Abstract
The (n + l)-point sequential secant method is prone to unstable behaviour. No satisfactory convergence results can be given for this method, even though it requires the least amount of computation for every step. The chapter presents the results for multistep methods and for one-step iterations that are either nonstationary or those in which xk +1 depends upon xk in a complex manner. A major example of the multistep methods are the secant-type methods. The chapter discusses the generalized Ostrowski theorem, consistent approximations, the discrete Newton theorem, the two-point Secant-Steffensen theorem, the general secant method, the general secant theorem, and the (n + 1)-point Secant–Steffensen theorem. A comparison can be attempted of the relative efficiencies of the various secant and Steffensen methods on the basis of the rate-of-convergence results.
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