Recent progress in exact geometric computation

AbstractComputational geometry has produced an impressive wealth of efficient algorithms. The robust implementation of these algorithms remains a major issue. Among the many proposed approaches for solving numerical non-robustness, Exact Geometric Computation (EGC) has emerged as one of the most successful. This survey describes recent progress in EGC research in three key areas: constructive zero bounds, approximate expression evaluation and numerical filters

Basis expansions in applied mathematics

Basis expansions are an extremely useful tool in applied mathematics. By using them, we can express a function representing a physical quantity as a linear combination of simpler ``modules'' with well-known properties. They are particularly useful for the applications described in this thesis. Perhaps the best known expansion of this type is the Fourier series of a periodic function, as decomposition into the infinite sum of simple sinusoidal and cosinusoidal elements, originally proposed by Fourier to study heat transfer. This dissertation employs some mathematical tools on problems taken from various areas of Engineering, exploiting their expansion properties: 1) Non-integer bases, which are applied to mathematical models in Robotics (Chapter 2). In this Chapter we study, in particular, a model for snake-like robots based on the Fibonacci sequence. It includes an investigation of the reachableworkspace, a more general analysis of the control system underlying the model, its reachability and local controllability properties. 2) Orthonormal bases, Riesz bases: exponential and cardinal Riesz basis with perturbations (Chapter 3). In this Chapter we obtain also a stability result for cardinal Riesz basis in the case of complex perturbations of the integers. We also consider a mathematical model for energy of the signal at the output of an ideal DAC, in presence of sampling clock jitter. When sampling clock jitter occurs, the energy of the signal at the output of ideal DAC does not satisfies a Parseval identity. Nevertheless, an estimation of the signal energy is here shown by a direct method involving cardinal series. 3) Orthogonal polynomials (Chapter 4). In this Chapter we introduce a new sequence of polynomials, which follow the same recursive rule of the well-known Lucas-Lehmer integer sequence. We show the most important properties of this sequence, relating them to the Chebyshev polynomials of the first and second kind. We discuss some relations between zeros of Lucas-Lehmer polynomials and Gray code. We study nested square roots of 2 applying a "binary code" that associates bits 0 and 1 to + and - signs in the nested form. This gives the possibility to obtain an ordering for the zeros of Lucas-Lehmer polynomials, which take the form of nested square roots of 2. These zeros are used to obtain two new formulas for Pi

Lazy exact real computation

EThOS - Electronic Theses Online ServiceGBUnited Kingdo

Algorithmic strategies for applicable real quantifier elimination

One of the most important algorithms for real quantifier elimination is the quantifier elimination by virtual substitution introduced by Weispfenning in 1988. In this thesis we present numerous algorithmic approaches for optimizing this quantifier elimination algorithm. Optimization goals are the actual running time of the implementation of the algorithm and the size of the output formula. Strategies for obtaining these goals include simplification of first-order formulas,reduction of the size of the computed elimination set, and condensing a new replacement for the virtual substitution. Local quantifier elimination computes formulas that are equivalent to the input formula only nearby a given point. We can make use of this restriction for further optimizing the quantifier elimination by virtual substitution. Finally we discuss how to solve a large class of scheduling problems by real quantifier elimination. To optimize our algorithm for solving scheduling problems we make use of the special form of the input formula and of additional information given by the description of the scheduling problemEines der bedeutendsten Verfahren zur reellen Quantorenelimination ist die Quantorenelimination mittels virtueller Substitution, die von Weispfenning 1988 eingeführt wurde. In der vorliegenden Arbeit werden zahlreiche algorithmische Strategien zur Optimierung dieses Verfahrens präsentiert. Optimierungsziele der Arbeit waren dabei die tatsächliche Laufzeit der Implementierung des Algorithmus sowie die Größe der Ausgabeformel. Zur Optimierung werden dabei die Simplifikation vonFormeln erster Stufe, die Reduktion der Größe der Eliminationsmenge sowie das Condensing, ein Ersatz für die virtuelle Substitution,untersucht. Lokale Quantorenelimination berechnet Formeln, die nur inder Nähe eines gegebenen Punktes äquivalent zur Eingabeformel ist. Diese Einschränkung erlaubt es, das Verfahren weiter zu verbessern.Als Anwendung des Eliminationsverfahren diskutieren wir abschließend, wie man eine große Klasse von Schedulingproblemen mittels reeller Quantorenelimination lösen kann. In diesem Fall benutzen wir die spezielle Struktur der Eingabeformel und zusätzliche Informationen über das Schedulingproblem, um die Quantorenelimination mittels virtueller Substitution problemspezifisch zu optimieren

