As computers keeping advancing, many difficult problems which were unable to compute formerly now have the chance to get answered. It is always the goal of mathematicians and computer scientists to compute and get the answers of the linear systems. Since 1950s, there have been a lot of published papers discussing the issue. As the linear systems larger and larger, the computer efficiency required is higher and higher, so that it is very difficult to get the answers of large linear systems. Now, the problems are showing aurora. In this dissertation, several mathematical calculations to compute the linear systems will be discussed, as well as their background and theory. Moreover, they will also be practiced.

No description available.

No description available.

No description available.

De nos jours, afin d'améliorer la sécurité routière, de plus en plus de systèmes d'assistance à la conduite, appelé ADAS (Advanced Driver Assistance System) sont embarqués dans les véhicules. Leur augmentation rend le développement des véhicules toujours plus complexes. Pour parer à ces difficultés, dans un premier temps, ma thèse propose l'élaboration d'un simulateur de véhicule, capable d'aider le développeur. Afin de résoudre les problèmes de décisions et de synchronisation, l'état de l'art a été considéré pour choisir une architecture adaptée aux ADAS. En dernier lieu, un algorithme de prise de décision a été développé, pour optimiser l'intégrité du véhicule. Pour modéliser le véhicule, un simulateur émule le comportement planaire de celui-ci et des actionneurs qui agissent sur sa dynamique, tels que le moteur ou les freins. Une fois la base du véhicule réalisée, j'ai concentré mon travail sur les ADAS. Comme actuellement aucune solution concrète n’existe pour la stratégie de décision, afin de choisir l’aide la plus adaptée à la situation, le dernier point traité dans ma thèse a été le développement d'une décision assurant l'intégrité du véhicule. Celle-ci couple un calcul de trajectoire avec un ensemble invariant de Lyapunov, obtenu par un problème d'optimisation avec contraintes sous forme de d'inégalités matricielles bilinéaires. Elle permet d’évaluer l'activation des fonctions et de fournir un avertissement au conducteur dans les situations critiques. Pour illustrer le fonctionnement de cette décision, un exemple de contrôle longitudinal a été choisi, comprenant un régulateur automatique de vitesse et un freinage d'urgence. / Nowadays, to enhance traffic safety, more and more Advance Driver Assistance Systems (ADAS) are embedded in mass-production vehicles. Their increase renders development of vehicles more and more complex, especially to design Electronic and Electric (E/E) architecture, to synchronize the different embedded ADAS and decide which ADAS should be engaged. To cope with E/E architecture issues, my PhD thesis proposes a vehicle simulator, which is able to support architect designers. Then, to solve synchronization and decision problems, ADAS architecture has been chosen, according to the state of the art. Finally, a decision algorithm has been developed to optimise vehicle safety. To model the vehicle, a simulator emulates its plane motion according to embedded actuators acting on dynamic, like engine and brakes. Once the vehicle basis has been performed, I focus my work on ADAS. Because nowadays no generic solution exists to decide which ADAS to engage, last focus of my PhD has been the design of a decision method, optimizing vehicle safety. This latter couples a path-planning witch a Lyapunov invariant set, obtained through optimization problem constraints by bilinear matrix inequality. This strategy enables to assess embedded ADAS-functions and to warn the driver in critical situations. In order to illustrate this former, it has been illustrated with 2 longitudinal functions, a Adaptive Cruise Control and an Emergency Brake.

The modelling of physical processes gives rise to mathematical systems of increasing complexity. Good mathematical models have to reproduce the physical process as precisely as possible while the computing time and the storage resources needed to simulate the mathematical model are limited. As a consequence, there must be a tradeoff between accuracy and computational constraints. At the present time, one is often faced with systems that have an unacceptably high level of complexity. It is then desirable to approximate such systems by systems of lower complexity. This is the Model Reduction Problem. This thesis focuses on the study of new model reduction techniques for linear systems. Our objective is twofold. First, there is a need for a better understanding of Krylov techniques. With such techniques, one can construct a reduced order transfer function that satisfies a set of interpolation conditions with respect to the original transfer function. A study of the generality of such techniques and their extension for MIMO systems via the concept of tangential interpolation constitutes the first part of this thesis. This also led us to study the generality of the projection technique for model reduction. Most large scale systems have a particular structure. They can be modelled as a set of subsystems that interconnect to each other. It then makes sense to develop model reduction techniques that preserve the structure of the original system. Both interpolation-based and gramian-based structure preserving model reduction techniques are developed in a unified way. Second order systems that appear in many branches of engineering deserve a special attention. This constitutes the second part of this thesis.

