A variant of Newton’s method with accelerated third-order convergence. (English) Zbl 0973.65037

Authors’ summary: In the given method, we suggest an improvement to the iteration of Newton’s method. Derivation of Newton’s method involves an indefinite integral of the derivative of the function, and the relevant area is approximated by a rectangle. In the proposed scheme, we approximate this indefinite integral by a trapezoid instead of a rectangle, thereby reducing the error in the approximation. It is shown that the order of convergence of the new method is three, and computed results support this theory. Even though we have shown that the order of convergence is three, in several cases, computational order of convergence is even higher. For most of the functions we tested, of convergence in Newton’s method was less than two and for our method, it was always close to three.


65H05 Numerical computation of solutions to single equations
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