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A block QR factorization algorithm using restricted pivoting

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J. R. BischofAuthors Info & Claims

Supercomputing '89: Proceedings of the 1989 ACM/IEEE conference on Supercomputing

Pages 248 - 256

https://doi.org/10.1145/76263.76290

Published: 01 August 1989 Publication History

Abstract

This paper presents a new algorithm for computing the QR factorization of a rank-deficient matrix on high-performance machines. The algorithm is based on the Householder QR factorization algorithm with column pivoting. The traditional pivoting strategy is not well suited for machines with a memory hierarchy since it precludes the use of matrix-matrix operations. However, matrix-matrix operations perform better on those machines than matrix-vector or vector-vector operations since they involve significantly less data movement per floating point operation. We suggest a restricted pivoting strategy which allows us to formulate a block QR factorization algorithm where the bulk of the work is in matrix-matrix operations. Incremental condition estimation is used to ensure the reliability of the restricted pivoting scheme. Implementation results on the Cray 2, Cray X-MP and Cray Y-MP show that the new algorithm performs significantly better than the traditional scheme and can more than halve the cost of computing the QR factorization.

References

[1]

Ake Bjhrck. Difference Meihods- Solutions of Equations in Rn, chapter Least Squares Methods. Volume III of Handbook of Numerical Analysis, Elsevier Publishers, 1989. to appear.

[2]

Zhajoun Bat and Jim Demmel. LAPACK Working Note #8; On a Block Implementation of Hessenberg Mullishifl QR Iteration. Technical Report ANL-MCS-TM-127, Argonne National Laboratory, Mathematics and Computer Sciences Division, 1989.

[3]

Michael Berry, Kyle Gallivan, William IIarrod, William Jalby, Sy-Shin Lo, Ulrike Meier, Bernard Philippe, and Ahmed Sameh. Parallel algorithms on the Cedar system. In W. H~indler, editor, Proceedings of CONPAR 86, pages 25-39, Springer Verlag, New York, 1986.

Digital Library

[4]

Christian Bischof, James Demmel, Jack Dongarra, Jeremy Du C:roz, Anne Greenbaum, Sven Hammarling, and Danny Sorensen. LAPACK Working Note #5: Provisional Contents. Technical Report ANL- 88-38, Argonne National Laboratory, Mathematics and Computer Sciences Division, September 1988.

[5]

Christian H. Bischof. Adaptive Blocking. Technical Report ANL/MCS-P39-1288, Argonne National Laboratory, Mathematics and Computer Sciences Division, 1988. to appear in The Journal of Supercomputing.

[6]

Christian H. Bischof. Computing the Singular Value Decomposition on a Distributed System of Vector Processors. Technical Report 87-869, Cornell UniversiW, Department of Computer Science, 1987. To appear in Parallel Computing.

Digital Library

[7]

Christian H. Bischof. Incremental Condition Estimation. Technical Report ANL/MCS-P15-1088, Argonne National Laboratory, Mathematics and Computer Sciences Division, 1988. to appear in SIAM Journal on Matrix Analysis and Applications.

Digital Library

[8]

Christian H. Bischof. A parallel QR faclorization algorithm with conlrolled local pivoting. Technical Report ANL/MCS-P21-1088, Argonne National Laboratory, Mathematics and Computer Sciences Division, 1988. submitted to SIAM Journal on Scientific .and Statistical Computing.

Digital Library

[9]

Christian H. Bischof. A pipelined block QR decomposition algorithm. In Garry Rodrigue, editor, Parallel Processing for Scientific Computing, pages 3-7, SIAM Press, Philadelphia, 1989.

Digital Library

[10]

Christian H. Bischof and :lack 3. Dongarra. A project for developing a linear algebra library for high-performance computers. In Graham Carey, editor, Parallel and Vector Supercom~,u~iug: Methods and Algorithms, John ~~iley ~z Sons, Somerset, NS, 1989. to appear.

Digital Library

[11]

Christian H. Bischof and Charles F. Van Loan. The WY representation for products of Householder matrices. SIAM Journal on Scientific and Statistical Computing, 8:s2-s13, 1987.

Digital Library

[12]

P. A. Businger and G. Ill. Golub. Linear least squares solution by Householder transformation. Numerische Mathematik, 7:269-276, 1965.

Digital Library

[13]

Tony F. Chan. Rank revealing QR factorizations. Linear Algebra and lts Applications, 88/89:67-82, 1987.

[14]

Thomas F. Coleman. Large Sparse Numerical Optimization. Volume 165 of Lecture Notes in Computer Science, Springer Verlag, New York, 1984.

Digital Library

[15]

Jim Demmel, Jack Dongarra, Jeremy Du Croz, Anne Greenbaum, Sven Hammarling, and Danny Sorensen. Prospectus for the Development of a Linear Algebra Library for High-Performance Computers. Technical Report ANI,-MCS-TM97, Argonne National Laboratory, Mathematics and Computer Sciences Division, September 1987.

[16]

J. J. Dongarra, 3. R. Bunch, C. B. Moler, and G. W. Stewart. LINPACK Users' Guide. SIAM Press, 1979.

[17]

J. J. Dongarra, F. G. Gustavson, and A. K arp. Implementing linear algebra algorithms for dense matrices on a vector pipeline machine. SIA M Review, 26:91-112, 1984.

