{"id":{"repo_id":"vt","oai_identifier":"oai:vtechworks.lib.vt.edu:10919/84924"},"canonical_url":"https://search.dev.ndltd.org/etd/vt/oai:vtechworks.lib.vt.edu:10919/84924","repository":{"repo_id":"vt","name":"Virginia Tech","base_url":"https://vtechworks.lib.vt.edu/oai/request"},"display":{"title":"Rational Interpolation Methods for Nonlinear Eigenvalue Problems","abstract":"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.","abstract_html":"This thesis investigates the numerical treatment of nonlinear eigenvalue problems. These problems are defined by the condition $T(lambda) v = boldsymbol{0}$, with <span class=\"etd-inline-math\">T: C to C<sup>n times n</sup></span>, where we seek to compute the scalar-vector pairs, $lambda in C$ and nonzero <span class=\"etd-inline-math\"> v in C<sup>n</sup></span>. 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 <span class=\"etd-inline-math\">T(z)<sup>-1</sup></span>, 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&#x27;s method for a particular scalar function.","abstract_has_math":true,"creators":["Brennan, Michael C."],"institution":"Virginia Tech","degree_name":"Master of Science","degree_level":"masters","degree_discipline":"Mathematics","degree_department":"Mathematics","school":null,"contributors":[],"advisors":[],"committee_chairs":["Gugercin, Serkan"],"committee_members":["Beattie, Christopher A.","Embree, Mark P."],"year":2018,"date_issued":"2018-08-27","date_published":"2018-08-27","updated_at":"2026-07-22T22:19:03Z","subjects":["Nonlinear Eigenvalue Problems","Contour Integration Methods","Iterative Methods","Dynamical Systems"],"languages":[],"rights":["In Copyright"],"rights_urls":["http://rightsstatements.org/vocab/InC/1.0/"],"identifier_entries":[{"key":"dc:identifier.other","label":"Dc Identifier Other","values":["vt_gsexam:16856"],"render_values":[{"text":"vt_gsexam:16856","href":null,"code":true}]}]},"links":{"outbound_url":"http://hdl.handle.net/10919/84924","outbound_label":"Handle","outbound_source":"dc:identifier.uri"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor.committeechair","label":"Committee Chair","values":["Gugercin, Serkan"]},{"key":"dc:contributor.committeemember","label":"Committee Member","values":["Beattie, Christopher A.","Embree, Mark P."]},{"key":"dc:contributor.department","label":"Department","values":["Mathematics"]},{"key":"dc:creator","label":"Author","values":["Brennan, Michael C."]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.accessioned","label":"Dc Date Accessioned","values":["2018-08-28T08:00:45Z"]},{"key":"dc:date.available","label":"Dc Date Available","values":["2018-08-28T08:00:45Z"]},{"key":"dc:date.issued","label":"Date","values":["2018-08-27"]},{"key":"dc:publisher","label":"Institution","values":["Virginia Tech"]},{"key":"dc:type","label":"Dc Type","values":["Thesis"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Mathematics"]},{"key":"thesis:degree_level","label":"Degree Level","values":["masters"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Master of Science"]},{"key":"thesis:institution_name","label":"Thesis Institution Name","values":["Virginia Polytechnic Institute and State University"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Nonlinear Eigenvalue Problems","Contour Integration Methods","Iterative Methods","Dynamical Systems"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:rights","label":"Dc Rights","values":["In Copyright"]},{"key":"dc:rights.uri","label":"Rights URI","values":["http://rightsstatements.org/vocab/InC/1.0/"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier.other","label":"Dc Identifier Other","values":["vt_gsexam:16856"]},{"key":"dc:identifier.uri","label":"Identifier URI","values":["http://hdl.handle.net/10919/84924"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["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."]},{"key":"dc:description.abstractgeneral","label":"General Abstract","values":["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."]},{"key":"dc:description.degree","label":"Dc Description Degree","values":["Master of Science"]},{"key":"dc:format.medium","label":"Dc Format Medium","values":["ETD"]},{"key":"dc:title","label":"Title","values":["Rational Interpolation Methods for Nonlinear Eigenvalue Problems"]}]}],"canonical_facts":{"dc:contributor.committeechair":["Gugercin, Serkan"],"dc:contributor.committeemember":["Beattie, Christopher A.","Embree, Mark P."],"dc:contributor.department":["Mathematics"],"dc:creator":["Brennan, Michael C."],"dc:date.accessioned":["2018-08-28T08:00:45Z"],"dc:date.available":["2018-08-28T08:00:45Z"],"dc:date.issued":["2018-08-27"],"dc:description.abstract":["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."],"dc:description.abstractgeneral":["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."],"dc:description.degree":["Master of Science"],"dc:format.medium":["ETD"],"dc:identifier.other":["vt_gsexam:16856"],"dc:identifier.uri":["http://hdl.handle.net/10919/84924"],"dc:publisher":["Virginia Tech"],"dc:rights":["In Copyright"],"dc:rights.uri":["http://rightsstatements.org/vocab/InC/1.0/"],"dc:subject":["Nonlinear Eigenvalue Problems","Contour Integration Methods","Iterative Methods","Dynamical Systems"],"dc:title":["Rational Interpolation Methods for Nonlinear Eigenvalue Problems"],"dc:type":["Thesis"],"thesis:degree_discipline":["Mathematics"],"thesis:degree_level":["masters"],"thesis:degree_name":["Master of Science"],"thesis:institution_name":["Virginia Polytechnic Institute and State University"]},"updated_at":"2026-07-22T22:19:03Z"}