Many methods for computing eigenvalues of a large sparse matrix involve shift-invert transformations which require the solution of a shifted linear system at each step. This thesis deals with shift-invert iterative techniques for solving eigenvalue problems where the arising linear systems are solved inexactly using a second iterative technique. This approach leads to an inner-outer type algorithm. We provide convergence results for the outer iterative eigenvalue computation as well as techniques for efficient inner solves. In particular eigenvalue computations using inexact inverse iteration, the Jacobi-Davidson method without subspace expansion and the shift-invert Arnoldi method as a subspace method are investigated in detail. A general convergence result for inexact inverse iteration for the non-Hermitian generalised eigenvalue problem is given, using only minimal assumptions. This convergence result is obtained in two different ways; on the one hand, we use an equivalence result between inexact inverse iteration applied to the generalised eigenproblem and modified Newton's method; on the other hand, a splitting method is used which generalises the idea of orthogonal decomposition. Both approaches also include an analysis for the convergence theory of a version of inexact Jacobi-Davidson method, where equivalences between Newton's method, inverse iteration and the Jacobi-Davidson method are exploited. To improve the efficiency of the inner iterative solves we introduce a new tuning strategy which can be applied to any standard preconditioner. We give a detailed analysis on this new preconditioning idea and show how the number of iterations for the inner iterative method and hence the total number of iterations can be reduced significantly by the application of this tuning strategy. The analysis of the tuned preconditioner is carried out for both Hermitian and non-Hermitian eigenproblems. We show how the preconditioner can be implemented efficiently and illustrate its performance using various numerical examples. An equivalence result between the preconditioned simplified Jacobi-Davidson method and inexact inverse iteration with the tuned preconditioner is given. Finally, we discuss the shift-invert Arnoldi method both in the standard and restarted fashion. First, existing relaxation strategies for the outer iterative solves are extended to implicitly restarted Arnoldi's method. Second, we apply the idea of tuning the preconditioner to the inner iterative solve. As for inexact inverse iteration the tuned preconditioner for inexact Arnoldi's method is shown to provide significant savings in the number of inner solves. The theory in this thesis is supported by many numerical examples.

In this thesis, we present some contributions to the Markov jump linear systems theory in a context of partial information on the Markov chain. We consider that the state of the Markov chain cannot be measured, but instead there is only an observed variable that could model an asynchronous phenomenon between the application and the plant, or a simple fault detection and isolation device. In this formulation, we investigate the problem of designing controllers and filters depending only on the observed variable in the context of H2, H?, and mixed H2/H? control theory. Numerical examples and academic applications are presented for active-fault tolerant control systems and networked control systems. / Nesta tese, apresentamos algumas contribuições para a teoria de sistemas lineares com saltos markovianos em um contexto de observação parcial da cadeia de Markov. Consideramos que o estado da cadeia de Markov não pode ser medido, porém existe uma variável observada que pode modelar um fenômeno assíncrono entre a aplicação e a planta, ou ainda um dispositivo de detecção de falhas simples. Através desse modelo, investigamos o problema da síntese de controladores e filtros que dependem somente da variável observada no contexto das teorias de controle H2, H?, e misto H2/H?. Exemplos numéricos e aplicações acadêmicas são apresentadas no âmbito dos sistemas de controle tolerantes a falhas e dos sistemas de controle através da rede.

