A fast and efficient composite Newton–Chebyshev method for systems of nonlinear equations

Abstract

An iterative method with fifth order of convergence for solving systems of nonlinear equations is presented. The scheme is composed of three steps, of which the first two steps are that of double Newton’s method with frozen derivative and third step is second derivative-free modification of Chebyshev’s method. The semilocal convergence of the method in Banach spaces is established by using a system of recurrence relations. Then an existence and uniqueness theorem is given to show the R-order of the method to be five and a priori error bounds. Computational complexity is discussed and compared with existing methods. Numerical results are included to confirm theoretical results. A comparison with the existing methods shows that the new method is more efficient than existing ones and hence use the minimum computing time in multiprecision arithmetic.