On algorithmically boosting fixed-point computations

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.