We compare the convergence behavior of Gauss quadrature with that of its younger brother, Clenshaw-Curtis. Seven-line MATLAB codes are presented that implement both methods, and experiments show that the supposed factor-of-2 advantage of Gauss quadrature is rarely realized. Theorems are given to explain this effect. First, following O'Hara and Smith in the 1960s, the phenomenon is explained as a consequence of aliasing of coefficients in Chebyshev expansions. Then another explanation is offered based on the interpretation of a quadrature formula as a rational approximation of $\log((z+1)/(z-1))$ in the complex plane. Gauss quadrature corresponds to Pade approximation at $z=\infty$. Clenshaw-Curtis quadrature corresponds to an approximation whose order of accuracy at $z=\infty$ is only half as high, but which is nevertheless equally accurate near $[-1,1]$.

/pdf/is-gauss-quadrature-better-than-clenshaw-curtis-44wkn57gls.pdf

Numerical analysis

Is Gauss Quadrature Better than Clenshaw-Curtis?

From the Publisher:
Do you spend too much time creating the building blocks of your graphics applications or finding and correcting errors? Geometric Tools for Computer Graphics is an extensive, conveniently organized collection of proven solutions to fundamental problems that you'd rather not solve over and over again, including building primitives, distance calculation, approximation, containment, decomposition, intersection determination, separation, and more.
If you have a mathematics degree, this book will save you time and trouble. If you don't, it will help you achieve things you may feel are out of your reach. Inside, each problem is clearly stated and diagrammed, and the fully detailed solutions are presented in easy-to-understand pseudocode. You also get the mathematics and geometry background needed to make optimal use of the solutions, as well as an abundance of reference material contained in a series of appendices.
KEY FEATURES: * Filled with robust, thoroughly tested solutions that will save you time and help you avoid costly errors.
* Covers problems relevant for both 2D and 3D graphics programming.
* Presents each problem and solution in stand-alone form allowing you the option of reading only those entries that matter to you.
* Provides the math and geometry background you need to understand the solutions and put them to work.
* Clearly diagrams each problem and presents solutions in easy-to-understand pseudocode.
* Resources associated with the book are available at the companion Web site.
Author Biography: Philip Schneider leads a modeling and dynamic simulation software group at Walt Disney Feature Animation. Prior to that, his work at Apple and Digital Equipment Corporation in 3D graphics ranged from low-level interfaces to graphics libraries and interactive applications. He holds an M.S. in Computer Science from the University of Washington.
Dave Eberly is the president of Magic Software, Inc., a company known for its free source code and documentation for computer graphics, image analysis, and numerical methods. Previously, he was the director of engineering at Numerical Design Limited, the company responsible for the real-time 3D game engine, NetImmerse. His background includes a B.A. in mathematics from Bloomsburg University, M.S. and Ph.D. degrees in mathematics from the University of Colorado at Boulder, and M.S. and Ph.D. degrees in computer science from the University of North Carolina at Chapel Hill. Dave is author of 3D Game Engine Design, co-author with Philip Schneider of Geometric Tools for Computer Graphics, and author of the forthcoming Game Physics (Spring 2003).

Geometric Tools for Computer Graphics

Accurate estimation of software development effort is critical in software engineering. Underestimates lead to time pressures that may compromise full functional development and thorough testing of software. In contrast, overestimates can result in noncompetitive contract bids and/or over allocation of development resources and personnel. As a result, many models for estimating software development effort have been proposed. This article describes two methods of machine learning, which we use to build estimators of software development effort from historical data. Our experiments indicate that these techniques are competitive with traditional estimators on one dataset, but also illustrate that these methods are sensitive to the data on which they are trained. This cautionary note applies to any model-construction strategy that relies on historical data. All such models for software effort estimation should be evaluated by exploring model sensitivity on a variety of historical data. >

Machine learning approaches to estimating software development effort

We present a novel, computerized method of examining cerebral cortical thickness. The normal cortex varies in thickness from 2 to 4 mm, reflecting the morphology of neuronal sublayers. Cortical pathologies often manifest abnormal variations in thickness, with examples of Alzheimer's disease and cortical dysplasia as thin and thick cortex, respectively. Radiologically, images are 2-D slices through a highly convoluted 3-D object. Depending on the relative orientation of the slices with respect to the object, it is impossible to deduce abnormal cortical thickness without additional information from neighboring slices. We approach the problem by applying Laplace's Equation (∇2ψ = 0) from mathematical physics. The volume of the cortex is represented as the domain for the solution of the differential equation, with separate boundary conditions at the gray-white junction and the gray-CSF junction. Normalized gradients of ψ form a vector field, representing tangent vectors along field lines connecting both boundaries. We define the cortical thickness at any point in the cortex to be the pathlength along such lines. Key advantages of this method are that it is fully three-dimensional, and the thickness is uniquely defined for any point in the cortex. We present graphical results that map cortical thickness everywhere in a normal brain. Results show global variations in cortical thickness consistent with known neuroanatomy. The application of this technique to visualization of cortical thickness in brains with known pathology has broad clinical implications. Hum. Brain Mapping 11:12–32, 2000. © 2000 Wiley-Liss, Inc.

/pdf/three-dimensional-mapping-of-cortical-thickness-using-dbvpydzxuv.pdf

Three-dimensional mapping of cortical thickness using Laplace's equation.

Legless figures such as snakes and worms are modelled as mass-spring systems. Muscle contractions are simulated by animating the spring tensions. Directional friction due to the surface structure is included in the dynamic model and legless figure locomotion results. Various modes of locomotion are described.

Numerical analysis

Citations

Is Gauss Quadrature Better than Clenshaw-Curtis?

Geometric Tools for Computer Graphics

Machine learning approaches to estimating software development effort

Three-dimensional mapping of cortical thickness using Laplace's equation.

The motion dynamics of snakes and worms

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