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

Summary

For a square matrixT∈ℂ^n,n, where (I−T) is possibly singular, we investigate the solution of the linear fixed point problemx=T x+c by applying semiiterative methods (SIM's) to the basic iterationx ₀∈ℂⁿ,x _k ≔T c _k−1+c(k≧1). Such problems arise if one splits the coefficient matrix of a linear systemA x=b of algebraic equations according toA=M−N (M nonsingular) which leads tox=M ⁻¹ N x+M ⁻¹ b≕T x+c. Even ifx=T x+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=T x+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.