A shift for accurate singular values¶in the differential qd algorithm

The differential qd algorithm with the incorporation of shifts computes the singular values of bidiagonal matrices preserving high accuracy [4]. In this paper we propose an algorithm for the shift which uses only the squares of the elements of the matrix, so that the complete algorithm is implemented in terms of squares until final blocks of sizes 1 × 1 or 2 × 2 are reached. This shift is always lower than the smallest singular value and yields quadratic convergence as in [4]. The algorithm which computes the shift uses some parameters also useful for convergence criteria. Moreover a slight modification in the single step is proposed which admits a larger domain of the matrices to which the differential qd algorithm can be applied, using IEEE arithmetic.