On the use of incomplete semiiterative methods for singular systems and applications in Markov chain modeling

摘要

There are several methods for solving linear fixed point problem x=Tx+c, where is a square matrix and I−T is possibly singular. Such problems arise if one splits the coefficient matrix of a singular system Ax=b of algebraic equations according to A=M−N (M nonsingular) which leads to x=M−1Nx+M−1b=:Tx+c. The basic iteration x0∈CN, requires the modulus of every eigenvalue of the iteration matrix T except 1 is less than 1 and q=index(I−T), the index of I−T is equal to 1 for convergence. In this paper, we try to use the incomplete semiiterative methods (ISIM) to solve x=Tx+c when . Usually the special semiiterative methods are convergent even when the spectral radius of the iteration matrix is greater than 1 and q⩾1. Then the use of the ISIM in the Markov chain modeling is considered. Finally, numerical examples are reported.