Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / In this work, we are treated some elements of functional analysis such as Banach spaces, inner product spaces and Hilbert spaces, also studied Fourier series and at the end briefly consider the equation of the vibrating string. With this, you realize that you do not need a lot of theory in order to get significant results. / Neste trabalho, são tratados alguns elementos da análise funcional como espaços de Banach, espaços com produto interno e espaços de Hilbert, estudamos também séries de Fourier e no final consideramos brevemente a equação da corda vibrante. Com isso, percebe-se que não se precisa de muita teoria para conseguirmos resultados significativos.

The results presented in this dissertation refer to the geometry of Minkowski spaces, i.e., of real finite-dimensional Banach spaces. First we study geometric properties of radial projections of bisectors in Minkowski spaces, especially the relation between the geometric structure of radial projections and Birkhoff orthogonality. As an application of our results it is shown that for any Minkowski space there exists a number, which plays somehow the role that $\sqrt2$ plays in Euclidean space. This number is referred to as the critical number of any Minkowski space. Lower and upper bounds on the critical number are given, and the cases when these bounds are attained are characterized. Moreover, with the help of the properties of bisectors we show that a linear map from a normed linear space $X$ to another normed linear space $Y$ preserves isosceles orthogonality if and only if it is a scalar multiple of a linear isometry. Further on, we examine the two tangent segments from any exterior point to the unit circle, the relation between the length of a chord of the unit circle and the length of the arc corresponding to it, the distances from the normalization of the sum of two unit vectors to those two vectors, and the extension of the notions of orthocentric systems and orthocenters in Euclidean plane into Minkowski spaces. Also we prove theorems referring to chords of Minkowski circles and balls which are either concurrent or parallel. All these discussions yield many interesting characterizations of the Euclidean spaces among all (strictly convex) Minkowski spaces. In the final chapter we investigate the relation between the length of a closed curve and the length of its midpoint curve as well as the length of its image under the so-called halving pair transformation. We show that the image curve under the halving pair transformation is convex provided the original curve is convex. Moreover, we obtain several inequalities to show the relation between the halving distance and other quantities well known in convex geometry. It is known that the lower bound for the geometric dilation of rectifiable simple closed curves in the Euclidean plane is $\pi/2$, which can be attained only by circles. We extend this result to Minkowski planes by proving that the lower bound for the geometric dilation of rectifiable simple closed curves in a Minkowski plane $X$ is analogously a quarter of the circumference of the unit circle $S_X$ of $X$, but can also be attained by curves that are not Minkowskian circles. In addition we show that the lower bound is attained only by Minkowskian circles if the respective norm is strictly convex. Also we give a sufficient condition for the geometric dilation of a closed convex curve to be larger than a quarter of the perimeter of the unit circle.

In a Minkowski three dimensional space, whose metric is based on a strictly convex and centrally symmetric unit ball , we deal with ruled surfaces Φ in the sense of E. Kruppa. This means that we have to look for Minkowski analogues of the classical differential invariants of ruled surfaces in a Euclidean space. Here, at first – after an introduction to concepts of a Minkowski space, like semi-orthogonalities and a semi-inner-product based on the so-called cosine-Minkowski function - we construct an orthogonal 3D moving frame using Birkhoff’s left-orthogonality. This moving frame is canonically connected to ruled surfaces: beginning with the generator direction and the asymptotic plane of this generator g we complete this flag to a frame using the left-orthogonality defined by ; ( is described either by its supporting function or a parameter representation). The plane left-orthogonal to the asymptotic plane through generator g(t) is called Minkowski central plane and touches Φ in the striction point s(t) of g(t). Thus the moving frame defines the Minkowski striction curve S of the considered ruled surface Φ similar to the Euclidean case. The coefficients occurring in the Minkowski analogues to Frenet-Serret formulae of the moving frame of Φ in a Minkowski space are called “M-curvatures” and “M-torsions”. Here we essentially make use of the semi-inner product and the sine-Minkowski and cosine-Minkowski functions. Furthermore we define a covariant differentiation in a Minkowski 3-space using a new vector called “deformation vector” and locally measuring the deviation of the Minkowski space from a Euclidean space. With this covariant differentiation it is possible to declare an “M-geodesicc parallelity” and to show that the vector field of the generators of a skew ruled surface Φ is an M-geodesic parallel field along its Minkowski striction curve s. Finally we also define the Pirondini set of ruled surfaces to a given surface Φ. The surfaces of such a set have the M-striction curve and the strip of M-central planes in common

