Abstract
Many problems in practice require the solution of very large systems of linear equations Ax = b in which the matrix A, fortunately, is sparse, i.e., has relatively few nonvanishing elements. Systems of this type arise, e.g., in the application of difference methods or finite-element methods to the approximate solution of boundary-value problems in partial differential equations. The usual elimination methods (see Chapter 4) cannot normally be applied here, since without special precautions they tend to lead to the formation of more or less dense intermediate matrices, making the number of arithmetic operations necessary for the solution much too large, even for present-day computers, not to speak of the fact that the intermediate matrices no longer fit into the usually available computer memory.
This is a preview of subscription content, log in via an institution to check access.
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 for Chapter 8
Axelsson, O.: Solution of linear systems of equations: Iterative methods. In: Barker (1977).
Barker, V. A. (Ed.): Sparse Matrix Techniques. Lecture Notes in Mathematics 572. Berlin, Heidelberg, New York: Springer-Verlag (1977).
Brandt, A.: Multi-level adaptive solutions to boundary value problems. Math. of Comput. 31, 333–390 (1977).
Briggs, W. L.: A Multigrid Tutorial. Philadelphia: SIAM (1987).
Buneman, O.: A compact non-iterative Poisson solver. Stanford University, Institute for Plasma Research Report No. 294, Stanford, CA (1969).
Buzbee, B. L., Dorr, F. W.: The direct solution of the biharmonic equation on rectangular regions and the Poisson equation on irregular regions. SIAM J. Numer. Anal. 11, 753–763 (1974).
Buzbee, B. L., Dorr, F. W., George, J. A., Golub, G. H.: The direct solution of the discrete Poisson equation on irregular regions. SIAM J. Numer. Anal. 8, 722–736 (1971).
Buzbee, B. L., Golub, G. H., Nielson, C. W.: On direct methods for solving Poisson’s equations. SIAM J. Numer. Anal. 7, 627–656 (1970).
Chan, T. F., Glowinski, R., Periaux, J., Widlund, O. (Eds.): Proceedings of the Second International Symposium on Domain Decomposition Methods. Philadelphia: SIAM (1989).
Forsythe, G. E., Moler, C. B.: Computer Solution of Linear Algebraic Systems. Series in Automatic Computation. Englewood Cliffs, N.J.: Prentice-Hall (1967).
Glowinski, R., Golub, G. H., Meurant, G. A., Periaux, J. (Eds.): Proceedings of the First International Symposium on Domain Decomposition Methods for Partial Differential Equations. Philadelphia: SIAM (1988).
Hackbusch, W.: Multigrid Methods and Applications. Berlin, Heidelberg, New York: Springer-Verlag (1985).
Hackbusch, W., Trottenberg, U. (Eds.): Multigrid Methods. Lecture Notes in Mathematics 960. Berlin, Heidelberg, New York: Springer-Verlag (1982).
Hestenes, M. R., Stiefel, E.: Methods of conjugate gradients for solving linear systems. Nat. Bur. Standards J. Res. 49, 409–436 (1952).
Hockney, R. W.: The potential calculation and some applications, Methods of Computational Physics 9, 136–211. New York, London: Academic Press (1969).
Householder, A. S.: The Theory of Matrices in Numerical Analysis. New York: Blaisdell Publ. Co. (1964).
Keyes, D. E., Gropp, W. D.: A comparison of domain decomposition techniques for elliptic partial differential equations and their parallel implementation. SIAM J. Sci. Statist. Comput. 8, s166 — s202 (1987).
McCormick, S.: Multigrid Methods. Philadelphia: SIAM (1987).
Meijerink, J. A., van der Vorst, H. A.: An iterative solution method for linear systems of which the coefficient matrix is a symmetric M-matrix. Math. Comp. 31, 148162 (1977).
O’Leary, D. P., Widlund, O.: Capacitance matrix methods for the Helmholtz equation on general three-dimensional regions. Math. Comp. 33, 849–879 (1979).
Proskurowski, W., Widlund, O.: On the numerical solution of Helmholtz’s equation by the capacitance matrix method. Math. Comp. 30, 433–468 (1976).
Reid, J. K. (Ed.): Large Sparse Sets of Linear Equations. London, New York: Academic Press (1971).
Reid, J. K. (Ed.): On the method of conjugate gradients for the solution of large sparse systems of linear equations. In: Reid (1971), 231–252.
Rice, J. R., Boisvert, R. F.: Solving Elliptic Problems Using ELLPACK. Berlin, Heidelberg, New York: Springer (1984).
Schröder, J., Trottenberg, U.: Reduktionsverfahren für Differenzengleichungen bei Randwertaufgaben I. Numer. Math. 22, 37–68 (1973).
Schröder, J., Trottenberg, U., Reutersberg, H.: Reduktionsverfahren für Differenzengleichungen bei Randwertaufgaben II. Numer. Math. 26, 429–459 (1976).
Swarztrauber, P. N.: The methods of cyclic reduction, Fourier analysis and the FACR algorithm for the discrete solution of Poisson’s equation on a rectangle. SIAM Rev. 19, 490–501 (1977).
Varga, R. S.: Matrix Iterative Analysis. Series in Automatic Computation. Englewood Cliffs, N.J.: Prentice-Hall (1962).
Wachspress, E. L.: Iterative Solution of Elliptic Systems and Application to the Neutron Diffusion Equations of Reactor Physics. Englewood Cliffs, N.J.: Prentice-Hall (1966).
Wilkinson, J. H., Reinsch, C.: Linear Algebra. Handbook for Automatic Computation, Vol. II. Grundlehren der mathematischen Wissenschaften in Einzeldarsstellungen, Bd. 186. Berlin, Heidelberg, New York: Springer (1971).
Wittmeyer, H.: Über die Lösung von linearen Gleichungssystemen durch Iteration. Z. Angew. Math. Mech. 16, 301–310 (1936).
Young, D. M.: Iterative Solution of Large Linear Systems. Computer Science and Applied Mathematics. New York: Academic Press (1971).
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 1993 Springer Science+Business Media New York
About this chapter
Cite this chapter
Stoer, J., Bulirsch, R. (1993). Iterative Methods for the Solution of Large Systems of Linear Equations. Some Further Methods. In: Introduction to Numerical Analysis. Texts in Applied Mathematics, vol 12. Springer, New York, NY. https://doi.org/10.1007/978-1-4757-2272-7_8
Download citation
DOI: https://doi.org/10.1007/978-1-4757-2272-7_8
Publisher Name: Springer, New York, NY
Print ISBN: 978-1-4757-2274-1
Online ISBN: 978-1-4757-2272-7
eBook Packages: Springer Book Archive