Convergence Theory of Iterative Methods based on Proper Splittings and Proper Multisplittings for Least Squares Problems

Abstract

Multisplitting methods are useful to solve differential algebraic equations. In this connection,
we discuss 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.