# Functions of Matrices: Theory and Computation

## Final Report Summary - MATFUN (Functions of Matrices: Theory and Computation)

This project was about making advances in theory and algorithms for matrix functions, which are mathematical tools finding increasing use in science, engineering and the social sciences. An important example of a matrix function is the matrix exponential.

We have obtained new theoretical results about many aspects of matrix functions. For example we obtained new results on the existence and characterization of roots of stochastic matrices and introduced the unwinding matrix function and showed how it aids treatment of identities involving multivalued functions, such as the matrix logarithm and fractional matrix powers. New results have also been obtained on Frechet derivatives of matrix functions, including their Jordan canonical forms, existence and continuity of higher derivatives, and relations between the level-1 and level-2 condition numbers of matrix functions.

Algorithms have been derived for computing matrix functions and their Frechet derivatives that are either completely new or improve on existing algorithms. These include:
- Matrix logarithm and its Frechet derivative.
- Inverse trigonometric and inverse hyperbolic functions.
- Matrix Lambert W function.
- Arbitrary real matrix power and its Frechet derivative.
- Matrix unwinding function and argument reduction for the matrix exponential.
- Blocked algorithms for the matrix square root.
- Action of the matrix exponential on a vector.
- Spectral divide and conquer algorithms for the symmetric eigenvalue
decomposition and the SVD.

Most of the new algorithms have been made available as open source MATLAB code, and are now being widely in academia and in industry.