Preconditioned Barzilai-Borwein method for the numerical solution of partial differential equations. (English) Zbl 0861.65025

The preconditioned method of J. Barzilai and J. M. Borwein [IMA J. Numer. Anal. 8, No. 1, 141-148 (1988; Zbl 0638.65055)] is introduced to solve large, sparse, symmetric and positive definite linear systems. A set of well-known preconditioning techniques are combined with this method, and numerical examples illustrate the effectiveness of this approach.


65F10 Iterative numerical methods for linear systems
65F35 Numerical computation of matrix norms, conditioning, scaling


Zbl 0638.65055
Full Text: DOI


[1] O. Axelsson,Iterative Solution Methods (Cambridge University Press, New York, 1994). · Zbl 0795.65014
[2] O. Axelsson, A survey of preconditioned iterative methods for linear systems of algebraic equations, BIT 25 (1985) 166–187. · Zbl 0566.65017
[3] O. Axelsson and V. A. Barker,Finite Element Solution of Boundary Value Problems, Theory and Computation (Academic Press, New York, 1984). · Zbl 0537.65072
[4] J. Barzilai and J. M. Borwein, Two point step size gradient methods, IMA J. Numer. Anal. 8 (1988) 141–148. · Zbl 0638.65055
[5] P. Concus, G. H. Golub and G. Meurant, Block preconditioning for the conjugate gradient method, SIAM J. Sci. Stat. Comp. 6 (1985) 220–252. · Zbl 0556.65022
[6] R. Fletcher, Low storage methods for unconstrained, optimization, Lectures in Applied Mathematics (AMS) 26 (1990) 165–179. · Zbl 0699.65052
[7] A. Friedlander, J. M. Martinez and M. Raydan, A new method for large-scale box constrained convex quadratic minimization problems, Opt. Methods and Software 5 (1995) 57–74.
[8] W. Glunt, T. L. Hayden and M. Raydan, Molecular conformations from distance matrices, J. Comp. Chem. 14 (1993) 114–120.
[9] G. H. Golub and C. F. Van Loan,Matrix Computations (Johns Hopkins University Press, Baltimore, 1989). · Zbl 0733.65016
[10] G. H. Golub and D. P. O’Leary, Some history of the conjugate gradient and Lanczos methods, SIAM Review 31 (1989) 50–102. · Zbl 0673.65017
[11] I. Gustafsson, A class of first order factorization methods, BIT 18 (1978) 142–156. · Zbl 0386.65006
[12] L. A. Hageman and D. M. Young,Applied Iterative Methods (Academic Press, New York, 1981). · Zbl 0459.65014
[13] J. M. Ortega,Introduction to Parallel and Vector Solution of Linear Systems (Plenum Press, New York, 1988). · Zbl 0669.65017
[14] M. Raydan, On the Barzilai and Borwein choice of steplength for the gradient method, IMA J. Numer. Anal. 13 (1993) 321–326. · Zbl 0778.65045
[15] H. A. Van der Vorst and K. Dekker, Conjugate gradient type methods and preconditioning, J. Comp. Appl. Math. 24 (1988) 73–87. · Zbl 0659.65033
[16] D. M. Young, A historical overview of iterative methods, Computer Physics Communications 53 (1989) 1–17. · Zbl 0798.65035
[17] D. M. Young,Iterative Solution of Large Linear Systems (Academic Press, New York, 1971). · Zbl 0231.65034
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