Efficient implementation of minimal polynomial and reduced rank extrapolation methods

Abstract The minimal polynomial extrapolation (MPE) and reduced rank extrapolation (RRE) are two very effective techniques that have been used in accelerating the convergence of vector sequences, such as those that are obtained from iterative solution of linear and nonlinear systems of equations. Their definitions involve some linear least-squares problems, and this causes difficulties in their numerical implementation. In this work timewise efficient and numerically stable implementations for MPE and RRE are developed. A computer program written in FORTRAN 77 is also appended and applied to some model problems, among them a hypersonic flow problem involving chemical reactions.

[1]  D. Faddeev,et al.  Computational Methods of Linear Algebra , 1959 .

[2]  David A. Smith,et al.  Erratum: correction to “extrapolation methods for vector sequences” , 1988 .

[3]  S. Eisenstat,et al.  Variational Iterative Methods for Nonsymmetric Systems of Linear Equations , 1983 .

[4]  Y. Saad,et al.  GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems , 1986 .

[5]  David A. Smith,et al.  Acceleration of convergence of vector sequences , 1986 .

[6]  M. Mesina.  Convergence acceleration for the iterative solution of the equations X = AX + f , 1977 .

[7]  A. Sidi.  On extensions of the power method for normal operators , 1989 .

[8]  A. Sidi.  Extrapolation vs. projection methods for linear systems of equations , 1988 .

[9]  D. Faddeev,et al.  Computational methods of linear algebra , 1959 .

[10]  Avram Sidi,et al.  Convergence and stability analyses for some vector extrapolation methods on the presence of defective iteration matrices , 1988 .

[11]  Y. Saad.  Krylov subspace methods for solving large unsymmetric linear systems , 1981 .

[12]  Recursive algorithms for vector extrapolation methods , 1988 .

[13]  O. Axelsson.  Conjugate gradient type methods for unsymmetric and inconsistent systems of linear equations , 1980 .

[14]  S. Cabay,et al.  A Polynomial Extrapolation Method for Finding Limits and Antilimits of Vector Sequences , 1976 .

[15]  P. Wynn,et al.  Acceleration techniques for iterated vector and matrix problems : (mathematics of computation, _1_6(1962), nr 79, p 301-322) , 1962 .

[16]  J. Gillis,et al.  Matrix Iterative Analysis , 1961 .

[17]  R. P. Eddy.  EXTRAPOLATING TO THE LIMIT OF A VECTOR SEQUENCE , 1979 .

[18]  A. Sidi,et al.  Extrapolation methods for vector sequences , 1987 .

[19]  Kang C. Jea,et al.  Generalized conjugate-gradient acceleration of nonsymmetrizable iterative methods , 1980 .

[20]  Gene H. Golub,et al.  Matrix computations , 1983 .

[21]  Convergence acceleration for vector sequences and applications to computational fluid dynamics , 1990 .

[22]  Avram Sidi.  Convergence and stability properties of minimal polynomial and reduced rank extrapolation algorithms , 1986 .

[23]  Avram Sidi,et al.  Application of vector extrapolation methods to consistent singular linear systems , 1990 .