Abstract
Many applications require the solution of very large systems of linear equations Ax = b in which the matrix A is fortunately sparse, i.e., has only relatively few nonzero elements. Such systems arise, for instance, if difference methods or finite element methods arc being used for solving boundary value problems in partial differential equations. The classical elimination methods [see Chapter 4] are not suitable in this context since they tend to lead to the formation of dense intermediate matrices, making the number of arithmetic operations necessary for the solution too large even for present-day computers, not to mention the fact that the memory requirements for such intermediate matrices exceed available space.
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References for Chapter 8
Arnoldi, W.E. (1951): The principle of minimized iteration in the solution of the matrix eigenvalue problem. Quart. Appl. Math. 9, 17–29.
Axelsson, O. (1977): Solution of linear systems of equations: Iterative methods. In: Barker (1977).
Axelsson, O. (1994): Iterative Solution Methods. Cambridge, UK: Cambridge University Press.
Barker, V.A. (Ed.) (1977): Sparse Matrix techniques. Lecture Notes in Mathematics Vol. 572, Berlin, Heidelberg, New York: Springer-Verlag.
Braess, D. (1997): Finite Elemente. Berlin, Heidelberg, New York: Springer-Verlag.
Bramble, J.H. (1993): Multigrid Methods. Harlow: Longman.
Brandt, A. (1977): Multi-level adaptive solutions to boundary value problems. Math. of Comput. 31, 333–390.
Briggs, W.L. (1987): A Multiyrid Tutorial. Philadelphia: SIAM.
Buneman, O. (1969): A compact non-iterative Poisson solver. Stanford University, Institute for Plasma Research Report Nr. 294, Stanford, CA.
Buzbee, B.L., Dorr, F.W. (1974): The direct solution of the biharmonic equation on rectangular regions and the Poisson equation on irregular regions. SIAM J. Numer. Anal. 11, 753–763.
Buzbee, B.L., F.W., George, J.A., Golub, G.H. (1971): The direct solution of the discrete Poisson equation on irregular regions. SIAM J. Numer. Anal. 8, 722–736.
Buzbee, B.L., Golub, G.H., Nielson, C.W. (1970): On direct methods for solving Poisson’s equations. SIAM J. Numer. Anal. 7, 627–656.
Chan, T.F., Glowinski, R., Periaux, J., Widlund, O. (Eds.) (1989): Proceedings of the Second International Symposium on Domain Decomposition Methods. Philadelphia: SIAM.
Fletcher, R. (1974). Conjugate gradient methods for indefinite systems. In: G.A. Watson (ed.), Proceedings of the Dundee Biennial Conference on Numerical Analysis 1974, p. 73–89. New York: Springer-Verlag 1975.
Forsythe, G.E., Moler, C.B. (1967): Computer Solution of Linear Algebraic Systems. Series in Automatic Computation. Englewood Cliffs, N.J.: Prentice Hall.
Freund, R.W., Nachtigal, N.M. (1991): QMR: a quasi-minimal residual method for non-Hermitian linear systems. Numerische Mathematik 60, 315–339
George, A. (1973): Nested dissection of a regular finite element mesh. SIAM J. Numer. Anal. 10, 345–363.
Glowinski, R., Golub, G.H., Meurant, G.A., Periaux, J. (Eds.) (1988): Proceedings of the First International Symposium on Domain Decomposition Methods for Partial Differential Equations. Philadelphia: SIAM.
Hackbusch, W. (1985): Multigrid Methods and Applications. Berlin, Heidelberg, New York: Springer-Verlag.
Hackbusch, W., Trottenberg, U. (Eds.) (1982): Multigrid Methods. Lecture Notes in Mathematics. Vol. 960. Berlin, Heidelberg, New York: Springer-Verlag.
Hestenes, M.R., Stiefel. E. (1952): Methods of conjugate gradients for solving linear systems. Nat. Bur. Standards, J. of Res. 49, 409–436.
Hockney, R.W. (1969): The potential calculation and some applications. Methods of Computational Physics 9, 136–211.
Householder, A.S. (1964): The Theory of Matrices in Numerical Analysis. New York: Blaisdell Publ. Comp.
