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Scaling by Binormalization

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Abstract

We present an iterative algorithm (BIN) for scaling all the rows and columns of a real symmetric matrix to unit 2-norm. We study the theoretical convergence properties and its relation to optimal conditioning. Numerical experiments show that BIN requires 2–4 matrix–vector multiplications to obtain an adequate scaling, and in many cases significantly reduces the condition number, more than other scaling algorithms. We present generalizations to complex, nonsymmetric and rectangular matrices.

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References

  1. F.L. Bauer, Optimally scaled matrices, Numer. Math. 5 (1963) 73–87.

    Google Scholar 

  2. R.B. Bellman, Introduction to Matrix Analysis (SIAM, Philadelphia, PA, 1997).

    Google Scholar 

  3. A. Brandt, Guide to multigrid development, in: Multigrid Methods, Proc. of Conf., Köln-Porz, 23–27 November, 1981, eds. W. Hackbusch and U. Trottenberg, Lecture Notes in Mathematics, Vol. 960 (Springer, New York, 1982).

    Google Scholar 

  4. A. Brandt, Algebraic multigrid theory: The symmetric case, Appl. Math. Comput. 19 (1986) 23–56.

    Google Scholar 

  5. A. Brandt, Barriers to achieving textbook multigrid efficiency in CFD, Preprint, ICASE Interim Report No. 32, NASA/CR-198–207647 (appears as appendix C in [24]).

  6. A. Brandt, Multiscale scientific computation: Review 2001, in: Multiscale and Multiresolution Methods: Theory and Applications, eds. T.J. Barth, T.F. Chan and R. Haimes, Lecture Notes in Computer Science and Engineering, Vol. 20 (Springer, New York, 2001) pp. 1–91.

    Google Scholar 

  7. A. Brandt and D. Ron, Multigrid solvers and multilevel optimization strategies, in: Multilevel Optimization and VLSICAD, eds. J. Cong and J.R. Shinnerl (Kluwer Academic, Dordrecht, 2002) pp. 1–69.

    Google Scholar 

  8. A.R. Curtis and J.K. Reid, On the automatic scaling of matrices for Gaussian elimination, J. Inst. Math. Appl. 10 (1972) 118–124.

    Google Scholar 

  9. S.C. Eisenstat, J.W. Lewis and M.H. Schultz, Optimal block diagonal scaling of block 2-cyclic matrices, Linear Algebra Appl. 44 (1982) 181–186.

    Google Scholar 

  10. G.E. Forsythe and E.G. Straus, On best conditioned matrices, Proc. Amer. Math. Soc. 6 (1955) 340–345.

    Google Scholar 

  11. W. Gander, G.H. Golub and U. von Matt, A constrained eigenvalue problem, Linear Algebra Appl. 114/115 (1989) 815–839.

    Google Scholar 

  12. D. Goldberg, What every computer scientist show know about floating-point arithmetic, ACM Comput. Surv. 23 (1991) 5–48.

    Google Scholar 

  13. G.H. Golub and C. Van Loan, Matrix Computations, 3rd ed. (Johns Hopkins Univ. Press, Baltimore, MD, 1996).

    Google Scholar 

  14. G.H. Golub and J.M. Varah, On the characterization of the best l 2-scaling of a matrix, SIAM J. Numer. Anal. 11 (1974) 472–479.

    Google Scholar 

  15. http://www.nag.co.uk/.

  16. http://www.cise.ufl.edu/research/sparse/umfpack/.

  17. http://www.cise.ufl.edu/research/sparse/matrices/.

  18. O.E. Livne, Multiscale eigenbasis algorithms, Ph.D. thesis, Weizmann Institute of Science, Israel (2000).

    Google Scholar 

  19. O.E. Livne and A. Brandt, Local mode analysis of multicolor and composite relaxation schemes, Preprint, Comput. Math. Appl. (2002).

  20. O.E. Livne and G.H. Golub, Scaling by binormalization, SCCM Technical Report, Stanford University (2003); available at http://www-sccm.stanford.edu/~livne/publications.html.

  21. J.M. Ortega and W.C. Rheinboldt, Iterative Solution of Nonlinear Equations in Several Variables (SIAM, Philadelphia, PA, 2000).

    Google Scholar 

  22. G.W. Stewart, Matrix Algorithms, Vol. I: Basic Decompositions (SIAM, Philadelphia, PA, 1998).

    Google Scholar 

  23. W.J. Stewart, Introduction to the Numerical Solution of Markov Chains (Princeton Univ. Press, Princeton, 1994).

    Google Scholar 

  24. U. Trottenberg, C.W. Oosterlee and A. Schüller, Multigrid (Academic Press, New York, 2000).

    Google Scholar 

  25. A. van der Sluis, Condition numbers and equilibration of matrices, Numer. Math. 14 (1969) 14–23.

    Google Scholar 

  26. R.S. Varga, Matrix Iterative Analysis (Springer, Berlin, 2000).

    Google Scholar 

  27. J.H. Wilkinson, The Algebraic Eigenvalue Problem (Oxford Univ. Press, Oxford, 1965).

    Google Scholar 

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Livne, O.E., Golub, G.H. Scaling by Binormalization. Numerical Algorithms 35, 97–120 (2004). https://doi.org/10.1023/B:NUMA.0000016606.32820.69

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  • DOI: https://doi.org/10.1023/B:NUMA.0000016606.32820.69

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