Abs Algorithms for General Linear Systems

  • E. Spedicato
Conference paper
Part of the NATO ASI Series book series (volume 77)

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.

Keywords

Cond 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. ABAFFY J. (1979) A Lineàris Egyenletrendszerek Altalànos Megoldàsànak Egy Modszerosztàlya, Alkalmaz. Mat. Lapok 5, 223–240.MathSciNetGoogle Scholar
  2. ABAFFY J. (1990) Superlinear Convergence Theorems for Newton Type Methods for Nonlinear Systems of Equations, Report DMSIA 90/10, University of BergamoGoogle Scholar
  3. ABAFFY J., BROYDEN C.G. and SPEDICATO E. (1984) A Class of Direct Methods for Linear Equations, Numer. Math. 45, 361–376MathSciNetMATHCrossRefGoogle Scholar
  4. ABAFFY J. and SPEDICATO E. (1989) ABS Projection Algorithms, Mathematical Techniques for Linear and Nonlinear Equations, Ellis Horwood, ChichesterMATHGoogle Scholar
  5. ADACHI N. (1971) On Variable Metric Algorithms, J.Optim.Theory Appl. 7, 391–410MathSciNetMATHCrossRefGoogle Scholar
  6. 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 BergamoGoogle Scholar
  7. PROYDEN C.G. (1985) On the Numerical Stability of Huang’s Update and Related Methods, J. Optim. Theory Appl. 47, 401–412MathSciNetCrossRefGoogle Scholar
  8. BROYDEN C.G. (1989) On the Numerical Stability of Huang’s Update, Report DMSIA 89/19, University of BergamoGoogle Scholar
  9. BODON E. (1988) Globally Optimally Conditioned Updates in the ABS Class for Linear Systems, Preprint, University of BergamoGoogle Scholar
  10. BODON E. (1989) Biconjugate Algorithms in the ABS Class I: Alternative Formulation and Theoretical Properties, Report DMSIA 89/3, University of BergamoGoogle Scholar
  11. BODON E. (1990) Derivation and Numerical Performance of Factorized ABS Algorithms for Solving Linear Equations, Preprint, University of BergamoGoogle Scholar
  12. BODON E. (1991) On some Conjugate Gradient Type ABS Algorithms for Nonsymmetric Linear Systems of Equations, Preprint, University of BergamoGoogle Scholar
  13. CRAIG E.J. (1955) The N-Step Iterations Procedures, J. Math. Phys. 34, 64–73MathSciNetMATHGoogle Scholar
  14. 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, 772–795MathSciNetMATHGoogle Scholar
  15. DENG N. and CHEN Z. (1990) Truncated Nonlinear ABS Algorithm and its Convergence Property, Computing 45, 169–173MathSciNetMATHCrossRefGoogle Scholar
  16. GALANTAI A. (1990) A Global Convergence Theorem for Nonlinear ABS Methods, Report DMSIA 90/9, University of BergamoGoogle Scholar
  17. HUANG H.Y. (1975) A Direct Method for the General Solution of a System of Linear Equations, J. Optim. Theory Appl. 16, 429–445MATHCrossRefGoogle Scholar
  18. HUANG Z. (1990) On the Convergence Analysis of the Nonlinear Block Scaled ABS Methods, Report DMSIA 90/14, University of BergamoGoogle Scholar
  19. SPEDICATO E. (1988) Optimal Conditioning Parameter Selection in the ABS Class for Linear Algebraic Systems, Mat. Applic. Computac. 3, 187–199MathSciNetGoogle Scholar
  20. 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, 195–205MathSciNetMATHCrossRefGoogle Scholar
  21. 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 BergamoGoogle Scholar
  22. SPEDICATO E. and XIA Z. (1990a) Application of ABS Algorithms to Constrained Optimization I: Linear Equality Constraints, Report DMSIA 90/17, University of BergamoGoogle Scholar
  23. SPEDICATO E. and XIA Z. (1990b) On some ABS Methods for the Computation of the Pseudoinverse, Report DMSIA 90/24, University of BergamoGoogle Scholar
  24. SPEDICATO E. and YANG Z. (1990) Optimally Conditioned Scaled ABS Algorithms for Linear Systems, J. Optim. Theory Appl. 67, 141–150MathSciNetMATHCrossRefGoogle Scholar
  25. SPEDICATO E., XIA Z. and GAO Z. (1990) The Basic Block ABS Method I: Formulation and Theoretical Properties, Report DMSIA 90/15, University of BergamoGoogle Scholar
  26. XIA Z. (1990) Application of ABS Algorithms to Constrained Optimization II: Linear Inequality Constraints, Report DMSIA 90/29, University of BergamoGoogle Scholar
  27. XIA Z. (1991a) The ABS Class and Quadratic Programming I: Equality Constraints, Report DMSIA 91/1, University of BergamoGoogle Scholar
  28. XIA Z. (1991b) The ABS Class and Quadratic Programming II: General Constraints, Report DMSIA 91/5, University of BergamoGoogle Scholar
  29. XIA Z. and GAO Z. (1990) The Basic Block ABS Method II: Application to the Rutheford Matrix Equation, Report DMSIA 90/3, University of BergamoGoogle Scholar
  30. ZHU M. (1990) Finding General Solution to Perturbed Linear Systems and its Application in Optimization, Report DMSIA 90/5, University of BergamoGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1991

Authors and Affiliations

  • E. Spedicato
    • 1
  1. 1.Department of MathematicsUniversity of BergamoBergamoItaly

Personalised recommendations