Skip to main content

Solution of linear systems of equations: Direct methods (general)

Part of the Lecture Notes in Mathematics book series (LNM,volume 572)

Keywords

  • Gaussian Elimination
  • Triangular Form
  • Sparsity Pattern
  • Fortran Subroutine
  • Variability Type

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 is a preview of subscription content, access via your institution.

Buying options

Chapter
USD   29.95
Price excludes VAT (Canada)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   39.99
Price excludes VAT (Canada)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   54.99
Price excludes VAT (Canada)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  • Bunch, J.R. (1974). Partial pivoting strategies for symmetric matrices. SIAM J. Numer. Anal. 11, 521–528.

    CrossRef  MathSciNet  MATH  Google Scholar 

  • Curtis, A.R. and Reid, J.K. (1971a). Fortran subroutines for the solution of sparse sets of linear equations. AERE Report R.6844. HMSO, London.

    Google Scholar 

  • Curtis, A.R. and Reid, J.K. (1971b). The solution of large sparse unsymmetric systems of linear equations. J. Inst. Math. Appl. 8, 344–353.

    CrossRef  MATH  Google Scholar 

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

    CrossRef  MATH  Google Scholar 

  • Duff, I.S. (1974). Pivot selection and row ordering in Givens reduction on sparse matrices. Computing 13, 239–248.

    CrossRef  MathSciNet  MATH  Google Scholar 

  • Duff, I.S. (1976a). A survey of sparse matrix research. Harwell report CSS 28. To appear in Proc. I.E.E.E.

    Google Scholar 

  • Duff, I.S. (1976b). On permutations to block triangular form. Harwell report CSS 27. To appear in J. Inst. Math. Appl.

    Google Scholar 

  • Duff, I.S. (1976c). On algorithms for obtaining a maximum transversal. To appear.

    Google Scholar 

  • Duff, I.S. and Reid, J.K. (1974). A comparison of sparsity orderings for obtaining a pivotal sequence in Gaussian elimination. J. Inst. Math. Appl. 14, 281–291.

    CrossRef  MathSciNet  MATH  Google Scholar 

  • Duff, I.S. and Reid, J.K. (1975). A comparison of some methods for the solution of sparse overdetermined systems of linear equations. Harwell report CSS 12. J. Inst. Math. Appl. 17, 267–280.

    CrossRef  MathSciNet  MATH  Google Scholar 

  • Duff, I.S. and Reid, J.K. (1976). An implementation of Tarjan's algorithm for the block triangularization of a matrix. Harwell report CSS 29. To appear in ACM Transactions on Mathematical Software.

    Google Scholar 

  • Erisman, A.M. and Reid, J.K. (1974). Monitoring the stability of the triangular factorization of a sparse matrix. Numer. Math. 22, 183–186.

    CrossRef  MathSciNet  MATH  Google Scholar 

  • Fulkerson, D.R. and Wolfe, P. (1962). An algorithm for scaling matrices. SIAM Rev. 4, 142–146.

    CrossRef  MathSciNet  MATH  Google Scholar 

  • Gentleman, W.M. (1973). Least squares computations by Givens transformations without square roots. J. Inst. Math. Appl. 12, 329–336.

    CrossRef  MathSciNet  MATH  Google Scholar 

  • Gustavson, F.G. (1972). Some basic techniques for solving sparse systems of linear equations. In Rose and Willoughby (1972), 41–52.

    Google Scholar 

  • Hachtel, G.D. (1972). Vector and matrix variability type in sparse matrix algorithms. In Rose and Willoughby (1972), 53–64.

    Google Scholar 

  • Hall, M. (1956). An algorithm for distinct representatives. Amer. Math. Monthly 63, 716–717.

    CrossRef  MathSciNet  MATH  Google Scholar 

  • Hamming, R.W. (1971). Introduction to applied numerical analysis. McGraw-Hill, New York.

    MATH  Google Scholar 

  • Hopcroft, J.E. and Karp, R.M. (1973). An n5/2 algorithm for maximum matchings in bipartite graphs. SIAM J. Comput. 2, 225–231.

    CrossRef  MathSciNet  MATH  Google Scholar 

  • Markowitz, H.M. (1957). The elimination form of the inverse and its application to linear programming. Management Science 3, 255–269.

    CrossRef  MathSciNet  MATH  Google Scholar 

  • Munro, I. (1971a). Efficient determination of the transitive closure of a directed graph. Information Processing Lett., 1, 55–58.

    CrossRef  MATH  Google Scholar 

  • Munro, I. (1971b). Some results in the study of algorithms. Ph.D. Thesis, Dept. of Comp. Sci., Toronto. Report # 32.

    MATH  Google Scholar 

  • Peters, G. and Wilkinson, J.H. (1970). The least squares problem and pseudoinverses. Comput. J. 13, 309–316.

    CrossRef  MATH  Google Scholar 

  • Reid, J.K. (1971). A note on the stability of Gaussian elimination. J. Inst. Math. Appl. 8, 374–375.

    CrossRef  MathSciNet  MATH  Google Scholar 

  • Reid, J.K. (1972). Two Fortran subroutines for direct solution of linear equations whose matrix is sparse, symmetric and positive definite. AERE Report R.7119. HMSO, London.

    Google Scholar 

  • Reid, J.K. (1976). Fortran subroutines for handling sparse linear programming bases. AERE Report R.8269. HMSO, London.

    Google Scholar 

  • Rose, D.J. and Willoughby, R.A. (Ed.) (1972). Sparse matrices and their applications. Proc. of Conf. at IBM Research Center, N.Y. Sept. 9th-10th 1971. Plenum Press, New York.

    Google Scholar 

  • Sargent, R.W.H. and Westerberg, A.W. (1964). \s`Speed-up\s` in chemical engineering design. Trans. Inst. Chem. Engrgs. 42, 190–197.

    Google Scholar 

  • Stewart, G.W. (1974). Modifying pivot elements in Gaussian elimination. Math. Comp. 28, 537–542.

    CrossRef  MathSciNet  MATH  Google Scholar 

  • Tarjan, R.E. (1972). Depth first search and linear graph algorithms. SIAM J. Comput. 1, 146–160.

    CrossRef  MathSciNet  MATH  Google Scholar 

  • Tarjan, R.E. (1975). Efficiency of a good but not linear set union algorithm. J. ACM 22, 215–225.

    CrossRef  MathSciNet  MATH  Google Scholar 

  • Tomlin, J.A. (1972). Pivoting for size and sparsity in linear programming inversion routines. J. Inst. Math. Appl. 10, 289–295.

    CrossRef  MathSciNet  MATH  Google Scholar 

  • Wilkinson, J.H. (1961). Error analysis of direct methods of matrix inversion. J. ACM 8, 281–330.

    CrossRef  MathSciNet  MATH  Google Scholar 

  • Zlatev, Z. and Thomsen, P.G. (1976). ST — A FORTRAN IV SUBROUTINE for the solution of large systems of linear algebraic equations with real coefficients by use of sparse technique. Report 76-05, Numerisk Institut, Danmarks Tekniske Højskole.

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and Permissions

Copyright information

© 1977 Springer-Verlag

About this paper

Cite this paper

Reid, J.K. (1977). Solution of linear systems of equations: Direct methods (general). In: Barker, V.A. (eds) Sparse Matrix Techniques. Lecture Notes in Mathematics, vol 572. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0116616

Download citation

  • DOI: https://doi.org/10.1007/BFb0116616

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-08130-2

  • Online ISBN: 978-3-540-37430-5

  • eBook Packages: Springer Book Archive