Function regula Falsi method

The regula falsi method is guaranteed to converge to a root, but it may or may not be faster than the bisection method, depending on how long it takes to calculate the slope of the line and the shape of the function. 

The regula falsi algorithm is very similar to the previous one. The difference is in the support points adopted to linearize the function the last two values at each iteration are used in the secant method, whereas the boundaries of the interval of uncertainty are adopted in the regula falsi method. 

Other authors attempt to improve the efficiency of the regula falsi method with another device. Typically the method would not work satisfactorily when the function has very different absolute values at the boundaries of the interval. If we select these boundaries as support points for linearization, we can guarantee convergence to the solution, but the interval reduction might be very small. The device consists of the selection of a smaller value (i.e., dividing by 2) in correspondence with the boundary where the function has its maximum absolute value. A reduction of this kind in the ordinate can also be proposed in successive iterations. 

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. 

Quasi-Newton methods start out by using two points xP and jfl spanning the interval of jc, points at which the first derivatives of fix) are of opposite sign. The zero of Equation (5.9) is predicted by Equation (5.10), and the derivative of the function is then evaluated at the new point. The two points retained for the next step are jc and either xP or xP. This choice is made so that the pair of derivatives / ( ), and either/ (jc ) or/ ( ), have opposite signs to maintain the bracket on jc. This variation is called regula falsi or the method of false position. In Figure 5.3, for the (k + l)st search, x and xP would be selected as the end points of the secant line. 

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