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Fast and approximate computation of Laplace and Fourier transforms / Schnelle und approximative Berechnung von Laplace- und Fourier-TransformationenMelzer, Ines 04 April 2016 (has links)
In this thesis, we treat the computation of transforms with asymptotically smooth and oscillatory kernels. We introduce the discrete Laplace transform in a modern form including a generalization to more general kernel functions. These more general kernels lead to specific function transforms. Moreover, we treat the butterfly fast Fourier transform. Based on a local error analysis, we develop a rigorous error analysis for the whole butterfly scheme. In the final part of the thesis, the Laplace and Fourier transform are combined to a fast Fourier transform for nonequispaced complex evaluation nodes. All theoretical results on accuracy and computational complexity are illustrated by numerical experiments.
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The condition number of Vandermonde matrices and its application to the stability analysis of a subspace method / Die Konditionzahl von Vandermondematrizen und ihre Anwendung für die Stabilitätsanalyse einer UnterraummethodeNagel, Dominik 19 March 2021 (has links)
This thesis consists of two main parts. First of all, the condition number of rectangular Vandermonde matrices with nodes on the complex unit circle is studied. The first time quantitative bounds for the extreme singular values are proven in the multivariate setting and when nodes of the Vandermonde matrix form clusters. In the second part, an optimized presentation of the deterministic stability analysis of the subspace method ESPRIT is given and results from the first part are applied.
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