Abstract
The effects of rounding errors on algorithms in numerical linear algebra have been much-studied for over fifty years, since the appearance of the first digital computers. The subject continues to occupy researchers, for several reasons. First, not everything is known about established algorithms. Second, new algorithms are continually being derived, and their behaviour in finite precision arithmetic needs to be understood. Third, new error analysis techniques lead to different ways of looking at and comparing algorithms, requiring a reassessment of conventional wisdom.
Keywords
- Singular Value Decomposition
- Cholesky Factorization
- Jacobi Method
- Numerical Linear Algebra
- Partial Pivoting
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
This work was supported by Engineering and Physical Sciences Research Council grant GR/L76532.
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Bibliography
Jan Ole Aasen. On the reduction of a symmetric matrix to tridiagonal form. BIT, 11: 233–242, 1971.
E. Anderson, Z. Bai, C. H. Bischof, J. W. Demmel, J. J. Dongarra, J. J. Du Croz, A. Greenbaum, S. J. Hammarling, A. McKenney, S. Ostrouchov, and D. C. Sorensen. LAPACK Users’ Guide, Release 2.0. Second edition, Society for Industrial and Applied Mathematics, Philadelphia, PA, USA, 1995. xix+325 pp. ISBN 0–89871–345–5.
E. Anderson, Z. Bai, and J. Dongarra. Generalized QR factorization and its applications. Linear Algebra and Appl., 162–164: 243–271, 1992.
Cleve Ashcraft, Roger G. Grimes, and John G. Lewis. Accurate symmetric indefinite linear equation solvers. To appear in SIAM J. Matrix Anal. Appl., September 1998. 51 pp.
Zhaojun Bai, James W. Demmel, and Ming Gu. Inverse free parallel spectral divide and conquer algorithms for nonsymmetric eigenproblems. Numer. Math., 76: 279–308, 1997.
J. L. Barlow, N. K. Nichols, and R. J. Plemmons. Iterative methods for equality-constrained least squares problems. SIAM J. Sci. Stat. Comput., 9 (5): 892–906, 1988.
Victor Barwell and Alan George. A comparison of algorithms for solving symmetric indefinite systems of linear equations. ACM Trans. Math. Software, 2 (3): 242–251, 1976.
Ake Björck. Numerical Methods for Least Squares Problems. Society for Industrial and Applied Mathematics, Philadelphia, PA, USA, 1996. xvii+408 pp. ISBN 0–89871–360–9.
Ake Björck and Gene H. Golub. Iterative refinement of linear least squares solutions by Householder transformation. BIT, 7: 322–337, 1967.
James R. Bunch. Partial pivoting strategies for symmetric matrices. SIAM J. Numer. Anal., 11 (3): 521–528, 1974.
James R. Bunch and Linda Kaufman. Some stable methods for calculating inertia and solving symmetric linear systems. Math. Comp., 31 (137): 163–179, 1977.
James R. Bunch and Beresford N. Parlett. Direct methods for solving symmetric indefinite systems of linear equations. SIAM J. Numer. Anal., 8 (4): 639–655, 1971.
Sheung Hun Cheng. Symmetric Indefinite Matrices: Linear System Solvers and Modified Inertia Problems. PhD thesis, University of Manchester, Manchester, England, January 1998. 150 pp.
Sheung Hun Cheng and Nicholas J. Higham. A modified Cholesky algorithm based on a symmetric indefinite factorization. Numerical Analysis Report No. 289, Manchester Centre for Computational Mathematics, Manchester, England, April 1996. 18 pp. To appear in SIAM J. Matrix Anal. Appl.
Anthony J. Cox and Nicholas J. Higham. Accuracy and stability of the null space method for solving the equality constrained least squares problem. Numerical Analysis Report No. 306, Manchester Centre for Computational Mathematics, Manchester, England, August 1997. 20 pp. To appear in BIT, 39 (1): 1999.
