Chebysheff-Halley-like methods in Banach spaces

Chebysheff-Halley methods are probably the best known cubically convergent iterative procedures for solving nonlinear equations. These methods however require an evaluation of the second Fréchet-derivative at each step which means a number of function evaluations proportional to the cube of the dimension of the space. To reduce the computational cost we replace the second Fréchet derivative with a fixed bounded bilinear operator. Using the majorant method and Newton-Kantorovich type hypotheses we provide sufficient conditions for the convergence of our method to a locally unique solution of a nonlinear equation in Banach space. Our method is shown to be faster than Newton’s method under the same computational cost. Finally we apply our results to solve nonlinear integral equations appearing in radiative transfer in connection with the problem of determination of the angular distribution of the radiant-flux emerging from a plane radiation field.