A Fourth-Order Iterative Formula Based on Thiele's Continued Fraction for Root-Finding Problems

Abstract： In this paper, we present a new one-step iterative method based on Thiele's continued fraction for solving non-linear equations. By applying the truncated Thiele's continued fraction twice, the iterative method is obtained. Analysis of convergence shows that the new method is fourth-order convergent. Numerical tests verifying the theory are given and based on the method, a new one-step iteration is developed.