Abstract
Using a strict bound of Spedicato to the condition number of bordered positive-definite matrices, we show that the scaling parameter in the ABS class for linear systems can always be chosen so that the bound of a certain update matrix is globally minimized. Moreover, if the scaling parameter is so chosen at every iteration, then the condition number itself is globally minimized. The resulting class of optimally conditioned algorithms contains as a special case the class of optimally stable algorithms in the sense of Broyden.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Abaffy, J., Broyden, C. G., andSpedicato, E.,A Class of Direct Methods for Linear Equations, Numerische Mathematik, Vol. 45, pp. 361–376, 1984.
Abaffy, J., andSpedicato, E.,A Generalization of the ABS Algorithm for Linear Systems, University of Bergamo, DMSIA Report No. 85/4, 1985.
Abaffy, J., andSpedicato, E.,ABS Projection Algorithms: Mathematical Techniques for Linear and Nonlinear Equations, Ellis Horwood, Chichester, England, 1989.
Huang, H. Y.,A Direct Method for the General Solution of a System of Linear Equations, Journal of Optimization Theory and Applications, Vol. 16, pp. 429–445, 1975.
Abaffy, J., andSpedicato, E.,Numerical Experiments with the Symmetric Algorithm in the ABS Class for Linear Systems, Optimization, Vol. 18, pp. 197–212, 1987.
Spedicato, E., andVespucci, M. T.,Computational Performance of the Implicit Gram-Schmidt-Huang Algorithm for Linear Algebraic Systems, University of Bergamo, DMSIA Report No. 87/4, 1987.
Yang, Z.,On the Numerical Stability of the Huang and the Modified Huang Algorithms and Related Topics, Dalian University of Technology, RABSCA Report No. 5, 1988.
Broyden, C. G.,On the Numerical Stability of Huang's Update, University of Bergamo, DMSIA Report No. 89/19, 1989.
Broyden, C. G.,On the Numerical Stability of Huang's and Related Methods, Journal of Optimization Theory and Applications, Vol. 47, pp. 401–412, 1985.
Spedicato, E.,Optimal Conditioning Parameter Selection in the ABS Class for Linear Systems, University of Würzburg, Angewandte Mathematik, Report No. 203, 1987.
Bodon, E.,Globally Optimally Conditioned Updates in the ABS Class for Linear Systems, University of Bergamo, DMSIA Report No. 88/1, 1988.
Deng, N. Y., andSpedicato, E.,Optimal Conditioning Parameter Selection in the ABS Class through a Rank-Two Update Formulation, University of Bergamo, DMSIA Report No. 88/18, 1988.
Spedicato, E.,A Bound to the Condition Number of Bordered Positive-Definite Matrices, University of Bergamo, DMSIA Report No. 87/1, 1987.
Spedicato, E., andBurmeister, W.,A Strict Bound to the Condition Number of Bordered Positive-Definite Matrices, Computing, Vol. 40, pp. 181–183, 1988.
Author information
Authors and Affiliations
Additional information
Communicated by L. C. W. Dixon
This work was done in the framework of research supported by MPI, Rome, Italy, 60% Program.
Rights and permissions
About this article
Cite this article
Spedicato, E., Yang, Z. Optimally conditioned scaled ABS algorithms for linear systems. J Optim Theory Appl 67, 141–150 (1990). https://doi.org/10.1007/BF00939740
Issue Date:
DOI: https://doi.org/10.1007/BF00939740