Method of interval halving

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. 

Discuss the methods of interval halving, successive substitution, and New-ton-Raphson for solving nonlinear algebraic equations. What are their relative advantages and disadvantages ... 

The method of interval halving is a two-step, first-order method. Before it is applied, two approximations must be known, such that x e... 

Calculation of pH titn. curves and end-points. Iterative method with interval halving... 

The golden section search guarantees that each new function evaluation will reduce the uncertainty interval to a length of >. times the previous interval. This is comparable to, but not as good as interval halving in the bisection method of solving a nonlinear equation. You can easily calculate that to attain an error tolerance EP we need... 

We know Qcol, U, Acoil, TR and TCm. Combining the two equations above gives one equation in one unknown, the temperature of the coolant leaving the coil Tc,out. However, the log term precludes an analytic solution, so an iterative interval halving solution method is used. 

It is more advantageous to apply methods based on minimalisation of the Gibbs function. In this case we may proceed as follows We determine the equilibrium composition, e.g. by means of the method of Lagrangian multipliers for several temperature values in the vicinity of the expected Tg value. To do this, we must know the dependence of c,- = G]jRT + In P values on temperature. At every temperature, for which we have calculated the equilibrium composition, we determine the values of AH = HE D START- Let AH < 0 apply for the temperature and AH > 0 for the temperature The required temperature Tg will then lie in the interval (Te Furthermore we can apply e.g. the interval halving method or the regula falsi method (see Appendix 3). The c = Cf(T) i = 1, 2,. ..,iV relationship is determined as follows values of — (G — Hp)IT, are tabulated in the literature for various values of T. The standard temperature is usually OK or 298.15 K. The quantity is independent of temperature, and polynomial development to at most the third or fourth degree will usually suffice to elucidate the value of —(Gy — Hy)/T. 

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. 

A straightforward derivative-free method is that of halving the interval of uncertainty. Consider a monotonic function /(x), shown in Figure 2.10, which is continuous from x = a to x = 6 where the values /(o) and f b) have opposite signs. For the equation... 

If it is (somehow) known that a root lies in the interval [a, b], then by simply halving the interval in which the root lies, the interval can be reduced to an acceptable level. This idea is at the heart of the bisection method as shown in Figure 1.2. 

---