# Isolating Block Validation in Matlab

This web page contains supplemental material for the article Rigorous validation of isolating blocks for flows and their Conley indices by Thomas Stephens and Thomas Wanner, which will be published in the SIAM Journal on Applied Dynamical Systems. The Matlab code for validating three-dimensional isolating blocks using the interval arithmetic package Intlab can be downloaded here as a gzipped tar archive. The software contains a few sample scripts, which are described below.
• The script scriptExitSet.m demonstrates the basic usage of the nodal domain validation algorithm at the heart of our method. Upon running the script, the nodal domains of the exit set test function u are validated for the isolating block C1 in Section 4.3 of the paper. The script also generates numerically computed images of the nodal domains of u, as well as of the associated modified tangency test function w. After its successful completion, the code will produce the three images below. Notice that the algorithm is a randomized method, so the specific adaptive grid might be different.
• The script scriptIsoBlock.m uses the isolating block algorithm to rigorously validate the set C1 in Section 4.3 of the paper as an isolating block for the flow. Upon running the script, it first produces an adaptive grid which can be used to validate the nodal domains of the exit set test function u, and thus furnishes an image similar to the first one below. It then moves through the boxes in the grid in a number of sweeps to ensure the positivity of the modified tangency test function w on the nodal lines of u. This usually requires additional box refinements and should lead to an image to the second one below. Finally, it overlays the final adaptive grid on numerically computed nodal domains of u, as well as the black zero set of w. Since the algorithm is a randomized method, the specific structure of the produced grids might be different.
• The Mathematica notebook createUWfiles.nb produces the Matlab m-files for the exit set test function u, the modified tangency test function w, as well as for their partial derivatives with respect to x and y.

Thomas Wanner, September 28, 2014.