# A naive question about single precision

**URL:** <https://fortran-lang.discourse.group/t/a-naive-question-about-single-precision/3935>\
**Category:** Help\
**Created:** [July 6, 2022, 12:05am UTC](https://fortran-lang.discourse.group/t/a-naive-question-about-single-precision/3935 "2022-07-06T00:05:40Z")\
**Posts on this page:** 1\
**Showing post:** 5

<div class="post-metadata">

**Author:** ![ivanpribec](https://yyz2.discourse-cdn.com/free1/user_avatar/fortran-lang.discourse.group/ivanpribec/32/3290_2.png) [@ivanpribec](https://fortran-lang.discourse.group/u/ivanpribec)\
**Post date:** [July 6, 2022, 2:22am UTC](https://fortran-lang.discourse.group/t/a-naive-question-about-single-precision/3935/5 "2022-07-06T02:22:16Z")

</div>

Later in the 70’s Gerald J. Bierman from JPL did a bunch of work on stable factorization algorithms for Kalman filtering. Some of his work is described in

- [Thornton, C. L., & Bierman, G. J. (1976). _A numerical comparison of discrete Kalman filtering algorithms: An orbit determination case study_ (No. NASA-CR-148278).](https://ntrs.nasa.gov/api/citations/19760019842/downloads/19760019842.pdf)
- [Bierman, G. J. (1977). Factorization methods for discrete sequential estimation. Academic Press, Inc., New York.](https://hpiers.obspm.fr/combinaison/documentation/articles/Bierman.pdf)

Notably, the routines Gerald Bierman and his brother Keith used in their consultancy company, Factorized Estimation Applications, Inc., were made available via Netlib:

> **1.** [a/esl.tgz](http://www.netlib.org/a/esl.tgz)  
> **for:** estimation and smoothing by UDU\*\*T and square root information filter SRIF for Kalman filtering  
> **by:** Keith and Gerald Bierman  
> **lang:** Fortran  
> **ref:** “Factorization Methods for Discrete Sequential Estimation”, Academic Press 1977. Republished by Dover

You can read in one of the Fortran source comments:

```fortran
C Introduction
C
C The Estimation Subroutine Library (ESL) was originally authored by Gerald J.
C Bierman and Keith H. Bierman. The algorithms largely correspond closely
C to those found the Research Monograph "Factorization Methods for Discrete
C Sequential Estimation" originally published by Academic Press. It has
C been reprinted by Dover with ISBN-10: 0486449815 and ISBN-13: 978-0486449814
C
C Originally published as commerical software by Factorized Estimation
C Applications Inc. it has largely been unavailable to the general public
C for over 20 years. This library is now licensed under the Creative Commons 
C Attribution 3.0 Unported License. 
C To view a copy of this license, visit http://creativecommons.org/licenses/by/3.0/ 
C or send a letter to Creative Commons, 171 Second Street, Suite 300, 
C San Francisco, California, 94105, USA.
C
C As most are aware, Gerald J. Bierman (Ph.D., IEEE Fellow) died suddenly in 
C his prime in 1987. This library is re-released in his memory.
C
C Keith H. Bierman 2008
C khbkhb@gmail.com
C Cherry Hills Village, CO 

```

For those interested, I’ve been able to locate the [_in memoriam_ for Gerald Jack Bierman.](http://www.malcolmdshuster.com/Pub_1988a_J_GJB-obit_scan.pdf) I would also bring to your attention his involvement in the [Voyager orbit determination at Jupiter (1983)](https://ieeexplore.ieee.org/stamp/stamp.jsp?arnumber=1103223&casa_token=jl29ZfuV6mYAAAAA:hRLxleXFz8jNfPMMIiy62_PI800OYLx__rDuS-o5-A95Rfb83_CMfN6mzw6q4jt0dqBtHWbvPg&tag=1). As he wrote there:

> Numerous tests were made throughout the mission to  
> test estimate consistency and accuracy of the SRIF/SRIS  
> and U-D algorithm implementations. The navigation  
> estimation software performed flawlessly. Indeed, the  
> estimation software performed so well that most of the  
> time it was taken for granted by the navigation team, and  
> that is the ultimate compliment.

I would note that [Voyager 1 & Voyager 2](https://voyager.jpl.nasa.gov/) remain in contact with Earth. Maybe Bierman’s Fortran factorizations are still doing their job on some NASA computer just like they did 50 years ago?

---

_[View the full topic](https://fortran-lang.discourse.group/t/a-naive-question-about-single-precision/3935)._