APL tutorial in mathematical modelling

ABSTRACT

The software presented below is a computer tutorial in mathematical modeling. The term "Mathematical Modeling" does not have a precise definition. We use the term to imply problem description in mathematical terms and the use of mathematical tools to investigate the given problem. For example, by using differential equations it is possible to describe a vast variety of different phenomena in the areas of physics, biology, sociology, etc. Some types of differential equations can be solved analytically, others require numerical methods and a computer. So today's mathematical modeler should be prepared to use a computer as an accessory.There are a great number of software packages allowing mathematical models to be studied numerically as well as analytically. Among these packages are MatLab, MathCad, Maple, Mathematica and others. However, students of Applied Mathematics should not only know how to apply numerical methods but also understand their principles and the limits to their application. Therefore, any student class in Numerical Methods should include some coding of numerical methods by using a programming language.Classical programming languages are overburdened with ritual operations (data description, cycles, manual distribution of memory) to such an extent that after programming the numerical method very little time is left to investigate the method itself: its behavior with different parameters and in different sorts of problem. Moreover, the object known as a "program" in a classical programming language is something monolithic; there are very few useful operations defined over such objects. Therefore APL enjoys a considerable advantage over classical programming. It allows students to concentrate on the algorithm itself and not on the data description.But one faces some disadvantages when using APL. APL does not provide for alternative representations of numbers or for the creation of standalone executable modules.The tutorial presented below is a program having a Windows interface which lets students familiarize themselves with the Theory of Calculation and problem solving in connection with numerical methods or actual models.The tutorial consists of lectures, problems, hints and solutions, plus a library of numerical methods. But most importantly, when using this tutorial, students can solve problems in the actual tutorial environment, i.e. the APL session and the GRAN subsystem for interactive graph plotting are available for the student's use.The tutorial below is not up to commercial standards, but is intended only as an illustration of APL's capabilities as a teaching tool. Clearly, this tutorial could be adapted to a variety of disciplines and this is a course of action we hope to pursue. In accordance with APL's history, this tutorial is an individualized and personalized study-aid and not a product for mass consumption.

INDEX TERMS

Primary Classification:
  D. Software
  D.3 PROGRAMMING LANGUAGES
      D.3.2 Language Classifications

          Nouns: APL

Additional Classification:
  G. Mathematics of Computing
  G.1 NUMERICAL ANALYSIS
      G.1.0 General

  K. Computing Milieux
  K.3 COMPUTERS AND EDUCATION
      K.3.1 Computer Uses in Education
          Subjects: Computer-assisted instruction (CAI)

General Terms:
Languages