Made available in DSpace on 2017-07-21T20:56:31Z (GMT). No. of bitstreams: 1 AnaElizaGFerreira2.pdf: 1359078 bytes, checksum: 9cfb391fa1094682956d06431e9b36fd (MD5) Previous issue date: 2013-02-28 / This work aims to show the importance of the study of linear systems in High School.Therefore, we surveyed prevous works that were already expressing concern with this order. We studied the historical emergence of linear systems in the development of mathematics, the definition and concepts in textbooks for High School and also in the literature of Numerical Calculus and Linear Algebra. There was a survey of methods for solving linear systems, stood out Gaussian elimination in direct methods and the methods of Jacobi and Gauss-Seidel between the iterative. Reflect the conditioning of linear systems and approximation of solution for incompatible systems that are usually not covered in the classroom and have great practical utility in applications. Observe curriculum guidelines on the topic confronting the approach of textbooks. It was found that the literature limits the content of safe handling Algebraic and does not offer diverse activities with use of computer resources and examples depth and clarifying consistent with the importance of the content. Shows an approach of linear systems that interrelates the content of linear systems with Analytic Geometry so offer to pupils able to visualize the behavior of solutions geometrically systems and thereby improve the understanding by different content representations semiotic. Similarly,applications of the content exposed at work, contributing to exemplify the use of linear systems in many areas of engineering, economics, biology, among others. We suggest the use of computational resources in mathematics lessons for reducing the time spent in algebraic manipulation teachers can deepen the concepts involved and seek larger systems that increase the prospect of working with linear systems, and motivate students find more meaning in their study and update. The work includes software GeoGebra, Winplot and Maxima, all free and disclosure applicable to the content of linear systems. Thus, from the reflections on the theme, are suggestions of issues and enriching to help overcome the shallowness of current approaches and offers alternatives to this. / Este trabalho tem por objetivo mostrar a importância do estudo dos sistemas lineares no Ensino Médio. Para tanto foram pesquisados trabalhos anteriores que já manifestavam preocupação com esse intuito. Estudou-se o surgimento histórico dos sistemas lineares no desenvolvimento da matemática, a definição e conceitos em livros didáticos para o Ensino Médio e também na literatura de Álgebra Linear e Cálculo Numérico. Fez-se uma pesquisa dos métodos de resolução de sistemas lineares, destacou-se a eliminação de Gauss nos métodos diretos e os métodos de Jacobi e Gauss-Seidel entre os iterativos. Refletiu-se o condicionamento de sistemas lineares e a aproximação de solução para sistemas incompatíveis que usualmente não são abordados em sala de aula e têm grande utilidade prática nas aplicações. Observou-s as orientações curriculares sobre o tema confrontando com a abordagem dos livros didáticos. Verificou-se que a literatura limita o conteúdo às condições de manipulação algébrica e não oferece atividades diversificadas com utilização de recursos computacionais e exemplos aprofundados e esclarecedores condizentes com a importância do conteúdo. Mostra-se uma abordagem de sistemas lineares que inter-relaciona os conteúdos de sistemas lineares com a Geometria Analítica de forma que ofereça aos alunos condições de visualizar geometricamente o comportamento das soluções dos sistemas e,assim, aprimorar a compreensão por diferentes representações semióticas do conteúdo. Da mesma forma, as aplicações do conteúdo expostas no trabalho, contribuem para exemplificar a utilização dos sistemas lineares em diversas áreas da engenharia, economia, biologia, entre outras. Sugere-se a utilização de recursos computacionais nas aulas de matemática para que reduzindo o tempo empregado na manipulação algébrica o professor possa aprofundar os conceitos envolvidos e buscar sistemas de maior porte que ampliem a perspectiva de trabalho com os sistemas lineares, além de motivar o aluno a encontrar mais sentido e atualização em seu estudo. O trabalho contempla os softwares GeoGebra, Winplot e Maxima, todos de divulgação livre e aplicáveis ao conteúdo de sistemas lineares. Sendo assim, a partir das reflexões sobre o tema, apresentam-se sugestões de assuntos relacionados e enriquecedores que ajudem superar as superficialidades das abordagens atuais e oferece alternativas para isso.

