A classical problem in approximation theory is to approximate a (real or complex) function f by a polynomial p. Combining beautiful theory and reliable algorithms, univariate polynomial approximation has reached a mature stage, as implemented in the Chebfun software package. For example for analytic functions on [-1, 1], polynomial approximation converges geometrically. Approximation theory is used virtually everywhere in scientific computing.
A relatively uncharted question is to approximate f by a composite poly- nomial, of the form q = pk(pk-1...(p2(p1))...). Composing polynomials is a highly efficient way of generating high-degree polynomials, so they can po- tentially be much more powerful than plain polynomials, with respect to the degrees of freedom. In fact they are the crucial tool for most algorithms for computing matrix functions (see Higham 2008), and one can understand deep learning as a composition of large number of piecewise polynomials. This project aims to investigate the power and limitations of composite polynomials as a tool for approximating exponential functions.
We focus particularly on Scaling and Squaring Method based on Composite Taylor Polynomials to approximate matrix exponential. We carry out some original research especially in the later sections of the dissertation. All the code related to the report is provided in this repository, however, the actual report is not attached yet. Once it is assessed, we will upload it here (if permission is given).
An interactive jupyter notebooks can be seen here (in HTML):
- ContourPlots: https://htmlpreview.github.io/?https://github.com/anmolaggarwal98/Scaling-and-Squaring-Method-Dissertation/blob/master/Code/Contour_plots.html
- Error Analysis Lines: https://htmlpreview.github.io/?https://github.com/anmolaggarwal98/Scaling-and-Squaring-Method-Dissertation/blob/master/Code/Error_Analysis_curves.html
- Matrix Exponential Approximation with SSM: https://htmlpreview.github.io/?https://github.com/anmolaggarwal98/Scaling-and-Squaring-Method-Dissertation/blob/master/Code/Matrix_error_analysis.html
- Oral Presentation https://htmlpreview.github.io/?https://github.com/anmolaggarwal98/Scaling-and-Squaring-Method-Dissertation/blob/master/Code/Presentation.html
It is possible that these link do not run since I have to make everything anonymous. Hence, the jupyter notebook files are also available in folder named Code. There respective graphs are available in the folder plots which might also not open because of the same anonymous reason.
Please feel free to download the code and run it on your local machine so you can tweak different parameters.
We hope you enjoy it.