Rate of convergence

Repeating the procedure, the method will converge to the exact solution Xi = 0.5 and X2 — 0.5.  

A weakness of Newton s method is the analytical evaluation of the Jacobian matrix. In many practical situations this is somewhat inconvenient and often tedious. This can be overcome by using finite difference to approximate the partial derivative that is. 

We have presented several techniques to handle nonlinear algebraic equations, but nothing has been said about the rate of convergence that is, how fast the iteration process yields a convergent solution. We discuss this point in the next section. 

Let be a sequence that converges to a and thereby define the error as e = a — a for each n 0. If positive constants A and /3 exist with 

A sequence with a large order of convergence will converge more rapidly than that with low order. The asymptotic constant A will affect the speed of convergence, but it is not as important as the order. However, for first order methods, A becomes quite important. 

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As the feed composition approaches a plait point, the rate of convergence of the calculation procedure is markedly reduced. Typically, 10 to 20 iterations are required, as shown in Cases 2 and 6 for ternary type-I systems. Very near a plait point, convergence can be extremely slow, requiring 50 iterations or more. ELIPS checks for these situations, terminates without a solution, and returns an error flag (ERR=7) to avoid unwarranted computational effort. This is not a significant disadvantage since liquid-liquid separations are not intentionally conducted near plait points. 

The perfomiance of the penalty fiinction algoritlun is heavily influenced by the value chosen for a.. The larger the value of o. the better the constraints are satisfied but the slower the rate of convergence. Optimizations with very high values of a, encounter severe convergence problems. However, the method is very general and... 

Order 2 minimization algorithms, which use the second derivative (curvamre) as well as the first derivative (slope) of the potential function, exhibit in many cases improved rate of convergence. For a molecule of N atoms these methods require calculating the 3N X 3N Hessian matrix of second derivatives (for the coordinate set at step k)... 

Figure 1 The course of energy minimization of a DNA duplex with different choices of coordinates. The rate of convergence is monitored by the decrease of the RMSD from the final local minimum structure, which was very similar in all three cases, with the number of gradient calls. The RMSD was normalized by its initial value. CC, IC, and SG stand for Cartesian coordinates, 3N internal coordinates, and standard geometry, respectively.

Thus, provided the rate of convergence is not less than that for L2 jDt — n, all terms other than the first in the series may be neglected. Equation 10.139 will converge more rapidly than this for L2/Dt > n, and equation 10.141 will converge more rapidly for L2/Dt < n. 

With these relations established, we conclude that if the scheme is stable and approximates the original problem, then it is convergent. In other words, convergence follows from approximation and stability and the order of accuracy and the rate of convergence are connected with the order of approximation. 

So far we have established an estimate for the rate of convergence in a very simple problem. It is possible to obtain a similar result for this problem by means of several other methods that might be even much more simpler. However, the indisputable merit of the well-developed method of energy inequalities is its universal applicability it can be translated without essential changes to the multidimensional case, the case of variable coefficients, difference schemes for parabolic and hyperbolic equations and other situations. 

On convergence and accuracy. The results obtained in the preceding two sections may be of help in establishing the rate of convergence for scheme ... 

The accurate account of the error z can be done as in Section 4, leading to the same rate of convergence. No progress is achieved for a = a in line with approved rules, because the choice of the coefficient should not cause the emergence of a higher-order accuracy. From the formula = 9 Q+0 h) it is easily seen that i]n+i — 0(h) and, hence, z = 0(h +T" ) if = 0, meaning that the heat source is located at one of the nodal points. 

The rate of convergence of iterations can be evaluated by means of the difference... 

The rate of convergence of expansions in the basis (1.2) has received little attention except for purely numerical studies [3,7,8,9,16] which indicated that the convergence is at least (unlike for bais set of type) not frustratingly slow. Rather detailed studies were performed for the even-tempered basis set, i.e. for exponents constructed from two parameters and /di (for each /)... 

There are hints [9,10,18] that the rate of convergence for basis sets of type (1.2) is even better than (1.4), if one uses better optimized basis sets than those of even tempered type (1.3),... 

For quantum chemistry the expansion of e in a Gaussian basis is, of course, much more important than that of 1/r. The formalism is a little more lengthy than for 1/r, but the essential steps of the derivation are the same. For an even-tempered basis one has a cut-off error exp(—n/i) and a discretization error exp(-7//i), such that results of the type (2.15) and (2.16) result. Of course, e is not well represented for r very small and r very large. This is even more so for 1/r, but this wrong behaviour has practically no effect on the rate of convergence of a matrix representation of the Hamiltonian. This is very different for basis set of type (1.1). Details will be published elsewhere. 

The relative rate of convergence, pre/> for a given synthesis is given by the ratio of slopes of the lines R2P and R2Pmcr... 

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