This research project is concerned with the development and analysis of novel methods for the approximation of matrix functions and integrals defined by matrix functions. The research blends linear algebra and approximation theory. New rational Lanczos methods for Hermitian matrices with short recursion formulas are being developed. The existence of short recursion relations is of significant theoretical and practical interest. It speeds up the computations because each new rational Lanczos vector only has to be orthogonalized against a few of the most recently computed Lanczos vectors. The number of required explicit orthogonalizations is bounded independently of the number of rational Lanczos steps. The recursion relations define the orthogonal projection of a given matrix onto a rational Krylov subspace. The projection is represented by a pentadiagonal matrix, which also has a block structure. This matrix is analogous to the symmetric tridiagonal matrix associated with a (standard) Gauss quadrature rule, and is the basis for new algorithms for the evaluation of rational Gauss, Gauss-Radau rules, and Gauss-Lobatto quadrature rules. The research is an extension of the very important work by Gene Golub, and his collaborators, on matrices, moments, orthogonal polynomials, and quadrature. There may be connections to the structured matrices, such as the CMV-matrices, which arise in the context of orthogonal polynomials and orthogonal rational functions on the unit circle, as well as to to semi- and quasi-separable matrices. When the given matrix is non-Hermitian, rational Arnoldi and rational non-Hermitian Lanczos processes can be applied. The projections onto the rational Krylov subspace again have a structure, the exploitation of which is an important topic in linear algebra. The need to estimate matrix functionals arises in many applications, including the investigation of social networks and in Tikhonov regularization of ill-posed inverse problems.
The principal investigator is studying developments of faster numerical methods for the evaluation, bounding or estimation of complicated nonlinear expressions that involve large symmetric or nonsymmetric matrices. The gain in speed is achieved by exploiting structure that until now has been ignored. The development of fast methods is important when solving large-scale problems of interest to scientists and engineers. These methods are applicable in the investigation of social networks, whose properties recently have received considerable attention, not only by scientists, but also by the New York Times. A major hurdle in the investigation of social networks is their large size. The research provides improved tools for investigating these kinds of networks. The methods can also be applied in solution methods for Maxwell's equation in three space-dimensions. Essentially, one has to compute the exponential of very large matrices. These matrices are much too large to allow the use of standard software. The methods to be developed within the framework of this proposal speed up the computations by exploiting inherent structure of these problems. Work on the problems of this proposal requires background in linear algebra, orthogonal polynomials and rational functions, approximation theory, and Gauss-type quadrature. Hence the research is well suited for doctoral students working on this project.