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The rate of convergence of a class of block Jacobi schemes. (English) Zbl 0631.65026

This paper determines the rate of convergence of a class of block Jacobi schemes, by determination of their spectrum, when these schemes are used to solve linear systems of the form \(Ax=b\) with \(A=trid(-I,T,-I),\) a tridiagonal square matrix of block order n. I denotes the identity matrix of order m and T is a symmetric positive definite matrix of order m whose eigenvalues are all greater than two. Such problems arise naturally when elliptic and parabolic partial differential equations whose coefficients do not depend on one of the spatial variables are discretized.
The author studies the case \(n=pq\) where \(p\geq 2\) and \(q\geq 2\) are integers. He gives the equations for determining the eigenvalues, and in particular the equation which determines the spectra value \(\rho\). The block Jacobi matrices considered here are consistently ordered 2-cyclic matrices, then the knowledge of \(\rho\) allows one to determine the optimal relaxation parameter \(\omega\) for block SOR methods naturally related to this block Jacobi scheme.
Reviewer: S.El Bernoussi

MSC:

65F10 Iterative numerical methods for linear systems
65F15 Numerical computation of eigenvalues and eigenvectors of matrices
15A42 Inequalities involving eigenvalues and eigenvectors
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