False-position method

If f(x)f(XL) 0 then the new lower limit will be x, otherwise x will replace the upper limit. 

The convergence is of order p, where p is slightly larger than 1. 

Indeed, the method usually perforins better then the bisection method, while having the same robustness. Therefore, it is recommended for solving problems with little information available on the form of the function f. The only requirement is sufficient smoothness of f near the root. 

2 REQUIRED ACCURACY NOT ATTAINED ESTIMATE OF THE ROOT FUNCTION VALUE F(X) 

Example 2.1.5 Molar volume by false position method 

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As a consequence of ion suppression, the analytical method is affected in different ways [38]. Eirst of all, it has an effect on the capability to detect analytes due to signal decrease. As a consequence, the true concentration of the analyte can be underestimated to the point of a false negative. If signal suppression involves an internal standard only, one may be induced to overestimate analyte concentration to the point of a false positive. Method precision response linearity can be affected because the degree of suppression can vary significantly in different samples. The presence of co-eluting compounds can modify the mass spectra, thus makiug the database search more complicated. 

REM EX. 2.1.3 MOLAR VOLUME BY FALSE POSITION METHOD 104 REM MERBE M22... 

The basic idea is the same as in the false position method, i.e., local linear approximation of the function. The starting interval [x x2J does not, however, necessarily include the root. Then the straight line through the... 

Retaining the latest estimates for x and X2, the slope of the line follows more closely the form of the function than in the false position method. The order of convergence can be shown to be 1.61B, the "golden ratio", which we will encounter in Section 2.2.1. The root, however, is not necessarily bracketed, and the next estimate x3 may be far away if the function value ffx ) is close to f(x2>. Therefore we may run into trouble when starting the search in a region where the function is not monotonic. 

Table 2.4 shows the SAS NLIN specifications and the computer output. You can choose one of the four iterative methods modified Gauss-Newton, Marquardt, gradient or steepest-descent, and multivariate secant or false position method (SAS, 1985). The Gauss-Newton iterative methods regress the residuals onto the partial derivatives of the model with respect to the parameters until the iterations converge. You also have to specify the model and starting values of the parameters to be estimated. It is optional to provide the partial derivatives of the model with respect to each parameter, b. Figure 2.9 shows the reaction rate versus substrate concentration curves predicted from the Michaelis-Menten equation with parameter values obtained by four different... 

The value of J3 = y(y ) is now calculated. It is extremely unlikely that 73 is zero, as it would be if A3 is the root we seek. The zero in 7 will presumably occur at an a value on one side or the other of A3, hi the second iteration, we repeat the above procedure, interpolating either between Aj and A3 or between A3 and A2, depending on which range includes the zero in 7, The procedure is repeated in subsequent iterations to obtain A4, A5, and so on until the change between iterations becomes insignificant and we can say that the process has effectively converged on the root to the desired degree of precision. This method is known as the false position method. 

Satisfactory convergence has been obtained, somewhat faster than with the false position method. Usnally the Newton-Raphson method is indeed the one that converges faster, bnt in some cases it can diverge in the early iterations. It may be helpful to use the false position method for the first one or two iterations, and then switch over to the Newton-Raphson method for further iterations leading to satisfactory convergence. This, and other root-finding procednres, are commonly used by the Solver or Optimizer operation of spreadsheet programs, whose nse is described in Chapter III. 

The above formula is also used in the secant method, but the secant method always retains the last two computed points, while the false position method retains two points that bracket a root. On the other hand, the only difference between the false position method and the bisection method is that the latter uses c,. = (a + )/2. 

It can be detected from the output that xR tends to be stagnant. Each time RepeatR is greater than 1, the fxR value is halved in Equation 2.1. This has a marked effect on the rate of convergence. Exercise 2.1 at the end of this chapter involves programming the unmodified false position method. Upon executing that program for the same test problem as shown in this example, the stagnation of xR will be observed. 

Exercise 2.2 In order to appreciate the increased efficiency of the modified false position method, redesign and reprogram Example 2.6 so that the original false position method is implemented. 

The false-position method is similar to the bisection method but improves on the iterative algorithm by making use of the magnitudes of the function at the upper and lower position values. The iterative algorithm is ... 

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