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Movie Title Year Distributor Notes Rev Formats Ballgame 1980 Caballero Home Video BJOnly 1 DRO Plato's The Movie 1980 Essex Video / Electric Hollywood Analysis Since the initial end-points a0 and b0 are chosen such that f?(a0) and f?(b0) are of opposite signs, at each step, one of the end-points will get closer to a root of f. If the second derivative of f is of constant sign (so there is no inflection point) in the interval, then one endpoint (the one where f also has the same sign) will remain fixed for all subsequent iterations while the converging endpoint becomes updated. As a result, unlike the bisection method, the width of the bracket does not tend to zero (unless the zero is at an inflection point around which sign(f?) = -sign(f?")). As a consequence, the linear approximation to f?(x), which is used to pick the false position, does not improve as rapidly as possible.
One example of this phenomenon is the function {\displaystyle f(x)=2x^{3}-4x^{2}+3x}f(x)=2x^{3}-4x^{2}+3x on the initial bracket [-1,1]. The left end, -1, is never replaced (it does not change at first and after the first three iterations, f?" is negative on the interval) and thus the width of the bracket never falls below 1. Hence, the right endpoint approaches 0 at a linear rate (the number of accurate digits grows linearly, with a rate of convergence of 2/3).[citation needed] For discontinuous functions, this method can only be expected to find a point where the function changes sign (for example at x = 0 for 1/x or the sign function). In addition to sign changes, it is also possible for the method to converge to a point where the limit of the function is zero, even if the function is undefined (or has another value) at that point (for example at x = 0 for the function given by f?(x) = abs(x) - x2 when x ? 0 and by f?(0) = 5, starting with the interval [-0.5, 3.0]). It is mathematically possible with discontinuous functions for the method to fail to converge to a zero limit or sign change, but this is not a problem in practice since it would require an infinite sequence of coincidences for both endpoints to get stuck converging to discontinuities where the sign does not change, for example at x = ±1 in

{\displaystyle f(x)={\frac {1}{(x-1)^{2}}}+{\frac {1}{(x+1)^{2}}}.}{\displaystyle f(x)={\frac {1}{(x-1)^{2}}}+{\frac {1}{(x+1)^{2}}}.} The method of bisection avoids this hypothetical convergence problem. Improvements in regula falsi Though regula falsi always converges, usually considerably faster than bisection, there are situations that can slow its convergence – sometimes to a prohibitive degree. That problem isn't unique to regula falsi: Other than bisection, all of the numerical equation-solving methods can have a slow-convergence or no-convergence problem under some conditions. Sometimes, Newton's method and the secant method diverge instead of converging – and often do so under the same conditions that slow regula falsi's convergence. But, though regula falsi is one of the best methods, and even in its original un-improved version would often be the best choice; for example, when Newton's isn't used because the derivative is prohibitively time-consuming to evaluate, or when Newton's and Successive-Substitutions have failed to converge. Regula falsi's failure mode is easy to detect: The same end-point is retained twice in a row. The problem is easily remedied by picking instead a modified false position, chosen to avoid slowdowns due to those relatively unusual unfavorable situations. A number of such improvements to regula falsi have been proposed; two of them, the Illinois algorithm and the Anderson–Björk algorithm, are described below. The Illinois algorithm The Illinois algorithm halves the y-value of the retained end point in the next estimate computation when the new y-value (that is, f?(ck)) has the same sign as the previous one (f?(ck - 1)), meaning that the end point of the previous step will be retained. Hence: {\displaystyle c_{k}={\frac {{\frac {1}{2}}f(b_{k})a_{k}-f(a_{k})b_{k}}{{\frac {1}{2}}f(b_{k})-f(a_{k})}}}{\displaystyle c_{k}={\frac {{\frac {1}{2}}f(b_{k})a_{k}-f(a_{k})b_{k}}{{\frac {1}{2}}f(b_{k})-f(a_{k})}}} or {\displaystyle c_{k}={\frac {f(b_{k})a_{k}-{\frac {1}{2}}f(a_{k})b_{k}}{f(b_{k})-{\frac {1}{2}}f(a_{k})}},}{\displaystyle c_{k}={\frac {f(b_{k})a_{k}-{\frac {1}{2}}f(a_{k})b_{k}}{f(b_{k})-{\frac {1}{2}}f(a_{k})}},} down-weighting one of the endpoint values to force the next ck to occur on that side of the function.