Time and Location
Many numerical algorithms involve feedback loops in which the results of each step of the algorithm act as inputs to the following steps.
Examples include numerical solution of differential equations, filtering of sound (or other signals) using IIR (Infinite Impulse Response) filters, and control of device motion. Inevitably, small numerical errors are introduced at each step. In an unstable algorithm, the effects of those errors grow in subsequent feedback steps, eventually dominating over the “right” answer you are looking for. In a stable system, the effects of errors shrink, or at least do not grow, in subsequent steps. The consequences of instability vary from “Huh?” through “aw, &%@%)$^&%*” to death.
In this talk, I will show examples of unstable and stable algorithms, investigate the differences, and introduce some methods for making sure your algorithms are stable. The presentation will include some college-level mathematics, but the emphasis will be on understanding why instability happens and how to avoid it.
The examples will be in C++, but the principles are not language-specific. This seminar will build on the general principles outlined in my previous NWCPP talk in November 2011, entitled “Introduction to Scientific Computing.” This talk begins where that one ends: with a demonstration of an unstable algorithm for solving a differential equation. (Sorry about the sound in that video. We have improved our technology since then, with your help. Thanks.)
Robert Goddard is a physicist and software developer. For the last 32 years, he has worked at the Applied Physics Laboratory of the University of Washington, mainly on computer modeling of underwater sound. He is the architect and team leader for the Sonar Simulation Toolset (SST), which produces simulated underwater sound, suitable as input to sophisticated signal processing systems (including human ears and brains), based on user-specified descriptions of the undersea environment, the listening system, and the sound sources and targets that might be out there. He has also developed systems for control of measurement devices, data analysis, visualization, modeling of quantum mechanical scattering, and optimization of parameter values to fit observations.
Bob has been an active participant in NWCPP for most of its existence, and is currently Treasurer.