Method Seidel

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]. 

The Gauss-Seidel Iterative Method. The Gauss-Seidel iterative method uses substitution in a way that is well suited to maehine eomputation and is quite easy to eode. One guesses a solution for xi in Eqs. (2-44)... 

The purpose of this projeet is to gain familiarity with the strengths and limitations of the Gauss-Seidel iterative method (program QGSEID) of solving simultaneous equations. 

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. 

The choice ya = ra is the method of steepest descent. If the ya are taken to be the vectors et in rotation the method turns out to be the Gauss-Seidel iteration. If each ya is taken to be that e, for which e ra is greatest, the method is the method of relaxation (often attributed to Southwell but actually known to Gauss). An alternative choice is the et for which the reduction Eq. (2-10) in norm is greatest. 

Seidel method. As we have mentioned above, implicit schemes are rather stable in comparison with explicit ones. Seidel method, being the simplest implicit iterative one, is considered first. The object of investigation here is the system of linear algebraic equations... 

Within these notations, Seidel method can be written as follows ... 

On account of the basic theorem proved in Section 1 of the present chapter Seidel method converges if the operator A is self-adjoint and positive. More specifically, the sufficient stability condition (11) for the convergence of iterations in scheme (3 ) with a non-self-adjoint operator B takes the form... 

Within the framework of Seidel method we thus have B = A D, r = and... 

The upper relaxation method. In order to accelerate the iteration process in view, we are forced to revise Seidel method by inserting in (5) the iteration parameter u> so that... 

This method falls within the category of relaxation methods and gives rise to Seidel method in one particular case where w = 1. In the modern literature the iteration process (9) with w > 1 is known as the upper relaxation method. 

As shown above, Seidel method is quite applicable for any operator A = A >0. However, the extra restriction 0 < w < 2 is necessary for the convergence of the upper relaxation method. This is certainly true under condition (8) with a known operator Bq. Along these lines, it is straightforward to verify that B = u> A + ), r, = w and... 

As a matter of fact, the upper relaxation method and Seidel method are nothing more than the implicit scheme (6) with B E incorporated. Still using the usual framework of implicit iterative methods, the value yk+i is determined from the equation... 

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. 

O. Lisec, P. Hugo and A. Seidel-Morgenstein, Frontal analysis method to determine competitive adsorption isotherms. J. Chromatogr.A 908 (2001) 19-34. 

Seidel, D., Alaupovic, P., and Furman, R. H., A lipoprotein characterizing obstructive jaundice. I. Method for quantitative separation and identification of lipoproteins in jaundiced subjects. J. Clin, hwest. 48, 1211-1223 (1969). 

The detailed 3D model of porous catalyst is solved in pseudo-steady state. A large set of non-linear algebraic equations is obtained after equidistant discretization of spatial derivatives. This set can be solved by the Gauss-Seidel iteration method (cf. Koci et al., 2007a). 

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