Plamen Koev:
Accurate Eigenvalues and SVDs of (singular) totally nonnegative matrices
with applications to Computer Aided Geometric Design.
Abstract:
In Computer Aided Geometric Design, totally nonnegative bases present good
shape properties and B-bases correspond to the bases with optimal shape
preserving properties.
In this talk we will present the algorithms for performing all matrix
computations with totally nonnegative matrices to high relative accuracy in
floating point arithmetic. The complexity is comparable to that of the
traditional LAPACK algorithms, which deliver no such accuracy.
In particular, these algorithms compute all eigenvalues, SVDs, various
decompositions, and totally-nonnegative preserving transformations, such as
products, submatrices, etc. Most significantly, we are also able to compute
exactly the Jordan blocks corresponding to zero eigenvalues--which we
believe is the first example of a Jordan structure being computed
accurately.
Our algorithms are based on the fact that when the computations are
performed in a way that preserved the total nonnegativity, no subtractions
are encountered and the relative accuracy is preserved.
References:
[1] Carnicer J.M., Pe??a J.M. (1996) Total positivity and optimal bases. In:
Gasca M., Micchelli C.A. (eds) Total Positivity and Its Applications.
Mathematics and Its Applications, vol 359. Springer, Dordrecht.
[2] Farin, G., Curves and Surfaces for CAGD, Elsevier, 2001.
[3] Goodman, T.N.T., Siad, H.B., Shape preserving properties of the
generalized Ball basis, Computer-Aided geometric design 8 (1991), 115-121.
[4] Karlin, S., Total Positivity, Vol. I, Stanford, 1968.
[5] Koev, P., Accurate computations with totally nonnegative matrices
SIAM J. Matrix Anal. Appl. 29 (2007), 731-751.
[6] Koev, P., Accurate eigenvalues and zero Jordan blocks of totally
nonnegative matrices, Preprint.