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Le Calcul numérique présente deux aspects apparemment distincts :<br /><br /><br />- La partie consacrée à la recherche de méthodes ou de procédés alorithmiques, partie algébrique. <br /><br /><br />- La partie consacrée au contrôle des calculs et des approximations, partie topologque. <br /><br /><br />Sur le plan théorique, l'Analyse Numérique présuppose donc l'algèbre et la topologie. La raison profonde en est<br />la définition même des nombres réels en laquelle ce double point de vue se fait déjà sentir.<br />Ce travail se propose d'utiliser deux idées dirigées dans les deux voies dont nous venons de parler. La première est<br />de montrer le rôle important joué dans la plupart des processus de calcul linéaire par des matrices de polynomes minimaux du<br />second degré, dont l'inverse est, par ce fait, immédiatement calculable. La seconde est de montrer comment la considération<br />de normes générales peut permettre de définir des notions importantes comme les conditionnements numériques et d'obtenir un<br />contrôle plus fin des erreurs de calcul.

Relays can be used in wireless communication systems to provide cell coverage extension, reduce coverage holes and increase throughput. Full duplex (FD) relays, which transmit and receive in the same time slot, can have a higher transmission rate compared with half duplex (HD) relays. However, FD relays suffer from self interference (SI) problems, which are caused by the transmitted relay signal being received by the relay receiver. This can reduce the performance of FD relays. In the literature, the SI channel is commonly nulled and removed as it simplifies the problem considerably. In practice, complete nulling is impossible due to channel estimation errors. Therefore, in this thesis, we consider the leakage of the SI from the FD relay. Our goal is to reduce the SI and increase the signal to noise ratio (SNR) of the relay system. Hence, we propose different precoder and weight vector designs. These designs may increase the end to end (e2e) signal to interference and noise ratio (SINR) at the destination. Here, a precoder is multiplied to a signal before transmission and a weight vector is multiplied to the received signal after reception. Initially, we consider an academic example where it uses a two path FD multiple input and multiple output (MIMO) system. The analysis of the SINR with the implementation of precoders and weight vectors shows that the SI component has the same underlying signal as the source signal when a relay processing delay is not being considered. Hence, to simulate the SI problem more realistically, we alter our relay design and focus on a one path FD MIMO relay system with a relay processing delay. For the implementation of precoders and weight vectors, choosing the optimal scheme is numerically challenging. Thus, we design the precoders and weight vectors using ad-hoc and near-optimal schemes. The ad-hoc schemes for the precoders are singular value decomposition (SVD), minimising the signal to leakage plus noise ratio (SLNR) using the Rayleigh Ritz (RR) method and zero forcing (ZF). The ad-hoc schemes for the weight vectors are SVD, minimum mean squared error (MMSE) and ZF. The near-optimal scheme uses an iterative RR method to compute the source precoder and destination weight vector and the relay precoder and weight vector are computed using the ad-hoc methods which provide the best performance. The average power and the instantaneous power normalisations are the two methods to constrain the relay precoder power. The average power normalisation method uses a novel closed form covariance matrix with an optimisation approach to constrain the relay precoder. This closed form covariance matrix is mathematically derived using matrix vectorization techniques. For the instantaneous power normalisation method, the constraint process does not require an optimisation approach. However, using this method the e2e SINR is difficult to calculate, therefore we use symbol error rate (SER) as a measure of performance. The results from the different precoder and weight vector designs suggest that reducing the SI using the relay weight vector instead of the relay precoder results in a higher e2e SINR. Consequently, to increase the e2e SINR, performing complicated processing at the relay receiver is more effective than at the relay transmitter.

Test bias is a current focus of concern in both education and psychology. No where are the polemics of test bias stronger than in the area of ethnic/racial group differences on measures of intellectual functioning. Th problem of test bias is exacerbated when verbal intelligence tests are administered to students in a language in which they have limited facility and/or cultural reference. This problem is most evident in the United States with Hispanics. This study was designed to evaluate the Raven's Progressive Matrices (RPM) as a measure of non-biased assessment for Mexican-Americans. The RPM was selected because it is a non-verbal intelligence measure. The subjects were 230 Anglo and 230 Mexican-American elementary school children from Douglas, Arizona. The data were evaluated by analysis of regression with the Total Reading and Total Math scores of the Metropolitan Achievement Test (MAT). Both groups were analyzed together to identify any significant slope or intercept differences. All regression analyses were by grade level. Group validity coefficients were also computed for both math and reading by grade level and across grade level. The results of regression analysis indicated no test bias against Mexican-Americans. Slopes were equal at all grade levels. Some intercept variance occurred which would result in bias against Anglos, rather than Mexican-Anericans, if common regression lines were used. Also, validity coefficients of academic prediction for the RPM compared favorably to coefficients of the WISC-R Performance Scale for Mexican-Americans. Limitations of the study and areas of further research are discussed.