Anthony J. Cox and Nicholas J. Higham. Row-wise backward stable elimination methods for the equality constrained least squares problem. Numerical Analysis Report No. 319, Manchester Centre for Computational Mathematics, Manchester, England, March 1998. 18 pp. To appear in SIAM J. Matrix Anal. Appl.
Anthony J. Cox and Nicholas J. Higham. Stability of Householder QR factorization for weighted least squares problems. In Numerical Analysis 1997, Proceedings of the 17th Dundee Biennial Conference, D. F. Griffiths, D. J. Higham, and G. A. Watson, editors, volume 380 of Pitman Research Notes in Mathematics, Addison Wesley Longman, Harlow, Essex, UK, 1998, pages 57–73.
James W. Demmel. Applied Numerical Linear Algebra. Society for Industrial and Applied Mathematics, Philadelphia, PA, USA, 1997. xi+419 pp. ISBN 0–89871–389–7.
James W. Demmel and Kresimir Veselic. Jacobi’s method is more accurate than QR. SIAM J. Matrix Anal. Appl., 13 (4): 1204–1245, 1992.
J. J. Dongarra, J. R. Bunch, C. B. Moler, and G. W. Stewart. LINPACK Users’ Guide. Society for Industrial and Applied Mathematics, Philadelphia, PA, USA, 1979. ISBN 0–89871–172–X.
Zlatko Drmac. Computing the Singular and the Generalized Singular Values. PhD thesis, Lehrgebiet Mathematische Physik, Fernuniversität Hagen, 1994. 193 pp.
Zlatko Drmac. Implementation of Jacobi rotations for accurate singular value computation in floating point arithmetic. SIAM J. Sci. Comput., 18 (4): 1200–1222, 1997.
Stanley C. Eisenstat and Ilse C. F. Ipsen. Relative perturbation techniques for singular value problems. SIAM J. Numer. Anal., 32 (6): 1972–1988, 1995.
G. E. Forsythe and P. Henrici. The cyclic Jacobi method for computing the principal values of a complex matrix. Trans. Amer. Math. Soc., 94: 1–23, 1960.
Gene H. Golub and Charles F. Van Loan. Matrix Computations. Third edition, Johns Hopkins University Press, Baltimore, MD, USA, 1996. xxvii+694 pp. ISBN 0–8018–5413–X (hardback), 0–8018–5414–8 (paperback).
Ming Gu. Backward perturbation bounds for linear least squares problems. SIAM J. Matrix Anal. Appl., 1998. To appear.
Sven J. Hammarling. The numerical solution of the general Gauss-Markov linear model. In Mathematics in Signal Processing, T. S. Durrani, J. B. Abbiss, and J. E. Hudson, editors, Oxford University Press, 1987, pages 451–456.
Richard J. Hanson. Aasen’s method for linear systems with self-adjoint matrices. Visual Numerics, Inc., http://www.vni.com/books/whitepapers/Aasen. html, July 1997.
Vjeran Hari. On sharp quadratic convergence bounds for the serial Jacobi methods. Numer. Math., 60: 375–406, 1991.
Magnus R. Hestenes. Inversion of matrices by biorthogonalization and related results. J. Soc. Indust. Appl. Math., 6 (1): 51–90, 1958.
Nicholas J. Higham. Computing the polar decomposition with applications. SIAM J. Sci. Stat. Comput., 7 (4): 1160–1174, October 1986.
Nicholas J. Higham. Accuracy and Stability of Numerical Algorithms. Society for Industrial and Applied Mathematics, Philadelphia, PA, USA, 1996. xxviii+688 pp. ISBN 0–89871–355–2.
Nicholas J. Higham. Stability of block LDLT factorization of a symmetric tridiagonal matrix. Numerical Analysis Report No. 308, Manchester Centre for Computational Mathematics, Manchester, England, September 1997. 8 pp. To appear in Linear Algebra and Appl.
