An optimal homotopy continuation method: Convergence and visual analysis

To avoid divergence in the traditional iterative root-finding methods the homotopy continuation approach is commonly used in the literature. However, neither their theoretical analysis in terms of local and semilocal convergence nor their stability is explored in the present literature. In this paper, we describe the homotopy continuation (HC) version of a fourth-order accurate optimal iterative algorithm. The local and semilocal convergence of the HC-based algorithm, including the basins of attraction, are being examined for the first time in the literature. These basins are used to demonstrate that the HC variant is more stable than the traditional iterative approach, which is widely held to be advantageous. The usual iterative method with first-order derivative is shown to be replaceable by its equivalent HC counterpart to achieve better stability for several numerical problems selected from academia and industry.