On the solution of singular linear systems of algebraic equations by semiiterative methods

SummaryFor a square matrixT∈ℂn,n, where (I−T) is possibly singular, we investigate the solution of the linear fixed point problemx=Tx+c by applying semiiterative methods (SIM's) to the basic iterationx0∈ℂn,xk≔Tck−1+c(k≧1). Such problems arise if one splits the coefficient matrix of a linear systemAx=b of algebraic equations according toA=M−N (M nonsingular) which leads tox=M−1Nx+M−1b≕Tx+c. Even ifx=Tx+c is consistent there are cases where the basic iteration fails to converge, namely ifT possesses eigenvalues λ≠1 with |λ|≧1, or if λ=1 is an eigenvalue ofT with nonlinear elementary divisors. In these cases — and also ifx=Tx+c is incompatible — we derive necessary and sufficient conditions implying that a SIM tends to a vector $$\hat x$$ which can be described in terms of the Drazin inverse of (I−T). We further give conditions under which $$\hat x$$ is a solution or a least squares solution of (I−T)x=c.