A new parallel solver based on SPIKE-TA algorithm has been developed using OpenMP API for solving diagonally-dominant banded linear systems on shared memory architectures. The results of the numerical experiments carried out for different test cases demonstrate high-performance and scalability on current multi-core platforms and highlight the time savings that SPIKE-TA OpenMP offers in comparison to the LAPACK BLAS-threaded LU model. By exploiting algorithmic parallelism in addition to threaded implementation, we obtain greater speed-ups in contrast to the threaded versions of sequential algorithms. For non-diagonally dominant systems, we implement the SPIKE-RL scheme and a new Spike-calling-Spike (SCS) scheme using OpenMP. The timing results for solving the non-diagonally dominant systems using SPIKE-RL show extremely good scaling in comparison to LAPACK and modified banded-primitive library.

No description available.

Submitted by Rosivalda Pereira (mrs.pereira@ufma.br) on 2017-08-25T20:02:47Z No. of bitstreams: 1 BrunoCosta.pdf: 2190874 bytes, checksum: a6789eff5c1a3161c551aa9d384917f4 (MD5) / Made available in DSpace on 2017-08-25T20:02:47Z (GMT). No. of bitstreams: 1 BrunoCosta.pdf: 2190874 bytes, checksum: a6789eff5c1a3161c551aa9d384917f4 (MD5) Previous issue date: 2017-05-30 / In this work we present some applications of the theory of matrices and linear systems, in which the Scilab software is used as an auxiliary tool in the calculation of matrix multiplication, inverse matrix and matrix scaling by the Gauss-Jordan method, which constitute the theoretical basis of Linear Algebra for analysis and discussion of the applications presented. It is intended that the results of this work be adapted as a complement in the teaching-learning process of matrices and linear systems. / Neste trabalho s˜ao apresentadas algumas aplica¸c˜oes da teoria de matrizes e sistemas lineares, em que o software Scilab ´e utilizado como ferramenta auxiliar no c´alculo de multiplica¸c˜ao de matrizes, matriz inversa e escalonamento de matriz pelo m´etodo de Gauss-Jordan, os quais constituem a base te´orica da Algebra Linear para an´alise e ´ discuss˜ao das aplica¸c˜oes apresentadas. Pretende-se que os resultados deste trabalho sejam adaptados como complemento no processo de ensino-aprendizagem de matrizes e sistemas lineares.

本篇論文主要在探討熱帶線性系統(tropical linear system) A x = b 與雙邊齊次熱帶線性系統(two-sided homogeneous tropical linear system) A x = B y 的求解方法。我們將明確的描述任何熱帶線性系統與雙邊齊次熱帶線性系統的解。 如同古典的論述, 當求解線性系統 A x = b 時, 我們首先會先找到對應的 ``齊次'' 系統 A x = 0 來求解。而對於雙邊齊次熱帶線性系統, 我們將利用勝序列的概念, 將雙邊齊次熱帶線性系統轉化為 k 組古典熱帶線性系統: 含等式系統 S: C[x^t -y^t 1]^t = 0 與不等式系統 T: D[x^t -y^t 1]^t <= 0 。除此之外, 利用相容性條件來減少 k 的數量。 過程中我們處理的 S, T 均為雙變量的系統, 係數分別為 1 與 -1, 對於 S 我們以高斯-喬登消去法(Gauss–Jordan elimination)處理。對於 T 我們將以類似高斯-喬登消去法的方式進行列運算, 因此我們定義次特殊矩陣(sub-special matrix), 而進行的過程我們稱之為次特殊化(sub–specialization)。 最後將以 MATLAB 作為工具來求解出這兩類的熱帶線性系統。 / The thesis mainly discusses the methods of finding solutions of tropical linear systems A x = b and two-sided homogeneous tropical linear systems A x = B y. We are able to give explicit descriptions of all solutions of any tropical linear systems A x = b and two-sided homogeneous tropical linear systems A x = B y. As the classical situations, when solving the linear systems of the form A x = b, we first find the solutions for the corresponding ``homogeneous'' case A x = 0. For two-sided homogeneous tropical linear systems A x = B y, we use the concept of win sequence to convert it into a finite number k of classical linear systems: either a system S: C[x^t -y^t 1]^t = 0 of equations or a system T: D[x^t -y^t 1]^t <= 0 of inequalities. Moreover, we used so called ``compatibility conditions'' to reduce the number of k. The particular feature of both S and T is that each item (equation or inequality) is bivariate. It involves exactly two variables; one variable with coefficient 1, and the other one with -1. S is solved by Gauss-Jordon elimination. We explain how to solve T by a method similar to Gauss-Jordon elimination. To achieve this, we introduce the notion of sub–special matrix. The procedure applied to T is called sub–specialization. Finally, we will use MATLAB to solve tropical linear systems of these two types.