Numerical Computation of an Analytic Singular Value Decomposition of a Matrix Valued Function Partial Support Received from Sfb 343, Diskrete Strukturen in Der Mathematik, Universitt at Bielefeld

This paper extends the singular value decomposition to a path of matrices E(t). An analytic singular value decomposition of a path of matrices E(t) is an analytic path of factorizations E(t) = X(t)S(t)Y (t)T where X(t) and Y (t) are orthogonal and S(t) is diagonal. To maintain di erentiability the diagonal entries of S(t) are allowed to be either positive or negative and to appear in any order. This paper investigates existence and uniqueness of analytic SVD's and develops an algorithm for computing them. We show that a real analytic path E(t) always admits a real analytic SVD, a full-rank, smooth path E(t) with distinct singular values admits a smooth SVD. We derive a di erential equation for the left factor, develop Euler-like and extrapolated Euler-like numerical methods for approximating an analytic SVD and prove that the Euler-like method converges.