### Acceleration of the Modified \(S\)-Algorithm to Search for a Fixed Point of a Nonexpansive Mapping

#### Abstract

The purpose of this paper is to present accelerations of the \(S\)-algorithm. We first apply the Picard algorithm to the smooth convex minimization problem and point out that the Picard algorithm is the steepest descent method for solving the minimization problem. Next, we provide the accelerated Picard algorithm by using the ideas of conjugate gradient methods that accelerated the steepest descent method. Then, based on the accelerated Picard algorithm, we present accelerations of the \(S\)-algorithm. Under certain assumptions, our algorithm strongly converges to a fixed point with the S-algorithm and show that it dramatically reduces the running time and iteration needed to find a fixed point compared with that algorithm.

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