 1985.

Google Scholar

[4]

DUFF, I. S., GRIMES, R., LEWIS, J., AND POOLE, B. Sparse matrix test problems. SIGNUM Newsl., (June 1982), p. 22.

Crossref

Google Scholar

[5]

DUFF, I. S., AND REID, J.K. The multifrontal solution of indefinite sparse symmetric linear equations. ACM Trans. Math. So{tw. 9, 3 (1983), 302-325.

Crossref

Google Scholar

[6]

EISENSTAT, S. C., SCHULTZ, M. H., AND SHERMAN, A.H. Applications of an element model for Gaussian elimination. In Sparse Matrix Computations, J. R. Bunch and D. J. Rose, Eds., Academic Press, New York, N.Y. 1976, pp. 85-96.

Google Scholar

[7]

EISENSTAT, S. C., SCHULTZ, M. H., AND SHERMAN, A. H. Software for sparse Gaussian elimination with limited core storage. In Sparse Matrix Proceedings, I. S. Duff and G. W. Stewart, Eds., SIAM Press, Philadelphia, Pa. 1979, pp. 135-153.

Google Scholar

[8]

EISENSTAT, S. C., SCHULTZ, M. H., AND SHERMAN, A.H. Algorithms and data structures for sparse symmetric Gaussian elimination. SIAM J. Sci. Stat. Comput., 2, (1981), 225-237.

Google Scholar

[9]

GEORGE, J.A. Nested dissection of a regular finite element mesh. SIAM J. Numer. Anal., 10, (1973), 345-363.

Google Scholar

[10]

GEORGE, J. A., AND LIU, Z. W.H. A note on fill for sparse matrices. SIAM J. Numer. Anal., 12, (1975), 452-454.

Google Scholar

[11]

GEORGE, J. A., AND LIE, J. W.H. An optimal algorithm for symbolic factorization of symmetric matrices. SIAM J. Cornput. 9, (1980), 583-593.

Google Scholar

[12]

GEORGE, J. A., AND LIu, J. W.H. Computer Solution of Large Sparse Positive Definite Systems. Prentice-Hall Inc., Englewood Cliffs, N.J., 1981.

Crossref

Google Scholar

[13]

JESS, J. A. G., AND KEES, H. G.M. A data structure for parallel L/U decomposition. IEEE Trans. Comput., C-31, (1982), 231-239.

Google Scholar

[14]

LIU, J. W.H. On general row merging schemes for sparse Givens transformations. Tech. Rep. CS-83-04, Dept. of Computer Science, York University, Ontario, Canada, 1983. To appear in SIAM J. Sci. Stat. Comput.

Google Scholar

[15]

L,U, J. W.H. Computational models and task scheduling for parallel sparse Cholesky factorization. Tech. Rep. CS-85-01, Dept. of Computer Science, York University, Ontario, Canada, 1985. To appear in Parallel Computing.

Google Scholar

[16]

LIv, J. W.H. Modification of the minimum-degree algorithm by multiple elimination. A CM Trans. Math So{tw. 11, 2 (1985), 141-153.

Crossref

Google Scholar

[17]

PARTER, S. The use of linear graphs in Gaussian elimination. SIAM Rev. 3, (1961), 119-130.

Google Scholar

[18]

PETERS, F.J. Sparse matrices and substructures. Mathematical Centre Tracts, 119, Mathematisch Centrum, Amsterdam, The Netherlands, 1980.

Google Scholar

[19]

ROSE, D.J. A graph-theoretic study of the numerical solution of sparse positive definite systems of linear equations. In Graph Theory and Computing, R. C. Read, Ed., Academic Press, New York, N.Y. 1973, 183-217.

Google Scholar

[20]

ROSE, D. J., WHITTEN, G. F., SHERMAN, A. l'I., AND TARJAN, R.E. Algorithms and software for in-core factorization of sparse symmetri~: positive definite matrices. Comput. Struct. 11, (1980), 597-608.

Google Scholar

[21]

SCHREIBER, R. A new implementation of sparse Gaussian elimination. ACM Trans. Math Softw. 8, (1982), 256-276.

