A method for the improvement of the convergence rates of stationary iterative schemes for the solution of complex linear systems
This paper presents some theorems and lemmas which may prove useful in improving the convergence rates of stationary iterative schemes for complex linear systems. The coefficient matrix of the system \(Ax=b\) is written in the form (k-part splitting) \(A=A_ 0-A_ 1-...-A_ k\) leading to the stationary iterative scheme (1) \(A_ 0x_{n+k}=A_ 1x_{n+k-1}+...+A_ kx_ n+b\). \textit{J. de Pillis} and \textit{M. Neumann} [IMA J. Numer. Anal. 1, 65-79 (1981; Zbl 0476.65017)] have shown how the asymptotic behaviour of (1) may be obtained from that of the equivalent first-order scheme \(y_{n+1}=Wy_ n+C\) with approximate choice of W. The present authors use a modification of the power method to derive a polynomial whose roots, are, asymptotically, the m eigenvalues of W greatest in modulus. A discussion of implementation of this result is given and an ad hoc `dynamic process' to approximate the solution of the equations from current estimates of the eigenvalues which makes substantial savings on the number of iterations for slowly convergent problems is described. The paper concludes with some numerical applications based on a Dirichlet boundary value problem.
scientific article; zbMATH DE number 4013863
scientific article