NUMERICAL GRADIENT ALGORITHMS FOR EIGENVALUE AND SINGULAR-VALUE CALCULATIONS

Recent work has shown that the algebraic question of determining the e igenvalues, or singular values, of a matrix can be answered by solving certain continuous-time gradient flows on matrix manifolds. To obtain computational methods based on this theory, it is reasonable to devel op algorithms that iteratively approximate the continuous-time flows. In this paper the authors propose two algorithms, based on a double Li e-bracket equation recently studied by Brockett, that appear to be sui table for implementation in parallel processing environments. The algo rithms presented achieve, respectively, the eigenvalue decomposition o f a symmetric matrix and the singular value decomposition of an arbitr ary matrix. The algorithms have the same equilibria as the continuous- time flows on which they are based and inherit the exponential converg ence of the continuous-time solutions.