E i g e n v a l u e C l i n i c
VIGRE Seminar, CAAM 699.010
Monday 11am-12pm (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.
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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
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| 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
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| 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 advection-diffusion examples do not perform well!
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| 8. |
Mar 18: Pseudospectra and convergence rates for linear solvers
- Spectra and Pseudospectra: Trefethen and Embree, 2005.
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| 7. |
Mar 11: Eigenvalue computation for non-self-adjoint operators
- make_davies.m: Chebyshev pseudospectral discretization of Davies' anharnmoic oscillator (and chebab.m) from Trefethen
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| 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 Clenshaw-Curtis rule (clencurtab.m from Trefethen)
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| 5. |
Feb 18: Spectral pollution, continued
- Davies, Wild spectral behavior of anharmonic oscillators, 1999
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| 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
(Heriot-Watt University, Edinburgh)
- pollute.m: demonstration with a multiplication operator
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| 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
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| 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
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| 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: Kelvin-Voigt damping
makeAmag.m: magnetic damping
chebab.m, clencurtab.m:
auxiliary routines adapted from Trefethen
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