In this thesis, we propose different CORDIC architectures which solve the problems of long-latency in traditional pipeline CORDIC and the large-area cost in table-based CORDIC. The original table-based CORDIC can be divided into two stages, coarse stage and fine stage. We also propose the three-stage architectures, composed of traditional pipeline CORDIC, Rom/Multiplier architecture and linear approximation. Detailed analysis and estimation in area and latency of these different two-stage and three-stage architectures with different bit accuracy are given in order to determine the best architecture design for a particular precision. Finally, we choose one of the architectures to implement, compare the results, and show its applications.

Αποδεικνύουμε την ύπαρξη καθολικών σειρών taylor στην περίπτωση συγκεκριμένου μη απλά συνεκτικού τόπου, καθώς και την ύπαρξη ενός πυκνού διανυσματικού υποχώρου. / We prove the existence of universal taylor series in the case of a specific non simply connected domain. We also prove the existence of a dense vector subspace.

Dados de contagem podem ser considerados, em geral, como provenientes de uma distribuição de Poisson. Neste contexto, a análise de tais dados apresenta certas dificuldades, pois não segue algumas pressuposições básicas para o ajuste de um modelo matemático. Desse modo, algumas transformações são sugeridas, mas nem sempre bons resultados são obtidos. No enfoque de Modelos Lineares Generalizados, a estatística que mede a qualidade do ajuste do modelo para os dados é chamada deviance. Porém, a distribuição da deviance é, em geral, desconhecida. No entanto, para dados com distribuição de Poisson, pode-se mostrar que a distribuição da deviance se aproxima de uma distribuição ?2, mas tal aproximação não é boa para tamanhos pequenos de amostra. Para melhorar essa aproximação, alguns fatores de correção para os dados são sugeridos, mas os resultados obtidos ainda não são satisfatórios. Assim, o objetivo deste trabalho é propor um novo fator de correção para os dados seguindo uma distribuição de Poisson, de modo a se obter uma melhora na distribuição da deviance para qualquer tamanho de amostra. Para isto, será adicionada uma constante à variável resposta e, através do valor esperado da deviance, calcula-se tal constante de modo a reduzir o erro cometido na aproximação. Para verificar a melhora na aproximação da distribuição da deviance a uma distribuição qui-quadrado, dados de uma distribuição de Poisson são simulados e o valor da deviance é calculado. QQ-plots são construídos para a comparação com a distribuição qui-quadrado. / Analysis of count data presents, in general, can be supposed coming from a Poisson distribution. The analysis of such data have some problems once the underlying distribution of them does not follow the basic assumptions to fit a model. Some tranformations can be suggested, but good results are not always obtained. In the approach of the Generalized Linear Models, the deviance is the statistics that measures the goodness of fit, but its distribution is unknown. Furthermore, considering Poisson distribution data, it is possible to approximate the distribution of the deviance for a chi-square distribution, but such approximation is not good for small sample size. In order of improve this approximation, corrections for the data are suggested, but the results are not good yet. Then, the aim of this work is to propose a new correction factor for data following a Poisson distribution in order to obtain an improvement in the distribution of the deviance for any sample size. For this, just adding a constant at the response variable and, through the expected value of the deviance, such constant is obtained in order to reduce the error in the aproximation. Simulated data from the Poisson distribution were made to calculate the deviance with and without the correction and QQ-plots were used to compare the values of the deviance with the chi-square distribution.

In this thesis, an efficiency method for reducing the rotation ROM size in table-based architecture is proposed. The original rotation can be divided into two stages, coarse stage and fine stage. Our approach modifies the previous two-stage rotation method and proposes a multi-stage architecture and discuses three-stage phase calculation. The effect of table reduction is more apparently for higher accuracy requirement in the three-stage architecture. The total area of the previous two-stage architecture is larger than the proposed table-reduced three-stage architecture because the table size takes a significant ratio of the total area especially when the required bit accuracy is large. In the proposed three-stage design, there are two different types of architectures, depending on the rotation angles in the first and second rotation stages. Comparison of different types of architecture with the previous method shows that our designs indeed reduce the table size and the total area significantly.

