Lloyd N. Trefethen
Professor
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 SchwarzChristoffel mapping and
algorithms for computing highprecision eigenmodes for regions such as
those that arise in the problem, "Can one hear the shape of a drum?"
Another ongoing 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 nonnormal matrices and operatorsÑmatrices 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 Kimchuan 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 nonnormality, 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 exgraduate student David Bau, I am writing a graduate
textbook, "Numerical Linear Algebra", to be published by SIAM in 1996.
University Activities
 Graduate Faculty Representative, Computer Science Department
 Sage Fellowship Committee
 College of Engineering Teaching Awards Committee
Professional Activities
 Editorial Board: SIAM Review, SIAM Journal on Numerical Analysis,
Journal of Computational and Applied Mathematics, Japan Journal
of Industrial and Applied Mathematics, Numerische Mathematik
 Board of Directors, ACM SIGNUM (Special Interest Group on Numerical
Mathematics)
Lectures
 What eigenvalues don't tell you. Dynamical Systems Colloquium,
University of Bremen, Bremen, Germany, July 1994.
 ___. CWI, Amsterdam, Netherlands, July 1994.
 Hydrodynamic stability without eigenvalues. MaxPlanck Institute
for Plasma Physics, Garching, Germany, July 1994.
 ___. Swiss Federal Institute of Technology, Lausanne, Switzerland,
July 1994.
 Evolution processes: linear and nonlinear, normal and nonnormal,
stable and unstable. Invited lecture, SIAM Annual Meeting,
San Diego, July 1994.
 Nonnormal operators in applied mathematics. Conference on Applications
of Operator Theory, Winnipeg, Canada, October 1994.
 Why Gaussian elimination works even though it is unstable. Invited
lecture, Seaway Section Meeting, Mathematical Association of America,
November 1994.
 ___. Computation and Complexity Seminar, IBM TJ Watson Research Center,
November 1994.
 Spectra, pseudospectra, and evolution problems. Invited lecture,
Workshop on Krylov Methods, North Carolina State University, Raleigh,
North Carolina, March 1995.
 Approximating pseudospectra by the Arnoldi iteration. Invited lecture,
Workshop on Krylov Methods, North Carolina State University,
Raleigh, North Carolina, March 1995.
 ___. 16th Biennial Conference on Numerical Analysis, Dundee, Scotland,
June 1995.
 Linearnonlinear interactions in transition to turbulence.
Invited lecture, 39th Meeting, Society for Natural Philosophy,
Blacksburg, Virginia, April 1995.
 Computing eigenmodes of isospectral drums. Mathematics Colloquium,
New Jersey Institute of Technology, Newark, New Jersey, April 1995.
 Spectra and pseudospectra for pipe and channel flows. SIAM Conference
on Applications of Dynamical Systems, Snowbird, Utah, May 1995.
Publications
 Pseudozeros of polynomials and pseudospectra of companion matrices.
Numerische Mathematik 68 (1994), 403425 (with K. Toh).
 Pseudospectra of the convectiondiffusion operator. SIAM Journal on
Applied Mathematics 54 (1994), 16341649 (with S. C. Reddy).
 A mostly linear model of transition to turbulence. Physics of
Fluids A 7 (1995), 833838 (with J. S. Baggett and T. A. Driscoll).
Software
 SchwarzChristoffel Conformal Mapping Matlab Toolbox, with graduate
student Toby Driscoll
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