Our algorithms for (singular) totally nonnegative matrices compute exactly the zero eigenvalues and their corresponding Jordan blocks in what we believe is the first example of Jordan blocks being computed stably in floating point arithmetic. This solves completely the eigenvalue problem for the class of irreducible totally nonnegative matrices. See our paper and software.
We transcribed Ian G. Macdonald's unpublished manuscripts:
Numerical linear algebra, computational mathematics, applied/computational
multivariate statistical analysis and random matrix theory, computational
algebra.
In particular, the
development of efficient algorithms for accurate computations as they
pertain to the these fields.
Items of interest:
Developed theory and algorithms for accurate computations with sign
regular, totally nonnegative, polynomial Vandermonde, generalized
Vandermonde, semiseparable, and M-matrices
Developed the first algorithm for computing the eigenvalues of a
nonsymmetric matrix to high relative accuracy in
O(n^{3})
time
Developed the first stable algorithm to stably compute a Jordan block (for irreducible totally nonnegative matrices)
Developed the fastest algorithm for computing the
hypergeometric
function of a matrix argument
Awards
Householder Award (for ``Best Dissertation in Numerical Linear
Algebra in the
period 2002-2004''), honorable mention
Outstanding Graduate Student Instructor, University of California, Berkeley,
1999
Second Prize, International Mathematics Olympiad, Beijing, China, 1990