1. The Doctoral Dissertation
How Can We Determine
Numerical Quality
in Software?
The Doctoral Dissertation
of Thomas Rowan

2. “Functional Stability Analysis Of Numerical Algorithms”
Thomas Harvey Rowan:
“Functional Stability Analysis
Of Numerical Algorithms”
Doctoral Dissertation
Department of Computer Sciences
The University of Texas at Austin
1990

3. What is meant by numerical quality?

4. (i.e. having slightly different input)
What is meant by numerical quality?
The computed solution
(i.e. the output)
is the exact solution
to a close problem
(i.e. having slightly different input)

5. (i.e. having slightly different input)
What is meant by numerical quality?
The computed solution
(i.e. the output)
is the exact solution
to a close problem
(i.e. having slightly different input)
This is called “numerical stability”

6. Input: a pair of lines in the plane

7. Input: a pair of lines in the plane
Output: the intersection (if any)

14. well conditioned problem
This is what we seek:
A poor solution to a
well conditioned problem

15. Notice the difficulty: We have to guarantee that ? ?
Notice the difficulty:
We have to guarantee that
the approximate solution is not
the solution to
ANY close problem

16. Idea: Search the input space attempting to maximize the ratio
of the forward error to the
condition number.

17. Idea: Search the input space attempting to maximize the ratio
of the forward error to the
condition number.
Details, details:
How do you get the forward error?
How do you get the condition number?

18. Would you like to find the
maximum of this?

19. Nelder Mead Simplex Algorithm:
The crawling spider

20. high
high
high
high

23. Simplex difficulties:
Huge use of storage (n2)
Equivalent weight to all variables

24. Simplex difficulties: Huge use of storage (n2)
Equivalent weight to all variables
Subplex:
Determine small subspaces
likely to be most useful for
determining minimum

25. the Subplex Algorithm to locate minima of
The idea is to use
the Subplex Algorithm to
locate minima of
(the negative of) the backward error
INSTAB
SUBPLEX
Backward Error
code
code
extra precision

26. Does it work?

27. Does it work? Tried on known unstable algorithms
Tried on known stable algorithms
Tried on unknown algorithms

28. Does it work? Tried on known unstable algorithms
Identified all as unstable
Tried on known stable algorithms
Found no false positives
Tried on unknown algorithms
Discovered unstable algorithms

29. Backword error estimate for Gaussian elimination

30. Anything left to do? Two papers to write
Translation into C++ or Matlab
More experiments
Refinements to INSTAB or SUBPLEX