An iterative solution method for linear systems of which the coefficient matrix is a symmetric -matrix

Authors:
J. A. Meijerink and H. A. van der Vorst

Journal:
Math. Comp. **31** (1977), 148-162

MSC:
Primary 65F10

MathSciNet review:
0438681

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Abstract: A particular class of regular splittings of not necessarily symmetric *M*-matrices is proposed. If the matrix is symmetric, this splitting is combined with the conjugate-gradient method to provide a fast iterative solution algorithm. Comparisons have been made with other well-known methods. In all test problems the new combination was faster than the other methods.

**[1]**James W. Daniel,*The conjugate gradient method for linear and nonlinear operator equations*, SIAM J. Numer. Anal.**4**(1967), 10–26. MR**0217987****[2]**Ky Fan,*Note on 𝑀-matrices*, Quart. J. Math. Oxford Ser. (2)**11**(1960), 43–49. MR**0117242****[3]**Magnus R. Hestenes,*The conjugate-gradient method for solving linear systems*, Proceedings of Symposia in Applied Mathematics. Vol. VI. Numerical analysis, McGraw-Hill Book Company, Inc., New York, for the American Mathematical Society, Providence, R. I., 1956, pp. 83–102. MR**0084178****[4]**H. S. PRICE & K. H. COATS, "Direct methods in reservoir simulation,"*Soc. Petroleum Engrs. J.*, v. 14, 1974, pp. 295-308.**[5]**J. K. Reid,*The use of conjugate gradients for systems of linear equations possessing “Property A”*, SIAM J. Numer. Anal.**9**(1972), 325–332. MR**0305567****[6]**Herbert L. Stone,*Iterative solution of implicit approximations of multidimensional partial differential equations*, SIAM J. Numer. Anal.**5**(1968), 530–558. MR**0238504****[7]**Richard S. Varga,*Matrix iterative analysis*, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1962. MR**0158502****[8]***Handbook for automatic computation. Vol. II*, Springer-Verlag, New York-Heidelberg, 1971. Linear algebra; Compiled by J. H. Wilkinson and C. Reinsch; Die Grundlehren der Mathematischen Wissenschaften, Band 186. MR**0461856****[9]**J. H. Wilkinson,*The algebraic eigenvalue problem*, Clarendon Press, Oxford, 1965. MR**0184422**

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DOI:
https://doi.org/10.1090/S0025-5718-1977-0438681-4

Article copyright:
© Copyright 1977
American Mathematical Society