Το αντικείμενο αυτής της εργασίας είναι η μελέτη κάποιων κλάσεων καθολικών συναρτήσεων. Οι κλάσεις αυτές περιέχουν συναρτήσεις μιας μιγαδικής μεταβλητής, οι οποίες πραγματοποιούν εντυπωσιακές προσεγγίσεις πάνω σε συμπαγή υποσύνολα του μιγαδικού επιπέδου. Πιο συγκεκριμένα, θα ασχοληθούμε με δύο κλάσεις καθολικών σειρών Taylor, και με μία κλάση καθολικών συναρτήσεων ως προς τις παραγώγους. Καθολική σειρά Taylor, με την έννοια του Β. Νεστορίδη, ονομάζουμε μία συνάρτηση f , ολόμορφη σε κάποιο ανοιχτό σύνολο Ω ⊂ 􀀀 , η οποία με τη βοήθεια των μερικών αθροισμάτων του αναπτύγματος Taylor γύρω από ένα κέντρο ζ∈Ω, προσεγγίζει όλα τα πολυώνυμα ομοιόμορφα στα συμπαγή υποσύνολα του Ωc , με συνεκτικό συμπλήρωμα. Αυτή την κλάση συναρτήσεων την συμβολίζουμε με U(Ω,ζ). Επιπλέον υπάρχει η ασθενέστερη κλάση 1 U (Ω,ζ) , των καθολικών σειρών Taylor με την έννοια του Luh, η οποία περιέχει συναρτήσεις που πραγματοποιούν του ίδιου τύπου προσεγγίσεις, αλλά μόνο σε συμπαγή υποσύνολα του c Ω . Στο πρώτο κεφάλαιο, παρουσιάζουμε κάποια γενικά αποτελέσματα, που είναι προαπαιτούμενα για ότι θα ακολουθήσει. Πιο συγκεκριμένα, διατυπώνουμε και αποδεικνύουμε το κλασσικό θεώρημα Baire που ισχύει σε πλήρεις χώρους καθώς και κάποια τοπολογικά λήμματα που ισχύουν στο μιγαδικό επίπεδο. Το πρώτο λήμμα είναι ένα κλασσικό αποτέλεσμα που μας εξασφαλίζει την ύπαρξη εξαντλούσας ακολουθίας συμπαγών συνόλων ενός ανοιχτού υποσυνόλου του 􀀀 , με κατάλληλες ιδιότητες. Το δεύτερο λήμμα είναι ένα πιο ειδικό και τεχνικό αποτέλεσμα, οφείλεται στον Β.Νεστορίδη και θεωρείται σημαντικό βήμα στην μελέτη των καθολικών σειρών Taylor. Επίσης, αναφέρουμε τα γνωστά προσεγγιστικά θεωρήματα των Runge και Mergelyan με την βοήθεια των οποίων, μπορούμε να μελετάμε μόνο τα πολυώνυμα και να έχουμε αποτελέσματα που ισχύουν σε γενικότερες συναρτήσεις. Στο δεύτερο κεφάλαιο θα παρουσιάσουμε την απόδειξη του Β. Νεστορίδη, ότι η κλάση U(Ω,ζ) είναι Gδ και πυκνό υποσύνολο του Η(Ω), με την τοπολογία που αναφέραμε παραπάνω. Το αποτέλεσμα αυτό είναι ιδιαίτερα εντυπωσιακό, διότι μας αποκαλύπτει ότι με μερικά αθροίσματα αναπτύγματος της ίδιας συνάρτησης, μπορούμε να προσεγγίσουμε όλα τα πολυώνυμα πάνω σε μια πολύ μεγάλη κλάση συμπαγών υποσυνόλων. Στο κεφάλαιο 3, θα παρουσιάσουμε την απόδειξη ότι η κλάση (Ω) der U είναι Gδ και πυκνό υποσύνολο στο Η(Ω). Όπως το παραπάνω αποτέλεσμα, έτσι και αυτό έχει ιδιαίτερο ενδιαφέρον, αφού μας εξασφαλίζει την ύπαρξη πολλών συναρτήσεων στην κλάση (Ω) der U . Το τελευταίο κεφάλαιο αποτελεί το κύριο μέρος της παρούσας εργασίας. Βασικός στόχος μας είναι να αποδείξουμε την ύπαρξη 1-1 συναρτήσεων στην τομή των κλάσεων 1( , ) ( ) der UΩζ IU Ω (βλ.[5]). Σημειώνουμε ότι για αυτό το αποτέλεσμα πρέπει το Ω να είναι ειδικότερα χωρίο Jordan. Στην 3 βιβλιογραφία έχει αποδειχτεί ότι η κλάση U(D,0) δεν περιέχει 1-1 συναρτήσεις, και μάλιστα είναι ξένη με την κλάση Nevanlinna οπότε δεν περιμένουμε από τις συναρτήσεις αυτές να έχουν καλές ιδιότητες (βλ.[14]). Αντίθετα το αποτέλεσμα που παρουσιάζουμε υποδηλώνει ότι το φαινόμενο αυτό δεν εμφανίζεται στις υπόλοιπες δύο κλάσεις καθολικών συναρτήσεων και δείχνει πόσο διαφορετική είναι η κλάση U(Ω,ζ) από την κλάση U1(Ω,ζ ). / -

