Fields of values and the ADI method for non-normal matrices
Generalizing the results of \textit{M. Eiermann} [Lin. Algebra Appl. 180, 167-197 (1993; Zbl 0784.65022)], the author obtains an upper bound for the error reduction of the ADI method in terms of the fields of values of the matrices which define splitting of the matrix in the iterations. Then it is shown that if \(A\) is an irreducible tridiagonal matrix and if \(B\) is the symmetric matrix similar to \(A\), then the field of values of \(B\) is included in that of \(A\), and the result is generalized to Kronecker sums of tridiagonal matrices. This result may be considered as an alternative to preconditioning for non-normal matrices. The estimate is tested on a model boundary value problem. Numerical results demonstrate the power of analysis based on the field of values for considered problems and indicate that the resulting estimates may be useful for an a priori prediction of the behaviour of the process in the finite stage of computation.
scientific article; zbMATH DE number 179139
scientific article