## Functions of Matrices and Exponential Integrators

by N. Higham and M. Hochbruck

Functions of matrices are widely used in science, engineering and the social sciences, due to the succinct and insightful way they allow problems to be formulated and solutions to be expressed. New applications involving matrix functions are regularly being found, ranging from small but difficult problems in medicine to huge, sparse systems arising in exponential integrators for the solution of partial differential equations. This course will treat the underlying theory of matrix functions, describe a variety of algorithms, give an overview of some applications, and briefly discuss software issues. It will also treat exponential integrators, giving an introduction to the topic from theory to software and emphasizing the use of matrix functions in the methods.

An outline for the part on matrix functions is as follows:

- theory of matrix functions
- small scale problems
- large scale problems
- approximation f(A)b by (rational) Krylov methods and contour integration
- applications
- software issues.

For exponential integrators, the following topics will be covered:

- construction of exponential integrators: basic principles
- error analysis in a framework suitable for pdes
- efficient implementation of the matrix functions arising in exponential integrators
- applications
- software issues.

The two parts will be closely linked together, providing synergy between the linear algebra and the differential equations. We will offer theoretical as well as programming tutorials, so that at the end of these two parts, the students should be able to solve time dependent odes and pdes by simple versions of exponential integrators.

*Contact email address: summerschool AT fudan.edu.cn*