ON DOMAINS OF SUPERIOR CONVERGENCE OF THE SSOR METHOD TO THAT OF THE SOR METHOD

Let nu denote the spectral radius of J(B)A, the block Jacobi iteration matrix. For the classes of (1) nonsingular M-matrices and (2) p-cycli c, p greater-than-or-equal-to 3, consistently ordered matrices, we stu dy domains in the (nu, omega) plane when nu < 1, where the block SSOR iteration method has at least as favorable an asymptotic rate of conve rgence as the block SOR method. Let L(omega)A and T(omega)A denote, re spectively, the block SOR and SSOR iteration matrices. For the class o f nonsingular M-matrices A, we determine conditions when the spectral radii satisfy rho(T(omega)A)) less-than-or-equal-to rho(L(omega)A) for -all 0 < omega less-than-or-equal-to 2/1 + nu and for-all 0 less-than- or-equal-to nu < 1. Under these conditions we also show that the optim al SOR iteration parameter is omega(b) = 1. For the class of p-cyclic, p greater-than-or-equal-to 3, consistently ordered matrices A we dete rmine for which omega's and nu's. rho(T(omega)A) < \omega - 1\ [less-t han-or-equal-to rho(L(omega)A)]. Our investigations make use of the eq uality case in Wielandt's inequality between the spectral radii of a c omplex matrix and its nonnegative and irreducible majorizers and of Ro uche's theorem for the location of zeros of complex functions.