Convergence rate order

On the subject of comparing iterative methods a word of caution is in order. Clearly in any quantitative comparison, the termination criteria should be comparable and the benchmark problems should be run on the same computer. Yet even for simple problems and methods, these two requirements prove to be difficult to enforce and insufficient to ensure meaningful comparisons. To allow for the fact that different methods do not terminate at exactly the same point even when the same termination criterion is used, Broyden (B13) introduced a mean convergence rate, R, which is... 

Therefore the method has excellent convergence properties near the root (with order of convergence p = 2), but may result in meaningless estimates otherwise. In addition, the number of equivalent function evaluations is usually larger than in the secant method, which does not require the derivative but has almost the same convergence rate. Neither the Newton-Raphson, nor the secant method are recommended if the function f has an extremum near the root. You can easily construct pathological cases to understand this rule. 

Suppose you want to solve the equation x = e z on a pocket calculator. Suggest a simple method, and verify that it works by working out its asymptotic convergence properties What is the convergence order If this is a first-order procedure, what is the convergence rate A d= lim 001j Show that the NR method, applied as f(x) = x - e x = 0, takes the form... 

The choice of optimization scheme in practical applications is usually made by considering the convergence rate versus the time needed for one iteration. It seems today that the best convergence is achieved using a properly implemented Newton-Raphson procedure, at least towards the end of the calculation. One full iteration is, on the other hand, more time-consuming in second order methods, than it is in more approximative schemes. It is therefore not easy to make the appropriate choice of optimization method, and different research groups have different opinions on the optimal choice. We shall discuss some of the more commonly implemented methods later. 

A method similar to the iterative, is the partial closure method [37], It was formulated originally as an approximated extrapolation of the iterative method at infinite number of iterations. A subsequent more general formulation has shown that it is equivalent to use a truncated Taylor expansion with respect to the nondiagonal part of T instead of T-1 in the inversion method. An interpolation of two sets of charges obtained at two consecutive levels of truncations (e.g. to the third and fourth order) accelerates the convergence rate of the power series [38], This method is no longer in use, because it has shown serious numerical problems with CPCM and IEFPCM. 

The main disadvantages of FVMs are low accuracy and low convergency rates. Compared to FDMs, the main disadvantage of the FVM is that methods of order higher than second are more difficult to develop in 3D because the FVM approach requires two levels of approximation considering the interpolation and integration processes. The FDM only requires approximations of the derivatives and interpolation. 

The order of accuracy of the upwind scheme can be improved by using a higher-order accurate scheme such as QUICK (quadratic upwind interpolation for convective kinematics).The concentration at an interface is interpolated by means of a parabola instead of a straight line. The use of QUICK or similar methods may, however, complicate implementation of boundary conditions or lessen the convergence rate of the solution algorithm. 

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