Dados de contagem podem ser considerados, em geral, como provenientes de uma distribuição de Poisson. Neste contexto, a análise de tais dados apresenta certas dificuldades, pois não segue algumas pressuposições básicas para o ajuste de um modelo matemático. Desse modo, algumas transformações são sugeridas, mas nem sempre bons resultados são obtidos. No enfoque de Modelos Lineares Generalizados, a estatística que mede a qualidade do ajuste do modelo para os dados é chamada deviance. Porém, a distribuição da deviance é, em geral, desconhecida. No entanto, para dados com distribuição de Poisson, pode-se mostrar que a distribuição da deviance se aproxima de uma distribuição ?2, mas tal aproximação não é boa para tamanhos pequenos de amostra. Para melhorar essa aproximação, alguns fatores de correção para os dados são sugeridos, mas os resultados obtidos ainda não são satisfatórios. Assim, o objetivo deste trabalho é propor um novo fator de correção para os dados seguindo uma distribuição de Poisson, de modo a se obter uma melhora na distribuição da deviance para qualquer tamanho de amostra. Para isto, será adicionada uma constante à variável resposta e, através do valor esperado da deviance, calcula-se tal constante de modo a reduzir o erro cometido na aproximação. Para verificar a melhora na aproximação da distribuição da deviance a uma distribuição qui-quadrado, dados de uma distribuição de Poisson são simulados e o valor da deviance é calculado. QQ-plots são construídos para a comparação com a distribuição qui-quadrado. / Analysis of count data presents, in general, can be supposed coming from a Poisson distribution. The analysis of such data have some problems once the underlying distribution of them does not follow the basic assumptions to fit a model. Some tranformations can be suggested, but good results are not always obtained. In the approach of the Generalized Linear Models, the deviance is the statistics that measures the goodness of fit, but its distribution is unknown. Furthermore, considering Poisson distribution data, it is possible to approximate the distribution of the deviance for a chi-square distribution, but such approximation is not good for small sample size. In order of improve this approximation, corrections for the data are suggested, but the results are not good yet. Then, the aim of this work is to propose a new correction factor for data following a Poisson distribution in order to obtain an improvement in the distribution of the deviance for any sample size. For this, just adding a constant at the response variable and, through the expected value of the deviance, such constant is obtained in order to reduce the error in the aproximation. Simulated data from the Poisson distribution were made to calculate the deviance with and without the correction and QQ-plots were used to compare the values of the deviance with the chi-square distribution.

No description available.

We propose two efficient automatic differentiation (AD) schemes to compute incrementally Taylor series and System Jacobian for solving differential-algebraic equations (DAEs) by Taylor series. Our schemes are based on topological ordering of a DAE's computational graph and then partitioning the topologically sorted nodes using structural information obtained from the DAE. Solving a DAE by Taylor series is carried out in stages. From one stage to another, partitions of the computational graph are incrementally activated so that we can reuse Taylor coefficients and gradients computed in previous stages. As a result, the computational complexity of evaluating a System Jacobian is independent of the number of stages. We also develop a common subexpression elimination (CSE) method to build a compact computational graph through operator overloading. The CSE method is of linear time complexity, which makes it suitable as a preprocessing step for general operator overloaded computing. By applying CSE, all successive overloaded computation can save time and memory. Furthermore, the computational graph of a DAE reveals its internal sparsity structure. Based on it, we devise an algorithm to propagate gradients in the forward mode of AD using compressed vectors. This algorithm can save both time and memory when computing the System Jacobian for sparse DAEs. We have integrated our approaches into the \daets solver. Computational results show multiple-fold speedups against two popular AD tools, \FAD~and ADOL-C, when solving various sparse and dense DAEs. / Thesis / Master of Science (MSc)

