This paper sketches the main research developments in the area of iterative methods for solving linear systems during the 20th century. Although iterative methods for solving linear systems find their origin in the early nineteenth century (work by Gauss), the field has seen an explosion of activity spurred by demand due to extraordinary technological advances in engineering and sciences. The past five decades have been particularly rich in new developments, ending with the availability of large toolbox of specialized algorithms for solving the very large problems which arise in scientific and industrial computational models. As in any other scientific area, research in iterative methods has been a journey characterized by a chain of contributions building on each other. It is the aim of this paper not only to sketch the most significant of these contributions during the past century, but also to relate them to one another. 1
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842
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GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems
– Saad, Schultz
- 1986
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754
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The Algebraic Eigenvalue Problem
– Wilkinson
- 1965
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721
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Iterative Methods for Sparse Linear Systems
– Saad
- 2003
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560
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Multigrid Methods and Applications
– Hackbusch
- 1985
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410
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E.: Methods of conjugate gradients for solving linear systems
– Hestenes, Stiefel
- 1952
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402
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Matrix Iterative Analysis
– Varga
- 1962
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374
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A multigrid tutorial
– Briggs
- 1987
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373
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der Vorst. Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods
– Barrett, Berry, et al.
- 1994
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303
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Iterative Solution methods
– Axelsson
- 1994
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284
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QMR: a quasi-minimal residual method for non-Hermitian linear systems
– Freund, Nachtigal
- 1991
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271
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An iteration method for the solution of the eigenvalue problem of linear differential and integral operators
– Lanczos
- 1950
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240
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Iterative Solution of Large Linear Systems
– Young
- 1971
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237
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An Introduction to Multigrid Methods
– WESSELING
- 1991
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196
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der Vorst. An iterative solution method for linear systems of which the coefficient matrix is a M-matrix
– Meijerink, Van
- 1977
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188
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The principle of minimized iterations in the solution of the matrix eigenvalue problem
– Arnoldi
- 1951
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183
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Iterative Methods for Solving Linear Systems
– Greenbaum
- 1997
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180
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A flexible inner-outer preconditioned GMRES algorithm
– Saad
- 1993
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174
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der Vorst, Bi-CGSTAB: A fast and smoothly converging variant of Bi-CG for the solution of nonsymmetric linear systems
– van
- 1992
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170
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LSQR: An algorithm for sparse linear equations and sparse least squares
– Paige, Saunders
- 1982
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166
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Solution of sparse indefinite systems of linear equations
– Paige, Saunders
- 1975
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141
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An introduction to algebraic multigrid
– Stuben
- 2001
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141
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CGS, a fast Lanczos-type solver for nonsymmetric linear systems
– Sonneveld
- 1989
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140
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Solution of systems of linear equations by minimized iterations
– Lanczos
- 1952
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136
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Nachtigal, An implementation of the look-ahead Lanczos algorithm for non-hermitian matrices, part II, math. numerical analysis report 90-11
– Freund, M
- 1990
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132
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T.: Parallel preconditioning with sparse approximate inverses
– Grote, Huckle
- 1997
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131
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Applied Iterative Methods
– Hageman, Young
- 1981
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123
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Conjugate gradient methods for indefinite systems
– Fletcher
- 1975
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119
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M.: A sparse approximate inverse preconditioner for the conjugate gradient method
– Benzi, Meyer, et al.
- 1996
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119
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Variable iterative methods for nonsymmetric systems of linear equations
– Eisenstat, Elman, et al.
- 1983
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113
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A transpose-free quasi-minimal residual algorithm for non-Hermitian linear systems
– Freund
- 1993
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109
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The Theory of Matrices in Numerical Analysis
– Householder
- 1964
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103
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Multilevel adaptive solution to boundary value problems
– BRANDT
- 1977
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89
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Factorized sparse approximate inverse preconditionings. IV: Simple approaches to rising efficiency
– Kolotilina, Yeremin
- 1999
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84
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How fast are nonsymmetric matrix iterations
– Nachtigal, Reddy, et al.
- 1992
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81
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Vassilevski: Algebraic multilevel preconditioning methods
– Axelsson, S
- 1989
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78
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A new method of solving nonlinear simultaneous equations
– Broyden
- 1969
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76
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der Vorst. Numerical Linear Algebra for High-Performance Computers
– Dongarra, Duff, et al.
- 1998
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72
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Block preconditioning for the conjugate gradient method
– Concus, Golub, et al.
- 1985
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72
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A completed theory of the unsymmetric Lanczos process and related algorithms,Part I
– Gutknecht
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72
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The effect of ordering on preconditioned conjugate gradients. BIT: Nordisk Tijdskrift for Informationbehandling
– Duff, Meurant
- 1989
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71
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Approximate inverse preconditioners via sparse-sparse iterations
– Chow
- 1998
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63
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Krylov subspace methods on supercomputers
– Saad
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61
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A generalized conjugate gradient method for the numerical solution of elliptic pa.rtia1 difFerentia.1 equations
– Concus, Goluh, et al.
- 1976
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60
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Necessary and sufficient conditions for the existence of a conjugate gradient method
– Faber, Manteuffel
- 1984
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60
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A class of first order factorization methods
– Gustafsson
- 1978
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58
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Approximate inverse preconditioning for sparse linear systems
– Cosgrove, Díaz, et al.
- 1992
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57
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Practical use of polynomial preconditionings for the conjugate gradient method
– Saad
- 1985
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55
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Chaotic relaxation
– Chazan, Miranker
- 1969
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52
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Reduction to tridiagonal form and minimal realizations
– Parlett
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52
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der Vorst. The rate of convergence of conjugate gradients
– Sluis, van
- 1986
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