# Mathematical Resources

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

### Surendra Singh, Klaus Halterman, and J. Merle ElsonNaval Air Warfare Center Weapons Division TP 8590September 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. LewisUniversity of Michigan1970

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. TrossetNASA/CR-2000-210125ICASE Report No. 2000-26May 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. SmithermanTexas A&M UniversityNASA Research Grant No. NGR-44-001-024

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

## Numerical Analysis I

### Mark EmbreeRice University18 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.

## Power Series Methods III: The Wave Equation

### Robert D. SmallUniversity of Wisconsin at MadisonU.S. Army Research Office/National Research Council of CanadaMRC Technical Summary Report #1925February 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.