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Mathematical Resources

Bi-Conjugate Gradient Algorithm for Solution of Integral Equations Arising in Electromagnetic Scattering Problems

Surendra Singh, Klaus Halterman, and J. Merle Elson
Naval Air Warfare Center Weapons Division TP 8590
September 2004

The bi-conjugate gradient (bi-CG) algorithm is applied to numerically solve linear equation systems resulting from integral equations arising in electromagnetic scattering problems. The basic advantage of using this algorithm over traditional methods, such as matrix inversion, is that the algorithm is iterative in nature . The iterative nature allows the user to control the residual error in the final solution. Also, the algorithm can be implemented without storing the coefficient matrix, thus providing huge saving in storage requirements. It was realized that the existing code that utilized matrix inversion to solve the linear equation system was limited to a coarse discretization of the geometry. This code could not handle very fine geometry discretization due to storage limitations. With the implementation of the bi-CG algorithm, this limitation was overcome . The present code can very easily handle very fine discretizations, thereby vastly improving the utility of the computer code. This report outlines the bi-CG algorithm and provides its implementation to solve an electromagnetic scattering problem of a nanowire illuminated by a plane wave. The report also includes a complete FORTRAN listing of the code.

Books by Ray C. Bowen

Ray C. Bowen was President of Texas A&M University from 1994 to 2002. He also wrote or co-authored several books which dealt with topics at the borders of mathematics and mechanical engineering. Those available here are as follows:

Calculus and Linear Algebra

Wilfred Kaplan and Donald J. Lewis
University of Michigan

The principal purpose of the book was to provide an integration of linear algebra and calculus. Chapters and topics are as follows:

  • 0: Introduction, Review of Algebra, Geometry and Trigonometry
  • 1: Two-Dimensional Vector Geometry
  • 2: Limits
  • 3: Differential Calculus
  • 4: Integral Calculus
  • 5: Elementary Transcendental Functions
  • 6: Applications of Differential Calculus
  • 7: Applications of Integral Calculus
  • 8: Infinite Series
  • 9: Vector Spaces
  • 10: Matrices and Determinants
  • 11: Linear Euclidean Geometry
  • 12: Differential Calculus of Functions of Several Variables
  • 13: Integral Calculus of Functions of Several Variables
  • 14: Ordinary Differential Equations

Direct Search Methods: Then and Now

Robert Michael Lewis, Virginia Torczon and Michael W. Trosset
ICASE Report No. 2000-26
May 2000

We discuss direct search methods for unconstrained optimization. We give a modern perspective on this classical family of derivative-free algorithms, focusing on the development of direct search methods during their golden age from 1960 to 1971. We discuss how direct search methods are characterized by the absence of the construction of a model of the objective. We then consider a number of the classical direct search methods and discuss what research in the intervening years has uncovered about these algorithms. In particular, while the original direct search methods were consciously based on straightforward heuristics, more recent analysis has shown that in most—but not all—cases these heuristics actually suffice to ensure global convergence of at least one sub-sequence of the sequence of iterates to a first-order stationary point of the objective function.

Equations for Runge-Kutta Formulas Through the Eighth Order

H.A. Luther and J.A. Smitherman
Texas A&M University
NASA Research Grant No. NGR-44-001-024

For a reminiscence about Hubert Luther, and his most important work, click here.

The work’s title is pretty much self-explanatory.

