Climent and Perea [Journal of Computational and Applied Mathematics 58:43–48, 2003; MR2013603] proposed first the convergence theory of two-stage iterative scheme for solving real rectangular linear systems. In this article, we revisit the same theory. The first main result provides some sufficient conditions which guarantee that the induced splitting from a two-stage iterative scheme is a proper weak regular splitting. We then establish a few comparison results. Out of these, many are even new in nonsingular matrix setting. Further, we study the monotone convergence theory of the two-stage iterative method. Besides these, we also prove the uniqueness of a proper splitting of a rectangular matrix under certain assumptions.
Multisplitting methods are useful to solve differential-algebraic equations.
In this connection, we discuss the theory of matrix splittings and
multisplittings, which can be used for finding the iterative solution of a
large class of rectangular (singular) linear system of equations of the form
Ax = b. In this direction, many convergence results are proposed for
different subclasses of proper splittings in the literature. But, in some
practical cases, the convergence speed of the iterative scheme is very slow.
To overcome this issue, several comparison results are obtained for different
subclasses of proper splittings. This paper also presents a few such
results. However, this idea fails to accelerate the speed of the iterative
scheme in finding the iterative solution. In this regard, Climent and Perea
[J. Comput. Appl. Math. 158 (2003), 43-48: MR2013603] introduced the notion
of proper multisplittings to solve the system Ax = b on parallel and vector
machines, and established convergence theory for a subclass of proper
multisplittings. With the aim to extend the convergence theory of proper
multisplittings, this paper further adds a few results. Some of the results
obtained in this paper are even new for the iterative theory of nonsingular
linear systems.
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