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Numerical Solution of Equations of a Single Variable
Published in Ramin S. Esfandiari, Numerical Methods for Engineers and Scientists Using MATLAB®, 2017
In many cases, the curve representing f(x) happens to be concave up or concave down. In these situations, when regula falsi is employed, one of the endpoints of the interval remains the same through all iterations, while the other endpoint advances in the direction of the root. For instance, in Figure 3.7, the function is concave up, the left endpoint remains unchanged, and the right endpoint moves toward the root. The regula falsi method can be modified such that both ends of the interval move toward the root, thus improving the rate of convergence. Among many proposed modifications, there is one that is presented here. Reconsider the scenario portrayed in Figure 3.7 now shown in Figure 3.8. If endpoint aremains stagnant after, say, three consecutive iterations, the usual straight line is replaced with one that is less steep, going through the point at 12f(a) instead of f(a), which causes the x-intercept to be closer to the actual root. It is possible that this still does not force the endpoint a to move toward the root. In that event, if endpoint a remains the same after three more iterations, the modified line will be replaced with yet a less steep line going through 14f(a), and so on; See Problem Set.
Numerical Modeling and Simulation
Published in Yogesh Jaluria, Design and Optimization of Thermal Systems, 2019
In the bisection method, an approximation to the root x3 is obtained by taking an average of the two end points, x1 and x2, of the interval in which the root lies, i.e., x3 = (x1 + x2)/2. At each iteration, the new, reduced interval in which the root lies is determined and a new approximation to the root computed. In the regula falsi method, interpolation is used, employing the end points of the interval at a given iteration, to approximate the root as x3 = [x1f (x2) – x2f (x1)]/[f (x2) – f (x1)]. Sign change between x3 and x1 or between x3 and x2 is used to choose the interval containing the root. The iterative process is continued, reducing the interval at each step, until the change in the approximation to the root from one iteration to the next is less than a chosen convergence criterion, as given by x(l+1)−x(l)≤εorx(l+1)−x(l)x(l)≤ε
Analytical Discrete Ordinates Solution for a 1D Model of Particle Transport in Ducts that Includes Wall Migration
Published in Nuclear Science and Engineering, 2022
Having located the distinct eigenvalues in well-defined intervals, we now summarize our computational procedure to determine them. A sufficiently high but finite value replaces as the upper limit of the interval that contains . We begin by using the Anderson-Björk iterative technique37 to refine the bracketing interval of each eigenvalue. The Anderson-Björk technique is a modified regula falsi technique that in general provides much faster convergence. As we have found the Anderson-Björk algorithm to hamper on a few occasions, we switch to another improved variant of the regula falsi technique, the Illinois algorithm,38 when this happens. Once we obtain an interval smaller than a prescribed size, we take the central point of that interval as our initial estimate of an eigenvalue and switch to Newton’s method to further improve that estimate. For eigenvalues with magnitude , we use Eq. (65) for that purpose; otherwise, we use Eq. (67).