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References
R.S. Varga, Matrix iterative analysis, Prentice-Hall, New Jersey, 1962.
M.R. Hestenes, Historical papers: Iterative methods for solving linear equations, JOTA 11 (1973), 323–334 and The solution of linear equations by minimization, JOTA 11 (1973), 335–359. Completed on December 12, 1951. The first paper originally appeared as NAML Report No. 52-9, 1951.
M.R. Hestenes and E. Stiefel, Methods of conjugate gradients for solving linear systems, J. Res. Nat. Bur. Stand. B49 (1952), 409–436.
R. Fletcher and C.M. Reeves, Function minimization by conjugate gradients, Computer Journal 7 (1964), 149–154.
E. Polak et G. Ribiére, Note sur la convergence de methodes de directions conjuguees, R. I. R. O. 16-R1 (1969), 35–43.
J.W. Daniel, The conjugate gradient method for linear and nonlinear operator equations, SIAM J. Numer. Anal. 4 (1967), 10–26.
R. Bartels and J.W. Daniel, A conjugate gradient approach to nonlinear elliptic boundary value problems in irregular regions, CNA 63, The University of Texas at Austin, 1973.
D.J. Evans, The use of preconditioning in iterative methods for solving linear equations with symmetric positive definite matrices, J. Inst. Maths. Applics. (1968) 4, 295–314.
E.G. D'Yakonov, On an iterative method for the solution of finite difference equations, Dokl. Akad. Nauk SSSR, 138 (1961), 522–525.
G.J. Habetler and E.L. Wachspress, Symmetric successive overrelaxation in solving diffusion difference equations, Math. Comp. 15 (1961), 356–362.
J.E. Gunn, The solution of elliptic difference equations by semi-explicit iterative techniques, SIAM J. Numer. Anal. Ser. B 2 (1964), 24–45.
T. Dupont, R.P. Kendall and H.H. Rachford, Jr., An approximate factorization procedure for solving self-adjoint elliptic difference equations, SIAM J. Numer. Anal. 5 (1968), 559–573.
O. Axelsson, A generalized SSOR method, BIT 13 (1972), 443–467.
O. Axelsson, Iterative methods for elliptic problems, CERN DD/72/13, Geneva, 1972.
O. Axelsson, On preconditioning and convergence acceleration in sparse matrix problems, CERN 74-10, Geneva, 1974.
O. Axelsson, A class of iterative methods for finite element equations, Computer methods in applied mechanics and engineering 7 (1976) (to appear).
D.M. Young, Iterative solution of large linear systems, Academic Press, 1971.
M.J.D. Powell, Restart procedures for the conjugate gradient method. C.S.S. 24 (1975), Harwell, England.
J.K. Reid, On the method of conjugate gradients for the solution of large sparse systems of linear equations, in Large sparse sets of linear equations (ed. Reid), Academic Press, 1971.
L.F. Richardson, The approximate arithmetical solution by finite differences of physical problems involving differential equations, with an application to the stresses in a masonry dam, Trans. Roy. Soc. London A210 (1911), 307–357
L.F. Richardson, How to solve differential equations approximately by arithmetic, Math. Gazette, 12 (1925), 415–421.
V.I. Lebedev, S.A. Finogenov, On the order of choice of the iteration parameters in the Chebyshev cyclic iteration method, Zh. Vychislit. Mat. i Mat. Fiz. 11 (1971), 425 and 13 (1973), 18.
O. Axelsson, Lecture notes on Iterative methods, Report 72.04, Department of Computer Sciences, Chalmers University of Technology, Göteborg, Sweden.
D.M. Young, Second degree iterative methods for the solution of large linear systems, J. Approx. Theory 5 (1972), 137.
V.I. Lebedev, Iterative methods for the solution of operator equations with their spectrum on several intervals, Zh. Vychislit. Mat. i Mat. Fiz. 9 (1969), 1247–1252.
L. Andersson, SSOR preconditioning of Toeplitz matrices, Thesis, Chalmers University of Technology, Göteborg, Sweden, 1976.
J. Todd, Introduction to the constructive theory of functions, ISNM vol. 1, Birkhäuser, Basel, 1963.
M. Engeli, Th. Ginsburg, H. Rutishauser, E. Stiefel, Refined iterative methods for computation of the solution and the eigenvalues of self-adjoint boundary value problems, Mitteilungen aus dem Institut für angewandte Mathematik, nr. 8, ETH, Zürich, 1959.
H. Wozniakowski, Numerical stability of iterations for solution of nonlinear equations and large linear systems, Department of Computer Science, Carnegie — Mellon University, 1975.
