Carlos Tomei

The bidiagonal singular value decomposition and Hamiltonian mechanics

We consider computing the singular value decomposition of a bidiagonal matrix B. This problem arises in the singular value decomposition of a general matrix, and in the eigenproblem for a symmetric positive definite tridiagonal matrix. We show that if the entries of B are known with high relative accuracy, the singular values and singular vectors of B will be determined to much higher accuracy than the standard perturbation theory suggests. We also show that the algorithm computes the singular vectors as well as the singular values to this accuracy. We also give a Hamiltonian interpretation of the algorithm and use differential equation methods to prove many of the basic facts. The Hamiltonian approach suggests a way to use flows to predict the accumulation of error in other eigenvalue algorithms as well. 36 refs., 2 figs., 2 tabs.

Toda flows with infinitely many variables

Abstract The Toda flow and related flows extend naturally to operators in Hilbert space and the purpose of this paper is to describe these flows and to analyse some of their special properties.

On the determinant of the adjacency matrix for a planar sublattice

Abstract It is proved that the determinant of the adjacency matrix for a finite subgraph G of Z × Z is 1, 0, or −1, provided that G has no “holes.”

A monotonicity property for Toda-type flows

If

Symplectic Aspects of Some Eigenvalue Algorithms

( X ( t ) )_r

The Toda flow on a generic orbit is integrable

is the leading

Direct and inverse scattering on the line

r times r

Ordinary Differential Equations and the Symmetric Eigenvalue Problem

submatrix of a matrix

Inverse scattering and the boussinesq equation

X ( t )

The toda flow on a generic orbit is integrable

undergoing a general Toda-type flow