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), 148162
MSC:
Primary 65F10
MathSciNet review:
0438681
Fulltext PDF Free Access
Abstract 
References 
Similar Articles 
Additional Information
Abstract: A particular class of regular splittings of not necessarily symmetric Mmatrices is proposed. If the matrix is symmetric, this splitting is combined with the conjugategradient method to provide a fast iterative solution algorithm. Comparisons have been made with other wellknown 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 (36 #1076)
 [2]
Ky
Fan, Note on 𝑀matrices, Quart. J. Math. Oxford Ser.
(2) 11 (1960), 43–49. MR 0117242
(22 #8024)
 [3]
Magnus
R. Hestenes, The conjugategradient method for solving linear
systems, Proceedings of Symposia in Applied Mathematics. Vol. VI.
Numerical analysis, McGrawHill Book Company, Inc., New York, for the
American Mathematical Society, Providence, R. I., 1956,
pp. 83–102. MR 0084178
(18,824c)
 [4]
H. S. PRICE & K. H. COATS, "Direct methods in reservoir simulation," Soc. Petroleum Engrs. J., v. 14, 1974, pp. 295308.
 [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
(46 #4697)
 [6]
Herbert
L. Stone, Iterative solution of implicit approximations of
multidimensional partial differential equations, SIAM J. Numer. Anal.
5 (1968), 530–558. MR 0238504
(38 #6780)
 [7]
Richard
S. Varga, Matrix iterative analysis, PrenticeHall, Inc.,
Englewood Cliffs, N.J., 1962. MR 0158502
(28 #1725)
 [8]
Handbook for automatic computation. Vol. II, SpringerVerlag, New
YorkHeidelberg, 1971. Linear algebra; Compiled by J. H. Wilkinson and C.
Reinsch; Die Grundlehren der Mathematischen Wissenschaften, Band 186. MR 0461856
(57 #1840)
 [9]
J.
H. Wilkinson, The algebraic eigenvalue problem, Clarendon
Press, Oxford, 1965. MR 0184422
(32 #1894)
 [1]
 J. W. DANIEL, "The conjugate gradient method for linear and nonlinear operator equations," SIAM J. Numer. Anal., v. 4, 1967, pp. 1026. MR 36 #1076. MR 0217987 (36:1076)
 [2]
 KY FAN, "Note on Mmatrices," Quart. J. Math. Oxford Ser. (2), v. 11, 1960, pp. 4349. MR 22 #8024. MR 0117242 (22:8024)
 [3]
 M. R. HESTENES, The ConjugateGradient Method for Solving Linear Systems, Proc. Sympos. Appl. Math., vol. VI, Numerical Analysis, McGrawHill, New York, 1956, pp. 83102. 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. 295308.
 [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. 325332. 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. 530558. MR 38 #6780. MR 0238504 (38:6780)
 [7]
 R. S. VARGA, Matrix Iterative Analysis, PrenticeHall, Englewood Cliffs, N.J., 1962. MR 28 #1725. MR 0158502 (28:1725)
 [8]
 J. H. WILKINSON & C. REINSCH, Linear Algebra, SpringerVerlag, 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)
Similar Articles
Retrieve articles in Mathematics of Computation
with MSC:
65F10
Retrieve articles in all journals
with MSC:
65F10
Additional Information
DOI:
http://dx.doi.org/10.1090/S00255718197704386814
PII:
S 00255718(1977)04386814
Article copyright:
© Copyright 1977
American Mathematical Society
