Abstract
Digital computation is central to almost all scientific endeavors and has become integral to university physics education. Students collect experimental data using digital devices, process data using spreadsheets and graphical software, and develop scientific programming skills for modeling, simulation, and computational work. Issues associated with the floating-point representation of numbers are rarely explored. In this article, problems of floating point are divided into three categories: significant-figure limits, propagation of floating-point representation error, and rounding. For each category, examples are presented of unexpected ways, in which the digital representation of floating-point numbers can impact the veracity of scientific results. These examples cover aspects of classical dynamics, numerical integration, cellular automata, statistical analysis, and digital timing. Suggestions are made for curriculum enhancement and project-style investigations that reinforce the issues covered at a level suitable for physics undergraduate students.
Highlights
Digital processing is ubiquitous in scientific investigation
Issues associated with the floating-point representation of numbers are rarely explored
Issues of the limitation of significant figures and of rounding have been discussed from the late 1950’s when numerical computation started to become widespread and a clear and detailed description of the IEEE single- and double-precision representation of numbers and precision limitations are presented by Landau et al
Summary
Digital processing is ubiquitous in scientific investigation. All numerical computation, computer simulation, processing of experimental data, and even gathering of the experimental data itself, normally utilise digital processing. Physics undergraduate curricula typically include a staged development of programming skills, and students have the opportunity to undertake extensive project work in later years. Magnetism, thermodynamics, classical dynamics and financial modeling are amongst many subject areas that lend themselves to numerical computational investigation. The inclusion of computational physics develops programming skills and reinforces the core curriculum. Undergraduate work in numerical simulation should address the issue of floating-point error because it is important to determine which errors are due to algorithmic and computational methods as opposed to limitations in the model itself. The problem is overcome by taking the absolute value of the difference and comparing with a suitable small number as in the expression if abs(a − b) < 0.000001 Beyond this example, issues with floating-point representation rarely appear and so may be overlooked when they do.
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