Abstract
Finding the roots of an equation is a fundamental problem in various fields, including numerical computing, social and physical sciences. Numerical techniques are used when an analytic solution is not available. There is not a single algorithm that works best for every function. We designed and implemented a new algorithm that is a dynamic blend of the bisection and regula falsi algorithms. The implementation results validate that the new algorithm outperforms both bisection and regula falsi algorithms. It is also observed that the new algorithm outperforms the secant algorithm and the Newton–Raphson algorithm because the new algorithm requires fewer computational iterations and is guaranteed to find a root. The theoretical and empirical evidence shows that the average computational complexity of the new algorithm is considerably less than that of the classical algorithms.
Highlights
Non-linear equations arise in most disciplines including computer science, natural sciences, biological engineering and social sciences, etc
Some researchers have implemented genetic algorithms to find the roots of equations and compared the results with the results of classical algorithms [12], it is not the intent of this paper to go into that area
The closer b is to r, the closer the secant line is to tangent, f0 (r), though, it is not required f(b)−f(a) f(b)−0 that f be differentiable
Summary
Non-linear equations arise in most disciplines including computer science, natural sciences (civil engineering, electrical engineering, mechanical engineering), biological engineering and social sciences (psychology, economics), etc. Newton–Raphson, secant and modified secant method, for finding roots of a non-linear equation f(x) = 0 [7,8,9,10,11] From these algorithms, the developer has to explore and exploit the algorithm suitable under specified constraints on the function and the domain. In applications, we may need to determine when will two functions g(x) and h(x) be equal, i.e., g(x) = h(x) This translates into finding roots of the classical equation f(x) = g(x) − h(x) = 0. The new blended algorithm is a tested technique that outperforms other existing methods by curtailing the number of iterations required for approximations (Table 1), while retaining the accuracy of the approximate root.
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