T-integration, which stands for "tunable numerical integration," is a fast, accurate, and numerically stable numerical integration formula given by
where is the integral, is the integrand, and are "phase" and "gain" tuning parameters, refers to the number of the iteration being evaluated, and is the integration step size.
The method was developed during the Apollo era to figure out how to simulate the digitally controlled Apollo command module during rendezvous and lunar landing operations. It was needed because none of the classical numerical integrators worked when trying to simulate the digital flight control systems maneuvering a space craft to a lunar landing.
For , varying from 0 to 2 gives many classical first-order integrators:
1. and : Euler integrator,
2. and : trapezoidal rule,
3. and : Rectangular rule,
4. and : Adams' method.
See alsoNumerical IntegrationPortions of this entry contributed by Jon Michael Smith
Explore with Wolfram|Alpha ReferencesFowler, M. "A New Numerical Method for Simulation." Simulation 6, 90-92, Feb. 1976.Smith, J. M. "Recent Developments in Numerical Integration." J. Dynam. Sys., Measurement and Control 96, Ser. G-1, No. 1, 61-70, Mar. 1974.Smith, J. M. "Zero-Order T-Integration and Its Relation to the Mean Value Theorem." In Proceedings of the Sixth Annual Pittsburgh Modeling and Simulation Conference, Part 1, April 24-25, 1975.Smith, J. M. "Modern Numerical Integration Methods." In Mathematical Modeling and Digital Simulation, 2nd ed. New York: John Wiley, 1988.Smith, J. M. "Fast T-Integration." J. Mech. Eng. Sys. 1, 27-31, Jul./Aug. 1990.Smith, J. M. "Jon Michael Smith on T-Integration: Trade Secrets in Numerical Analysis." http://members.aol.com/jsmith46ws/ni1.htm. Referenced on Wolfram|AlphaT-Integration Cite this as:Smith, Jon Michael and Weisstein, Eric W. "T-Integration." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/T-Integration.html
Subject classificationsRetroSearch is an open source project built by @garambo | Open a GitHub Issue
Search and Browse the WWW like it's 1997 | Search results from DuckDuckGo
HTML:
3.2
| Encoding:
UTF-8
| Version:
0.7.4