Existence and uniqueness of splittings for stationary iterative methods with applications to alternating methods |
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Authors: | Michele Benzi Daniel B Szyld |
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Institution: | (1) CERFACS, 42 Ave. G. Coriolis, F-31057 Toulouse Cedex, France; benzi@cerfacs.fr, FR;(2) Department of Mathematics, Temple University, Philadelphia, Pennsylvania 19122-2585, USA; szyld@math.temple.edu, US |
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Abstract: | Summary. Given a nonsingular matrix , and a matrix of the same order, under certain very mild conditions, there is a unique splitting , such that . Moreover, all properties of the splitting are derived directly from the iteration matrix . These results do not hold when the matrix is singular. In this case, given a matrix and a splitting such that , there are infinitely many other splittings corresponding to the same matrices and , and different splittings can have different properties. For instance, when is nonnegative, some of these splittings can be regular splittings, while others can be only weak splittings. Analogous results
hold in the symmetric positive semidefinite case. Given a singular matrix , not for all iteration matrices there is a splitting corresponding to them. Necessary and sufficient conditions for the existence of such splittings are
examined. As an illustration of the theory developed, the convergence of certain alternating iterations is analyzed. Different
cases where the matrix is monotone, singular, and positive (semi)definite are studied.
Received September 5, 1995 / Revised version received April 3, 1996 |
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Keywords: | Mathematics Subject Classification (1991):65F10 15A06 |
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