INCOMPLETE SEMI-ITERATIVE METHODS FOR SOLVING SINGULAR LINEAR OPERATOR EQUATIONS IN BANACH SPACE WITH APPLICATIONS IN MARKOV CHAIN MODELING

1　 IntroductionLet Cbe the open complex plane,let X be a complex Banach space.The set of allbounded linear operators from X into X is denoted by BX] which is also a Banach space.If X=Cn,the n-dimensional Euclidean space,then BX] is the set of all n×n matrices,denoted by Cn,n. We denote the spectrum of an operator T∈ BX] byσ( T) and its resol-vent operator R( λ,T) =( λI-T) - 1 ,where I is the identity operator andλ∈C.The spectral radius of T is denoted by r( T) .N( T) and…

INCOMPLETE SEMI-ITERATIVE METHODS FOR SOLVING SINGULAR LINEAR OPERATOR EQUATIONS IN BANACH SPACE WITH APPLICATIONS IN MARKOV CHAIN MODELING

We discuss the incomplete semi-iterative method (ISIM) for an approximate solution of a linear fixed point equations x＝ Tx+c with a bounded linear operator T acting on a complex Banach space X such that its resolvent has a pole of order k at the point 1. Sufficient conditions for the convergence of ISIM to a solution of x=Tx+c , where c belongs to the range space of (I-T)k, are established. We show that the ISIM has an attractive feature that it is usually convergent even when the spectral radius of the operator T is greater than 1 and Ind1T≥ 1. Applications in finite Markov chain is considered and illustrative examples are reported, showing the convergence rate of the ISIM is very high.