Some Real-Life Applications of a Newly Designed Algorithm for Nonlinear Equations and Its Dynamics via Computer Tools

In this article, we design a novel fourth-order and derivative free root-finding algorithm. We construct this algorithm by applying the finite difference scheme on the well-known Ostrowski’s method. The convergence analysis shows that the newly designed algorithm possesses fourth-order convergence. To demonstrate the applicability of the designed algorithm, we consider five real-life engineering problems in the form of nonlinear scalar functions and then solve them via computer tools. The numerical results show that the new algorithm outperforms the other fourth-order comparable algorithms in the literature in terms of performance, applicability, and efficiency. Finally, we present the dynamics of the designed algorithm via computer tools by examining certain complex polynomials that depict the convergence and other graphical features of the designed algorithm.

Truncating the exponential with a uniform distribution

For a sample of Exponentially distributed durations we aim at point estimation and a confidence interval for its parameter. A duration is only observed if it has ended within a certain time interval, determined by a Uniform distribution. Hence, the data is a truncated empirical process that we can approximate by a Poisson process when only a small portion of the sample is observed, as is the case for our applications. We derive the likelihood from standard arguments for point processes, acknowledging the size of the latent sample as the second parameter, and derive the maximum likelihood estimator for both. Consistency and asymptotic normality of the estimator for the Exponential parameter are derived from standard results on M-estimation. We compare the design with a simple random sample assumption for the observed durations. Theoretically, the derivative of the log-likelihood is less steep in the truncation-design for small parameter values, indicating a larger computational effort for root finding and a larger standard error. In applications from the social and economic sciences and in simulations, we indeed, find a moderately increased standard error when acknowledging truncation.

Modelling of Planetary Gravitational Microlensing Events

<p>This thesis describes and develops procedures for the generation of theoretical lightcurves that can be used to model gravitational microlensing events that involve multiple lenses. Of particular interest are the cases involving a single lens star with one or more orbiting planets, as this has proven to be an effective way of detecting extrasolar planets. Although there is an analytical expression for microlensing lightcurves produced by single lensing body, the generation of model lightcurves for more than one lensing body requires the use of numerical techniques. The method developed here, known as the semi-analytic method, involves the analytical rearrangement of the relatively simple ‘lens equation’ to produce a high-order complex lens polynomial. Root-finding algorithms are then used to obtain the roots of this ‘lens polynomial’ in order to locate the positions of the images and calculate their magnifications. By running example microlensing events through the root-finding algorithms, both the speed and accuracy of the Laguerre and Jenkins-Traub algorithms were investigated. It was discovered that, in order to correctly identify the image positions, a method involving solutions of several ‘lens polynomials’ corresponding to different coordinate origins needed to be invoked. Multipole and polygon approximations were also developed to include finite source and limb darkening effects. The semi-analytical method and the appropriate numerical techniques were incorporated into a C++ modelling code at VUW (Victoria University of Wellington) known as mlens2. The effectiveness of the semi-analytic method was demonstrated using mlens2 to generate theoretical lightcurves for the microlensing events MOA-2009-BLG-319 and OGLE-2006-BLG-109. By comparing these theoretical lightcurves with the observed photometric data and the published models, it was demonstrated that the semi-analytic method described in this thesis is a robust and efficient method for discovering extrasolar planets.</p>

Modelling of Planetary Gravitational Microlensing Events

<p>This thesis describes and develops procedures for the generation of theoretical lightcurves that can be used to model gravitational microlensing events that involve multiple lenses. Of particular interest are the cases involving a single lens star with one or more orbiting planets, as this has proven to be an effective way of detecting extrasolar planets. Although there is an analytical expression for microlensing lightcurves produced by single lensing body, the generation of model lightcurves for more than one lensing body requires the use of numerical techniques. The method developed here, known as the semi-analytic method, involves the analytical rearrangement of the relatively simple ‘lens equation’ to produce a high-order complex lens polynomial. Root-finding algorithms are then used to obtain the roots of this ‘lens polynomial’ in order to locate the positions of the images and calculate their magnifications. By running example microlensing events through the root-finding algorithms, both the speed and accuracy of the Laguerre and Jenkins-Traub algorithms were investigated. It was discovered that, in order to correctly identify the image positions, a method involving solutions of several ‘lens polynomials’ corresponding to different coordinate origins needed to be invoked. Multipole and polygon approximations were also developed to include finite source and limb darkening effects. The semi-analytical method and the appropriate numerical techniques were incorporated into a C++ modelling code at VUW (Victoria University of Wellington) known as mlens2. The effectiveness of the semi-analytic method was demonstrated using mlens2 to generate theoretical lightcurves for the microlensing events MOA-2009-BLG-319 and OGLE-2006-BLG-109. By comparing these theoretical lightcurves with the observed photometric data and the published models, it was demonstrated that the semi-analytic method described in this thesis is a robust and efficient method for discovering extrasolar planets.</p>

