BIT Numerical Mathematics

, 37:473 | Cite as

Matrix enlarging methods and their application

  • Fernando L. Alvarado


This paper explores several methods for matrix enlarging, where an enlarged matrixà is constructed from a given matrixA. The methods explored include matrix primitization, stretching and node splitting. Graph interpretations of these methods are provided. Solving linear problems using enlarged matrices yields the answer to the originalAx=b problem.à can exhibit several desirable properties. For example,à can be constructed so that the valence of any row and/or column is smaller than some desired number (≥4). This is beneficial for algorithms that depend on the square of the number of entries of a row or column. Most particularly, matrix enlarging can results in a reduction of the fill-in in theR matrix which occurs during orthogonal factorization as a result of dense rows. Numerical experiments support these conjectures.

AMS subject classification


Key words

Sparse matrices node splitting orthogonal factorization least squares QR factorization matrix stretching matrix primitization 


  1. 1.
    F. L. Alvarado,Experiments on assorted sparse prototyping and development environments, Presentation at the Linear Algebra Year Workshop, CERFACS, Toulouse, France, September 1995.Google Scholar
  2. 2.
    F. L. Alvarado,Formation of Y-node using the primitive Y-node concept, IEEE Transaction on Power Apparatus and Systems, PAS-101:12 (December, 1982), pp. 4563–4572.Google Scholar
  3. 3.
    F. L. Alvarado,Symmetric matrix primitization, Technical Report ECE-89-19, The University of Wisconsin, Madison, November 1989.Google Scholar
  4. 4.
    F. L. Alvarado,Manipulation and visualization of sparse matrices, ORSA (Operations Research Society of America) Journal on Computing, 2:2 (1990), pp. 186–207.zbMATHGoogle Scholar
  5. 5.
    F. L. Alvarado, M. K. Enns, and W. F. Tinney, Sparsity enhancement in mutually coupled networks, IEEE Transactions on Power Apparatus and Systems, PAS-103 (June 1984), pp. 1502–1508.Google Scholar
  6. 6.
    F. L. Alvarado, W. F. Tinney, and M. K. Enns,Sparsity in large-scale network computation, in C. T. Leondes, ed., Advances in Electric Power and Energy Conversion System Dynamics and Control, volume 41 of Control and Dynamic Systems, Academic Press, 1991, pp. 207–272.Google Scholar
  7. 7.
    F. L. Alvarado and Z. Wang,Direct sparse interval hull computations for thin non-m matrices, Interval Computations, March 1993, pp. 5–28.Google Scholar
  8. 8.
    K. D. Andersen,A modified Schur-complement methods for handling dense columns in interior-point methods for linear programming, ACM Transactions on Mathematical Software, 22:3 (1996), pp. 348–356.zbMATHMathSciNetCrossRefGoogle Scholar
  9. 9.
    Å. Björck,Numerical Methods for Least Squares Problems, SIAM, Philadelphia, PA, 1996.zbMATHGoogle Scholar
  10. 10.
    Å. Björck and M. Lundquist.Splitting rows in sparse least squares problems. Tech. Report, LiTH-MAT-R 97-17, Department of Mathematics, University of Linköping, 1997.Google Scholar
  11. 11.
    L. Chua, C. Desoer, and E. Kuh,Linear and Non-linear Circuits, McGraw-Hill, 1987.Google Scholar
  12. 12.
    T. F. Coleman, A. Edenbrandt, and J. R. Gilbert,Predicting fill for sparse orthogonal factorization, Journal of the Association for Computing Machinery, 33:3 (1986), pp. 517–532.zbMATHMathSciNetGoogle Scholar
  13. 13.
    E. Cuthill and J. McKee,Reducing the bandwidth of sparse symmetric matrices, in Proceedings of the 24th National Conference of the Association for Computing Machinery, Brandon Press, NJ, 1969, pp. 157–172.Google Scholar