Keyes, D.E., Gropp, W.D. (1987): A comparison of domain decomposition techniques for elliptic partial differential equations. SIAM J. Sci. Statist. Comput. 8, 166–202.
Lanczos, C. (1950): An iteration method for the solution of the eigenvalue problem of linear differential and integral equations. J. Res. Nat. Bur. Standards. 45, 255–282.
Lanczos, C. (1952): Solution of systems of linear equations by minimized iterations. J. Res. Nat. Bur. Standards. 49, 33–53.
McCormick, S. (1987): Multigrid Methods. Philadelphia: SIAM.
Meijerink, J.A., van der Vorst, H.A. (1977): An iterative solution method for linear systems of which the coefficient matrix is a symmetric M-matrix. Math. Comp. 31, 148–162.
O’Leary, D.P., Widlund, O. (1979): Capacitance matrix methods for the Helmholtz equation on general three-dimensional regions. Math. Comp. 33, 849–879.
Paige, C.C., Saunders, M.A. (1975): Solution of sparse indefinite systems of linear equations. SIAM J. Numer. Analysis 12, 617–624.
Proskurowski, W., Widlund, O. (1976): On the numerical solution of Helmholtz’s equation by the capacitance matrix method. Math. Comp. 30, 433–468.
Quarteroni, A., Valli, A. (1997): Numerical Approximation of Partial Differential Equations, 2d edition. Berlin, Heidelberg, New York: Springer-Verlag.
Reid, J.K. (1971a): Large Sparse Sets of Linear Equations. London, New York: Academic Press.
Reid, J.K. (1971): On the method of conjugate gradients for the solution of large sparse systems of linear equations. In: Reid ( 1971a ). 231–252.
Rice, J.R., Boisvert, R.F. (1984): Solving Elliptic Problems Using ELLPACK. Berlin, Heidelberg, New York: Springer-Verlag.
Saad, Y. (1996): Iterative Methods for Sparse Linear Systems. Boston: PWS Publishing Company.
Saad, Y., Schultz, M.H. (1986): GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems. SIAM J. Scientific and Statistical Computing, 7, 856–869.
Schröder, J., Trottenberg, U. (1973): Reduktionsverfahren für Differenzengleichungen bei Randwertaufgaben I. Numer.Math. 22, 37–68.
Schröder, J., Reutersberg, H. (1976): Reduktionsverfahren für Differenzengleichungen bei Randwertaufgaben II. Numer. Math. 26, 429–459.
Sonneveldt, P. (1989): CGS, a fast Lanczos-type solver for nonsymmetric linear systems. SIAM J. Scientific and Statistical Computing 10, 36–52.
Swarztrauber, P.N. (1977): The methods of cyclic reduction. Fourier analysis and the FACR. algorithm for the discrete solution of Poisson’s equation on a rectangle. SIAM Review 19, 490–501.
Vorst (1992): Bi-CGSTAB: A fast and smoothly converging variant of Bi-CG for the solution of non-symmetric linear systems. SIAM J. Scientific and Statistical Computing 12, 631–644.
Varga, R.S. (1962): Matrix Iterative Analysis. Series in Automatic Computation. Englewood Cliffs: Prentice Hall (2d revised and expanded edition 2000, Berlin, Heidelberg, New York: Springer-Verlag ).
Wachspress, E.L. (1966): Iterative Solution of Elliptic Systems and Application to the Neutron Diffusion Equations of Reactor Physics. Englewood Cliffs, N.J.: Prentice-Hall.
Wilkinson, J.H., Reinsch, C. (1971): Linear Algebra. Handbook for Automatic Computation, Vol. H. Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen, Bd. 186. Berlin, Heidelberg, New York: Springer-Verlag.
Young, D.M. (1971): Iterative Solution of Large Linear Systems. Computer Science and Applied Mathematics. New York: Academic Press.
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Stoer, J., Bulirsch, R. (2002). Iterative Methods for the Solution of Large Systems of Linear Equations. Additional Methods. In: Introduction to Numerical Analysis. Texts in Applied Mathematics, vol 12. Springer, New York, NY. https://doi.org/10.1007/978-0-387-21738-3_8
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DOI: https://doi.org/10.1007/978-0-387-21738-3_8
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