In a Minkowski three dimensional space, whose metric is based on a strictly convex and centrally symmetric unit ball , we deal with ruled surfaces Φ in the sense of E. Kruppa. This means that we have to look for Minkowski analogues of the classical differential invariants of ruled surfaces in a Euclidean space. Here, at first – after an introduction to concepts of a Minkowski space, like semi-orthogonalities and a semi-inner-product based on the so-called cosine-Minkowski function - we construct an orthogonal 3D moving frame using Birkhoff’s left-orthogonality. This moving frame is canonically connected to ruled surfaces: beginning with the generator direction and the asymptotic plane of this generator g we complete this flag to a frame using the left-orthogonality defined by ; ( is described either by its supporting function or a parameter representation). The plane left-orthogonal to the asymptotic plane through generator g(t) is called Minkowski central plane and touches Φ in the striction point s(t) of g(t). Thus the moving frame defines the Minkowski striction curve S of the considered ruled surface Φ similar to the Euclidean case. The coefficients occurring in the Minkowski analogues to Frenet-Serret formulae of the moving frame of Φ in a Minkowski space are called “M-curvatures” and “M-torsions”. Here we essentially make use of the semi-inner product and the sine-Minkowski and cosine-Minkowski functions. Furthermore we define a covariant differentiation in a Minkowski 3-space using a new vector called “deformation vector” and locally measuring the deviation of the Minkowski space from a Euclidean space. With this covariant differentiation it is possible to declare an “M-geodesicc parallelity” and to show that the vector field of the generators of a skew ruled surface Φ is an M-geodesic parallel field along its Minkowski striction curve s. Finally we also define the Pirondini set of ruled surfaces to a given surface Φ. The surfaces of such a set have the M-striction curve and the strip of M-central planes in common

Les méthodes d'algèbres linéaire classiques, pour le calcul de valeurs et vecteurs propres d'une matrice, ou des approximations de rangs inférieurs (low-rank approximations) d'une solution, etc..., ne tiennent pas compte des structures de matrices. Ces dernières sont généralement détruites durant le procédé du calcul. Des méthodes alternatives préservant ces structures font l'objet d'un intérêt important par la communauté. Cette thèse constitue une contribution dans ce domaine. La décomposition SR peut être calculé via l'algorithme de Gram-Schmidt symplectique. Comme dans le cas classique, une perte d'orthogonalité peut se produire. Pour y remédier, nous avons proposé deux algorithmes RSGSi et RMSGSi qui consistent à ré-orthogonaliser deux fois les vecteurs à calculer. La perte de la J-orthogonalité s'est améliorée de manière très significative. L'étude directe de la propagation des erreurs d'arrondis dans les algorithmes de Gram-Schmidt symplectique est très difficile à effectuer. Nous avons réussi à contourner cette difficulté et donner des majorations pour la perte de la J-orthogonalité et de l'erreur de factorisation. Une autre façon de calculer la décomposition SR est basée sur les transformations de Householder symplectique. Un choix optimal a abouti à l'algorithme SROSH. Cependant, ce dernier peut être sujet à une instabilité numérique. Nous avons proposé une version modifiée nouvelle SRMSH, qui a l'avantage d'être aussi stable que possible. Une étude approfondie a été faite, présentant les différentes versions : SRMSH et SRMSH2. Dans le but de construire un algorithme SR, d'une complexité d'ordre O(n³) où 2n est la taille de la matrice, une réduction (appropriée) de la matrice à une forme condensée (J(Hessenberg forme) via des similarités adéquates, est cruciale. Cette réduction peut être effectuée via l'algorithme JHESS. Nous avons montré qu'il est possible de réduire une matrice sous la forme J-Hessenberg, en se basant exclusivement sur les transformations de Householder symplectiques. Le nouvel algorithme, appelé JHSJ, est basé sur une adaptation de l'algorithme SRSH. Nous avons réussi à proposer deux nouvelles variantes, aussi stables que possible : JHMSH et JHMSH2. Nous avons constaté que ces algorithmes se comportent d'une manière similaire à l'algorithme JHESS. Une caractéristique importante de tous ces algorithmes est qu'ils peuvent rencontrer un breakdown fatal ou un "near