Accurate Computation of Singular Values in Terms of Shifted Integrable Schemes

Abstract

A new scheme with a shift of origin for computing singular values $\sigma_{k}$ is presented. A shift $\theta$ is introduced into the recurrence relation defined by the discrete integrable Lotka--Volterra system with variable step-size. A suitable shift strategy is given so that the singular value computation becomes numerically stable. It is proved that variables in the new scheme converge to $\sigma_{k}^{2}-\sum\theta^{2}$. A comparison of the zero-shift and the nonzero-shift routines is drawn. With respect to both the computational time and the numerical accuracy, it is shown that the nonzero-shift routine is more accurate and faster than a credible LAPACK routine for singular values at least in four different types of test matrices.