Made available in DSpace on 2016-04-27T16:57:29Z (GMT). No. of bitstreams: 1 Nilza Aparecida de Freitas.pdf: 5037486 bytes, checksum: bcfebcb3dddaae4a09e8eb366d4cf16c (MD5) Previous issue date: 2013-12-04 / Secretaria da Educação do Estado de São Paulo / This research aims to develop, implement and analyze a didactic sequence to investigate how the high school students of a public school in the city of São This research aims to develop, implement and analyze a didactic sequence to the proposed approach promotes the conversion and treatment of Registers of Semiotic Representation from Duval (2009). Therefore, we have presented an instructional sequence so that students can determine the solution of linear systems of two equations in the proposed activities and in connection with the different forms of representation: algebraic, graphical and natural language. On graphic record was used a software named Geogebra. After the application with methodological support of Didactic Engineering from Artigue (2010), we analyzed the protocols of six students participating in three sessions. The results reveal how evolution occurred from these students in solving linear systems, provided by the conversion activity and treatment the records of Semiotic Representation. It is hoped that this research will contribute to the learning of students in linear systems theme and encourage other studies of this mathematical object in research of Mathematics Education / Esta pesquisa tem como objetivo desenvolver, aplicar e analisar uma sequência didática para investigar como os alunos do terceiro ano do Ensino Médio de uma Escola Pública da Rede Estadual na cidade de São Bernardo do Campo no Estado de São Paulo resolvem sistemas de equações lineares 2X2 em que a abordagem proposta favorece a conversão e o tratamento de Registros de Representação Semiótica, aporte teórico de Duval (2009). Deste modo, elaborou-se uma sequência didática para que os alunos possam determinar a solução dos sistemas lineares de duas equações nas atividades propostas e em conexão com as diferentes formas de representação: algébrica, gráfica e na linguagem natural. No registro gráfico, utilizou-se o software Geogebra. Após a aplicação, com suporte metodológico nos pressupostos da Engenharia Didática de Artigue (2010), foram analisados os protocolos de seis alunos participantes nas três sessões. Os resultados revelam como ocorreu a evolução desses alunos na resolução dos sistemas lineares propiciado pela atividade de conversão e tratamento dos registros de Representação Semiótica. Espera-se que esta pesquisa possa contribuir para a aprendizagem dos alunos no tema sistemas lineares e incentivar outros estudos deste objeto matemático nas pesquisas em Educação Matemática

In this thesis, we present some contributions to the Markov jump linear systems theory in a context of partial information on the Markov chain. We consider that the state of the Markov chain cannot be measured, but instead there is only an observed variable that could model an asynchronous phenomenon between the application and the plant, or a simple fault detection and isolation device. In this formulation, we investigate the problem of designing controllers and filters depending only on the observed variable in the context of H2, H?, and mixed H2/H? control theory. Numerical examples and academic applications are presented for active-fault tolerant control systems and networked control systems. / Nesta tese, apresentamos algumas contribuições para a teoria de sistemas lineares com saltos markovianos em um contexto de observação parcial da cadeia de Markov. Consideramos que o estado da cadeia de Markov não pode ser medido, porém existe uma variável observada que pode modelar um fenômeno assíncrono entre a aplicação e a planta, ou ainda um dispositivo de detecção de falhas simples. Através desse modelo, investigamos o problema da síntese de controladores e filtros que dependem somente da variável observada no contexto das teorias de controle H2, H?, e misto H2/H?. Exemplos numéricos e aplicações acadêmicas são apresentadas no âmbito dos sistemas de controle tolerantes a falhas e dos sistemas de controle através da rede.

Submitted by Cássia Santos (cassia.bcufg@gmail.com) on 2017-07-17T11:02:06Z No. of bitstreams: 2 Dissertação - Messias Henrique Vieira Silva - 2017.pdf: 5254101 bytes, checksum: a686ed3b5fe20282056da703d94dd863 (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) / Approved for entry into archive by Luciana Ferreira (lucgeral@gmail.com) on 2017-07-17T11:26:29Z (GMT) No. of bitstreams: 2 Dissertação - Messias Henrique Vieira Silva - 2017.pdf: 5254101 bytes, checksum: a686ed3b5fe20282056da703d94dd863 (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) / Made available in DSpace on 2017-07-17T11:26:29Z (GMT). No. of bitstreams: 2 Dissertação - Messias Henrique Vieira Silva - 2017.pdf: 5254101 bytes, checksum: a686ed3b5fe20282056da703d94dd863 (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) Previous issue date: 2017-07-04 / This dissertation address the use of mathematical software Scilab and Maxima as instruments of study and teaching of linear systems. Initially, it is presented an introductory study on: Linear Algebra, linear systems and some method applied to solve linear systems. Following, it is used Scilab to solve possible and determined systems and it is applied Maxima to calculate determinants of matrices of order higher than three to solve possible and indeterminate systems. In the end,it is presented a approach on the solution of linear systems through of the graphical representation of their equations, starting in systems of two equations with graphs in the xOy plane and ending with systems of three equations with graphics on three-dimensional space, all of this assited by Scilab and Maxima. / Esta dissertação trata da utilização dos softwares matemáticos Scilab e Maxima como instrumentos de estudo e ensino de sistemas lineares. Inicialmente, é apresentado um estudo introdutório a respeito da Álgebra Linear, dos tipos de sistemas lineares e de formas manuais de resolução. Em seguida, aborda-se a utilização do Scilab para resolver sistemas possíveis e determinados e do Maxima para calcular determinantes de matrizes de ordem superior a três e ainda resolver sistemas possíveis e indeterminados. Por fim, apresenta-se uma abordagem acerca da solução de sistemas lineares através da representação grá#ca de suas equações, começando em sistemas de duas equações com gráficos no plano xOy e encerrando com sistemas de três equações com gráficos no espaço tridimensional, tudo isso com a utilização do Scilab e do Maxima.

