An inverse subspace iteration for computing q smallest singular values of a matrix

Consider a matrix A E R-nxn with singular values sigma (1) greater than or equal to sigma (2) greater than or equal to ... greater than or equal to sigma (p) > sigma (p+1) greater than or equal to ... greater than or equal to sigma (p+q), p = n - q. For computing the singular values Sigma (2) = diag (sigma (p+1), ...,sigma (p+q)), i.e., for finding U-2, V-2 is an element of R-nxq with (U2U2)-U-T = (V2V2)-V-T = I-q and AV(2) = U(2)Sigma (2), A(T)U(2) = V(2)Sigma (2), an iteration is proposed which requires per step to solve alternately a linear system with matrix B-2l = B(Y-2l, X2l-1, Ohm (2l)) or B-2l+1(T) = B(Y-2l, X2l+1, Ohm (2l+1))(T), resp., l greater than or equal to 0, where [GRAPHICS] The B-k used have uniformly bounded condition numbers, and under weak assumptions the singular vector approximations extracted from X2l-1. and Y-2e converge linearly With factor kappa := sigma (p+1)/sigma (p) < 1 whereas certain approximations to Sigma (2) have the factor kappa (2). The theoretical results are confirmed by some numerical examples.