Abstract
The paper presents a comparative analysis of various preconditioning procedures and proposes a new approach to the construction of preconditioning methods for the solution of large systems of linear algebraic equations with sparse matrices. Efficient implementation of preconditioning methods on MIMD multiprocessor computing systems is considered under constraints on computer resources and technical possibilities.
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S. M. Belotserkovskii and B. K. Skripach, Aerodynamic Aircraft and Airfoil Derivatives at Subsonic Velocities [in Russian], Nauka, Moscow (1975).
V. V. Voevodin and Yu. A. Kuznetsov, Matrices and Computation [in Russian], Nauka, Moscow (1984).
V. M. Glushkov and I. N. Molchanov, “On some issues of problem solving by computer with parallel computation,” Kibernetika, No. 4, 82–88 (1981).
A. Yu. Eremin and I. E. Kaporin, “Spectral optimization of explicit iteration methods, I,” in: Numerical Methods and Issues of Computational Optimization [in Russian], Zapiski Nauchn. Seminarov LOMI,138, 51–60 (1984).
I. I. Lyashko, G. E. Mistetskii, and A. Ya. Oleinik, Computation of Seepage in Hydroconstruction Zone [in Russian], Naukova Dumka, Kiev (1977).
G. I. Marchuk, Methods of Computational Mathematics [in Russian], Nauka, Moscow (1977).
V. S. Mikhalevich and I. N. Molchanov, Trends in Supercomputer Development and Associated Problems [in Russian], Preprint 84-4, Akad. Nauk UkrSSR, Inst. Kiber., Kiev (1984).
I. N. Molchanov, Numerical Methods of Solution of Some Problems in Elasticity Theory [in Russian], Naukova Dumka, Kiev (1979).
A. A. Samarskii and E. S. Nikolaev, Methods of Solution of Grid Equations [in Russian], Nauka, Moscow (1978).
D. K. Faddeev and V. N. Faddeeva, Computational Methods of Linear Algebra [in Russian], Fizmatgiz, Moscow (1960).
O. Axelsson, “A generalized SSOR method,” BIT,13, 443–467 (1972).
A. Greenbaum, “Comparison of splittings used with the conjugate gradient algorithm,” Numer. Math.,33, 181–194 (1979).
A. Jennings and G. A. Malik, “Partial elimination,” J. Inst. Math. Applic.,20, 308–316 (1977).
O. G. Johnson, C. A. Micchelli, and G. Paul, “Polynomial preconditioners for conjugate gradient calculations,” SIAM J. Numer. Anal.,20, No. 2, 362–376 (1983).
T. A. Manteuffel, “An incomplete factorization technique for positive definite linear systems,” Math. Comput.,34, 473–497 (1980).
J. A. Meijerink and H. A. Van der Vorst, “Guidelines for usage of incomplete decompositions in solving sets of linear equations as they occur in practical problems,” J. Comp. Phys.,44, 134–155 (1981).
J. A. Meijerink and H. A. Van der Vorst, “An iterative solution method for linear systems of which the coefficient matrix is a symmetric M-matrix,” Math. Comput.,31, 148–162 (1977).
N. M. Missirlis and D. J. Evans, “On the convergence of some generalized preconditioned iterative methods,” SIAM J. Numer. Anal.,18, No. 4, 571–586 (1981).
N. Munksgaard, “Solving sparse symmetric sets of linear equations by preconditioned conjugate gradients,” ACM Trans. Math. Software,6, No. 2, 206–216 (1980).
R. S. Varga, “Factorization and normalized iterative methods,” in: Boundary Problems in Differential Equations, R. E. Langer (ed.), Univ. of Wisconsin Press, Madison (1960).
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Translated from Vychislitel'naya i Prikladnaya Matematika, No. 63, pp. 42–49, 1987.
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Molchanov, I.N., Ryabtsev, V.E. & Trigub, A.S. Construction and efficient implementation of implicit preconditioning methods. I. J Math Sci 66, 2159–2164 (1993). https://doi.org/10.1007/BF01098600
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DOI: https://doi.org/10.1007/BF01098600