Submitted by Erika Demachki (erikademachki@gmail.com) on 2014-08-28T21:24:23Z No. of bitstreams: 2 license_rdf: 23148 bytes, checksum: 9da0b6dfac957114c6a7714714b86306 (MD5) Vasconcelos, Eduardo Silva..pdf: 1832341 bytes, checksum: c00d3ca071a0f15e55eeef2079d52047 (MD5) / Made available in DSpace on 2014-08-28T21:24:23Z (GMT). No. of bitstreams: 2 license_rdf: 23148 bytes, checksum: 9da0b6dfac957114c6a7714714b86306 (MD5) Vasconcelos, Eduardo Silva..pdf: 1832341 bytes, checksum: c00d3ca071a0f15e55eeef2079d52047 (MD5) Previous issue date: 2013-04-12 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / This paper aims to describe the Simplex Method and Method of Resolution Graphics problems on Linear Programming, aiming at the teaching and learning of linear algebra in high school. And for this, presents some basic concepts in linear programming, it follows brie y on the Simplex Method and Method of Resolution Graphics and presents two resolutions of Linear Programming problems, a maximization and minimization another, both problems are solved by two methods cited. We understand the importance of this work is to present the Simplex Method eao Method Graphical resolution to high school students because we believe that together these two methods applied in teaching linear algebra could lead to increased motivation of students in learning mathematics. / Este trabalho tem o objetivo de descrever o Método Simplex e o Método de Resoluções Grá cas em problemas de Programação Linear, visando o ensino e a aprendizagem da álgebra linear no Ensino Médio. E, para tal, apresenta alguns conceitos básicos em Programação Linear, decorre sucintamente sobre o Método Simplex e o Método de Resolução Grá ca e apresenta duas resoluções de problemas de Programação Linear, uma de maximização e outra de minimização, ambos os problemas são resolvidos pelos dois métodos citados. Entendemos que a importância deste trabalho está em apresentar o Método Simplex e a o Método de Resolução Grá ca aos alunos do Ensino Médio pois, acreditamos que estes dois métodos juntamente aplicados no ensino de álgebra linear poderá levar a uma maior motivação destes alunos na aprendizagem da matemática.

Este trabalho consiste no estudo de métodos de otimização aplicados em um problema de controle para sistemas lineares com saltos markovianos (SLSM). SLSM formam uma importante classe de sistemas que têm sido muito úteis em aplicações envolvendo sistemas sujeitos a falhas e outras alterações abruptas de comportamento. Este estudo enfoca diferentes métodos para resolução deste problema. Comparamos o método variacional com o de Newton, sob o ponto de vista do número de problemas resolvidos e pelo nível de sub-otimalidade obtido (relação entre os custos obtidos por estes métodos). Também propomos um novo método, o qual pode ser inicializado com soluções de equações de Riccati acopladas, e o comparamos com o método variacional. Além disso, para a comparação dos métodos, propomos um algoritmo que gerou dez mil exemplos / This work addresses optimizations methods applied to a control problem for linear systems with markovian jumps, which form an important class of systems that have been very useful in applications involving systems subject to failures and other abrupt changes. This study focuses on different methods for solving this problem. We compare the variational approach with the Newton method, in terms of the number of solved problems and the level of sub-optimality (ratio between the costs obtained by these approaches). We also propose a new method, which can be initialized with solutions of coupled Riccati equations, and we compare it with the variational approach. We have proposed an algorithm for creating ten thousand examples for the comparisons

