An optimal class of fourth-order multiple-root finders of Chebyshev-Halley type and their basins of attraction

In this paper, we propose a family of optimal fourth-order of Chebyshev-Halley type methods free from second-order derivative for finding the multiple roots. The new methods are tested and compared with other well-known methods on different academical test functions. Further, for quantitative comparison, we have also computed total number of convergent points and convergent percentages, average number of iterations per convergent points and CPU time (in seconds) along with the basins of attraction on number of test problems to recommend the best optimal fourth-order method. We also consider a concrete variety of real life problems such as the trajectory of an electron in the air gap between two parallel plates, van der Waals equation 'which explains the behaviour of a real gas' by introducing in the ideal gas equation, in order to check the applicability and effectiveness of our proposed methods.