Successive overrelaxation method

In many problems, the iterated solutions approach the exact solutions in a monotonic fashion. Therefore, it is useful in this case to speed up the convergence process by overrelaxing the iterated solutions. The equation for the overrelaxation scheme is modified from the Gauss-Seidel equation 

There is no fast rule on how to choose the optimum w for a given problem. It must be found from numerical experiments. 

In this section, we will consider briefiy the eigenproblems, that is, the study of the eigenvalues and eigenvectors. Ilie study of coupled linear differential equations presented in the next section requires the analysis of the eigenproblems. 

Let us consider this linear equation written in compact matrix notation 

The homogeneous form of the equation is simply Ax = 0, where 0 is the zero vector, that is, a vector with all zero elements. If all equations in Eq. B.77 are 

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The successive overrelaxation method is a variant of the Gauss-Seidel method, wherein the ( H-l)th iteration is a weighted average of the Ganss Seidel h and ( -tl)th estimates x, and xf , respectively. The reader is referred to Jensen and Jeffreys (1977) for a detailed account on the matrices and solution methods. 

There are three commonly used iterative methods which we will briefly present here. They are Jacobi, Gauss-Seidel and the successive overrelaxation methods. 

Theorem The successive overrelaxation method with optimum relaxation factor converges at least twice as fast as the Chebyshev semi-iterative method with respect to the Jacobi method, and therefore at least twice as fast as any semi-iterative method with respect to the Jacobi method. Furthermore, as the number of iterations tends to infinity, the successive overrelaxation method becomes exactly twice as fast as the Chebyshev semi-iterative method. 

A comparison of the successive overrelaxation method and semi-iterative... 

Figure 6 Error reduction rate of the successive overrelaxation method. The smaller slope of the curve for small values of the relative error indicates the poor performance of solvers based on this method.

For successive overrelaxation, we understand Eq. (26) to incorporate the use of o(k+1 values in place of o k) values in the convolution product as soon as they are formed for preceding x values. This adaptation can be explicitly displayed by the appropriate use of the Heaviside step function in a modified version of Eq. (26). The method of Van Cittert is a special case of simultaneous relaxation in which C = 1. 

There are many iterative methods (Jacobi, Gauss—Seidel, successive overrelaxations, conjugate gradients, conjugate directions, etc.) characterized by various choices of the matrix M. However, very often the most successful iterative processes result from physico-chemical considerations and, hence, corresponding subroutines cannot be found in normal computer libraries. 

Similarly to the two-dimensional case, the convergence was improved by the method of successive overrelaxation (8). Between one-half to one-tenth of the original computer time for the successive displacement method was required to obtain an accuracy of 0.005%. Nevertheless, as... 

The proof of this last statement uses only the non-negative irreducible and convergent nature of the matrix M. In order to sharpen this last result, as well as introduce the basis for the successive overrelaxation iterative method of Young and Frankel [52 12], we make the following definition. 

The quantity o> in (4.11) is called the relaxation factor. We observe that, for CO = 1, this iterative method reduces to the Gauss-Seidel iterative method of (4.8)-(4.8 ). For reasons of brevity, we shall say that a matrix G, which is cyclic of index 2, is consistently ordered [52] if it is the form of (4.10). With the concept of a consistent ordering. Young [52] established the following general relationship between the eigenvalues A of the successive overrelaxation matrix... 

In terms of actual arithmetic operations, the direct inversion of large submatrices combined with an iterative method tends to increase the amount of arithmetic operations per mesh point. This is, of course, to be balanced by an increase in the rate of convergence. In the case of the iterative method SLOR, either in two or three space dimensions, it fortunately can be shown [6] by suitable normalization of equations, that no additional arithmetic operations are required for the successive line overrelaxation method (SLOR) over what is required by the successive point overrelaxation method, while an improvement in rate of convergence is always obtained. 

As in the case of successive overrelaxation, the efficiency of the application of Chebyshev polynomials in accelerating the outer iterations depends upon the accurate estimation of the particular constant, 5, the dominance ratio for the matrix T. A practical numerical method for estimating <7 is given in [45]. 

Because of symmetry, it is known that the values 0.2883 and 0.2924 shonld be the same, but even after 15 iterations, they are relatively far apart. To improve this, there is the method of successive overrelaxation or the SOR method. The idea here is that on each sweep, the newly calculated value is not used directly instead an interpolation/extrapolation formula as shown by the following eqnation is nsed ... 

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