<p>This thesis deals with the problem of detecting faults in an environment where the measurements are affected by additive noise. To do this, a residual sensitive to faults is derived and statistical methods are used to distinguish faults from noise. Standard methods for fault detection compare a batch of data with a model of the system using the generalized likelihood ratio. Careful treatment of the initial state of the model is quite important, in particular for short batch sizes. One method to handle this is the parity-space method which solves the problem by removing the influence of the initial state using a projection.</p><p>In this thesis, the case where prior knowledge about the initial state is available is treated. This can be obtained for example from a Kalman filter. Combining the prior estimate with a minimum variance estimate from the data batch results in a smoothed estimate. The influence of the estimated initial state is then removed. It is also shown that removing the influence of the initial state by an estimate from the data batch will result in the parity-space method. To model slowly changing faults, an efficient parameterization using Chebyshev polynomials is given.</p><p>The methods described above have been applied to an Inertial Measurement Unit, IMU. The IMU usually consists of accelerometers and gyroscopes, but has in this work been extended with a magnetometer. Traditionally, the IMU has been used to estimate position and orientation of airplanes, missiles etc. Recently, the size and cost has decreased making it possible to use IMU:s for applications such as augmented reality and body motion analysis. Since a magnetometer is very sensitive to disturbances from metal, such disturbances have to be detected. Detection of the disturbances makes compensation possible. Another topic covered is the fundamental question of observability for fault inputs. Given a fixed or linearly growing fault, conditions for observability are given.</p><p>The measurements from the IMU show that the noise distribution of the sensors can be well approximated with white Gaussian noise. This gives good correspondence between practical and theoretical results when the sensor is kept at rest. The disturbances for the IMU can be approximated using smooth functions with respect to time. Low rank parameterizations can therefore be used to describe the disturbances. The results show that the use of smoothing to obtain the initial state estimate and parameterization of the disturbances improves the detection performance drastically.</p>

This thesis deals with the problem of detecting faults in an environment where the measurements are affected by additive noise. To do this, a residual sensitive to faults is derived and statistical methods are used to distinguish faults from noise. Standard methods for fault detection compare a batch of data with a model of the system using the generalized likelihood ratio. Careful treatment of the initial state of the model is quite important, in particular for short batch sizes. One method to handle this is the parity-space method which solves the problem by removing the influence of the initial state using a projection. In this thesis, the case where prior knowledge about the initial state is available is treated. This can be obtained for example from a Kalman filter. Combining the prior estimate with a minimum variance estimate from the data batch results in a smoothed estimate. The influence of the estimated initial state is then removed. It is also shown that removing the influence of the initial state by an estimate from the data batch will result in the parity-space method. To model slowly changing faults, an efficient parameterization using Chebyshev polynomials is given. The methods described above have been applied to an Inertial Measurement Unit, IMU. The IMU usually consists of accelerometers and gyroscopes, but has in this work been extended with a magnetometer. Traditionally, the IMU has been used to estimate position and orientation of airplanes, missiles etc. Recently, the size and cost has decreased making it possible to use IMU:s for applications such as augmented reality and body motion analysis. Since a magnetometer is very sensitive to disturbances from metal, such disturbances have to be detected. Detection of the disturbances makes compensation possible. Another topic covered is the fundamental question of observability for fault inputs. Given a fixed or linearly growing fault, conditions for observability are given. The measurements from the IMU show that the noise distribution of the sensors can be well approximated with white Gaussian noise. This gives good correspondence between practical and theoretical results when the sensor is kept at rest. The disturbances for the IMU can be approximated using smooth functions with respect to time. Low rank parameterizations can therefore be used to describe the disturbances. The results show that the use of smoothing to obtain the initial state estimate and parameterization of the disturbances improves the detection performance drastically.