G. Forsythe, E.G. Straus, On best conditioned matrices, Proc. Amer. Math. Soc. 6 (1955), 340–345.
J.A. Meijerink, H.A. van der Vorst, An iterative solution method for linear systems of which the coefficient matrix is a symmetric M-matrix, TR-1, Academic Computer Centre, Utrecht, the Netherlands, 1976.
P. Concus, G.H. Golub, Use of fast direct methods for the efficient numerical solution of nonseparable elliptic equations, STAN-CS-72-278, Stanford University, 1972.
E.L. Wachspress, Iterative solution of elliptic systems and applications to the neutron diffusion equations of reactor physics, Prentice-Hall, 1966.
C. Lanczos, An iteration method for the solution of the eigenvalue problem of linear differential and integral operators, J. Res. Nat. Bur. Stand. B 45 (1950), 255–282.
H. Liebmann, Die angenäherte Ermittelung harmonischer Funktionen und konformer Abbildungen, Bayer. Akad. Wiss., Math-Phys. Klasse, Sitz. (1918), 385–416.
J.W. Sheldon, On the numerical solution of elliptic equations, Math. Tables and other Aids to Comp. 9 (1955), 101–111.
A.C. Aitken, On the iterative solution of a system of linear equations, Proc. Roy. Soc. Edinburgh A63 (1950), 52.
L.W. Ehrlich, The block symmetric successive overrelaxation method, J. Soc. Indust. Appl. Math. 12 (1964), 807–826.
D. Young, On Richardson's method for solving linear systems with positive definite matrices, J. Math. Phys. 32 (1954), 243–255.
E. Stiefel, Kernel polynomials in linear algebra and their numerical applications, NBS, Appl. Math. Series 49 (1958), 1–22.
E. Cuthill, J. Mc Kee, Reducing the bandwidth of sparse symmetric matrices, Proc. 24th Nat. Conf. of the ACM, ACM Publ. P-69, New York, (1969), 157–172.
W-H. Liu, A.H. Sherman, Comparative analysis of the Cuthill-Mc Kee and the reverse Cuthill-Mc Kee ordering algorithms for sparse matrices, SIAM J. Numer. Anal. 13 (1976), 198–213.
I. Fried, More on gradient iterative methods in finite-element analysis, AIAA Journal 7 (1969), 565–567.
J. Smith, The coupled equation approach to the numerical solution of the Biharmonic equation by finite differences I, II, SIAM J. Numer. Anal. 5 (1968), 323–339, 7 (1970), 104–111.
O. Axelsson, Notes on the numerical solution of the Biharmonic equation, J. Inst. Maths. Applics. 11 (1973), 213–226.
L.W. Ehrlich, Solving the Biharmonic equation as coupled finite difference equations, SIAM J. Numer. Anal. 8 (1971), 278–287.
J.W. Mc Laurin, A general coupled equation approach for solving the Biharmonic boundary value problem, SIAM J. Numer. Anal. 11 (1974), 14–23.
M.M. Gupta, Discretization error estimates for certain splitting procedures for solving first Biharmonic boundary value problems, SIAM J. Numer. Anal. 12 (1975), 364–377.
T.A. Manteuffel, An iterative method for solving nonsymmetric linear systems with dynamic estimation of parameters, UIUCDCS-R-75-758, University of Illinois at Urbana-Champaign, Illinois, 1975.
P.P. Fedorenko, The speed of convergence of one iterative method, Zh. Vychislit. Mat. Fiz. 4 (1964), 559–564.
P.O. Fredrickson, Fast approximate inversion of large sparse linear systems, Mathematics Report # 7-75, Lakehead University, 1975.
P. Concus, G.H. Golub, D.P. O'Leary, A generalized conjugate gradient method for the numerical solution of elliptic partial differential equations, STAN-CS-76-533, Stanford University, 1976.
A. George, Nested dissection of a regular finite element mesh, SIAM J. Numer. Anal. 10 (1973), 345–363.
A. George, Numerical experiments using dissection methods to solve n by n grid problems, Research Report CS-75-07, University of Waterloo, Canada.
R.L. Fox, E.L. Stanton, Developments in structured analysis by direct energy minimization, AIAA Journal 6 (1968), 1036–1042.
P. Pohl, Iterative improvement without double precision in a boundary value problem, BIT 14 (1974), 361–365.
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Axelsson, O. (1977). Solution of linear systems of equations: Iterative methods. In: Barker, V.A. (eds) Sparse Matrix Techniques. Lecture Notes in Mathematics, vol 572. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0116614
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