# Research in Numerical Linear Algebra

Numerical Linear Algebra is a flourishing research area, with worldwide activity including theoretical work, development of algorithms (for both serial and parallel computers) and software development. Our work is in all these areas, with current efforts focusing on generalized and quadratic eigenvalue problems and associated pseudospectra, analysis and computation of matrix functions, and analysis of the accuracy and numerical stability of algorithms.

The quadratic eigenvalue problem (QEP) has the form

λ^{2} M x + λ C x + K x = 0,

where M, C, K are n x n matrices. If x is nonzero then λ is an eigenvalue and x is the corresponding right eigenvector. A major application area for QEPs is in finite element modelling of structures such as bridges and automobiles, wherein the eigenvalues determine the stability of the structure (see, for example, The Millennium Bridge). Few numerical methods are available for solving the QEP in its original form, so the development of numerical methods is an active research topic. Peudospectra are sets of 'approximate eigenvalues' of QEPs and can be used to gain insight into eigenvalue sensitivity and stability. For example, the following plot is of pseudospectra of the Orr-Sommerfeld equation in a region containing the first few modes, for the Reynolds number R = 5772; it provides information about the spatial stability of the associated flow. The plot was computed in MATLAB using techniques developed by the group.

Other recent work has focused on matrix condition number estimation, the computation of matrix functions, and the solution of least squares problems. Software has been developed for the LAPACK Fortran linear algebra library and for MATLAB (e.g., functions condest and funm in MATLAB, and the Matrix Computation Toolbox).

Books include Accuracy and Stability of Numerical Algorithms (second edition, SIAM, 2002) and MATLAB Guide, (second edition, SIAM, 2005), and LAPACK Users' Guide (third edition, SIAM, 1999).

This work is supported mainly by research grants from the EPSRC, with studentship support also from NAG Ltd. An article *Mathematicians Make Tools for Industry* describing some of this work appeared in EPSRC's Newsline magazine, July 2001.

An interesting example of how research in numerical analysis can have an impact on the “real world” is the inclusion of translated versions of several LAPACK routines in the HP 48G calculator (512K ROM, multi-line LCD screen). The keypresses MTH MATR- NORM-COND invoke a condition estimation algorithm developed here at Manchester.

A variety of PhD and postdoctoral positions are available in the group. See the page Numerical Linear Algebra Group: Opportunities page for current details.