  14. 14.
    C. DeBoor and H.-O. Kreiss,On the condition number of the linear system associated with discretized VVPs of ODEs, SIAM Journal of Numerical Analysis, 23:5 (1986), pp. 936–939.MathSciNetCrossRefGoogle Scholar
  15. 15.
    I. S. Duff, R. Grimes, and J. Lewis,Sparse matrix test problems, ACM Transactions on Mathematical Software, 15:1 (1989), pp. 1–14.zbMATHCrossRefGoogle Scholar
  16. 16.
    I. S. Duff and J. K. Reid,A comparison of some methods for the solution of overdetermined systems of linear equations, Journal of the Institute of Mathematics and its Applications, 17 (1976), pp. 267–280.zbMATHMathSciNetCrossRefGoogle Scholar
  17. 17.
    A. George and M. T. Heath,Solution of sparse linear least squares problems using givens rotations, Linear Algebra and its Applications, 34 (1980), pp. 69–83.zbMATHMathSciNetCrossRefGoogle Scholar
  18. 18.
    A. George and E. Ng,On row and column ordering for sparse least squares problems, SIAM Journal of Numerical Analysis, 20 (1983), pp. 326–344.zbMATHMathSciNetCrossRefGoogle Scholar
  19. 19.
    J. R. Gilbert and T. Peierls,Sparse partial pivoting in time proportional to arithmetic operations, SIAM Journal of Scientific and Statistical Computing, 9 (1988), pp. 862–874.zbMATHMathSciNetCrossRefGoogle Scholar
  20. 20.
    P. E. Gill and W. Murray.Nonlinear least squares and nonlinearly constrained optimization, in Proceedings of the Dundee Conference on Numerical Analysis 1975, Springer-Verlag, New York, Lecture Notes in Mathematics No. 506, 1976, pp. 201–212.Google Scholar
  21. 21.
    P. E. Gill and W. Murray,The orthogonal factorization of a large sparse matrix, in Sparse Matrix Computations, J. R. Bunch and D. J. Rose, eds., Springer-Verlag, New York, 1976, pp. 201–212.Google Scholar
  22. 22.
    G. H. Golub,Numerical methods for solving linear least squares problems, Numerical Mathematics, 7 (1965), pp. 206–216.zbMATHMathSciNetCrossRefGoogle Scholar
  23. 23.
    J. F. Grcar,Matrix stretching for linear equations, Technical Report SAND90-8723, Sandia National Laboratories, November 1990.Google Scholar
  24. 24.
    M. T. Heath,Numerical methods for large sparse linear least squares problems, SIAM Journal on Scientific and Statistical Computing, 5 (1984), pp. 497–513.zbMATHMathSciNetCrossRefGoogle Scholar
  25. 25.
    Joseph W. H. Liu,Modification of the minimum degree algorithm by multiple elimination, ACM Transactions on Mathematical Software, 11 (1985), pp. 141–153.zbMATHCrossRefGoogle Scholar
  26. 26.
    P. Matstoms,Sparse QR factorization with applications to linear least squares problems, PhD thesis, Department of Mathematics, Linköping University, Sweden, 1994, Linköping Studies in Science and Technology Dissertations, No. 337.Google Scholar
  27. 27.
    W. F. Tinney, W. L. Powell, and N. M. Peterson,Sparsity-oriented network reduction, in Power Industry Computer Applications Conference, 1973, pp. 384–390.Google Scholar
  28. 28.
    R. J. Vanderbei,Splitting dense columns in sparse linear systems, Linear Algebra and its Applications, 152 (1991), pp. 107–117.zbMATHMathSciNetCrossRefGoogle Scholar
  29. 29.
    N. Vempati, I. W. Slutsker, and W. F. Tinney,Enhancements to Givens rotations for power system state estimation, IEEE Transactions on Power Systems, 6:2 (1991), pp. 842–849.CrossRefGoogle Scholar

Copyright information

© BIT Foundation 1997

Authors and Affiliations

  • Fernando L. Alvarado
    • 1
  1. 1.Department of Electrical and Computer EngineeringThe University of WisconsinMadison

Personalised recommendations