Digital Library

[18]

jack Dongarra, Jeremy Du Croz, Iain Duff, and Sven Hammarling. A Set of Level 3 Basic Linear Algebra Subprograms. Technical Report MCS-P1- 0888, Argonne National Laboratory, Mathematics and Computer Sciences Division, August 1988.

[19]

Jack Dongarra, Sven Hammarling, and Linda Kaufman. Squeezing the most out of eigenvalue solvers on high-performance computers. Linear Algebra and Its Applications, 77:113-136, 1986.

[20]

Jack J. Dongarra, Jeremy Du Croz, Sven Hammarling, and Richard J. Hanson. An extended set of Fortran basic linear algebra subprograms. A CM Transactions on Mathematical Software, 14(1):t- 17, 1988.

Digital Library

[21]

Jack J. Dongarra and Inn S. Duff. Advanced Computer Archilectures. Technical Report ANL-MCS- TM57, Argonne National Laboratory, Mathematics and Computer Sciences Division, 198}'. Revision 1.

Cited By

Dessole MDell'Orto MMarcuzzi F(2023)The Lawson‐Hanson algorithm with deviation maximization: Finite convergence and sparse recoveryNumerical Linear Algebra with Applications10.1002/nla.249030:5Online publication date: 13-Jan-2023
https://doi.org/10.1002/nla.2490
Dessole MMarcuzzi F(2022)Deviation maximization for rank-revealing QR factorizationsNumerical Algorithms10.1007/s11075-022-01291-191:3(1047-1079)Online publication date: 5-Apr-2022
https://doi.org/10.1007/s11075-022-01291-1
Li MAbsar JBougard Bvan der Perre LCatthoor F(2007)Systematic Optimization of Programmable QRD Implementation for Multiple Application Scenarios2007 IEEE Workshop on Signal Processing Systems10.1109/SIPS.2007.4387510(19-24)Online publication date: Oct-2007
https://doi.org/10.1109/SIPS.2007.4387510
Show More Cited By

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A block QR factorization algorithm using restricted pivoting

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cover image ACM Conferences

Supercomputing '89: Proceedings of the 1989 ACM/IEEE conference on Supercomputing

August 1989

849 pages

ISBN:0897913418

DOI:10.1145/76263

Chairman:
F. Ron Bailey

Copyright © 1989 ACM.

Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected]

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SIGARCH: ACM Special Interest Group on Computer Architecture
IEEE-CS: Computer Society

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Los Alamos National Labs: Los Alamos National Labs
NASA: National Aeronatics and Space Administration
Argonne Natl Lab: Argonne National Lab

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Association for Computing Machinery

New York, NY, United States

Publication History

Published: 01 August 1989

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SC '89

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IEEE-CS

SC '89: International Conference for High Performance Computing, Networking, Storage and Analysis

November 12 - 17, 1989

Nevada, Reno, USA

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Overall Acceptance Rate 1,516 of 6,373 submissions, 24%

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Cited By

Dessole MDell'Orto MMarcuzzi F(2023)The Lawson‐Hanson algorithm with deviation maximization: Finite convergence and sparse recoveryNumerical Linear Algebra with Applications10.1002/nla.249030:5Online publication date: 13-Jan-2023
https://doi.org/10.1002/nla.2490
Dessole MMarcuzzi F(2022)Deviation maximization for rank-revealing QR factorizationsNumerical Algorithms10.1007/s11075-022-01291-191:3(1047-1079)Online publication date: 5-Apr-2022
https://doi.org/10.1007/s11075-022-01291-1
Li MAbsar JBougard Bvan der Perre LCatthoor F(2007)Systematic Optimization of Programmable QRD Implementation for Multiple Application Scenarios2007 IEEE Workshop on Signal Processing Systems10.1109/SIPS.2007.4387510(19-24)Online publication date: Oct-2007
https://doi.org/10.1109/SIPS.2007.4387510
Tai YLo CPsarris K(2006)An FPGA-based computation model for blocked algorithmsProceedings of the 6th WSEAS International Conference on Applied Informatics and Communications10.5555/1366421.1366471(286-291)Online publication date: 18-Aug-2006
https://dl.acm.org/doi/10.5555/1366421.1366471
Bischof CQuintana-Ortí G(1998)Computing rank-revealing QR factorizations of dense matricesACM Transactions on Mathematical Software10.1145/290200.28763724:2(226-253)Online publication date: 1-Jun-1998
https://dl.acm.org/doi/10.1145/290200.287637
Bischof CHansen P(1992)A block algorithm for computing rank-revealing QR factorizationsNumerical Algorithms10.1007/BF021394752:3(371-391)Online publication date: 1-Oct-1992
https://dl.acm.org/doi/10.1007/BF02139475
Bischof C(1990)Fundamental Linear Algebra Computations on High-Performance ComputersSupercomputer ’9010.1007/978-3-642-75833-1_13(167-182)Online publication date: 1990
https://doi.org/10.1007/978-3-642-75833-1_13

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