Computational Physics is the study and implementation of numerical algorithm and the techniques which make calculations easy using computers.
Purpose and PhilosophyEdit
The purpose of this course is demonstrate to students how computers can enable us to both broaden and deepen our understanding of physics by vastly increasing the range of mathematical calculations which we can conveniently perform.
Our approach to computational physics is to write self-contained programs in a high-level scientific language--i.e., either FORTRAN or C++. Of course, there are many other possible approaches, each with their own peculiar advantages and disadvantages. It is instructive to briefly examine the available options.
Scientific Programming MethodologyEdit
Basically, there are three possible methods by which we could perform the numerical calculations which we are going to encounter during this course.
First, we could use a mathematical software package, such as MATHEMATICA, MAPLE or MATLAB. The main advantage of these packages is that they facilitate the very rapid coding up of numerical problems. The main disadvantage is that they produce executable code which is interpreted, rather than compiled. Compiled code is translated directly from a high-level language into machine code instructions, which, by definition, are platform dependent--after all, an Intel x86 chip has a completely different instruction set to a Power-PC chip. Interpreted code is translated from a high-level language into a set of meta-code instructions which are platform independent. Each meta-code instruction is then translated into a fixed set of machine code instructions which is peculiar to the particular hardware platform on which the code is being run. In general, interpreted code is nowhere near as efficient, in terms of computer resource utilization, as compiled code: i.e., interpreted code run a lot slower than equivalent compiled code. Thus, although MATHEMATICA, MAPLE, and MATLAB are ideal environments in which to perform relatively small calculations, they are not suitable for full-blown research projects, since the code which they produce generally runs far too slowly.
Second, we could write our own programs in a high-level language, but use calls to pre-written, pre-compiled routines in commonly available subroutine libraries, such as NAG,4 LINPACK,5 and ODEPACK,6 to perform all of the real numerical work. This is the approach used by the majority of research physicists.
Third, we could write our own programs--completely from scratch--in a high-level language. This is the approach used in this course. I have opted not to use pre-written subroutine libraries, simply because I want students to develop the ability to think for themselves about scientific programming and numerical techniques. Students should, however, realize that, in many cases, pre-written library routines offer solutions to numerical problems which are pretty hard to improve upon.
Choice of Scientific Programming LanguagesEdit
What is the best high-level language to use for scientific programming? This, unfortunately, is a highly contentious question. Over the years, literally hundreds of high-level languages have been developed. However, few have stood the test of time. Many languages (e.g., Algol, Pascal, Haskell) can be dismissed as ephemeral computer science fads. Others (e.g., Cobol, Lisp, Ada) are too specialized to adapt for scientific use. Let us examine the remaining options:
FORTRAN was the first high-level programming language to be developed: in fact, it predates the languages listed below by decades. Before the advent of FORTRAN. Moreover, FORTRAN was specifically designed for scientific computing. Indeed, in the early days of computers all computing was scientific in nature--i.e., physicists and mathematicians were the original computer scientists! FORTRAN's main advantages are that it is very straightforward, and it interfaces well with most commonly available, pre-written subroutine libraries (since these libraries generally consist of compiled FORTRAN code). FORTRAN's main disadvantages are all associated with its relative antiquity. For instance. FORTRAN's control statements are fairly rudimentary, whereas its input/output facilities are positively paleolithic.
This language was originally developed by computer scientists to write operating systems. Indeed, all UNIX operating systems are written in C. C is, consequently, an extremely flexible and powerful language. Amongst its major advantages are its good control statements and excellent input/output facilities. C's main disadvantage is that, since it was not specifically written to be a scientific language, some important scientific features (e.g., complex arithmetic) are missing. Although C is a high-level language, it incorporates many comparatively low-level features, such as pointers (this is hardly surprisingly, since C was originally designed to write operating systems). The low-level features of C--in particular, the rather primitive implementation of arrays--sometimes make scientific programming more complicated than need be the case, and undoubtedly facilitate programming errors. On the other hand, these features allow scientific programmers to write extremely efficient code. Since efficiency is generally the most important concern in scientific computing, the low-level features of C are, on balance, advantageous.
This language is a major extension of C whose main aim is to facilitate object-orientated programming. Object-orientation is a completely different approach to programming than the more traditional procedural approach: it is particularly well suited to large projects involving many people who are each writing different segments of the same code. However, object-orientation represents a large, and somewhat unnecessary, overhead for the type of straightforward, single person programming tasks considered in this course. Note, however, that C++ incorporates some non-object-orientated extensions to C which are extremely useful. Of the above languages, we can immediately rule out C++, because object-orientation is an unnecessary complication (at least, for our purposes), and FORTRAN 90, because of the absence of an inexpensive compiler. The remaining options are FORTRAN 77 and C. I have chosen to use C++ (because it not only can be object oriented it can also be procedural like C) in this course, simply because I find the complex features of FORTRAN 77 too embarrassing to teach students in the 21st century.