Machine Learning for Conservative-to-Primitive in Relativistic Hydrodynamics

The numerical solution of relativistic hydrodynamics equations in conservative form requires root-finding algorithms that invert the conservative-to-primitive variables map. These algorithms employ the equation of state of the fluid and can be computationally demanding for applications involving sophisticated microphysics models, such as those required to calculate accurate gravitational wave signals in numerical relativity simulations of binary neutron stars. This work explores the use of machine learning methods to speed up the recovery of primitives in relativistic hydrodynamics. Artificial neural networks are trained to replace either the interpolations of a tabulated equation of state or directly the conservative-to-primitive map. The application of these neural networks to simple benchmark problems shows that both approaches improve over traditional root finders with tabular equation-of-state and multi-dimensional interpolations. In particular, the neural networks for the conservative-to-primitive map accelerate the variable recovery by more than an order of magnitude over standard methods while maintaining accuracy. Neural networks are thus an interesting option to improve the speed and robustness of relativistic hydrodynamics algorithms.

A New Three-Step Root-Finding Numerical Method and Its Fractal Global Behavior

There is an increasing demand for numerical methods to obtain accurate approximate solutions for nonlinear models based upon polynomials and transcendental equations under both single and multivariate variables. Keeping in mind the high demand within the scientific literature, we attempt to devise a new nonlinear three-step method with tenth-order convergence while using six functional evaluations (three functions and three first-order derivatives) per iteration. The method has an efficiency index of about 1.4678, which is higher than most optimal methods. Convergence analysis for single and systems of nonlinear equations is also carried out. The same is verified with the approximated computational order of convergence in the absence of an exact solution. To observe the global fractal behavior of the proposed method, different types of complex functions are considered under basins of attraction. When compared with various well-known methods, it is observed that the proposed method achieves prespecified tolerance in the minimum number of iterations while assuming different initial guesses. Nonlinear models include those employed in science and engineering, including chemical, electrical, biochemical, geometrical, and meteorological models.

On iterative techniques for estimating all roots of nonlinear equation and its system with application in differential equation

In this article, we construct a family of iterative methods for finding a single root of nonlinear equation and then generalize this family of iterative methods for determining all roots of nonlinear equations simultaneously. Further we extend this family of root estimating methods for solving a system of nonlinear equations. Convergence analysis shows that the order of convergence is 3 in case of the single root finding method as well as for the system of nonlinear equations and is 5 for simultaneous determination of all distinct and multiple roots of a nonlinear equation. The computational cost, basin of attraction, efficiency, log of residual and numerical test examples show that the newly constructed methods are more efficient as compared to the existing methods in literature.

New Approach for Secant Update Generalized Version of PSB

Working with Quasi-Newton methods in optimization leads to one important challenge, being to find an estimate of the Hessian matrix as close as possible to the real matrix. While multisecant methods are regularly used to solve root finding problems, they have been little explored in optimization because the symmetry property of the Hessian matrix estimation is generally not compatible with the multisecant property. In this paper, we propose a solution to apply multisecant methods to optimization problems. Starting from the Powell-Symmetric-Broyden (PSB) update formula and adding pieces of information from the previous steps of the optimization path, we want to develop a new update formula for the estimate of the Hessian. A multisecant version of PSB is, however, generally mathematically impossible to build. For that reason, we provide a formula that satisfies the symmetry and is as close as possible to satisfy the multisecant condition and vice versa for a second formula. Subsequently, we add enforcement of the last secant equation to the symmetric formula and present a comparison between the different methods.

A Large-scale Benchmark and an Inclusion-based Algorithm for Continuous Collision Detection

We introduce a large-scale benchmark for continuous collision detection (CCD) algorithms, composed of queries manually constructed to highlight challenging degenerate cases and automatically generated using existing simulators to cover common cases. We use the benchmark to evaluate the accuracy, correctness, and efficiency of state-of-the-art continuous collision detection algorithms, both with and without minimal separation. We discover that, despite the widespread use of CCD algorithms, existing algorithms are (1) correct but impractically slow; (2) efficient but incorrect, introducing false negatives that will lead to interpenetration; or (3) correct but over conservative, reporting a large number of false positives that might lead to inaccuracies when integrated in a simulator. By combining the seminal interval root finding algorithm introduced by Snyder in 1992 with modern predicate design techniques, we propose a simple and efficient CCD algorithm. This algorithm is competitive with state-of-the-art methods in terms of runtime while conservatively reporting the time of impact and allowing explicit tradeoff between runtime efficiency and number of false positives reported.