Abstract
In this paper we present the main properties of the class of direct methods for solving linear equations proposed by Abaffy, Broyden and Spedicato in 1984 and called the ABS class.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
ABAFFY J. (1979) A Lineà ris Egyenletrendszerek Altalà nos Megoldà sà nak Egy Modszerosztà lya, Alkalmaz. Mat. Lapok 5, 223–240.
ABAFFY J. (1990) Superlinear Convergence Theorems for Newton Type Methods for Nonlinear Systems of Equations, Report DMSIA 90/10, University of Bergamo
ABAFFY J., BROYDEN C.G. and SPEDICATO E. (1984) A Class of Direct Methods for Linear Equations, Numer. Math. 45, 361–376
ABAFFY J. and SPEDICATO E. (1989) ABS Projection Algorithms, Mathematical Techniques for Linear and Nonlinear Equations, Ellis Horwood, Chichester
ADACHI N. (1971) On Variable Metric Algorithms, J.Optim.Theory Appl. 7, 391–410
BERTOCCHI M. and SPEDICATO E. (1990) Vectorizing the Implicit Gauss/Cholesky Algorithm of the ABS Class on the IBM 3090 VF, Technical Report 1/37, PF Sistemi Informatici e Calcolo Parallelo, University of Bergamo
PROYDEN C.G. (1985) On the Numerical Stability of Huang’s Update and Related Methods, J. Optim. Theory Appl. 47, 401–412
BROYDEN C.G. (1989) On the Numerical Stability of Huang’s Update, Report DMSIA 89/19, University of Bergamo
BODON E. (1988) Globally Optimally Conditioned Updates in the ABS Class for Linear Systems, Preprint, University of Bergamo
BODON E. (1989) Biconjugate Algorithms in the ABS Class I: Alternative Formulation and Theoretical Properties, Report DMSIA 89/3, University of Bergamo
BODON E. (1990) Derivation and Numerical Performance of Factorized ABS Algorithms for Solving Linear Equations, Preprint, University of Bergamo
BODON E. (1991) On some Conjugate Gradient Type ABS Algorithms for Nonsymmetric Linear Systems of Equations, Preprint, University of Bergamo
CRAIG E.J. (1955) The N-Step Iterations Procedures, J. Math. Phys. 34, 64–73
DANIEL J., GRAGG W.B., KAUFMAN L. and STEWART G.W. (1976) Reorthogonalization and Stable Algorithms for Updating the Gram-Schmidt QR Factorization, Math. Comput. 30, 772–795
DENG N. and CHEN Z. (1990) Truncated Nonlinear ABS Algorithm and its Convergence Property, Computing 45, 169–173
GALANTAI A. (1990) A Global Convergence Theorem for Nonlinear ABS Methods, Report DMSIA 90/9, University of Bergamo
HUANG H.Y. (1975) A Direct Method for the General Solution of a System of Linear Equations, J. Optim. Theory Appl. 16, 429–445
HUANG Z. (1990) On the Convergence Analysis of the Nonlinear Block Scaled ABS Methods, Report DMSIA 90/14, University of Bergamo
SPEDICATO E. (1988) Optimal Conditioning Parameter Selection in the ABS Class for Linear Algebraic Systems, Mat. Applic. Computac. 3, 187–199
SPEDICATO E. and BODON E. (1989) Solving Linear Least Squares by Orthogonal Factorization and Pseudoinverse Computation via the MHA in the ABS Class, Computing 42, 195–205
SPEDICATO E. and VESPUCCI M.T. (1989) Computational Performance of the Implicit Gram-Schmidt (Huang) Algorithm for Linear Algebraic Systems, Report DMSIA 89/6, University of Bergamo
SPEDICATO E. and XIA Z. (1990a) Application of ABS Algorithms to Constrained Optimization I: Linear Equality Constraints, Report DMSIA 90/17, University of Bergamo
SPEDICATO E. and XIA Z. (1990b) On some ABS Methods for the Computation of the Pseudoinverse, Report DMSIA 90/24, University of Bergamo
SPEDICATO E. and YANG Z. (1990) Optimally Conditioned Scaled ABS Algorithms for Linear Systems, J. Optim. Theory Appl. 67, 141–150
SPEDICATO E., XIA Z. and GAO Z. (1990) The Basic Block ABS Method I: Formulation and Theoretical Properties, Report DMSIA 90/15, University of Bergamo
XIA Z. (1990) Application of ABS Algorithms to Constrained Optimization II: Linear Inequality Constraints, Report DMSIA 90/29, University of Bergamo
XIA Z. (1991a) The ABS Class and Quadratic Programming I: Equality Constraints, Report DMSIA 91/1, University of Bergamo
XIA Z. (1991b) The ABS Class and Quadratic Programming II: General Constraints, Report DMSIA 91/5, University of Bergamo
XIA Z. and GAO Z. (1990) The Basic Block ABS Method II: Application to the Rutheford Matrix Equation, Report DMSIA 90/3, University of Bergamo
ZHU M. (1990) Finding General Solution to Perturbed Linear Systems and its Application in Optimization, Report DMSIA 90/5, University of Bergamo
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1991 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Spedicato, E. (1991). Abs Algorithms for General Linear Systems. In: Spedicato, E. (eds) Computer Algorithms for Solving Linear Algebraic Equations. NATO ASI Series, vol 77. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-76717-3_4
Download citation
DOI: https://doi.org/10.1007/978-3-642-76717-3_4
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-76719-7
Online ISBN: 978-3-642-76717-3
eBook Packages: Springer Book Archive