Nicholas J. Higham. Stability of the diagonal pivoting method with partial pivoting. SIAM J. Matrix Anal. Appl., 18 (1): 52–65, January 1997.
Nicholas J. Higham. QR factorization with complete pivoting and accurate computation of the SVD. Numerical Analysis Report 324, Manchester Centre for Computational Mathematics, Manchester, England, August 1998. 26 pp.
Ilse C. F. Ipsen. Relative perturbation results for matrix eigenvalues and singular values. In Acta Numerica, volume 7, Cambridge University Press, 1998, pages 151–201.
E. G. Kogbetliantz. Solution of linear equations by diagonalization of coefficients matrix. Quart. Appl. Math., 13 (2): 123–132, 1955.
Charles L. Lawson and Richard J. Hanson. Solving Least Squares Problems. Society for Industrial and Applied Mathematics, Philadelphia, PA, USA, 1995. xii+337 pp. Revised republication of work first published in 1974 by Prentice–Hall. ISBN 0–89871–356–0.
Roy Mathias. Accurate eigensystem computations by Jacobi methods. SIAM J. Matrix Anal. Appl., 16 (3): 977–1003, 1995.
Cleve Moler and Peter J. Costa. Symbolic Math Toolbox Version 2.0: User’s Guide. The MathWorks, Inc., Natick, MA, USA, 1997.
J. C. Nash. Compact Numerical Methods for Computers: Linear Algebra and Function Minimisation. Second edition, Adam Hilger, Bristol, 1990. xii+278 pp. ISBN 0–85274–319–X.
C. C. Paige. Some aspects of generalized QR factorizations. In Reliable Numerical Computation, M. G. Cox and S J Hammarling, editors, Oxford University Press, 1990, pages 73–91.
Beresford N. Parlett. Analysis of algorithms for reflections in bisectors. SIAM Review, 13 (2): 197–208, 1971.
Beresford N. Parlett. The Symmetric Eigenvalue Problem. Society for Industrial and Applied Mathematics, Philadelphia, PA, USA, 1998. xxiv+398 pp. Unabridged, amended version of book first published by Prentice–Hall in 1980. ISBN 0–89871–402–8.
M. J. D. Powell and J. K. Reid. On applying Householder transformations to linear least squares problems. In Proc. IFIP Congress 1968, North-Holland, Amsterdam, The Netherlands, 1969, pages 122–126.
William H. Press, Saul A. Teukolsky, William T. Vetterling, and Brian P. Flannery. Numerical Recipes in FORTRAN: The Art of Scientific Computing. Second edition, Cambridge University Press, 1992. xxvi+963 pp. ISBN 0 521 43064 X.
G. W. Stewart and Ji–guang Sun. Matrix Perturbation Theory. Academic Press, London, 1990. xv+365 pp. ISBN 0–12–670230–6.
Lloyd N. Trefethen and David Bau III. Numerical Linear Algebra. Society for Industrial and Applied Mathematics, Philadelphia, PA, USA, 1997. xii+361 pp. ISBN 0–89871–361–7.
A. van der Sluis. Condition numbers and equilibration of matrices. Numer. Math., 14: 14–23, 1969.
Kregimar Veselie and Vjeran Hari. A note on a one-sided Jacobi algorithm. Numer. Math., 56: 627–633, 1989.
Hongyuan Zha and Zhenyue Zhang. Fast parallelizable methods for the Hermitian eigenvalue problem. Technical Report CSE-96–041, Department of Computer Science and Engineering, Pennsylvania State University, University Park, PA, May 1996. 19 pp.
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Higham, N.J. (1999). Notes on Accuracy and Stability of Algorithms in Numerical Linear Algebra. In: Ainsworth, M., Levesley, J., Marletta, M. (eds) The Graduate Student’s Guide to Numerical Analysis ’98. Springer Series in Computational Mathematics, vol 26. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-03972-4_2
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