Computing the common zeros of two bivariate functions via Bezout resultants

The common zeros of two bivariate functions can be computed by finding the common zeros of their polynomial interpolants expressed in a tensor Chebyshev basis. From here we develop a bivariate rootfinding algorithm based on the hidden variable resultant method and B�ezout matrices with polynomial entries. Using techniques including domain subdivision, B�ezoutian regularization and local refinement we are able to reliably and accurately compute the simple common zeros of two smooth functions with polynomial interpolants of very high degree (�$\ge$ 1000). We analyze the resultant method and its conditioning by noting that the B�ezout matrices are matrix polynomials. Our robust algorithm is implemented in the roots command in Chebfun2, a software package written in object-oriented MATLAB for computing with bivariate functions

Computing the common zeros of two bivariate functions via Bézout resultants

The common zeros of two bivariate functions can be computed by finding the common zeros of their polynomial interpolants expressed in a tensor Chebyshev basis. From here we develop a bivariate rootfinding algorithm based on the hidden variable resultant method and Bézout matrices with polynomial entries. Using techniques including domain subdivision, Bézoutian regularization, and local refinement we are able to reliably and accurately compute the simple common zeros of two smooth functions with polynomial interpolants of very high degree (≥ 1000). We analyze the resultant method and its conditioning by noting that the Bézout matrices are matrix polynomials. Two implementations are available: one on the Matlab Central File Exchange and another in the roots command in Chebfun2 that is adapted to suit Chebfun’s methodology

A robust and scalable implementation of the Parks-McClellan algorithm for designing FIR filters

Preliminary version accepted for publicationInternational audienceWith a long history dating back to the beginning of the 1970s, the Parks-McClellan algorithm is probably the most well-known approach for designing finite impulse response filters. Despite being a standard routine in many signal processing packages, it is possible to find practical design specifications where existing codes fail to work. Our goal is twofold. We first examine and present solutions for the practical difficulties related to weighted minimax polynomial approximation problems on multi-interval domains (i.e., the general setting under which the Parks-McClellan algorithm operates). Using these ideas, we then describe a robust implementation of this algorithm. It routinely outperforms existing minimax filter design routines

Doctor of Philosophy

dissertationWhile boundary representations, such as nonuniform rational B-spline (NURBS) surfaces, have traditionally well served the needs of the modeling community, they have not seen widespread adoption among the wider engineering discipline. There is a common perception that NURBS are slow to evaluate and complex to implement. Whereas computer-aided design commonly deals with surfaces, the engineering community must deal with materials that have thickness. Traditional visualization techniques have avoided NURBS, and there has been little cross-talk between the rich spline approximation community and the larger engineering field. Recently there has been a strong desire to marry the modeling and analysis phases of the iterative design cycle, be it in car design, turbulent flow simulation around an airfoil, or lighting design. Research has demonstrated that employing a single representation throughout the cycle has key advantages. Furthermore, novel manufacturing techniques employing heterogeneous materials require the introduction of volumetric modeling representations. There is little question that fields such as scientific visualization and mechanical engineering could benefit from the powerful approximation properties of splines. In this dissertation, we remove several hurdles to the application of NURBS to problems in engineering and demonstrate how their unique properties can be leveraged to solve problems of interest

Trajectory generation for autonomous unmanned aircraft using inverse dynamics

The problem addressed in this research is the in-flight generation of trajectories for autonomous unmanned aircraft, which requires a method of generating pseudo-optimal trajectories in near-real-time, on-board the aircraft, and without external intervention. The focus of this research is the enhancement of a particular inverse dynamics direct method that is a candidate solution to the problem. This research introduces the following contributions to the method. A quaternion-based inverse dynamics model is introduced that represents all orientations without singularities, permits smooth interpolation of orientations, and generates more accurate controls than the previous Euler-angle model. Algorithmic modifications are introduced that: overcome singularities arising from parameterization and discretization; combine analytic and finite difference expressions to improve the accuracy of controls and constraints; remove roll ill-conditioning when the normal load factor is near zero, and extend the method to handle negative-g orientations. It is also shown in this research that quadratic interpolation improves the accuracy and speed of constraint evaluation. The method is known to lead to a multimodal constrained nonlinear optimization problem. The performance of the method with four nonlinear programming algorithms was investigated: a differential evolution algorithm was found to be capable of over 99% successful convergence, to generate solutions with better optimality than the quasi- Newton and derivative-free algorithms against which it was tested, but to be up to an order of magnitude slower than those algorithms. The effects of the degree and form of polynomial airspeed parameterization on optimization performance were investigated, and results were obtained that quantify the achievable optimality as a function of the parameterization degree. Overall, it was found that the method is a potentially viable method of on-board near- real-time trajectory generation for unmanned aircraft but for this potential to be realized in practice further improvements in computational speed are desirable. Candidate optimization strategies are identified for future research.EThOS - Electronic Theses Online ServiceGBUnited Kingdo

Parametric and nonparametric identification of nonlinearity in structural dynamics

The work described in this thesis is concerned with procedures for the identification of nonlinearity in structural dynamics. It begins with a diagnostic method which uses the Hubert transform for detecting nonlinearity and describes the neccessary conditions for obtaining a valid Hubert transform. The transform is shown to be incapable of producing a model with predictive power. A method based on the identification of nonlinear restoring forces is adopted for extracting a nonlinear model. The method is critically examined; various caveats, modifications and improvements are obtained. The method is demonstrated on time data obtained from computer simulations. It is shown that a parameter estimation approach to restoring force identification based on direct least—squares estimation theory is a fast and accurate procedure. In addition, this approach allows one to obtain the equations of motion for a multi—degree—of—freedom system even if the system is only excited at one point. The data processing methods for the restoring force identification including integration and differentiation of sampled time data are developed and discussed in some detail. A comparitive study is made of several of the most well—known least—squares estimation procedures and the direct least —squares approach is applied to data from several experiments where it is shown to correctly identify nonlinearity in both single— and multi—degree--of—freedom systems. Finally, using both simulated and experimental data, it is shown that the recursive least—squares algorithm modified by the inclusion of a data forgetting factor can be used to identify time—dependent structural parameters.Science and Engineering Research Counci