Mathematical Computing

In the twenty-first century science flows in a river of computing technology. Scientists store their measurements in sophisticated databases that they access and manage across networks. They use powerful software to analyze and model those data. They publish the results in a finished form, both in print and on the World Wide Web. A scientist or mathematician who is ignorant of all the tools of this computing technology will soon drown. Certainly, there are computing support personnel that can throw a life ring, perhaps graduate students can be hired to manage a raft of computers, but in the end, a scientist who does not have at least a rudimentary understanding of the tools of modern scientific computing will not navigate the river successfully.

In this document we discuss some of the most common tools needed by a modern investigator in mathematical and scientific disciplines. The topics are quite varied, ranging from tools used in reporting results to the tools actually used in the analysis and modeling of data. In most cases we present software and techniques that are actually in use now. In a few cases, we try to do some foreshadowing, bringing in tools we think will be used heavily in the relatively near future.

To illustrate this information we will examine the tasks required of a person who wants to analyze data from a database somewhere on the Internet, and then present the results in a mathematical paper or on the World Wide Web. We will necessarily consider a very simple example, so that anyone with a beginning undergraduate background in mathematics can follow it. The stress will not be on the science or mathematics, but instead on the computing tools required to make the final report happen.

The proper order of the topics could be debated. It might be argued that analysis comes first, and writing and reporting comes later. On the other hand, one might need to use the Internet extensively early in order to download data from a remote site, or in monitoring equipment at that site. We will not worry excessively about the order in which the tools would be used in actually producing the science and reports about it. We instead begin by discussing the Internet itself and the ideas that make it work. We then discuss the World Wide Web and the tools that make it a powerful instrument for retrieving and disseminating data, and particularly scientific and mathematical information. Following that we will spend some time discussing the defacto standard for typesetting mathematical documents. Many areas of science do not use \LaTeX, but since it remains the most sophisticated software for typesetting documents that use mathematical notation, and because its syntax is widely used among other programs such as Microsoft Word and LibreOffice, we will present it in detail.

We finish with a discussion of several commonly-used mathematical/statistical packages: Matlab, Maple, Python/Scipy/Sympy, and R. These represent a wide array of modern scientific software packages. We cannot possibly do justice to all of the software in current use for statistical and mathematical analysis, but these packages and languages use relatively standard syntax and are very widely available.