Projected iterative algorithms for complex symmetric systems arising in magnetized multicomponent transport
The problem is to solve a constrained linear system \({\mathcal G}a=b\), \(a\in{\mathcal C}\) with \({\mathcal G}=G+iG'\) a \(n\times n\) complex matrix, symmetric (not hermitian!) with \(G\) positive semi-definite, and a nullspace \(N(G)\) satisfying \(G'N(G)=0\). The linear space \({\mathcal C}=C+iC\) is a complexification of \(C\) with \(C\oplus N(G)={\mathbb R}^n\). The problem has been studied before in the case \(G'=0\) [cf. \textit{A. Ern} and \textit{V. Giovangigli}, Linear Algebra Appl. 250, 289--315 (1997; Zbl 0867.65017)]. The natural solution of the problem involves a generalized inverse with prescribed range and nullspace. First it is shown that this generalized inverse can be chosen to be symmetric. The symmetry and positive definiteness allows to use a complex Cholesky factorization method. Next it is shown that the convergence of a projected stationary iterative method for an appropriate splitting of the matrix \({\mathcal G}\) is similar to the case \(G'=0\). On the other hand, the (preconditioned) orthogonal projected residual method, makes use of the symmetry and positivity properties of the real part, whence it has better convergence properties in general. Applications in magnetized multicomponent transport are included.
scientific article; zbMATH DE number 5505637
scientific article