Computational methods for generalized inverse matrices arising from proper splittings
Let \(A\) be a real rectangular matrix. \(A = M-N\) is a proper splitting of \(A\) if \(A\) and \(M\) have the same ranges and null spaces. If \(M^-\) is a g-inverse of \(M\) then the iteration process \(x^{(k+1)}=M^-Nx^{(k)} +M^- b\) converges to \(A^-b\) for every choice of \(x^{(0)}\) if and only if the spectral radius of \(M^-N\) is less than 1. \(A^-\) will be the same kind of g-inverse of \(A\) as \(M^-\) is of \(M\). For a positive cone \(K\) in \(\mathbb R^n\), \(\rho(M^-N)<1\) if and only if \(A^-NK\subseteq K\), where \(A^-\) is the g-inverse of \(A\) given by \(A^- = (I -M^-N)^{-1}M^-\). Stronger results are obtained when \(M^-\) is the minimum norm g-inverse of \(M\). For two proper splittings \(A=M_1-N_1=M_2 - N_2\) such that \(A^-=(I -M_1^-N_1)^{-1}M_1^-= (I -M_2^-N_2)^{-1}M_2^-\) and \(0\leq N_1\leq N_2\), if \(A^- >0\), \(M_i^-N_i \geq 0\) and \(M_i^-\geq 0\) for \(i=1,2\) then \(0 < \rho(M_1^-N_1) < \rho(M_2^-N_2) <1\) from which it follows that the above iteration using the proper splitting \(A =M_2 - N_2\) converges more rapidly than the iteration based on \(A =M_1 - N_1\). Some numerical examples are provided.
scientific article; zbMATH DE number 3523757