Gauss-Seidel method

The equation system of eq.(6) can be used to find the input signal (for example a crack) corresponding to a measured output and a known impulse response of a system as well. This way gives a possibility to solve different inverse problems of the non-destructive eddy-current testing. Further developments will be shown the solving of eq.(6) by special numerical operations, like Gauss-Seidel-Method [4]. 

If P = I, this is the Gauss-Seidel method. If > I, it is overrelaxation if P < I it is underrelaxation. The value of may be chosen empirically, 0 < P < 2, but it can be selected theoretically tor simple problems hke this (Refs. 106 and 221). In particular, these equations can be programmed in a spreadsheet and solved using the iteration feature, provided the boundaries are all rectangular. 

An iterative solution method for linear algebraic systems which damps the shortwave components of the iteration error very fast and, after a few iterations, leaves predominantly long-wave components. The Gauss-Seidel method [85] could be chosen as a suitable solver in this context. 

Note that the first two terms on the right-hand side of the preceding equation are from iteration n and the last two terms are from the current iteration. The Gauss-Seidel method... 

The Gauss-Seidel method is one which we shall discuss later. An old method suitable for hand calculations with a small number of nodes is called the relaxation method. In this technique the nodal equation is set equal to some residual q , and the following calculation procedure followed ... 

The Gauss-Seidel method can be accelerated by calculating the next approximation and then deliberately overshooting it. Chebyshev acceleration (polynomial extrapolation) may also be used to improve the rate of convergence. 

The iteration equation for the Gauss-Seidel method is obtained employing the last available values within the iteration process ... 

The convergence rate of the Jacobi and Gauss-Seidel methods depends on the properties of the iteration matrix. By experience, it has been found that these methods can be improved by the introduction of a relaxation parameter a. Consider the Gauss-Seidel method, it can be rewritten as ... 

When 0=1, the original Gauss-Seidel method is recovered. Other values of the parameter a yields different iterative sequences. If 0 < a < 1 then the procedure is an under-relaxation method, else with a > 1 we have obtained an approach that is called the successive over-relaxation (SOR) technique. 

Set up the material balance equations for each of the four reactors, and use the Gauss-Seidel method to determine the exit concentration from each reactor. 

SOLUTION OF SIMULTANEOUS LINEAR EQUATIONS BY GAUSS-SEIDEL METHOD ... 

Iterative methods are sometimes used due to ease of computer coding and lesser computational storage requirements. The Jacobi method is the simplest iterative method but has slower convergence in comparison with the Gauss-Seidel method. In the Gauss-Seidel method, the (A -tl)th iteration of the value of the unknown x, is given by... 

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. 

When the conditions given by Eqs. (15-24) and (15-25) are satisfied, convergence of the Gauss-Seidel method can be assured see Prob. 15-7. 

Cramer s rule is usually sufficient for solving two equations in two unknowns or three equations in three unknowns. However, for larger sets of equations, other solution procedures are preferred, such as Gauss-Jordan reduction and the Gauss-Seidel method. But in most cases, the best method is LU decomposition, in which the coeffi-... 

Like the Jacobi method, the Gauss-Seidel method requires diagonal dominance for the convergence of iterated solutions. 

A class of procedures that are not as sensitive to round-off errors is the so-called iterative procedures. One such procedure is the Gauss—Seidel method, briefly described below [3,5,9,14]. 

In procedures such as the Jacobi and Gauss-Seidel methods, a residual vector r... 

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