Lloyd N. Trefethen
PhD Stanford University, 1982

My field is numerical analysis or scientific computing: the study of constructive methods for solving the problems of continuous applied mathematics. Areas of particular interest over the years have been numerical conformal mapping, approximation theory, finite difference and spectral methods for partial differential equations, linear algebra, and applications in fluid mechanics.

One ongoing project, with graduate student Toby Driscoll, concerns solution of conformal mapping and eigenvalue problems on polygonal regions. Driscoll has developed a Matlab Toolbox for Schwarz-Christoffel mapping and algorithms for computing high-precision eigenmodes for regions such as those that arise in the problem, "Can one hear the shape of a drum?"

Another on-going project, with graduate students Arun Verma and Divakar Viswanath, concerns the stability of Gaussian elimination with partial pivoting. By focusing on column spaces of matrices, we have obtained the first theorems that explain, at least in part, why Gaussian elimination is resoundingly stable in practice even though it is unstable for certain matrices.

My central topic of research in recent years has been the analysis and algorithmic treatment of non-normal matrices and operators—matrices and operators whose eigenvectors are not orthogonal. The theme of this work is an attempt to go "beyond eigenvalues" in treating these problems, since eigenvalue analysis frequently fails. Currently I am involved in two areas of application. One, jointly with graduate student Kim-chuan Toh, is the study of iterative methods in linear algebra. How can we imitate for nonsymmetric matrices the successes of conjugate gradient and Lanczos iterations in the symmetric case? The second, jointly with graduate student Jeff Baggett, is the study of transition to turbulence of fluid flows. Our work on non-normality, together with that of colleagues at Cornell and elsewhere, is leading to a new understanding of why it is that flow of water in a pipe, for example, invariably becomes turbulent if the speed is high enough.

Together with ex-graduate student David Bau, I am writing a graduate textbook, "Numerical Linear Algebra", to be published by SIAM in 1996.

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Last modified: 26 November 1995 by Denise Moore (denise@cs.cornell.edu).