Regula Falsi

Eq. (18) is now a function of temperature only. It can be solved by any trial and error procedure such as successive substitution or regula falsi (Gerald (1978)). Having obtained the optimum temperature from Eq. (18), the corresponding minimum time can be calculated from Eq. (17). 

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. 

This method is called regula falsi, and it has superlinear convergence, which means that 70 > 1. In this one-dimensional case, 70 = 1.618. 

For a given value of O and y in the log-normal distribution function, the mean and variance of the distribution function were computed and compared with the mean and variance of the measured bubble lengths. A regula falsi technique was used to minimize the difference between observed and calculated mean and variance. The values of O and y that minimized the difference between observed and calculated mean and variance were then employed in Equation (1) to describe the local bubble diameter distribution. The Sauter mean bubble diameter was evaluated from the second and third moments of Equation (4). 

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)... 

One numerical method for solving such a problem is called interpolation regula falsi (see Probs. 1-7 and 1-8). This method consists of the linear interpolation between the most recent pair of points (7> , Sn) and (TF n+l, <5n+1) by use of the following formula... 

Extend this straight line until it intersects that x axis at the point (xk+2, 0). [This procedure amounts to the assumption that the function is linear over the range of the extrapolation.] Show that at, v = xk + 2 and f(x) = 0, Eq. (A) may be solved to give the interpolation regula falsi formula... 

FORMAT ( O, REGULA-FALSI DID NOT CONVERGE IN 20 ITERATIONS ) 93 STOP... 

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 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 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. 

The zero root of the function can be found by an application of the Regula Falsi, taking care of the possible range of values. The algorithm [6] is defined by... 

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. 

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