jbs@WATSON.IBM.COM (06/21/91)
David Hough and myself have been disagreeing about the condition
number of uniform random matrices. It appears this is because we have
been using different matrix norms. My results were for the L1 norm while
David Hough's were using the L2 norm. Somewhat to my surprise it appears
that uniform random matrices are much better conditioned with respect to
the L2 norm than the L1 norm. For example for the size 1000 linpack
matrix I compute a L2 norm of 3623. as compared to a L1 norm of 223642.
(in an earlier post I reported an L1 norm of 368554, this was an error).
10 random 1000X1000 matrices had L2 norms ranging from 2398. to 27667.
while as I posted earlier the L1 norms ranged from 97627. to 1124910.
The median L2 norms appear to grow roughly like 5*n. I guess this goes
to show condition numbers do not provide terribly precise estimates of
the error to be expected.
James B. Shearer