Crossref

Google Scholar

[22]

SHERMAN, A.H. On the efficient solution of sparse systems of linear and nonlinear equations. Ph.D. dissertation, Dept. of Computer Science., Yale University, New Haven, Conn., 1975.

Crossref

Google Scholar

[23]

TARJAN, R.E. Efficiency of a good but not linear set union algorithm. J. ACM 22, 2, (1975), 215-225.

Crossref

Google Scholar

[24]

TARJAN, R. E. Data Struc~ures and Network Algorithms. CBMS-NSF Regional Conference Series in Applied Math, SIAM Press, Philade}phia, Pa. 1983.

Crossref

Google Scholar

Cited By

View all

Sundram STariq MKjolstad F(2024)Compiling Recurrences over Dense and Sparse ArraysProceedings of the ACM on Programming Languages10.1145/36498208:OOPSLA1(250-275)Online publication date: 29-Apr-2024
https://dl.acm.org/doi/10.1145/3649820
Scott JTůma MScott JTůma M(2023)Sparse Cholesky Solver: The Symbolic PhaseAlgorithms for Sparse Linear Systems10.1007/978-3-031-25820-6_4(53-72)Online publication date: 11-Jan-2023
https://doi.org/10.1007/978-3-031-25820-6_4
Scott JTůma MScott JTůma M(2023)Introduction to Matrix FactorizationsAlgorithms for Sparse Linear Systems10.1007/978-3-031-25820-6_3(31-51)Online publication date: 11-Jan-2023
https://doi.org/10.1007/978-3-031-25820-6_3
Show More Cited By

Index Terms

A compact row storage scheme for Cholesky factors using elimination trees

Recommendations

Row Modifications of a Sparse Cholesky Factorization

Given a sparse, symmetric positive definite matrix C and an associated sparse Cholesky factorization LDL$\tr$, we develop sparse techniques for updating the factorization after a symmetric modification of a row and column of C. We show how the ...
Row elimination in sparse matrices using rotations
A fully portable high performance minimal storage hybrid format Cholesky algorithm

We consider the efficient implementation of the Cholesky solution of symmetric positive-definite dense linear systems of equations using packed storage. We take the same starting point as that of LINPACK and LAPACK, with the upper (or lower) triangular ...

Reviews

Reviewer: Ian Gladwell

This paper surveys methods for a symbolic factorization of sparse symmetric positive definite matrices. A new method, based on a compact row-oriented storage scheme, is described. It is shown that the amount of overhead storage required for indexing in the elimination is bounded by the number of nonzeros in the original matrix. In this respect, the algorithm is superior to all its competitors, including the column-oriented scheme popularized by Sherman. Numerical experience is presented for a regular k-by- k grid model problem and for a number of problems from the Harwell-Boeing test matrix set. These experiments show clearly that the storage gains to be made in the symbolic factorization are significant. However, timings of the setup for the data structure involved in the proposed scheme (compared with the Sherman column-oriented scheme) are less clear cut. Indeed, in the implementations used here they seem to generally favor the column-oriented scheme.

Access critical reviews of Computing literature here

Become a reviewer for Computing Reviews.

Comments

Information & Contributors

Information

Published In

cover image ACM Transactions on Mathematical Software

ACM Transactions on Mathematical Software Volume 12, Issue 2

June 1986

96 pages

ISSN:0098-3500

EISSN:1557-7295

DOI:10.1145/6497

Editor:
John R. Rice

Issue’s Table of Contents

Publisher

Association for Computing Machinery

New York, NY, United States

Publication History

Published: 01 June 1986

Published in TOMS Volume 12, Issue 2

Permissions

Request permissions for this article.

