Secant‐type methods and nondiscrete induction

The celebrated nondiscrete mathematical induction has been used to improve error bounds of distances involved in the discrete case but not the sufficient convergence conditions for Secant–type methods. We show that using the same information as before, the following advantages can be obtained: weaker sufficient convergence conditions; tighter error bounds on the distances involved and a more precise information on the location of the solution. Numerical examples validating the theoretical conclusions are also provided in this study.