Non-divergent circular arc root algorithm

A new iteration algorithm for numerically locating the roots of nonlinear algebraic functions is proposed. The algorithm is non-divergent even in the vicinity of local extrema. The algorithm depends of drawing a tangent circular arc to the initial estimation point on the curve. The radius of curvature of the circular arc is equal to the functions radius of curvature at the estimated point. The intersection points of the circular arc with the x axis determine the first iteration for the roots. The iteration equation is derived first. The conditions for which the algorithm works are discussed. It is proven that the convergence rate of the new algorithm is quadratic. Using sample problems, the algorithm is contrasted with the well-known Newton-Raphson algorithm and the parabolic algorithm. It is shown that the algorithm requires less iterations, has a wider convergence interval in general and does not diverge in the vicinity of local extrema as compared to the Newton-Raphson method. For the example considered, the algorithm is better than the parabolic algorithm in terms of the iterations and computational times.

Keywords:

Root finding Algorithms , Newton-Raphson method , Non-Divergent Algorithm , Nonlinear equations , Convergence rate