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On the goodness of Global Optimisation Algorithms 131The algorithm departs from a starting interval [l, r] that is halved iteratively based on thesign of the derivative in the midpoint. This means that the method is only applicable whenthe derivative is available at the generated midpoints. The point x k converges to a minimumpoint within the interval [l, r]. If the interval contains only one minimum point, it convergesto that. In our test cases, several minima exist and one can observe the convergence to one ofthem. The algorithm is effective in the sense of converging to a local (nonglobal) minimumTable 1. Bisection for functions h and g, ɛ = 0.015function hfunction gk l k r k x k h ′ (x k ) h(x k ) l k r k x k g ′ (x k ) g(x k )0 3.00 7.00 5.00 -1.80 1.30 3.00 7.00 5.00 -1.80 1.301 5.00 7.00 6.00 3.11 0.76 5.00 7.00 6.00 3.11 0.762 5.00 6.00 5.50 -1.22 0.29 5.00 6.00 5.50 -1.22 0.293 5.50 6.00 5.75 0.95 0.24 5.50 6.00 5.75 0.95 0.244 5.50 5.75 5.63 -6.21 0.57 5.50 5.75 5.63 -0.21 0.205 5.63 5.75 5.69 -2.64 0.29 5.63 5.75 5.69 0.36 0.206 5.69 5.75 5.72 -0.85 0.24 5.63 5.69 5.66 0.07 0.197 5.72 5.75 5.73 0.05 0.23 5.63 5.66 5.64 -0.07 0.198 5.72 5.73 5.73 -0.40 0.23 5.64 5.66 5.65 0.00 0.19point for both cases. Another starting interval could have lead to another minimum point. Inthe end we are certain that the current iterate x k is not further away than ɛ from a minimumpoint. Many other stopping criteria like convergence of function value or of the derivative tozero could be used. The current stopping criterion is easy for analysis on the efficiency. Onequestion could be: How many iterations corresponding (derivative) function evaluations arenecessary to come closer than ɛ to a minimum point. The bisection algorithm is a typical caseof linear convergence with a convergence factor of 1 2 , |r k+1−l k+1 ||r k −l k |= 1 2. This means one candetermine the number of iterations necessary for reaching ɛ-convergence:| r k − l k | = ( 1 2 )k × | r 0 − l 0 | < ɛ( 1 2 )k ln ɛ − ln | r 0 − l 0 |ln 1 2The example case would require at least k = 12 iterations to reach an accuracy of ɛ = 0.01.An alternative for finding the zero point of an equation, in our case the derivative, is theso-called method of Newton. The idea is that its efficiency is known to be superlinear (e.g.[5]), so it should be faster than bisection. We will analyse its efficiency and effectiveness for thetwo test cases. In general, the aim of this algorithm is to converge to a point where the derivativeis zero. Depending on the starting point x 0 the method may converge to a maximum. Itmay also not converge at all, for instance when a minimum point does not exist. If x 0 is in theAlgorithm 3 Newt(x 0 , f, ɛ)Set k = 0,while (| f ′ (x k ) |> ɛ)x k+1 = x k − f ′ (x k )f ′′ (x k )k = k + 1End while