This report explores several applications of iteration and the various connections that can be made to different areas of mathematics. The ties iteration has to the Wada Property, bifurcation diagram, root finding, and applications in geometry are all investigated. Finally, a rationale for incorporating iteration into secondary mathematics courses to support a more robust curriculum is discussed. / text

As groundwater models are being used increasingly in the area of resource allocation, there has been an increase in the level of complexity in an attempt to capture heterogeneity, complex geometries and detail in interaction between the model domain and the outside hydraulic influences. As models strive to represent the real world in ever increasing detail, there is a strong likelihood that the boundary conditions will become nonlinear. Nonlinearities exist in the groundwater flow equation even in simple models when watertable (unconfined) conditions are simulated. This thesis is concerned with how these nonlinearities are treated numerically, with particular focus on the MODFLOW groundwater flow software and the nonlinear nature of the unconfined condition simulation. One of the limitations of MODFLOW is that it employs a first order fixed point iterative scheme to linearise the nonlinear system that arises as a result of the finite difference discretisation process, which is well known to offer slow convergence rates for highly nonlinear problems. However, Newton's method can achieve quadratic convergence and is more effective at dealing with higher levels of nonlinearity. Consequently, the main objective of this research is to investigate the inclusion of Newton's method to the suite of computational tools in MODFLOW to enhance its flexibility in dealing with the increasing complexity of real world problems, as well as providing a more competitive and efficient solution methodology. Furthermore, the underpinning linear iterative solvers that MODFLOW currently utilises are targeted at symmetric systems and a consequence of using Newton's method would be the requirement to solve non-symmetric Jacobian systems. Therefore, another important aspect of this work is to investigate linear iterative solution techniques that handle such systems, including the newer Krylov style solvers GMRES and BiCGSTAB. To achieve these objectives a number of simple benchmark problems involving nonlinearities through the simulation of unconfined conditions were established to compare the computational performance of the existing MODFLOW solvers to the new solution strategies investigated here. One of the highlights of these comparisons was that Newton's method when combined with an appropriately preconditioned Krylov solver was on average greater than 40% more CPU time efficient than the Picard based solution techniques. Furthermore, a significant amount of this time saving came from the reduction in the number of nonlinear iterations due to the quadratic nature of Newton's method. It was also found that Newton's method benefited more from improved initial conditions than Picard's method. Of all the linear iterative solvers tested, GMRES required the least amount of computational effort. While the Newton method involves more complexity in its implementation, this should not be interpreted as prohibitive in its application. The results here show that the extra work does result in performance increase, and thus the effort is certainly worth it.

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It is our purpose here to investigate the method of solving equations for real roots by Newton's Method and to indicate a generalization arising from this method.

Quantile is a basic and an important quantity of a random variable. In some distributions, their quantiles have closed-form expressions. However, for many continuous distributions, the closed-form expressions of their quantiles do not exist. Yu and Zelterman (2011) and Chang (2004) have proposed an approximation of quantiles. In this paper, we propose an improved method which is combined the Taylor expansion with Newton¡¦s method. Some examples are given to compare the computing time of the method we proposed with the methods in Yu and Zelterman (2011) and Chang (2004).

Iterative methods have been a very important area of study in numerical analysis since the inception of computational science. Their use ranges from solving algebraic equations to systems of differential equations and many more. In this thesis, we discuss several iterative methods, however our main focus is Newton's method. We present a detailed study of Newton's method, its order of convergence and the asymptotic error constant when solving problems of various types as well as analyze several pitfalls, which can affect convergence. We also pose some necessary and sufficient conditions on the function f for higher order of convergence. Different acceleration techniques are discussed with analysis of the asymptotic behavior of the iterates. Analogies between single variable and multivariable problems are detailed. We also explore some interesting phenomena while analyzing Newton's method for complex variables.

We present a practical algorithm for computing the minimum volume n-dimensional ellipsoid that must contain m given points al,...,am C Rn . This convex constrained problem arises in a variety of applied computational settings, particularly in data mining and robust statistics. Its structure makes it particularly amenable to solution by interior-point methods, and it has been the subject of much theoretical complexity analysis. Here we focus on computation. We present a combined interior-point and active-set method for solving this problem. Our computational results demonstrate that our method solves very large problem instances (m = 30, 000 and n = 30) to a high degree of accuracy in under 30 seconds on a personal computer.

Führungsgetriebe die durch Servomotoren angetrieben werden, benötigen für definierte Stellungen des Abtriebsglieds eine programmierte Funktion (elektronische Kurvenscheibe). Diese leitet sich aus dem möglicherweise fehlerbehafteten kinematischen Modell des Getriebes ab (inverse Kinematik). Zur Verbesserung der Genauigkeit der Führungsbewegung wird ein Verfahren zur Justierung der Übertragungsfunktion auf Basis des Newton-Verfahrens unter Nutzung der Singulärwertzerlegung vorgestellt. Dabei werden die realen Getriebeabmessungen anhand einer Messung berechnet und werden anschließend korrigiert zur Anpassung der Übertragungsfunktion verwendet.

Nonlinear algebraic equations (NAEs) occur in many areas of science and engineering. The process of solving these NAEs is generally difficult, from finding a good initial guess that leads to a desired solution to deciding on convergence criteria for the approximate solution. In practice, Newton's method is the only robust general-purpose method for solving a system of NAEs. Many variants of Newton's method exist. However, it is generally impossible to know a priori which variant of Newton's method will be effective for a given problem.<p>Many high-quality software libraries are available for the numerical solution of NAEs. However, the user usually has little control over many aspects of what the library does. For example, the user may not be able to easily switch between direct and indirect methods for the linear algebra. This thesis describes a problem-solving environment (PSE) called pythNon for studying the effects (e.g., performance) of different strategies for solving systems of NAEs. It provides the researcher, teacher, or student with a flexible environment for rapid prototyping and numerical experiments. In pythNon, users can directly influence the solution process on many levels, e.g., investigation of the effects of termination criteria and/or globalization strategies. In particular, to show the power, flexibility, and ease of use of the pythNon PSE, this thesis also describes the development of a novel forcing-term strategy for approximating the Newton direction efficiently in the pythNon PSE.

Nonlinear algebraic equations (NAEs) occur in many areas of science and engineering. The process of solving these NAEs is generally difficult, from finding a good initial guess that leads to a desired solution to deciding on convergence criteria for the approximate solution. In practice, Newton's method is the only robust general-purpose method for solving a system of NAEs. Many variants of Newton's method exist. However, it is generally impossible to know a priori which variant of Newton's method will be effective for a given problem.<p>Many high-quality software libraries are available for the numerical solution of NAEs. However, the user usually has little control over many aspects of what the library does. For example, the user may not be able to easily switch between direct and indirect methods for the linear algebra. This thesis describes a problem-solving environment (PSE) called pythNon for studying the effects (e.g., performance) of different strategies for solving systems of NAEs. It provides the researcher, teacher, or student with a flexible environment for rapid prototyping and numerical experiments. In pythNon, users can directly influence the solution process on many levels, e.g., investigation of the effects of termination criteria and/or globalization strategies. In particular, to show the power, flexibility, and ease of use of the pythNon PSE, this thesis also describes the development of a novel forcing-term strategy for approximating the Newton direction efficiently in the pythNon PSE.