Iterative computation of the smallest singular value and the corresponding singular vectors of a matrix

Influenced by some techniques used for computing singular points of nonlinear equations, a generalized inverse iteration method is proposed for approximating the smallest singular value σn and the associated left and right singular vectors u, v of a matrix . In the practically relevant case σn>0 the method is mathematically equivalent to inverse iteration with AAT. However, unlike classic inverse iteration, the new method works with matrices obtained by bordering A in such a way that the Bk have uniformly bounded condition numbers. This allows using iterative Krylov-type solvers for large problems. If σn−1>σn the singular vector approximations convergence linearly with factor κ=σn/σn−1<1. Moreover, a certain generalized Rayleigh quotient σ(k) obtained as a byproduct has a relative error (σ(k)−σn)/σn which goes to zero R-linearly with factor κ2. Some numerical examples confirm the theoretical results and show that the algorithm works reliable also for almost singular matrices and when using Krylov solvers.