[From Jason Reidy again....]
__________________
And Cleve Ashcraft writes:
- Consider a matrix where the degrees of freedom are mixed, e.g.,
- state and control variables, or translations and rotations. The
- scaling may be such that it is crucial to eliminate one type of
- variable before another within the supernode to avoid element growth.
So equilibration is the key here; I suppose
it's a
matter of perspective. Static pivoting require paying
explicit attention to equilibration/scaling, while
restricted pivotings solve many equilibation problems
for you.
Some people have recommended dynamic equilibration
as a supplement to static pivoting. I don't know if
dynamic equilibration buys any stability or if it can
be made practical, but it's another idea.
(Also, scaling is an over-extended term, so I'm abusing
equilibration to encompass any matrix scaling and not
just those that fix row and column norms to some number.
I suppose I should just specify matrix scaling when
there's any doubt.)
- Also think of a saddle point problem, where there
may be some zeros
- on the diagonals of the supernodes. It is better to pivot within
- the supernode than to modify the zero diagonal elements and carry on.
Good point, thank you! My observations are for
unsymmetric,
non-structurally-singular problems where you have the
freedom to permute non-zero entries onto the diagonal
initially. Over those matrices in the UF collection, it
seems that if a tiny pivot occurs in a larger supernode,
it was caused by catastrophic cancellation from an earlier
tiny pivot. (Or it doesn't matter because some other error
occurred in another path. hm. I haven't looked for that
possibility, just the first such problem. Element growth
from tiny pivots produces a chain of tiny pivots up the
etree, and those swamp other possible problems.)
And I'm not sure how well saddle point problems are
represented
in current matrix collections. I'm sure more would be accepted
happily...
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