Request Permissions

Check for updates

Qualifiers

Article

Contributors

Other Metrics

View Article Metrics

Bibliometrics & Citations

Bibliometrics

Article Metrics

90
Total Citations
View Citations
1,520
Total Downloads

Downloads (Last 12 months)104
Downloads (Last 6 weeks)12

Reflects downloads up to 13 Nov 2024

Other Metrics

View Author Metrics

Citations

Cited By

View all

Sundram STariq MKjolstad F(2024)Compiling Recurrences over Dense and Sparse ArraysProceedings of the ACM on Programming Languages10.1145/36498208:OOPSLA1(250-275)Online publication date: 29-Apr-2024
https://dl.acm.org/doi/10.1145/3649820
Scott JTůma MScott JTůma M(2023)Sparse Cholesky Solver: The Symbolic PhaseAlgorithms for Sparse Linear Systems10.1007/978-3-031-25820-6_4(53-72)Online publication date: 11-Jan-2023
https://doi.org/10.1007/978-3-031-25820-6_4
Scott JTůma MScott JTůma M(2023)Introduction to Matrix FactorizationsAlgorithms for Sparse Linear Systems10.1007/978-3-031-25820-6_3(31-51)Online publication date: 11-Jan-2023
https://doi.org/10.1007/978-3-031-25820-6_3
Bik AKoanantakool PShpeisman TVasilache NZheng BKjolstad F(2022)Compiler Support for Sparse Tensor Computations in MLIRACM Transactions on Architecture and Code Optimization10.1145/354455919:4(1-25)Online publication date: 16-Sep-2022
https://dl.acm.org/doi/10.1145/3544559
Lourenco CMoreno-Centeno E(2022)Exactly Solving Sparse Rational Linear Systems via Roundoff-Error-Free Cholesky FactorizationsSIAM Journal on Matrix Analysis and Applications10.1137/20M137159243:1(439-463)Online publication date: 14-Mar-2022
https://doi.org/10.1137/20M1371592
Gilbert JLi XNg EPeyton B(2022)Computing Row and Column Counts for Sparse QR and LU FactorizationBIT10.1023/A:102194390202541:4(693-710)Online publication date: 11-Mar-2022
https://dl.acm.org/doi/10.1023/A%3A1021943902025
Jacquelin MNg EPeyton B(2021)Fast Implementation of the Traveling-Salesman-Problem Method for Reordering Columns within SupernodesSIAM Journal on Matrix Analysis and Applications10.1137/20M136807042:3(1337-1364)Online publication date: 7-Sep-2021
https://doi.org/10.1137/20M1368070
Guo FLu WFeng ZYu YZeng H(2021)Study on algorithms for solving depletion sparse matrix based on DESCAR moduleAnnals of Nuclear Energy10.1016/j.anucene.2020.107976151(107976)Online publication date: Feb-2021
https://doi.org/10.1016/j.anucene.2020.107976
Thuerck D(2019)Stretching Jacobi: Two-Stage Pivoting in Block-Based Factorization2019 IEEE/ACM 9th Workshop on Irregular Applications: Architectures and Algorithms (IA3)10.1109/IA349570.2019.00014(51-58)Online publication date: Nov-2019
https://doi.org/10.1109/IA349570.2019.00014
Liu CYang HWang DWu TWu XLuo ZTang X(2018)New Parallel Algorithms for Direct Solution of Large Sparse Matrix Equations2018 10th International Conference on Measuring Technology and Mechatronics Automation (ICMTMA)10.1109/ICMTMA.2018.00092(354-357)Online publication date: Feb-2018
https://doi.org/10.1109/ICMTMA.2018.00092
Show More Cited By

View Options

View options

PDF

View or Download as a PDF file.

PDF

eReader

View online with eReader.

eReader

Get Access

Login options

Check if you have access through your login credentials or your institution to get full access on this article.

Abstract

References

Cited By

Index Terms

Recommendations

Row Modifications of a Sparse Cholesky Factorization

Row elimination in sparse matrices using rotations

A fully portable high performance minimal storage hybrid format Cholesky algorithm

Reviews

Access critical reviews of Computing literature here

Comments

Information

Published In

Publisher

Publication History

Permissions

Check for updates

Qualifiers

Contributors

Other Metrics

Bibliometrics

Article Metrics

Other Metrics

Citations

Cited By

View options

PDF

eReader

Get Access

Login options

Full Access

Figures

Other

Share

Share this Publication link

Share on social media

Affiliations