Report discusses several methods of performing numerical integration with computer. When data can be expressed as state vector that is dependent variable in a differential equation, self-starting integrators can be used to predict future data

Lear, W. M.

English

NASA Technical Reports Server

-	 October 1975	 B75-10273 
•	 NASA TECH BRIEF 
Lyndon B. Johnson Space Center 
NASA Tech Briefs announce new technology derived from the U.S. space program. They are issued to encourage 
commercial application. Tech Briefs are available on a subscription basis from the National Technical Information 
Service, Springfield, Virginia 22151. Requests for individual copies or questions relating to the Tech Brief program may 
be directed to the Technology Utilization Office, NASA, Code 1<1, Washington, D.C. 20546. 
A Study of Accuracy in Selected Numerical-Analysis Integration Techniques 
.
A report, "Some Self-Starting Integrators," has 
been published discussing several methods of 
performing numerical integration with a computer. 
The increased use of sequential data processors (e.g., 
Kalman filters) to handle large quantities of data 
requires software systems that can make accurate 
predictions of future data based on current data. 
When the data can be expressed as a state vector 
that is the dependent variable in a differential 
equation, self-starting integrators can be used to 
predict future data. An example is predicting the 
position of a moving vehicle. From a position/velocity 
measurement, a Kalman filter and the integrators can 
be used to help predict future positions, compare 
them with actual positions, and apply course 
corrections. 
A self-starting integrator is one which takes a 
current estimate of a state vector (Xi) (no previous-
position vectors are needed) and, using the equation 
= f(,t), propagates the estimate forward a time At. 
Taylor-series integrators and 2d and 4th order 
Runge-Kutta integrators are numerical-analysis tech-
niques that have been generally used for this purpose. 
For many applications the best particular technique 
and its degree of accuracy have not been well known. 
The report makes comparisons between Taylor-
series integrators and various types and orders of 
Runge-Kutta integrators. A numerical method of 
determining integrator accuracy over each step is 
shown. Taylor-series and 2d, 4th, and 5th order 
Runge-Kutta integrators are discussed. No exact 
solution to the 5th order Runge-Kutta is known. It is 
shown that with vector differential equations six 
derivatives must be evaluated for the 5th order 
integrator rather than the five that might be expected. 
Two sets of approximate 5th order integrator 
constants are shown and evaluated. When compared 
to 4th order Runge-Kutta integrators, they give only 
slightly more accurate solutions. For this reason it is
felt that the 4th order method is more suitable for 
most applications. 
Finally, a new set of equations is presented for 
integrating onboard accelerometer data. These 
accelerometers usually read out the integral of the 
sensed acceleration over a given period of time rather 
than the instantaneous acceleration. A set of 
equations is reported for integrating the acceler-
ometer data more accurately than the series of 
short-step integrations formerly used. 
Notes: 
1. This study is described in the following report: 
Some Self-Starting Integrators for 1 = f(x,t) 
NASA CR-141784 (N75-24419). 
Copies of this report may be obtained at cost from: 
Technology Application Center 
University of New Mexico 
Albuquerque, New Mexico 87131 
Telephone: (505) 277-3622 
Reference: B75-10273 
2. Specific technical questions may be directed to: 
Technology Utilization Officer 
Johnson Space Center 
Code AT3 
Houston, Texas 77058 
Reference: B75-10273 
Patent status: 
NASA has decided not to apply for a patent. 
Source: W. M. Lear of
TRW, Inc.
under contract to
Johnson Space Center 
(MSC-14802) 
Category: 09 (Mathematics and 
Information Sciences) 
This dqcument was prepared under the sponsorship of the National 	 Government assumes any liability resulting from the use of the 
Aeronautics and Space Administration. Neither the United States 	 information contained in this document, or warrants that such use 
Government nor any person acting on behalf of the United States 	 will be free from privately owned rights.
https://ntrs.nasa.gov/search.jsp?R=19750000273 2020-03-19T20:14:06+00:00Z


A study of accuracy in selected numerical-analysis integration techniques

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