Investigate the hybrids of the bisection and the secant methods Research Paper

Investigate the hybrids of the bisection and the secant methods - Research Paper Example The rate of convergence which records the number of iterations needed to attain a particular degree of accuracy, is not the key subject when assessing the computational effectiveness of the algorithm. The quantity of floating point operations (flops), for each iteration should also be considered. In case the iteration needs many flops, although an algorithm has a greater rate of convergence it might take more time to reach a required degree of precision. This method is therefore faster than Newton’s method and has an advantage since it only needs a single function evaluation for every iteration. This then serves as a compensation for the slower rate of convergence when the function and its derivative cost higher to evaluate. Another disadvantage of this method is that, similar to newton’s method, it lacks robustness, particularlty when the primary guesses are further from root. In addition, the method does not need differentiation. The bisection method is the modest and most robust algorithm for root-finding in a 1-dimensional continous function that has a closed interval. The basic principle of this technique is that if f(.) is a continous function expressed over an interval {a,b} and f(a) and f(b) with opposite signs, according to the theorem of intermediate value, at least a single r{a,b} exists making f(r) = 0. This technique is iterative and every iteration begins by breaching the existing interval forming brackets around the root(s) into two subintervals of matching lengths. The endpoint of one the subintervals must have different signs. This subinterval is now the new interval and the subsequent iteration starts. Therefore it is possible to define lesser and lesser intervals such that every interval has r by checking subintervals of the present interval and selecting the interval where f(.) changes signs. This is a continous process that ends when the width of the interval having a root