superconvergence

A sequence $x_{0},x_{1},\dots$ superconverges to 0 if, when the $x_{i}$ are written in base 2, then each number $x_{i}$ starts with $2^{i}-1\approx 2^{i}$ zeroes. For example, the following sequence is superconverging to 0.

 $\begin{array}[]{clcl}x_{n+1}&=x_{n}^{2}&(x_{n})_{10}&(x_{n})_{2}\\ x_{0}&=&\frac{1}{2}&.1\\ x_{1}&=&\frac{1}{4}&.01\\ x_{2}&=&\frac{1}{16}&.0001\\ x_{3}&=&\frac{1}{256}&.00000001\\ x_{4}&=&\frac{1}{65536}&.0000000000000001\end{array}$

In this case it is easy to see that the number of binary 0’s doubles each $x_{n}$.

A sequence $\{x_{i}\}$ superconverges to $x$ if $\{x_{i}-x\}$ superconverges to 0, and a sequence $\{y_{i}\}$ is said to be superconvergent if there exists a $y$ to which the sequence superconverges.

Title superconvergence Superconvergence 2013-03-22 11:58:12 2013-03-22 11:58:12 mathcam (2727) mathcam (2727) 15 mathcam (2727) Definition msc 41A25 superconverge NewtonsMethod KantorovitchsTheorem SuperincreasingSequence