A Method to Accelerate the Convergence of the Secant Algorithm

We present an acceleration technique for the Secant method. The Secant method is a root-searching algorithm for a general function f. We exploit the fact that the combination of two Secant steps leads to an improved, so-called first-order approximant of the root. The original Secant algorithm can be modified to a first-order accelerated algorithm which generates a sequence of first-order approximants. This process can be repeated: two nth order approximants can be combined in a (n+1)th order approximant and the algorithm can be modified to an (n+1)th order accelerated algorithm which generates a sequence of such approximants. We show that the sequence of nth order approximants converges to the root with the same order as methods using polynomial fits of f of degree n