A Method for Finding the Optimum Successive Over-Relaxation Parameter

Abstract

A proof is given here of the well-known relation between the eigenvalues of the Jacobi and S.O.R. iteration matrices in the case having Property A and consistent ordering. This proof also yields a relationship between the corresponding eigenvectors, and we use this relation to form a method of obtaining an approximation to the optimum relaxation parameter. Analytical results We consider here the solution by successive over-relaxation of the set of linear equations and suppose that Ax = b = D-L-U, 0) (2) where D is diagonal, L is strictly lower-triangular and U is strictly upper-triangular. We will assume that the matrix A possesses Property A and is consistently ordered, that is that for any positive scalar p there exists a diagonal matrix G p such that p U2 L (3) This is equivalent to the definitions given by Young (1954). He required the existence of an ordering vector (q { , q 2 ,.. . q n) with integer coefficients sucti that if the elements of A are a, 7 and if a 0-^ 0 and i ^ j then either q t = q.-f 1 and i > / o r q, = q s-1 and i U)y s = A,(£>-(5) and co(U+ \L)y, = (A,. + co-\)Dy h (6) which may be written as LO\)I 2 {\TV 2 U + X) l2 L)yi = (A,-+ co-l)Dy, (7) provided A,-=/ = 0. Using (3) and rearranging, we find D~ \L (8) provided co # 0 (and the case co = 0 is of no interest to us). Now D~ \L + U) is the Jacobi iteration matrix and (8) shows that it has G^V; a s a n eigenvector. If the corresponding eigenvalue is fi t we find the well-known relation (A, (9) It also holds for A,-= 0 since in this case we find from (5) that that is det{(l-co)D + coU} = 0 (1-co)" II d-, = 0 (10) (11) if the elements of D are d,. Now the iteration is not possible unless each d-t is non-zero. It follows that a) = 1 and (9) is still valid. Practical application for symmetric, positive-definite matrices In the case where A is symmetric, it is a well-known deduction from equation (9) that the spectral radius of M.,s is minimized if co is chosen as "opt 1 + (1-(12) = max This is shown by Varga (1962), where / for example. This is very satisfactory as it stands if a good a priori estimate for fx is available, but otherwise we need an algorithm that finds it without increasing unduly the total amount of work. Carre (1961) and Kulsrud (1961) each describe useful techniques based on examination of the displacement vectors S (l) which satisfy the relation = j|f gw. (13) Both rely on the use of a relaxation factor slightly less than co op , to ensure that the dominant A, corresponds to the dominant /x ;. Carre iterates a few times with parameter co k suggesting twelve times as suitable. He then makes an estimate, v k , of the dominant latent root of 200