Convergence iteration schemes

Successive Substitutions Let/(x) = 0 be the nonlinear equation to be solved. If this is rewritten as x = F x), then an iterative scheme can be set up in the form Xi + = F xi). To start the iteration an initial guess must be obtained graphically or otherwise. The convergence or divergence of the procedure depends upon the method of writings = F x), of which there will usually be several forms. However, if 7 is a root of/(x) = 0, and if IF ( 7)I < I, then for any initial approximation sufficiently close to a, the method converges to a. This process is called first order because the error in xi + is proportional to the first power of the error in xi for large k. 

The equation in cell B1 is copied into cells Cl though El. Then turn on the iteration scheme in the spreadsheet and watch the solution converge. Whether or not convergence is achieved can depend on how you write the equations, so some experimentation may be necessary. Theorems for convergence of the successive substitution method are useful in this regard. 

A stationary scheme. The main theorem on the convergence of iterations. Quite often, the iteration schemes such as... 

Implicit iteration schemes. Convergence of implicit iteration schemes was the subject of investigation in Section 2 for the special case... 

In this appendix we follow the treatment of Maday and Turinici [94], and show that the iterative scheme laid out in Eqs. (21.a-d) is guaranteed to converge. The convergence of the algorithm can be proved by evaluating the difference between the values of the objective functional between two successive iterations. Suppose that 5 0 and t 0, then,... 

It is straightforward to solve the hierarchy of equations iteratively. However, the simplest iteration schemes converge only linearly (i.e., poorly). One should better consider quadratic iteration schemes. 

Having replaced (1 + a)Su by (1 + Q )5j/i(. and au by ayk y, we try to adapt the explicit scheme, the parameter a of which needs to be selected by the approved rule in a minimal number of iterations. Unfortunately, more a detailed exploration on this point and the convergence of scheme (3) are not available in the present book. A final result can be obtained through such an analysis by utilizing the fact that the residual rk = A yk — f satisfies the homogeneous equations... 

Similar iterative schemes were used to determine the MO s for multiconfigurational wave functions, in the early implementations. Fock-like operators were constructed and diagonalized iteratively. The convergence problems with these methods are, however, even more severe in the MCSCF case, and modem methods are not based on this approach. The electronic energy is instead considered to be a function of the variational parameters of the wave function - the Cl coefficients and the molecular orbital coefficients. Second order (or approximate second order) iterative methods are then used to find a stationary point on the energy surface. 

The parameters a are obtained by inserting (4 27) into (4 24) and project onto each of the The process normally converges in less than ten iterations, although, as in all iterative schemes of this kind, convergence can be slowed down considerably in near singular situations (small eigenvalues to the Hessian... 

Neenan and Miller [118] reported the preparation of poly (arylester) dendrons using a convergent iterative sequence consisting of esterification and hydrolysis reactions as illustrated in Scheme 14. The key intermediate in the synthesis of these dendrons is 5-(te/V-butyldimethylsiloxy)isophthaloyl dichloride, 48, which is first converted to the diester with phenol, followed by hydrolysis to the diester phenol. This dendron is designated [3-OH] indicating it contains three phenyl... 

Overbeek s work was first extended by Wlersema et al. l. These results were an important step forward but still had some limitations because of an Iterative scheme, which failed to converge at high Consequently, applicability was kept down to about f 150 mV and f g 25 mV for (1-1) or (2-2) electrolytes, respectively. 

The important result of this chapter is that all iterative schemes, based on regularized minimal residual methods, always converge for any linear inverse problem The proofs of the corresponding convergence theorems are similar to Theorem 21. 

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