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.
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References for Chapter 8
Axelsson, O.: Solution of linear systems of equations—iterative methods. In: Barker (1977), 1–51 (1976).
Barker, V. A., Ed.: Sparse Matrix Techniques. Lecture Notes in Mathematics 572. Berlin, Heidelberg, New York: Springer-Verlag, 1977.
Buneman, O.: A compact non-iterative Poisson solver. Stanford University, Institute for Plasma Research, Report No. 294, Stanford, Calif. 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., Dorr, F. W., Golub, G. H., Nielson, C. W.: On direct methods for solving Poisson’s equations. SIAM J. Numer. Anal. 7, 627–656 (1970).
Forsythe, G. E., Moler, C. B.: Computer Solution of Linear Algebraic Systems. Series in Automatic Computation. Englewood Cliffs: Prentice-Hall 1967.
Hackbusch, W.: Ein iteratives Verfahren zur schnellen Auflösung elliptischer Randwertprobleme. Mathematical Institute of the University of Cologne, Report 76–12, 1976.
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).
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, Ed. (1971), 231–252 (1971).
Rose, D. J., Willoughby, R. A., Eds.: Sparse Matrices and Their Applications. New York: Plenum Press 1972.
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).
Tewarson, R. P.: Sparse Matrices, New York: Academic Press 1973.
Varga, R. S.: Matrix Iterative Analysis. Series in Automatic Computation. Englewood Cliffs: Prentice-Hall 1962.
Wachspress, E. L.: Iterative Solution of Elliptic Systems and Application to the Neutron Diffusion Equations of Reactor Physics. Englewood Cliffs: Prentice-Hall 1966.
Widlund, O., Proskurowski, W.: On the numerical solution of Helmholtz’s equation by the capacitance matrix method. ERDA Rep. C 00–3077–99, Courant Institute of Mathematical Sciences, New York University 1975.
Wilkinson, J. H., Reinsch, G.: Linear Algebra. Handbook for Automatic Computation, Vol. II. Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen, Bd. 186. Berlin, Heidelberg, New York: Springer-Verlag 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.
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Stoer, J., Bulirsch, R. (1980). Iterative Methods for the Solution of Large Systems of Linear Equations. Some Further Methods. In: Introduction to Numerical Analysis. Springer, New York, NY. https://doi.org/10.1007/978-1-4757-5592-3_8
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DOI: https://doi.org/10.1007/978-1-4757-5592-3_8
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