This paper is a thought experiment on exponentiating algorithms. One of the main contributions of this paper is to show that this idea finds material implementation in exponentiating fixed-point computation algorithms. Various problems in computer science can be cast as instances of computing a fixed point of a map. In this paper, we present a general method of boosting the convergence of iterative fixed-point computations that we call algorithmic boosting, which is a (slight) generalization of algorithmic exponentiation. We first define our method in the general setting of nonlinear maps. Secondly, we restrict attention to convergent linear maps and show that our algorithmic boosting method can set in motion exponential speedups in the convergence rate. Thirdly, we show that algorithmic boosting can convert a (weak) non-convergent iterator to a (strong) convergent one. We then consider a variational approach to algorithmic boosting providing tools to convert a non-convergent continuous flow to a convergent one. We, finally, discuss implementations of the exponential function, an important issue even for the scalar case.

本文探讨算法指数增长的思维实验，重点在于阐述该思想在指数增长定点计算算法中找到了实际应用。作者提供了一种名为算法增强的推进迭代定点计算收敛的通用方法，还介绍了该方法能够在收敛速度上实现指数级加速，最后讨论了算法实现有关的问题，特别是对于标量情形的指数函数。

关于算法提升固定点计算