Mathematical Methods for Chemists:
NUMERICAL METHODS: FORTRAN &MATHEMATICA

Introduction to Course Structure

This course comprises approximately 15 hours of lectures and 15 hours of problem solving. This time is spread over the next two weeks with 3 hours/day. My rough schedule is to deliver:

5 lectures on Basic Fortran programming
5 lectures on Fortran libraries
5 lectures on Mathematica

but this may vary depending on feedback, degree of difficulty and level of interest.

The course content will initially try to focus on the essential basics with some more advanced topics being introduced later. Examples and exercises will be given relating to the formal part of the mathematical methods course given previously. For example, the Fortran programming and Mathematica examples (and exercises and problems) will (may) include:

The numerical solution of the eigenvalue problem.
Fourier series and Fourier transforms.
Methods for solving ordinary differential equations.
Laplace transforms.
Numerical differentiation and integration.
Numerical simulation: molecular dynamics and Monte Carlo methods.

A Quick Tour of Fortran & Fortran Compilers

Fortran is a high level programming language that has been in continuous use in further evolving versions (Fortran 66, Fortran 77, Fortran 90) since the late 1950's. A program written as a series of Fortran commands (source code) is compiled by a sophisticated program known as a Fortran compiler to ultimately produce an executable program. This executable program can then be run (either interactively or in batch mode) on the computer that it was compiled for. Fortran compilers exist for a wide range of computer platforms (Macintosh, PC, Vax, Sun, Silicon Graphics, Cray).

At this point I will demonstrate the basic steps in compiling and running a number of simple Fortran programs.

A Quick Tour of Advanced Fortran Programming & Program Libraries

Once the basic programming concepts in Fortran are mastered it is possible to develop some rather sophisticated programs which may incorporate various algorithms that solve a particular mathematical problem. Often the development of such efficient algorithms and their translation into Fortran code is rather difficult and tedious (not to mention error prone). Furthermore, some algorithms are useful for many purposes (e.g. algorithms for matrix operations, solution of the eigenvalue problem, Fourier transforms, etc...).

Thus, it is worthwhile to maintain a library of such useful programs (subprograms e.g. functions or subroutines) that can be reused for different problems. Such libraries are available commercially (NAGLIB, IMSL) , as shareware and others may be in the public domain (freeware).

At this point, I will briefly look at some typical examples of complex Fortran programs and how such codes make use of program libraries.

A Quick Tour of Mathematica

While we might tend to associate computers and computing merely with numerical calculations, rapid relatively recent advances in sophisticated algorithms have made it feasible to do symbolic algebra on a computer.

Here we will briefly look at one such computer environment, known as Mathematica. It not only can perform symbolic algebra but it can also do numerical calculation and display results in a wide range of graphic formats.

A (Bonus) Quick Tour of the Internet & the WWW

While it is not directly related to the topic of Mathematical Methods, depending on demand I may take the opportunity to introduce the concepts of the Internet and the World Wide Web (WWW). These concepts respectively refer to the world-wide hyper-network and a recently developed paradigm for an integrated graphical means whereby one can (with relative ease) access information on the Internet.

I may quickly provide a brief overview of the Internet and describe some of the tools useful in exploiting information resources available on the Internet. Many of these resources can provide useful scientific information and assistance in scientific calculations and programming. For example, a large official Mathematica WWW site exists which provides considerable further information and program packages (some are freeware) over and above that provided in the usual release provided with the program.

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