Accuracy iterative method

Adams Closed Formulas require an iterative method to solve for y, since the right side of the expression requires a value for f,. The iteration of estimating y, evaluating f, and obtaining a new estimate of y is repeated until it converges to the desired accuracy. The general formula is... 

In some cases, the rearrangement of matrix A for improving accuracy is undesirable or even impossible, since it increases the number of tearing variables. In such cases, an iterative method can be used to improve the accuracy of the solution. 

Note that the accuracy of calculation by the iterative method for the Hall properties of a 3D composite with chaotic structure depends on the accuracy of the calculation of the probability function Y(p) (probability that the point belongs to a connecting set) which describes the random structure of a composite). 

There are several reasons for observing differences between the computed results and experimental data. Errors arise from the modeling, discretization and simulation sub-tasks performed to produce numerical solutions. First, approximations are made formulating the governing differential equations. Secondly, approximations are made in the discretization process. Thirdly, the discretized non-linear equations are solved by iterative methods. Fourthly, the limiting machine accuracy and the approximate convergence criteria employed to stop the iterative process also introduce errors in the solution. The solution obtained in a numerical simulation is thus never exact. Hence, in order to validate the models, we have to rely on experimental data. The experimental data used for model validation is representing the reality, but the measurements... 

As the dimension of the blocks of the Hessian matrix increases, it becomes more efficient to solve for the wavefunction corrections using iterative methods instead of direct methods. The most useful of these methods require a series of matrix-vector products. Since a square matrix-vector product may be computed in 2N arithmetic operations (where N is the matrix dimension), an iterative solution that requires only a few of these products is more efficient than a direct solution (which requires approximately floating-point operations). The most stable of these methods expand the solution vector in a subspace of trial vectors. During each iteration of this procedure, the dimension of this subspace is increased until some measure of the error indicates that sufficient accuracy has been achieved. Such iterative methods for both linear equations and matrix eigenvalue equations have been discussed in the literature . 

To examine the accuracy of the technique, the ratio of numerical to physical phase velocity, unum/ js examined for a medium of er = 0.1 S/m and a uniform lattice with 2 = 0.1 m. Solving (5.3) for knnm via Newtons iterative method, the results for three different grid resolutions, Nk = A/At, and 9 = jt/2, are shown in Figure 5.1(a), where the efficacy of the higher order schemes is obvious. The next example takes into account a parallel-plate waveguide... 

Hence, an iterative algorithm is used to find the optimal solution. Assuming an initial trial value for x and y, Equations 5.56 and 5.57 are used to update the initial values. These values, in turn, are used to update d , and get the next set of values for x and y. The process is repeated until two successive values of x and y are very close, within a user-specified accuracy of e%. The most common initial values are those obtained by setting d = for all n = 1,2,..., N (assuming that the initial location is at the centroid). The origin (0,0) can also be used as the initial value. We shall now illustrate the iterative method with an example. 

A numerical method is said to be direct when it finds a solution within a given precision and, with a given accuracy, in an initially known number of operations. The time required to solve a differential equation is then well known a priori, and it is independent of the initial or boundary conditions of the problem. Iterative methods, on the other hand, are based on a sequence of approximations to the required solution, starting from an initial guess that converges to the solution. The number of operations, and the time required by these latter methods, are initially unknown because they depend on the initial guess and may vary dramatically as a function of the parameters of the problem. 

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