Practical simplification of elementary functions using CAD

EThOS - Electronic Theses Online ServiceGBUnited Kingdo

Unconventional Coupled Cluster Theory: Single-Reference Methods for Multi-Reference Problems

Our overarching goal is the development of wavefunction-based quantum chemistry methods that give good results for both weakly- and strongly-correlated systems, that preserve as many fundamental properties of the exact wavefunction as possible, and that are computationally affordable and black-box for the end user. Many of our contributions in this area are based on the archetypal quantum chemistry method, coupled-cluster theory, and they have resulted from our efforts to understand and remedy the failure of symmetry-adapted coupled-cluster theory in the presence of strong correlation. In this thesis, we recount these investigations and discuss the key insights and results that they have produced. We describe in detail the novel wavefunction approaches we have devised, as well as the approximations and considerations one must make in order to arrive at useful equations. We believe that this work could be important for the continued development of affordable methods capable of accurately describing a wide variety of quantum chemical systems

Systematic construction of efficient six-stage fifth-order explicit Runge-Kutta embedded pairs without standard simplifying assumptions

This thesis examines methodologies and software to construct explicit Runge-Kutta (ERK) pairs for solving initial value problems (IVPs) by constructing efficient six-stage fifth-order ERK pairs without standard simplifying assumptions. The problem of whether efficient higher-order ERK pairs can be constructed algebraically without the standard simplifying assumptions dates back to at least the 1960s, with Cassity's complete solution of the six-stage fifth-order order conditions. Although RK methods based on the six-stage fifth-order order conditions have been widely studied and have continuing practical importance, prior to this thesis, the aforementioned complete solution to these order conditions has no published usage beyond the original series of publications by Cassity in the 1960s. The complete solution of six-stage fifth-order ERK order conditions published by Cassity in 1969 is not in a formulation that can easily be used for practical purposes, such as a software implementation. However, it is shown in this thesis that when the order conditions are solved and formulated appropriately using a computer algebra system (CAS), the generated code can be used for practical purposes and the complete solution is readily extended to ERK pairs. The condensed matrix form of the order conditions introduced by Cassity in 1969 is shown to be an ideal methodology, which probably has wider applicability, for solving order conditions using a CAS. The software package OCSage developed for this thesis, in order to solve the order conditions and study the properties of the resulting methods, is built on top of the Sage CAS. However, in order to effectively determine that the constructed ERK pairs without standard simplifying assumptions are in fact efficient by some well-defined criteria, the process of selecting the coefficients of ERK pairs is re-examined in conjunction with a sufficient amount of performance data. The pythODE software package developed for this thesis is used to generate a large amount of performance data from a large selection of candidate ERK pairs found using OCSage. In particular, it is shown that there is unlikely to be a well-defined methodology for selecting optimal pairs for general-purpose use, other than avoiding poor choices of certain properties and ensuring the error coefficients are as small as possible. However, for IVPs from celestial mechanics, there are obvious optimal pairs that have specific values of a small subset of the principal error coefficients (PECs). Statements seen in the literature that the best that can be done is treating all PECs equally do not necessarily apply to at least some broad classes of IVPs. By choosing ERK pairs based on specific values of individual PECs, not only are ERK pairs that are 20-30% more efficient than comparable published pairs found for test sets of IVPs from celestial mechanics, but the variation in performance between the best and worst ERK pairs that otherwise would seem to have similar properties is reduced from a factor of 2 down to as low as 15%. Based on observations of the small number of IVPs of other classes in common IVP test sets, there are other classes of IVPs that have different optimal values of the PECs. A more general contribution of this thesis is that it specifically demonstrates how specialized software tools and a larger amount of performance data than is typical can support novel empirical insights into numerical methods