Numerical linear algebra for functions of matrices

The proposed research project aims to address the challenge of efficiently and reliably computing matrix functions for large matrices with particular structures.

Matrix functions are essential in a variety of fields, including natural science, systems and control, signal processing, data science, and quantum chemistry.

The project will develop new and efficient algorithms with a focus on evaluating polynomials and rational functions using a minimal number of matrix multiplications and left divisions.

To achieve this, the project will characterize a complete set of approximations and develop novel approximation theory methods that generalize classical methods. The project will also provide error analysis for both approximation and rounding errors.

Furthermore, the project will explore efficient and reliable algorithms for specific matrix structures, including hierarchical matrices and sparse matrices with decay properties. The developed algorithms will be publicly available as software for use by researchers in the relevant fields.