[10] The factor ½ used above looks arbitrary, but it guarantees superlinear convergence (asymptotically, the algorithm will perform two regular steps after any modified step, and has order of convergence 1.442). There are other ways to pick the rescaling which give even better superlinear convergence rates.[11] The above adjustment to regula falsi is called the Illinois algorithm by some scholars.[10][12] Ford (1995) summarizes and analyzes this and other similar superlinear variants of the method of false position.[11] Anderson–Björck algorithm Suppose that in the k-th iteration the bracketing interval is [ak, bk] and that the functional value of the new calculated estimate ck has the same sign as f?(bk). In this case, the new bracketing interval [ak + 1, bk + 1] = [ak, ck] and the left-hand endpoint has been retained. (So far, that's the same as ordinary Regula Falsi and the Illinois algorithm.) But, whereas the Illinois algorithm would multiply f?(ak) by 1 / 2 , Anderson–Björck algorithm multiplies it by m, where m has one of the two following values: {\displaystyle m=1-{\frac {f(c_{k})}{f(b_{k})}}}{\displaystyle m=1-{\frac {f(c_{k})}{f(b_{k})}}} if that value of m is positive, otherwise, let {\displaystyle m={\frac {1}{2}}}{\displaystyle m={\frac {1}{2}}}. For simple roots, Anderson–Björck was the clear winner in Galdino's numerical tests.[13][14] For multiple roots, no method was much faster than bisection. In fact, the only methods that were as fast as bisection were three new methods introduced by Galdino. But even they were only a little faster than bisection. Practical considerations When solving one equation, or just a few, using a computer, the bisection method is an adequate choice. Although bisection isn't as fast as the other methods—when they're at their best and don't have a problem—bisection nevertheless is guaranteed to converge at a useful rate, roughly halving the error with each iteration – gaining roughly a decimal place of accuracy with every 3 iterations. For manual calculation, by calculator, one tends to want to use faster methods, and they usually, but not always, converge faster than bisection. But a computer, even using bisection, will solve an equation, to the desired accuracy, so rapidly that there's no need to try to save time by using a less reliable method—and every method is less reliable than bisection. An exception would be if the computer program had to solve equations very many times during its run. Then the time saved by the faster methods could be significant. Then, a program could start with Newton's method, and, if Newton's isn't converging, switch to regula falsi, maybe in one of its improved versions, such as the Illinois or Anderson–Björck versions. Or, if even that isn't converging as well as bisection would, switch to bisection, which always converges at a useful, if not spectacular, rate. When the change in y has become very small, and x is also changing very little, then Newton's method most likely will not run into trouble, and will converge. So, under those favorable conditions, one could switch to Newton's method if one wanted the error to be very small and wanted very fast convergence. Example code This example program, written in the C programming language, includes the Illinois Algorithm. To find the positive number x where cos(x) = x3, the equation is transformed into a root-finding form f?(x) = cos(x) - x3 = 0. #include #include double f(double x) { return cos(x) - x*x*x; } /* s,t: endpoints of an interval where we search e: half of upper bound for relative error m: maximal number of iterations */ double FalsiMethod(double s, double t, double e, int m) { double r,fr; int n, side=0; /* starting values at endpoints of interval */ double fs = f(s); double ft = f(t); for (n = 0; n < m; n++) { r = (fs*t - ft*s) / (fs - ft); if (fabs(t-s) < e*fabs(t+s)) break; fr = f(r); if (fr * ft > 0) { /* fr and ft have same sign, copy r to t */ t = r; ft = fr; if (side==-1) fs /= 2; side = -1; } else if (fs * fr > 0) { /* fr and fs have same sign, copy r to s */ s = r; fs = fr; if (side==+1) ft /= 2; side = +1; } else { /* fr * f_ very small (looks like zero) */ break; } } return r; } int main(void) { printf("%0.15f\n", FalsiMethod(0, 1, 5E-15, 100)); return 0; } After running this code, the final answer is approximately

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