Regula Falsi method

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. 

The advantage this has over the previous criteria is that the new iteration falls within the interval of uncertainty and thus method convergence is ensured. 

The new prevision replaces the bound where the function assumes the same sign. 

Its main disadvantage is that its convergence speed is slower than the secant method. 

To calculate the square root of 2 using this method 

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Several numerical techniques for accelerating triaLand-error searches of this sort are discussed in Appendix A.2. One of them, the regula-falsi method, is used by goalseek tools in many spreadsheet programs. This procedure has been used to generate the trial temperatures shown here for the third and subsequent trials. 

In this and the next subsection, we outline algorithms for finding roots of single-variable equations of the form f(x) = 0. The first procedure, termed the regula-falsi method, is appropriately used when an analytical expression for the derivative of / with respect to x is not available—as. for example, when f x) is obtained as the output of a computer program for an input value of x. The algorithm is as follows ... 

The regula-falsi method is the procedure used by many spreadsheet programs in their goalseek algorithms. 

Method of Linear Interpolation (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. 

Find the root of x + ln x) = 0 but this time use the regula falsi method. 

In some cases, the regula falsi method will take longer than the bisection method, depending on the shape of the curve. However, it generally worth trying for a couple of iterations due to the drastic speed increases possible. 

The efficiency of the regula falsi method can be improved by using the two best values from the previous iterations rather than the boundary values for the linear approximation. This device is implemented by checking the interval of uncertainty. 

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. 

This equation can be solved numerically using the Regula Falsi method, which does not need analytical derivatives in contrast to the Newton method. The interval of the search for the root should be specified. The polymer concentration of the polymer-rich phase must be above the critical concentration (Xg = 0.00756). The search interval is therefore between the critical concentration Xg = 0.00756 and a value at higher polymer concentration (xg 2 = 0.02). Using the Regula Falsi... 

Table 10.1 shows the iteration history when the Regula Falsi method is applied. 

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. 

A schematic description of the Regula-Falsi method is shown in Figure 1.7. 

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