Iterative bisection method

Calculations involving this equation and Eq. 4.139 are straightforward since no trial and error is involved. Further, the accuracy should be much better than that possible with only Eq. 4.139. In fact, the results obtained by the above predictor-corrector method for the example problem in Table 4.3 have been found to be identical with those obtained by solving Eq. 4.136 by an iterative bisection method. 

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... 

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). 

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... 

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. 

Two of the most popular iterative techniques are the bisection and the Newton-Raphson methods. The former has the advantage that it always converges to a solution. However, it is linearly convergent, which can imply an excessive number of iterations an intermediate solution may be closer to the correct answer than the final one. The Newton-Raphson technique is quadratically convergent and therefore requires fewer iterations than the bisection method. However, because of its small radius of convergence, it must be provided with good initial estimates for the density to avoid the problem of trivial... 

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 ... 

If we are certain that the optimum parameter estimates lie well within the constraint boundaries, the simplest way to ensure that the parameters stay within the boundaries is through the use of the bisection rule. Namely, during each iteration of the Gauss-Newton method, if anyone of the new parameter estimates lie beyond its boundaries, then vector Ak +I) is halved, until all the parameter constraints are satisfied. Once the constraints are satisfied, we proceed with the determination of the step-size that will yield a reduction in the objective function as already discussed in Chapters 4 and 6. 

At this point we can summarize the steps required to implement the Gauss-Newton method for PDE models. At each iteration, given the current estimate of the parameters, ky we obtain w(t,z) and G(t,z) by solving numerically the state and sensitivity partial differential equations. Using these values we compute the model output, y(t k(i)), and the output sensitivity matrix, (5yr/5k)T for each data point i=l,...,N. Subsequently, these are used to set up matrix A and vector b. Solution of the linear equation yields Ak(jH) and hence k°M) is obtained. The bisection rule to yield an acceptable step-size at each iteration of the Gauss-Newton method should also be used. 

In this section two iterative schemes for calculating the density given values for the pressure and temperature are described the bisection or interval-halving method, and the Newton-Raphson technique. These methods and others are described in more detail by Burden et al. (1978), who also give algorithms for these procedures. 

Once the bracket becomes sufficiently small that we feel that Newton s method or the secant method should be able to find the solution, we switch to one of those more efficient procedures. If this fails, we continue with bisection until the initial guess is sufficiently close for the iterative method to succeed. In MATLAB, the routine fzero takes such an approach. For further discussion of iterative methods to solve a single equation /(x) = 0, consult Press et al. (1992) and Quateroni et al. (2000). 

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