This dissertation is comprised of two separate topics within the domain of polarization optical engineering. The first topic is a Mueller matrix roots decomposition, and the second topic is polarization computer generated holography. The first four chapters of the dissertation are on the topic of the Mueller matrix roots decomposition. Recently, an order-independent Mueller matrix decomposition was proposed in an effort to organize the nine depolarization degrees of freedom. Chapter 1 discusses relevant Mueller matrix decomposition prior art and the motivation for this work. In chapter 2, the critical computational issues involved in applying this Mueller matrix roots decomposition are addressed, along with a review of the principal root and common methods for its calculation. The choice of the pth root is optimized at p = 10⁵, and computational techniques are proposed which allow singular Mueller matrices and Mueller matrices with a half-wave of retardance to be evaluated with the matrix roots decomposition. A matrix roots algorithm is provided which incorporates these computational results. In chapter 3, the Mueller matrix roots decomposition is reviewed and a set of Mueller matrix generators are discussed. The parameterization of depolarization into three families, each with three degrees of freedom is explained. Analysis of the matrix roots parameters in terms of degree of polarization maps demonstrates that depolarizers fall into two distinct classes: amplitude depolarization in one class, and phase and diagonal depolarization in another class. It is shown that each depolarization family and degree of freedom can be produced by averaging two non-depolarizing Mueller matrix generators. This is extended to provide further insight on two sample measurements, which are analyzed using the matrix roots decomposition. Chapter 4 discusses additional properties of the Mueller matrix roots generators and parameters, along with a pupil aberration application of the matrix roots decomposition. Appendix C, adapted from a conference proceedings paper, presents an application of the matrix roots depolarization parameters for estimating the orientation of a one-dimensionally textured object. The last two chapters are on the topic of polarization computer generated holography. In chapter 5, an interlaced polarization computer-generated hologram (PCGH) is designed to produce specific irradiance and polarization states in the image plane. The PCGH produces a tangentially polarized annular pattern with correlated speckle, which is achieved by a novel application of a diffuser optimization method. Alternating columns of orthogonal linear polarizations illuminate an interlaced PCGH, producing a ratio of polarization of 88% measured on a fabricated sample. In chapter 6, an etched calcite square-wave retarder is designed, fabricated, and demonstrated as an illuminator for an interlaced polarization computer generated hologram (PCGH). The calcite square-wave retarder enables alternating columns of orthogonal linear polarizations to illuminate the interlaced PCGH. Together, these components produce a speckled, tangentially polarized PCGH diffraction pattern with a measured ratio of polarization of 84% and a degree of linear polarization of 0.81. An experimental alignment tolerance analysis is also reported.

Matrix calculations underlie countless problems in science, mathematics, and engineering. When the involved matrices are highly structured, displacement operators can be used to accelerate fundamental operations such as matrix-vector multiplication. In this thesis, we provide an introduction to the theory of displacement operators and study the interplay between displacement and natural matrix constructions involving direct sums, Kronecker products, and blocking. We also investigate the algebraic behavior of displacement operators, developing results about invertibility and kernels.

Dans cette thèse, on aborde trois thèmes : problème de sélection de colonnes dans une matrice, distance de Banach-Mazur au cube et estimation de la covariance de matrices aléatoires. Bien que les trois thèmes paraissent éloignés, les techniques utilisées se ressemblent tout au long de la thèse. Dans un premier lieu, nous généralisons le principe d'invertibilité restreinte de Bourgain-Tzafriri. Ce résultat permet d'extraire un "grand" bloc de colonnes linéairement indépendantes dans une matrice et d'estimer la plus petite valeur singulière de la matrice extraite. Nous proposons ensuite un algorithme déterministe pour extraire d'une matrice un bloc presque isométrique c'est à dire une sous-matrice dont les valeurs singulières sont proches de 1. Ce résultat nous permet de retrouver le meilleur résultat connu sur la célèbre conjecture de Kadison-Singer. Des applications à la théorie locale des espaces de Banach ainsi qu'à l'analyse harmonique sont déduites. Nous donnons une estimation de la distance de Banach-Mazur d'un corps convexe de Rn au cube de dimension n. Nous proposons une démarche plus élémentaire, basée sur le principe d'invertibilité restreinte, pour améliorer et simplifier les résultats précédents concernant ce problème. Plusieurs travaux ont été consacrés pour approcher la matrice de covariance d'un vecteur aléatoire par la matrice de covariance empirique. Nous étendons ce problème à un cadre matriciel et on répond à la question. Notre résultat peut être interprété comme une quantification de la loi des grands nombres pour des matrices aléatoires symétriques semi-définies positives. L'estimation obtenue s'applique à une large classe de matrices aléatoires