Bisection method

The Regula Falsa method is a refinement of the bisection method, in which the new end point of a new interval is calculated from the old end points by... 

In the application of the bisection method it is assumed only that the function f(x) is continuous. It requires that two initial values of x, say xa and X >, be chosen so that they straddle the desired zero. Thus, /(xa) and fixb) will have opposite signs and their product will be negative. Now, take the midpoint xm = (xa + xb)/2 and calculate /(xm). If, for example, the product /(xtt)/(xm) < 0, the desired root lies between xa and xffl. The midpoint between these two limits is then calculated and the process is repeated to die desired degree of accuracy. Here again, the better the choice of the initial limits, the fewer the number of bisections that will be required. 

As the bisection method does not depend on the derivatives of the function in question, it can be applied with confidence, even if there are stationary points within the chosen limits, x and Xh. However, convergence is often... 

As a simple example, we can apply the bisection method to find a minimum... 

An alternative approach to our problem is Newton s method. The idea behind this method is illustrated in Fig. 3.5. If we define g(x) = f (x). then from a Taylor expansion g(x I It) = g(x) + hg (x). This expression neglects terms with higher orders of h, so it is accurate for small values of h. If we have evaluated our function at some position where g(x) A 0, our approximate equation suggests that a good estimate for a place where the function is zero is to define x = x I h = x g(x)/g x). Because the expression from the Taylor expansion is only approximate, x does not exactly define a place where g(.ri) = 0. Just as we did with the bisection method, we now have to repeat our calculation. For Newton s method, we repeat the calculation starting with the estimate from the previous step. 

Figure 3.6 Convergence analysis for the bisection method and Newton s method calculations described in the text.

There is no natural way to generalize the one-dimensional bisection method to solve this multidimensional problem. But it is possible to generalize Newton s method to this situation. The one-dimensional Newton method was derived using a Taylor expansion, and the multidimensional problem can be approached in the same way. The result involves a 3/V x 3/V matrix of derivatives, J, with elements 7y = dg, /dxj. Note that the elements of this matrix are the second partial derivatives of the function we are really interested in, E(x). Newton s method defines a series of iterates by... 

We showed how to find a minimum off(x) = e x cos x using the bisection method and Newton s method. Apply both of these methods to find the same minimum as was discussed above but using different initial estimates for the solution. How does this change the convergence properties illustrated in Fig. 3.6 This function has multiple minima. Use Newton s method to find at least two more of them. 

The same regression analysis methodology has been appHed for analyzing the model for aggregation of surfactants at the interface. The non-Hnear Eq. 28 has been numerically solved by the bisection method. The surface tension predicted by Eq. 27 has been fitted to the experimental data by min-... 

The standard deviation has been determined as ct = j where v is the number of degrees of freedom in the fit. The parameters for the molecular interaction /3, the maximum adsorption Too, the equilibrium constant for adsorption of surfactant ions Ki, and the equilibrium constant for adsorption of counterions K2, are thus obtained. The non-linear equations for the Frumkin adsorption isotherm have been numerically solved by the bisection method. 

Example 2.1.2 Molar volume of n-buthane by bisection method... 

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. 

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... 

Another method to solve scalar equations in one real variable x uses inclusion and bisection. Assume that for a given one variable continuous function / R —> R we know of two points X < xup G R with f xi) f(xup) < 0, i.e., / has opposite signs at X and xup. Then by the intermediate value theorem for continuous functions, there must be at least one value x included in the open interval (x ,xup) with f(x ) = 0. The art of inclusion/bisection root finders is to make judicious choices for the location of the root x G (x , xup) from the previously evaluated / values and thereby to bisect the interval of inclusion [x , xup] to find closer values v < u e [x , xup] with v — u < x — xup and f(v) f u) < 0, thereby closing in on the actual root. Inclusion and bisection methods are very efficient if there is a clear intersection of the graph of / with the x axis, but for slanted, near-multiple root situations, both Newton s method and the inclusion/bisection... 

The following experiments validate our assessment of troubles with Newton or bisection root finders for multiple roots. First we use the bisection method based MATLAB root finder f zero, followed by a simple Newton iteration code, both times using the chosen polynomial p x) of degree 9 in its extended form (1.6). 

Our first attempt involves MATLAB s built-in root finder fzero, which uses the bisection method and thereafter we introduce a new and more appropriate numerical method for solving equations with multiple roots. 

Figure 3.3 shows the typical limited range of useful output from the bisection method in the bifurcation range for this problem when (3 = 1 and 7 = 8.5. 

Limited bifurcation output from a bisection method Figure 3.3... 

Compared to solveadiabxy. m for the adiabatic CSTR case in Section 3.1, the above MATLAB function solveNadiabxy. m depends on the two extra parameters Kc and yc that were defined following equation (3.9). It uses MATLAB s built-in root finder fzero.m. As explained in Section 3.1, such root-finding algorithms are not very reliable for finding multiple steady states near the borders of the multiplicity region. The reason - as pointed out earlier in Section 1.2 - is geometric the points of intersection of the linear and exponential parts of equations such as (3.16) are very shallow, and their values are very hard to pin down via either a Newton or a bisection method, especially near the bifurcation points. 

The Nemst parameter, En(I), is afunctionof node current, /, through the consumption of reactants with I. The loss terms, r]j, are the ohmic, concentration, and electrochemical over-potential, all of which are functions of node current. A combination Newton and simple bisection method is used to converge to the desired solution. Once the current is known at each node, the dynamic equations are stepped forward one time step for all nodes. 

The MOLSIMIL program developed by Carbo and Calabuig [101] is written in FORTRAN-77, and therefore the source code is transferable. The Carbo-type SI is calculated from CNDO-like molecular electron density functions in the program. It is also possible to determine the maximum value of SI(C) with the help of the bisection method. 

Equation (13-14) is solved iteratively for V/F, followed by the calculation of values o(x,anAy, from Eqs. (13-12) and (13-13) and L from the total mole balance. Any one of a number of numerical root-finding procedures such as the Newton-Raphson, secant, false-position, or bisection method can be used to solve Eq. (13-14). Values of K, are constants if they are independent of liquid and vapor compositions. Then the resulting calculations are straightforward. Otherwise, the K, values must be periodically updated for composition effects, perhaps... 

Eigenfunction expansions as used in Ref. 168 are not accurate near the critical point. Instead, we developed a shooting point method in order to make a direct numerical integration of Eq. (110) with the condition Eq. (112). Real energies (bound and virtual) were found by bisection methods, and for complex energies it was necessary to combine the Newton-Raphson and grid methods. 

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