numerics

[Peter Cameron]

To summarise one point that arose from our discussions this morning:

Many (but probably not all) instances where numerical approximations are
made consist of calculating eigenvalues of rational matrices. Now there
are two things to note. First, we can compute any symmetric polynomial
of the eigenvalues precisely, even though the result of plugging the
computed eigenvalues into the polynomial will not usually give the
correct result.

But more interestingly, if we compute the eigenvalues by factorising
the characteristic polynomial into irreducible factors and then finding
the roots of these factors, we can make even the numerical statement
more informative. e.g. the list of eigenvalues could contain entries
like "0.357 (multiplicity 3), 0.357 (multiplicity 4)", where we can
give an absolute guarantee that the two numbers written as 0.357 are
not equal. This kind of list should be distinguished semantically from
a list which does not give such a guarantee.

Peter.