A computationally stable method for the general solution of a system of linear equations is given. The system isA Tx−B=0, where then-vectorx is unknown and then×q matrixA and theq-vectorB are known. It is assumed that the matrixA T and the augmented matrix [A T,B] are of the same rankm, wherem≤n, so that the system is consistent and solvable. Whenm<n, the method yields the minimum modulus solutionx m and a symmetricn ×n matrixH m of rankn−m, so thatx=x m+H my satisfies the system for ally, ann-vector. Whenm=n, the matrixH m reduces to zero andx m 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.
This is a preview of subscription content, log in to check access.
Buy single article
Instant access to the full article PDF.
Price includes VAT for USA
Subscribe to journal
Immediate online access to all issues from 2019. Subscription will auto renew annually.
This is the net price. Taxes to be calculated in checkout.
Penrose, R.,A Generalized Inverse for Matrices, Proceedings of the Cambridge Philosophical Society, Vol. 51, pp. 406–413, 1955.
Ben-Israel, A., andCharnes, A.,Contributions to the Theory of Generalized Inverses, SIAM Journal on Applied Mathematics, Vol. 11, No. 3, 1963.
Householder, A. S.,Unitary Triangularization of a Nonsymmetric Matrix, Journal of the Association for Computing Machinery, Vol. 5, No. 4, 1958.
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.
Golub, G.,Numerical Methods for Solving Linear Least Squares Problems, Numerische Mathematik, Vol. 7, No. 2, 1965.
Noble, B.,A Method for Computing the Generalized Inverse of a Matrix, SIAM Journal on Numerical Analysis, Vol. 3, No. 4, 1966.
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.
Tewarson, R. P.,A Computational Method for Evaluating Generalized Inverses, Computer Journal, Vol. 10, No. 4, 1968.
Peters, G., andWilkinson, J. H.,The Least Squares Problem and Pseudo-Inverses, Computer Journal, Vol. 13, No. 3, 1970.
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.
This research was supported by the National Science Foundation, Grant No. GP-41158.
About this article
Cite this article
Huang, H.Y. A direct method for the general solution of a system of linear equations. J Optim Theory Appl 16, 429–445 (1975). https://doi.org/10.1007/BF00933852
- Mathematical programming
- conjugate-gradient methods
- variable-metric methods
- linear equations
- numerical methods
- computing methods