Journal of Optimization Theory and Applications

, Volume 16, Issue 5–6, pp 429–445 | Cite as

A direct method for the general solution of a system of linear equations

  • H. Y. Huang
Contributed Papers

Abstract

A computationally stable method for the general solution of a system of linear equations is given. The system isATx−B=0, where then-vectorx is unknown and then×q matrixA and theq-vectorB are known. It is assumed that the matrixAT and the augmented matrix [AT,B] are of the same rankm, wheremn, so that the system is consistent and solvable. Whenm<n, the method yields the minimum modulus solutionxm and a symmetricn ×n matrixHm of ranknm, so thatx=xm+Hmy satisfies the system for ally, ann-vector. Whenm=n, the matrixHm reduces to zero andxm becomes the unique solution of the system.

The method is also suitable for the solution of a determined system ofn linear equations. When then×n coefficient matrix is ill-conditioned, the method can produce a good solution, while the commonly used elimination method fails.

Key Words

Mathematical programming conjugate-gradient methods variable-metric methods linear equations numerical methods computing methods 

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References

  1. 1.
    Penrose, R.,A Generalized Inverse for Matrices, Proceedings of the Cambridge Philosophical Society, Vol. 51, pp. 406–413, 1955.Google Scholar
  2. 2.
    Ben-Israel, A., andCharnes, A.,Contributions to the Theory of Generalized Inverses, SIAM Journal on Applied Mathematics, Vol. 11, No. 3, 1963.Google Scholar
  3. 3.
    Householder, A. S.,Unitary Triangularization of a Nonsymmetric Matrix, Journal of the Association for Computing Machinery, Vol. 5, No. 4, 1958.Google Scholar
  4. 4.
    Ben-Israel, A., andWersan, S. J.,An Elimination Method for Computing the Generalized Inverse of an Arbitrary Complex Matrix, Journal of the Association for Computing Machinery, Vol. 10, No. 4, 1963.Google Scholar
  5. 5.
    Golub, G.,Numerical Methods for Solving Linear Least Squares Problems, Numerische Mathematik, Vol. 7, No. 2, 1965.Google Scholar
  6. 6.
    Noble, B.,A Method for Computing the Generalized Inverse of a Matrix, SIAM Journal on Numerical Analysis, Vol. 3, No. 4, 1966.Google Scholar
  7. 7.
    Rust, B., Burrus, W. R., andSchneerberger, C.,A Simple Algorithm for Computing the Generalized Inverse of a Matrix, Communications of the ACM, Vol. 9, No. 5, 1966.Google Scholar
  8. 8.
    Tewarson, R. P.,A Computational Method for Evaluating Generalized Inverses, Computer Journal, Vol. 10, No. 4, 1968.Google Scholar
  9. 9.
    Peters, G., andWilkinson, J. H.,The Least Squares Problem and Pseudo-Inverses, Computer Journal, Vol. 13, No. 3, 1970.Google Scholar
  10. 10.
    Gregory, R. T., andKarney, D. L.,A Collection of Matrices for Testing Computational Algorithms, John Wiley and Sons (Interscience Publishers), New York, New York, 1969.Google Scholar

Copyright information

© Plenum Publishing Corporation 1975

Authors and Affiliations

  • H. Y. Huang
    • 1
  1. 1.Rice UniversityHouston

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