Walter Gander

Least-Squares Fitting of Circles and Ellipses

Fitting circles and ellipses to given points in the plane is a problem that arises in many application areas, e.g., computer graphics, coordinate meteorology, petroleum engineering, statistics. In the past, algorithms have been given which fit circles and ellipses insome least-squares sense without minimizing the geometric distance to the given points.In this paper we present several algorithms which compute the ellipse for which thesum of the squares of the distances to the given points is minimal. These algorithms are compared with classical simple and iterative methods.Circles and ellipses may be represented algebraically, i.e., by an equation of the formF(x)=0. If a point is on the curve, then its coordinates x are a zero of the functionF. Alternatively, curves may be represented in parametric form, which is well suited for minimizing the sum of the squares of the distances.

Adaptive quadrature-revisited

First, the basic principles of adaptive quadrature are reviewed. Adaptive quadrature programs being recursive by nature, the choice of a good termination criterion is given particular attention. Two Matlab quadrature programs are presented. The first is an implementation of the well-known adaptive recursive Simpson rule; the second is new and is based on a four-point Gauss-Lobatto formula and two successive Kronrod extensions. Comparative test results are described and attention is drawn to serious deficiencies in the adaptive routines quad and quad8 provided by Matlab.

Least squares with a quadratic constraint

SummaryWe present the theory of the linear least squares problem with a quadratic constraint. New theorems characterizing properties of the solutions are given. A numerical application is discussed.

Fitting of circles and ellipses least squares solution

Publisher Summary Fitting ellipses to given points in the plane is a problem that arises in many application areas, such as computer graphics, coordinate metrology, petroleum engineering, statistics. In the past, algorithms have been given which fit circles and ellipses in some least squares sense without minimizing the geometric distance to the given points. This chapter first presents algorithms that compute the ellipse, for which the sum of the squares of the distances to the given points is minimal. Note that the solution of this non-linear least squares problem is generally expensive. Further, the chapter gives an overview of linear least squares solutions which minimize the distance in some algebraic sense. Given only a few points, it can be seen that the geometric solution often differs significantly from algebraic solutions. The chapter also refines the algebraic method by iteratively solving weighted linear least squares. A criterion based on the singular value decomposition is shown to be essential for the quality of the approximation to the exact geometric solution.

The Remote Computation System

Today many high performance computers are reachable over some network However the access and use of these computers is often complicated This prevents many users to work on such machines The goal of our Remote Computation System RCS is to alleviate the usage of modern algorithms on high performance computers RCS has an easytouse mechanism for using computational resources remotely The computational resources available are used as eciently as possible in order to minimize the response time We report on experiments involving computations from highend workstations up to supercomputers

Algorithms for the polar decomposition

For the polar decomposition of a square nonsingular matrix, Higham [SIAM J. Sci. Statist. Comput., 7 (1986), pp. 1160–1174] has given a reliable quadratically convergent algorithm that is based on Newtons iteration. Motivated by Halleys iteration, the author constructs a new family of methods that contains both methods (Highams and Halleys) as special cases. These methods generalize to rectangular matrices and some of them are also useful in computing the polar decomposition of rank deficient matrices.

The remote computation system

Abstract Today many high performance computers are reachable over some network. However, the access and use of these computers is often complicated. This prevents many users from working on such machines. The goal of the remote computation system (RCS) is to provide easy access to modern parallel algorithms on supercomputers for the inexperienced user. RCS has an easy-to-use mechanism for using computational resources remotely. The computational resources available are used as efficiently as possible in order to minimize the response time.

Gram-Schmidt orthogonalization: 100 years and more

In 1907, Erhard Schmidt published a paper in which he introduced an orthogonalization algorithm that has since become known as the classical Gram-Schmidt process. Schmidt claimed that his procedure ...

Restricted rank modification of the symmetric eigenvalue problem: Theoretical considerations

The problem is considered how to obtain the eigenvalues and vectors of a matrix A+VVT where A is a symmetric matrix with known spectral decomposition and VVT is a positive semidefinite matrix of low rank. It is shown that the eigenvalues of A+VVT can easily be located to any desired accuracy by means of the inertia of the matrix I − VT(λ − A)-1V. The problem of determining the eigenvalues of A restricted to R(V)⊥ can be treated likewise.

Scientific Computing - An Introduction using Maple and MATLAB

Scientific computing is the study of how to use computers effectively to solve problems that arise from the mathematical modeling of phenomena in science and engineering. It is based on mathematics, numerical and symbolic/algebraic computations and visualization. This book serves as an introduction to both the theory and practice of scientific computing, with each chapter presenting the basic algorithms that serve as the workhorses of many scientific codes; we explain both the theory behind these algorithms and how they must be implemented in order to work reliably in finite-precision arithmetic. The book includes many programs written in Matlab and Maple Maple is often used to derive numerical algorithms, whereas Matlab is used to implement them. The theory is developed in such a way that students can learn by themselves as they work through the text. Each chapter contains numerous examples and problems to help readers understand the material hands-on.