Η παρούσα διπλωματική εργασία αποτελείται από δύο μέρη. Στο πρώτο μέρος θα μελετήσουμε βασικές έννοιες και θεωρήματα από την θεωρία δυναμικού. Έννοιες όπως το δυναμικό, τα polar σύνολα, η συνάρτηση Green ενός συνόλου και η χωρητικότητα ενός συνόλου είναι αναγκαίες ώστε να οδηγηθούμε στο περίφημο θεώρημα των Bernstein-Walsh, το οποίο δίνει την ταχύτητα της πολυωνυμικής προσέγγισης αναλυτικών συναρτήσεων σε συμπαγή σύνολα με συνεκτικό συμπλήρωμα. Στο δεύτερο μέρος, μελετάμε ένα αποτέλεσμα των Γ. Κωστάκη και Ν. Τσιρίβα, για μία έννοια σχετική με τις καθολικές σειρές Taylor, τις διπλά καθολικές σειρές Taylor. Συγκεκριμένα, για δοσμένη γνησίως αύξουσα ακολουθία φυσικών αριθμών (λn), μια ολόμορφη συνάρτηση f στον ανοιχτό μοναδιαίο δίσκο λέγεται διπλά καθολική σειρά Taylor, ως προς τις ακολουθίες (n),(λn), αν για κάθε συμπαγές σύνολο Κ, υποσύνολο του μιγαδικού επιπέδου, ξένο με τον δίσκο και με συνεκτικό συμπλήρωμα και για κάθε ζεύγος συναρτήσεων (g1,g2) συνεχών στο Κ, ολόμορφων στο εσωτερικό του Κ, υπάρχει υπακολουθία των φυσικών αριθμών (μn), τέτοια ώστε (S_{μn}(f,0),S_{λ_{μn}}(f,0)) προσεγγίζουν ομοιόμορφα τις (g_{1},g_{2}) (όπου S_{n}(f,0) το n-οστό μερικό άθροισμα του αναπτύγματος Taylor της f με κέντρο το 0). Το κεντρικό λοιπόν αποτέλεσμα είναι ότι για δοσμένη ακολουθία (λn), η οικογένεια των διπλά καθολικών σειρών Taylor, ως προς τις ακολουθίες (n),(λn), είναι Gδ και πυκνή στο σύνολο των ολόμορφων συναρτήσεων στον ανοιχτό μοναδιαίο δίσκο (ειδικότερα είναι μη-κενή) αν και μόνο αν το ανώτερο όριο limsup_{n}(λn/n) είναι άπειρο. Εργαλείο-κλειδί για το παραπάνω αποτέλεσμα είναι το Θεώρημα Bernstein-Walsh. / --

Orientador: Denis Dalmazi / Coorientador: Álvaro de Souza Dutra / Banca: Julio Marny Hoff da Silva / Banca: Rafael Augusto Couceiro Corrêa / Resumo: Certos sistemas físicos podem ser descritos por uma classe de equações não-lineares. Essas equações descrevem pacotes de onda chamado de sólitons que tem aplicações em diversas áreas, por exemplo, Óptica, Cosmologia, Matéria Condensada e Física de Partículas. Alguns métodos foram desenvolvidos ao longo dos anos para encontrar as soluções dessas equações. Buscaremos essas soluções usando o que chamamos de Método Iterativo de Taylor (MIT), que fornece uma solução aproximada em polinômio de Taylor de forma distinta do que se tem na literatura. Usaremos o MIT para calcular soluções por aproximantes de Padé que são razões entre dois polinômios e fornecem soluções melhores que o polinômio de Taylor que o gerou. Inicialmente resolveremos a equação de um modelo de um campo denominado λφ4 . Em seguida resolveremos um modelo com dois campos escalares acoplados e encontraremos uma solução analítica aproximada em casos onde não existe solução analítica, explorando a diversidade das soluções do modelo. Usando essa abordagem por aproximantes de Padé veremos que há algumas vantagens em relação a outros métodos / Abstract: Certain physical systems can be described by a class of non-linear differential equations. Those equations describe wave packets called solitons which have applications in several areas, for example, Optics, Cosmology, Condensed Matter, and Particle Physics. Some methods have been developed over the years to find solutions to these equations. We will look for those solutions using what we call the Taylor Iterative Method (TIM), which provides an approximate solution in terms of a Taylor's polynomial in a unusual way, regarding the present literature. We will use TIM to calculate solutions by Padé approximants, which are ratios between two polynomials and provide better solutions than the Taylor polynomial itself. We first solve the field equation of a model called λφ4 . Then we will solve a model with two coupled scalar fields and find an approximate analytic solution in cases where there is no known analytical solution, exploring the diversity of the solutions of the model. We will see that there are some advantages in using the Padè approximants as compared to other methods / Mestre