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Numerical Methods for Structured Matrix Factorizations13 June 2001 (has links)
This thesis describes improvements of the periodic QZ algorithm and several variants of the Schur algorithm for block Toeplitz matrices.
Documentation of the available software is included.
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Preconditioned iterative methods for a class of nonlinear eigenvalue problemsSolov'ëv, Sergey I. 31 August 2006 (has links)
In this paper we develop new preconditioned
iterative methods for solving monotone nonlinear
eigenvalue problems. We investigate the convergence
and derive grid-independent error estimates for
these methods. Numerical experiments demonstrate
the practical effectiveness of the proposed methods
for a model problem.
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Largest eigenvalues of the discrete p-Laplacian of trees with degree sequencesBiyikoglu, Türker, Hellmuth, Marc, Leydold, Josef 08 November 2018 (has links)
Trees that have greatest maximum p-Laplacian eigenvalue among all trees with a given degree sequence are characterized. It is shown that such extremal trees can be obtained by breadth-first search where the vertex degrees are non-increasing. These trees are uniquely determined up to isomorphism. Moreover, their structure does not depend on p.
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Attractors of autoencoders : Memorization in neural networks / Attractors of autoencoders : Memorization in neural networksStrandqvist, Jonas January 2020 (has links)
It is an important question in machine learning to understand how neural networks learn. This thesis sheds further light onto this by studying autoencoder neural networks which can memorize data by storing it as attractors.What this means is that an autoencoder can learn a training set and later produce parts or all of this training set even when using other inputs not belonging to this set. We seek out to illuminate the effect on how ReLU networks handle memorization when trained with different setups: with and without bias, for different widths and depths, and using two different types of training images -- from the CIFAR10 dataset and randomly generated. For this, we created controlled experiments in which we train autoencoders and compute the eigenvalues of their Jacobian matrices to discern the number of data points stored as attractors.We also manually verify and analyze these results for patterns and behavior. With this thesis we broaden the understanding of ReLU autoencoders: We find that the structure of the data has an impact on the number of attractors. For instance, we produced autoencoders where every training image became an attractor when we trained with random pictures but not with CIFAR10. Changes to depth and width on these two types of data also show different behaviour.Moreover, we observe that loss has less of an impact than expected on attractors of trained autoencoders.
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Diagonal Entry Restrictions in Minimum Rank Matrices, and the Inverse Inertia and Eigenvalue Problems for GraphsNelson, Curtis G. 11 June 2012 (has links) (PDF)
Let F be a field, let G be an undirected graph on n vertices, and let SF(G) be the set of all F-valued symmetric n x n matrices whose nonzero off-diagonal entries occur in exactly the positions corresponding to the edges of G. Let MRF(G) be defined as the set of matrices in SF(G) whose rank achieves the minimum of the ranks of matrices in SF(G). We develop techniques involving Z-hat, a process termed nil forcing, and induced subgraphs, that can determine when diagonal entries corresponding to specific vertices of G must be zero or nonzero for all matrices in MRF(G). We call these vertices nil or nonzero vertices, respectively. If a vertex is not a nil or nonzero vertex, we call it a neutral vertex. In addition, we completely classify the vertices of trees in terms of the classifications: nil, nonzero and neutral. Next we give an example of how nil vertices can help solve the inverse inertia problem. Lastly we give results about the inverse eigenvalue problem and solve a more complex variation of the problem (the λ, µ problem) for the path on 4 vertices. We also obtain a general result for the λ, µ problem concerning the number of λ’s and µ’s that can be equal.
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Soil-Structure Interaction of Pile Groups for High-Speed Railway BridgesStrand, Tommy, Severin, Johannes January 2018 (has links)
No description available.
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Finite Geometry Correction Factors for the Stress Field and Stress Intensities at Transverse Fillet WeldsRiggenbach, Kane Ryan 27 August 2012 (has links)
No description available.
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On the Hermitian Geometry of k-Gauduchon Orthogonal Complex StructuresKhan, Gabriel Jamil Hart 24 September 2018 (has links)
No description available.
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Generalized eigenvalue problem and systems of differential equations: Application to half-space problems for discrete velocity modelsEsinoye, Hannah Abosede January 2024 (has links)
In this thesis, we study the relationship between the generalized eigenvalue problem (GEP) $Ax=\lambda Bx$, and systems of differential equations. We examine both the Jordan canonical form and Kronecker's canonical form (KCF). The first part of this work provides an introduction to the fundamentals of generalized eigenvalue problems and methods for solving this problem. We discuss the QZ algorithm, which can be used to determine the generalized eigenvalues and also how it can be implemented on MATLAB with the built in function 'eig'. One essential facet of this work is the exploration of symmetric matrix pencils, which arise when A and B are both symmetric matrices. Furthermore we discuss discrete velocity models (DVMs) focusing specifically on a 12-velocity model on the plane. The results obtained are then applied to half-space problems for discrete velocity models, with a focus on planar stationary systems.
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The role of three-body forces in few-body systemsMasita, Dithlase Frans 25 August 2009 (has links)
Bound state systems consisting of three nonrelativistic particles are numerically
studied. Calculations are performed employing two-body and three-body forces as
input in the Hamiltonian in order to study the role or contribution of three-body
forces to the binding in these systems. The resulting differential Faddeev equations
are solved as three-dimensional equations in the two Jacobi coordinates and the
angle between them, as opposed to the usual partial wave expansion approach. By
expanding the wave function as a sum of the products of spline functions in each of
the three coordinates, and using the orthogonal collocation procedure, the equations
are transformed into an eigenvalue problem.
The matrices in the aforementioned eigenvalue equations are generally of large order.
In order to solve these matrix equations with modest and optimal computer memory
and storage, we employ the iterative Restarted Arnoldi Algorithm in conjunction
with the so-called tensor trick method. Furthermore, we incorporate a polynomial
accelerator in the algorithm to obtain rapid convergence. We applied the method
to obtain the binding energies of Triton, Carbon-12, and Ozone molecule. / Physics / M.Sc (Physics)
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