Recursive Gradient Algorithms for Eigenvalue and Singular Value Decompositions

Recent work has shown that the algebraic question of determining the eigenvalues, 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 necessary to develop recursive algorithms which achieve the same solutions as the continuous-time flows. In this paper we propose two recursive algorithms, based on a double Lie-bracket equation proposed by Brockett, which are suitable for implementation in parallel processing environments. The algorithms presented achieve, respectively, the eigenvalue decomposition of a symmetric matrix, and the singular value decomposition of an arbitrary matrix. The algorithms have the same equilibria as the continuous-time flows on which they are based, and for suitable choice of step-size, they also inherit the exponential convergence of the continuous-time solutions. A characterisation of suitable step-size selection schemes is given and two schemes are presented.