Interval-halving convergence

The method involves a simple iteration on only one variable, pH. Simple interval-halving convergence (see Chap. 4) can be used very effectively. The titration curves can be easily converted into simple functions to include in the computer program. For example, straight-line sections can be used to interpolate between data points. 

Successive linearisation has the advantage of relative simplicity and fast calculation. In addition, it can be modified to choose a step size that minimizes a prespecified penalty function. The step size is chosen by the method of interval halving (Pai and Fisher, 1988). However, variable bounds cannot be handled it may fail to converge to the desired minimum and it might oscillate when multiple minima exist. 

Clearly, the number of iterations to eonverge depends on how far the initial guess is from the correct value and the size of the initial step. Table 4.1 gives results for several initial guesses of temperature (TO) and several step sizes PTO). The interval-halving algorithm takes 10 to 20 iterations to converge to the eorrect temperature. 

Interval halving can also be used when more than one unknown must be found. For example, suppose there are two unknowns. Two interval-halving loops could be used, one inside the other. With a fixed value of the outside variable, the inside loop is converged first to find the inside variable. Then the outside variable is changed, and the inside loop is reeonverged. This procedure is repeated until both unknown variables are found that satisfy all the required equations. 

This convergence technique is a combination of Newton-Raphson and interval halving. An initial guess Tq is made, and the fimction , is evaluated. A step is taken in the correct direction to a new temperature T, and is evaluated. If... 

Compare convergence times, using interval halving, Newton-Raphson, and false position, for on ideal, four-component, vapor-liquid equilibrium system. The pure component vapor pressures are ... 

The interval-halving search method will alw ays converge since there is only one zero crossing of the charge balance if the pH is within the original search interval selected and if the arithmetic precision of the computer is sufficient for the pH system. 

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