Métodos Numéricos

Andrés Granados
Simon Bolívar University
November 2016

Una importante razón motivó la elaboración de este trabajo. En la literatura especializada de habla española no existe un texto que realice una introducción al Cálculo Numérico de una forma sencilla, resumida y completa, simultáneamente con aplicaciones al campo de la ingeniería mecánica. Un texto de Análisis Numérico sería demasiado tedioso para un curso enfocado para estudiantes de ingeniería. Un compendio de algoritmos sin la formalidad del análisis sería demasiado estéril para aquellos estudiantes que quieren un enfoque más general. En esta oportunidad se ha tratado de crear un híbrido de ambos aspectos aparentemente extremos. Esto se ha hecho mediante la estructuración de un recetario de métodos numéricos con inserciones de aspectos analíticos importantes, que en primera instancia pueden ser obviados. Sin embargo, para los estudiantes más curiosos, estos aspectos analíticos pueden ser revisados en una segunda o tercera lectura más detallada. Esta monografía en primera instancia fue desarrollada para estudiantes de pregrado, sin embargo, puede servir para un curso introductorio a nivel de postgrado, en donde se haga más énfasis a los aspectos analíticos. El curso se ha diseñado para completarse en un trimestre, pero puede fácilmente extenderse a un semestre si los dos últimos capítulos se estudian con mayor profundidad. Todo el temario de este texto se ha estructurado en cinco (5) capítulos y un (1) apéndice: I – Solución de Ecuaciones No Lineales. II – Solución de Sistemas de Ecuaciones. III – Interpolación, Integración y Aproximación. IV – Ecuaciones Diferenciales Ordinarias. V – Ecuaciones en Derivadas Parciales. A – Series de Taylor. Todos los temas tratados en este trabajo se han enfocado siguiendo un proceso de desarrollo de los temas de forma inductiva. Se comienzan los temas con problemas o algoritmos particulares y luego se generalizan dentro de un espacio de problemas o de algoritmos. Esto último, como se planteó antes, viene acompañado de su respectivo análisis, lo cual completa y formaliza las ideas vagamente planteadas en un principio.


Numerical Analysis I

Mark Embree
Rice University
18 January 2010

We model our world with continuous mathematics. Whether our interest is natural science, engineering,
even finance and economics, the models we most often employ are functions of real variables. The equations can be linear or nonlinear, involve derivatives, integrals, combinations of these and beyond. The tricks and techniques one learns in algebra and calculus for solving such systems exactly cannot tackle the complexities that arise in serious applications. Exact solution may require an intractable amount of work; worse, for many problems, it is impossible to write down an exact solution using elementary functions like polynomials, roots, trig functions, and logarithms.

This course tells a marvellous success story. Through the use of clever algorithms, careful analysis, and speedy computers, we are able to construct approximate solutions to these otherwise intractable problems with remarkable speed. Trefethen defines numerical analysis to be ‘the study of algorithms for the problems of continuous mathematics’. This course takes a tour through many such algorithms, sampling a variety of techniques suitable across many applications. We aim to assess alternative methods based on both accuracy and efficiency, to discern well-posed problems from ill-posed ones, and to see these methods in action through computer implementation.

Partial Differential Equations

Viktor Grigoryan
University of California, Santa Barbara
Fall 2010

These lecture notes arose from the course “Partial Differential Equations” – Math 124A taught by the author in the Department of Mathematics at UCSB in the fall quarters of 2009 and 2010. The selection of topics and the order in which they are introduced is based on Walter A. Strauss, Partial differential equations. John Wiley & Sons Ltd., Chichester, second edition, 2008. Most of the problems appearing in this text are also borrowed from Strauss. A list of other references that were consulted while teaching this course appears in the bibliography at the end.

Power Series Methods III: The Wave Equation

Robert D. Small
University of Wisconsin at Madison
U.S. Army Research Office/National Research Council of Canada
MRC Technical Summary Report #1925
February 1979

The power series method used by the author to generate highly accurate finite difference schemes for ordinary differential equations and for the heat equation is here applied to the wave equation . The analysis runs parallel to earlier work and involves semi-discrete approximations in t and in x before the totally discrete scheme is derived . The results differ from earlier work in that an arbitrarily accurate difference scheme is found for the wave equation that is stable and consistent with the differential equation. No such scheme exists for the heat equation. The step sizes in x and t must be equal for this difference scheme. Other difference schemes that do not restrict the step sizes are stable only when the order of accuracy in x is less than 5. The lowest order scheme is shown to coincide with
Keller’s Box Scheme.

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