The Algorithms of Gauss, Borchardt and Carlson

Abstract

We shall study several algorithms of the form

$${X_{n + 1}} = f\left( {{X_n},{Y_n}} \right)$$

$${Y_{n + 1}} = g\left( {{X_n},{Y_n}} \right)$$

where f, g are given functions and x₀, y₀ are given, from the point of view of the rates of convergence of the sequences y _n . In most cases x₀, y₀ will be non-negative, the sequences x _n , y _n will be monotonie and bounded and therefore convergent. The limits of x _n , y _n will be the same in each case but the rates of convergence to these limits will differ markedly from case to case.