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

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

MathSciNet review:
0438681

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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]**J. W. DANIEL, "The conjugate gradient method for linear and nonlinear operator equations,"*SIAM J. Numer. Anal.*, v. 4, 1967, pp. 10-26. MR**36**#1076. MR**0217987 (36:1076)****[2]**KY FAN, "Note on*M*-matrices,"*Quart. J. Math. Oxford Ser.*(2), v. 11, 1960, pp. 43-49. MR**22**#8024. MR**0117242 (22:8024)****[3]**M. R. HESTENES,*The Conjugate-Gradient Method for Solving Linear Systems*, Proc. Sympos. Appl. Math., vol. VI, Numerical Analysis, McGraw-Hill, New York, 1956, pp. 83-102. MR**18**, 824. MR**0084178 (18:824c)****[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.*, v. 9, 1972, pp. 325-332. MR**46**#4697. MR**0305567 (46:4697)****[6]**H. L. STONE, "Iterative solution of implicit approximations of multidimensional partial differential equations,"*SIAM J. Numer. Anal.*, v. 5, 1968, pp. 530-558. MR**38**#6780. MR**0238504 (38:6780)****[7]**R. S. VARGA,*Matrix Iterative Analysis*, Prentice-Hall, Englewood Cliffs, N.J., 1962. MR**28**#1725. MR**0158502 (28:1725)****[8]**J. H. WILKINSON & C. REINSCH,*Linear Algebra*, Springer-Verlag, Berlin and New York, 1971. MR**0461856 (57:1840)****[9]**J. H. WILKINSON,*The Algebraic Eigenvalue Problem*, Clarendon Press, Oxford, 1965. MR**32**#1894. MR**0184422 (32:1894)**

Retrieve articles in *Mathematics of Computation*
with MSC:
65F10

Retrieve articles in all journals with MSC: 65F10

Additional Information

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

Article copyright:
© Copyright 1977
American Mathematical Society