Successive approximation programs

A method of successive approximation (s.a. method) is one that would provide the solution as the limit of an infinite sequence of steps if these steps were carried out exactly. These steps are usually quite simple, and nearly identical, each to the next, so that programming is a relatively easy task. Most methods operate upon a given approximation to obtain a better one, hence they are self-correcting. Because of rounding, at some stage the computed correction will no longer be... 

Because of the convenient mathematical characteristics of the x -value (it is additive), it is also used to monitor the fit of a model to experimental data in this application the fitted model Y - ABS(/(x,. ..)) replaces the expected probability increment ACP (see Eq. 1.7) and the measured value y, replaces the observed frequency. Comparisons are only carried out between successive iterations of the optimization routine (e.g. a simplex-program), so that critical X -values need not be used. For example, a mixed logarithmic/exponential function Y=Al LOG(A2 + EXP(X - A3)) is to be fitted to the data tabulated below do the proposed sets of coefficients improve the fit The conclusion is that the new coefficients are indeed better. The y-column shows the values actually measured, while the T-columns give the model estimates for the coefficients A1,A2, and A3. The x -columns are calculated as (y- Y) h- Y. The fact that the sums over these terms, 4.783,2.616, and 0.307 decrease for successive approximations means that the coefficient set 6.499... yields a better approximation than either the initial or the first proposed set. If the x sum, e.g., 0.307,... 

Successive linear programming (SLP) methods solve a sequence of linear programming approximations to a nonlinear programming problem. Recall that if g,(x) is a nonlinear function and x° is the initial value for x, then the first two terms in the Taylor series expansion of gt(x) around x° are... 

Successive quadratic programming (SQP) methods solve a sequence of quadratic programming approximations to a nonlinear programming problem. Quadratic programs (QPs) have a quadratic objective function and linear constraints, and there exist efficient procedures for solving them see Section 8.3. As in SLP, the linear constraints are linearizations of the actual constraints about the selected point. The objective is a quadratic approximation to the Lagrangian function, and the algorithm is simply Newton s method applied to the KTC of the problem. 

This constrained nonlinear optimisation problem can be solved using a Successive Quadratic Programming (SQP) algorithm. In the SQP, at each iteration of optimisation a quadratic program (QP) is formed by using a local quadratic approximation to the objective function and a linear approximation to the nonlinear constraints. The resulting QP problem is solved to determine the search direction and with this direction, the next step length of the decision variable is specified. See Chen (1988) for further details. 

Hennig86 was written by James S. Farris (American Museum of Natural History, New York, NY 10024). It is a fast and effective parsimony program. It is often faster than PAUP but has many fewer features and options. However, Hennig86 does contain a routine for successive approximation a posteriori character weighting. 

A problem common to all iterative computational methods is knowing when to quit. An iterative method rarely yields a precise root, but rather gives successive approximations that (if the method converges) approach the root more and more closely. Whether you are doing the calculation by hand or writing a program to do it. you must specify how close is close enough. 

The computational effort in evaluating the Hessian matrix is significant, and quasi-Newton approximations have been used to reduce this effort. The Wilson-Han-Powell method is an enhancement to successive quadratic programming where the Hessian matrix, (q. ), is replaced by a quasi-Newton update formula such as the BEGS algorithm. Consequently, only first partial derivative information is required, and this is obtained from finite difference approximations of the Lagrangian function. 

The first term in the equation is the volume of a drop on a plane surface the remaining terms are corrections due to roughness. Since Equation 41 cannot be solved for either cj) or p explicitly, the computer is programmed to find them by successive approximations. For most of our calculations, we are satisfied when p and cf) are foimd which give the chosen volume to 1 part in 10 is then computed from Equa-... 

Hennig UG, Chatten LG, Moskalyk RE and Ediss C, Benzodiiadiazine dissociation constants. Part 1. Ultraviolet spectrophotometric pKa determinations. Analyst, 106, 557-564 (1981). NB Thermodynamic values for the dissociation constants of a number of benzothiadiazines, including chlorothiazide, mediyclodiiazide, polythiazide and diazoxide, were determined by UV spectrophotometry. Resolution of tfie overlapping acidity constants was achieved by employing a computer program of successive approximations. 

Using this iteratively from a known pose and updating the obtained position by the forward conversion, /r, a successive approximation of the exact path to a programmed point can be obtained. It assumed that the initial pose is... 

Farris (1990) pointed out that his program Hennig-86 uses an updated successive approximations approach which includes a corrected homoplasy index. The original index was shown to be biased by the number of taxa and number of characters used in a study (Archie, 1989a, Farris 1989). The revised homoplasy measure used, called the "retention index", is the same as Archie s "homoplasy excess ratio maximum" (Archie 1990, Farris 1990), which is an estimator of the "homoplasy excess ratio" used in his test of character state randomness (Archie 1989ab). The consistency index for individual characters can be rescaled to vary between zero and one by multiplying it by the retention index and this value can be used for character weighting (Farris 1990, Archie 1990). 

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