E i g e n v a l u e C l i n i c
VIGRE Seminar, CAAM 699.010
Monday 11am12pm (DH 2014)
The Eigenvalue Clinic is a weekly workshop where participants pose
research problems related to applications of spectral theory in
differential equations, numerical linear algebra, and control theory.
This semester, we are methodically working through the steps one makes
when moving from a physical application (e.g., vibration or stability problem)
to numerically computed eigenvalues.
For more information, see the syllabus
or contact Mark Embree.

Spring 2013
Reid Atcheson,
Jonathan Baker,
Russell Carden,
Mark Embree,
Jeffrey Hokanson,
Cosmin Ionita,
Paul Munger,
Charles Puelz
11. 
April 10: Relative accuracy in eigenvalue computations
 lapacc.m: Laplacian example (finite differences; bad!)
 beamacc.m: beam example (finite differences; even worse!)
 sm_lapacc.m: Laplacian example (spectral method; better), needs chebab.m from Trefethen
 sm_beamacc.m: beam example (spectral method; better?), needs chebab.m from Trefethen
 Demmel and Veselić, Jacobi's method is more accurate than QR, 1992
 Ye, Relative perturbation bounds for eigenvalues ... , 2009

 
 
10. 
Apr 1: Similarity transformations for improving eigenvalue computations
 morgan_ex.m: the perils of balancing, an example from Ron Morgan
 Parlett and Reinsch, Balancing a matrix for calculation of eigenvalues and eigenvectors, 1969.
 Higham, Mackey, Tisseur, Garvey,
Scaling, sensitivity and stability in the numerical
solution of QEPs, 2008

 
 
9. 
Mar 25: Arnoldi convergence for operators
 arnoldi_laplace.nb: inverse Arnoldi method for the Laplacian (Mathematica)
 arnoldi_advdiff_gf.nb: inverse Arnoldi method for an advection diffusion operator (Mathematica)
 cf_laplace.nb: inverse Arnoldi method for the Laplacian (Chebfun)
 cf_advdiff.nb: inverse Arnoldi method for an advection diffusion operator (Chebfun)
 The Chebfun software
Note: the advectiondiffusion examples do not perform well!

 
 
8. 
Mar 18: Pseudospectra and convergence rates for linear solvers
 Spectra and Pseudospectra: Trefethen and Embree, 2005.

 
 
7. 
Mar 11: Eigenvalue computation for nonselfadjoint operators
 make_davies.m: Chebyshev pseudospectral discretization of Davies' anharnmoic oscillator (and chebab.m) from Trefethen

 
 
6. 
Mar 4: An example of classical convergence theory
 Atkinson, Convergence rates for approximate eigenvalues
of compact integral operators, 1975
Some supporting work:
 Atkinson, The numerical solution of the eigenvalue problem
for compact integral operators, 1967
 Anselone, Collectively
compact operator approximations, 1967
See also the work of John Osborn, e.g.:
 Osborn, Spectral approximation for compact operators, 1975
Some MATLAB examples for an integral equation from laser theory:
 laser_trap.m: Nystrom discretization via the trapezoid rule
 laser_cc.m: Nystrom discretization via the ClenshawCurtis rule (clencurtab.m from Trefethen)

 
 
5. 
Feb 18: Spectral pollution, continued
 Davies, Wild spectral behavior of anharmonic oscillators, 1999

 
 
4. 
Feb 11: Spectral pollution
 Davies and Plum, Spectral pollution, 2004
 Deift and Hempel, On the existence of eigenvalues of
the Schrödinger operator..., 1986
 slides from Lyonell Boulton
(HeriotWatt University, Edinburgh)
 pollute.m: demonstration with a multiplication operator

 
 
3. 
Feb 4: Basics of spectral theory
 Basic results can be found in most functional analysis texts,
elementary presentations:
N. Young,
E. Kreyszig
see also:
Reed and Simon,
Davies,
Edmunds and Evans
 ex1trap.m: discretization of a compact integral operator

 
 
2. 
Jan 28: Nonlinear eigenvalue problems
 Michiels
and Niculescu on eigenvalue problems in delay differential operators
 nlcont.m, nlpoly.m: approximate
nonlinear EVP with polynomial EVP
 Betcke, Higham, Mehrmann, Schröder and Tisseur
NLEVP: A Collection of Nonlinear Eigenvalue Problems
Download the collection here.
 Higham, Mackey, and Tisseur,
The conditioning of linearizations of matrix polynomials, 2007
 Higham, Mackey, Tisseur, Garvey,
Scaling, sensitivity and stability in the numerical
solution of QEPs, 2008

 
 
1. 
Jan 21: From vibrations to eigenvalue problems
 Antman, Nonlinear Problems in Elasticity, 2005
 Three discretizations of damped wave operators:
makeAconst.m: viscous damping
makeAkvconst.m: KelvinVoigt damping
makeAmag.m: magnetic damping
chebab.m, clencurtab.m:
auxiliary routines adapted from Trefethen

