Rational Interpolation Methods for Nonlinear Eigenvalue Problems

This thesis investigates the numerical treatment of nonlinear eigenvalue problems. These problems are defined by the condition $T(lambda)v=boldsymbol0$, with $T:CtoCntimesn$, where we seek to compute the scalar-vector pairs, $lambdainC$ 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.