breakdown" rendant impossible la suite des calculs, ou débouchant sur une instabilité numérique, privant le résultat final de toute signification. Ce phénomène n'a pas d'équivalent dans le cas Euclidien. Nous avons réussi à élaborer une stratégie très efficace pour "guérir" le breakdown fatal et traîter le near breakdown. Les nouveaux algorithmes intégrant cette stratégie sont désignés par MJHESS, MJHSH, JHM²SH et JHM²SH2. Ces stratégies ont été ensuite intégrées dans la version implicite de l'algorithme SR lui permettant de surmonter les difficultés rencontrées lors du fatal breakdown ou du near breakdown. Rappelons que, sans ces stratégies, l'algorithme SR s'arrête. Finalement, et dans un autre cadre de matrices structurées, nous avons présenté un algorithme robuste via FFT et la matrice de Hankel, basé sur le calcul approché de plus grand diviseur commun (PGCD) de deux polynômes, pour résoudre le problème de la déconvolution d'images. Plus précisément, nous avons conçu un algorithme pour le calcul du PGCD de deux polynômes bivariés. La nouvelle approche est basée sur un algorithme rapide, de complexité quadratique O(n²), pour le calcul du PGCD des polynômes unidimensionnels. La complexité de notre algorithme est O(n²log(n)) où la taille des images floues est n x n. Les résultats expérimentaux avec des images synthétiquement floues illustrent l'efficacité de notre approche. / The classical linear algebra methods, for calculating eigenvalues and eigenvectors of a matrix, or lower-rank approximations of a solution, etc....do not consider the structures of matrices. Such structures are usually destroyed in the numerical process. Alternative structure-preserving methods are the subject of an important interest mattering to the community. This thesis establishes a contribution in this field. The SR decomposition is usually implemented via the symplectic Gram-Schmidt algorithm. As in the classical case, a loss of orthogonality can occur. To remedy this, we have proposed two algorithms RSGSi and RMSGSi, where the reorthogonalization of a current set of vectors against the previously computed set is performed twice. The loss of J-orthogonality has significantly improved. A direct rounding error analysis of symplectic Gram-Schmidt algorithm is very hard to accomplish. We managed to get around this difficulty and give the error bounds on the loss of the J-orthogonality and on the factorization. Another way to implement the SR decomposition is based on symplectic Householder transformations. An optimal choice of free parameters provided an optimal version of the algorithm SROSH. However, the latter may be subject to numerical instability. We have proposed a new modified version SRMSH, which has the advantage of being numerically more stable. By a detailes study, we are led to two new variants numerically more stables : SRMSH and SRMSH2. In order to build a SR algorithm of complexity O(n³), where 2n is the size of the matrix, a reduction to the condensed matrix form (upper J-Hessenberg form) via adequate similarities is crucial. This reduction may be handled via the algorithm JHESS. We have shown that it is possible to perform a reduction of a general matrix, to an upper J-Hessenberg form, based only on the use of symplectic Householder transformations. The new algorithm, which will be called JHSH algorithm, is based on an adaptation of SRSH algorithm. We are led to two news variants algorithms JHMSH and JHMSH2 which are significantly more stable numerically. We found that these algortihms behave quite similarly to JHESS algorithm. The main drawback of all these algorithms (JHESS, JHMSH, JHMSH2) is that they may encounter fatal breakdowns or may suffer from a severe form of near-breakdowns, causing a brutal stop of the computations, the algorithm breaks down, or leading to a serious numerical instability. This phenomenon has no equivalent in the Euclidean case. We sketch out a very efficient strategy for curing fatal breakdowns and treating near breakdowns. Thus, the new algorithms incorporating this modification will be referred to as MJHESS, MJHSH, JHM²SH and JHM²SH2. These strategies were then incorporated into the implicit version of the SR algorithm to overcome the difficulties encountered by the fatal breakdown or near-breakdown. We recall that without these strategies, the SR algorithms breaks. Finally ans in another framework of structured matrices, we presented a robust algorithm via FFT and a Hankel matrix, based on computing approximate greatest common divisors (GCD) of polynomials, for solving the problem pf blind image deconvolution. Specifically, we designe a specialized algorithm for computing the GCD of bivariate polynomials. The new algorithm is based on the fast GCD algorithm for univariate polynomials , of quadratic complexity O(n²) flops. The complexitiy of our algorithm is O(n²log(n)) where the size of blurred images is n x n. The experimental results with synthetically burred images are included to illustrate the effectiveness of our approach