How Can We Determine

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How Can We Determine Numerical Quality in
Software?
The Doctoral Dissertation of Thomas Rowan
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Thomas Harvey Rowan Functional Stability
Analysis Of Numerical Algorithms Doctoral
Dissertation Department of Computer Sciences The
University of Texas at Austin 1990
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What is meant by numerical quality?
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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)
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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
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Input a pair of lines in the plane
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Input a pair of lines in the plane
Output the intersection (if any)
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This is what we seek A poor solution to a well
conditioned problem
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?
?
Notice the difficulty We have to guarantee
that the approximate solution is not the
solution to ANY close problem
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Idea Search the input space attempting to
maximize the ratio of the forward error to the
condition number.
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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?
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Would you like to find the maximum of this?
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Nelder Mead Simplex Algorithm The crawling
spider
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high
high
high
high
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• Simplex difficulties
• Huge use of storage (n2)
• Equivalent weight to all variables

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• Simplex difficulties
• Huge use of storage (n2)
• Equivalent weight to all variables

Subplex Determine small subspaces likely to be
most useful for determining minimum
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The idea is to use the Subplex Algorithm
to locate minima of (the negative of) the
backward error
SUBPLEX
INSTAB
code
Backward Error
code extra precision
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Does it work?
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Does it work?

• Tried on known unstable algorithms

• Tried on known stable algorithms

• Tried on unknown algorithms

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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
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Backword error estimate for Gaussian elimination
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Anything left to do?

• Two papers to write

• Translation into C or Matlab

• More experiments

• Refinements to INSTAB or SUBPLEX