Over- and underrelaxation for linear systems with weakly cyclic jacobi matrices of index p

Abstract

D. Young's results from 1954 concerning the application of the successive-overrelaxation (SOR) method to linear systems Ax = b with matrices possessing "property A" were generalized by R.S. Varga in 1959 to systems with Jacobi matrices JA which are weakly cyclic of index p; if the eigenvalues of JpA are nonnegative, then for the optimal parameter ωopt there holds 1 ≤ ωopt < 1 + 1 (p - 1). Here a different proof of Varga's results is given and it is shown that 1 - 1 (p - 1) < ωopt ≤ 1 holds if the eigenvalues of JpA are nonpositive. Further exact intervals for those parameters ω which yield convergence of the SOR method are given in terms of Chebyshev polynomials. © 1987.