This thesis investigates the numerical treatment of nonlinear eigenvalue problems. These problems are defined by the condition $T(lambda) v = boldsymbol{0}$, with $T: C to C^{n times n}$, where we seek to compute the scalar-vector pairs, $lambda in C$ and nonzero $ v in C^{n}$. The first contribution of this work connects recent contour integration methods to the theory and practice of system identification. This observation leads us to explore rational interpolation for system realization, producing a Loewner matrix contour integration technique. The second development of this work studies the application of rational interpolation to the function $T(z)^{-1}$, where we use the poles of this interpolant to approximate the eigenvalues of $T$. We then expand this idea to several iterative methods, where at each step the approximate eigenvalues are taken as new interpolation points. We show that the case where one interpolation point is used is theoretically equivalent to Newton's method for a particular scalar function. / Master of Science / This thesis investigates the numerical treatment of nonlinear eigenvalue problems. The solutions to these problems often reveal characteristics of an underlying physical system. One popular methodology for handling these problems uses contour integrals to compute a set of the solutions. The first contribution of this work connects these contour integration methods to the theory and practice of system identification. This leads us to explore other techniques for system identification, resulting in a new method.
Another common methodology approximates the nonlinear problem directly. The second development of this work studies the application of rational interpolation for this purpose. We then use this idea to form several iterative methods, where at each step the approximate solutions are taken to be new interpolation points. We show that the case where one interpolation point is used is theoretically equivalent to Newton’s method for a particular scalar function.
| Identifer | oai:union.ndltd.org:VTETD/oai:vtechworks.lib.vt.edu:10919/84924 |
| Date | 27 August 2018 |
| Creators | Brennan, Michael C. |
| Contributors | Mathematics, Gugercin, Serkan, Beattie, Christopher A., Embree, Mark P. |
| Publisher | Virginia Tech |
| Source Sets | Virginia Tech Theses and Dissertation |
| Detected Language | English |
| Type | Thesis |
| Format | ETD, application/pdf |
| Rights | In Copyright, http://rightsstatements.org/vocab/InC/1.0/ |
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