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:
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Developments in Numerical Integration." J. Dynam. Sys., Measurement and Control96,
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.
![X_n=X_(n-1)+TG[P((dX)/(dt))_n+(1-P)((dX)/(dt))_(n-1)],](/images/equations/T-Integration/NumberedEquation1.svg)
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.
,
varying 
from 0 to 2 gives many classical first-order integrators:
and 
: Euler integrator,
and 
:
trapezoidal rule,
and 
:
Rectangular rule,
and 
:
Adams' method.
