Françoise Tisseur, University of Manchester
The tropical semiring consists of the real numbers and infinity along
with two binary operations: addition defined by the max or min operation and
multiplication. Tropical algebra is the tropical analogue of linear algebra,
working with matrices with entries on the extended real line. There are
analogues of eigenvalues and singular values of matrices, and matrix
factorizations in the tropical setting, and when combined with a
valuation map these analogues offer `order of magnitude' approximations to
eigenvalues and singular values, and factorizations of matrices in the usual algebra.
What makes tropical algebra a useful tool for numerical linear algebra is that these
tropical analogues are usually cheaper to compute than those in the
conventional algebra. They can then be used in the design of preprocessing steps to
improve the numerical behaviour of algorithms. In this talk I will review the
contributions of tropical algebra to numerical linear algebra and
discuss recent results on the selection of Hungarian scalings prior to solving linear
systems and eigenvalue problems.