An improved dqds type algorithm (Q1976415)

The standard set of the Fortran 77 routines called ``Lapack'' provides algorithms (``dlasq1'' to ``dlasq4'') for the computation of singular values of a matrix. Their key ingredients are Cholesky factorization and reduction to bidiagonal matrices. Recently, \textit{K. V. Fernando} and \textit{B. N. Parlett} [Numerical Math. 67, No. 2, 191-229 (1994; Zbl 0814.65036)] offered their sophistication and called it the ``differential qd'' algorithm. In the present paper, another step in the direction is made. Main attention is paid to an improvement of the upper bound of the relative variation of the singular values of two bidiagonal matrices. The result is recommended as a guide to the improved auxiliary ``splitting'' (i.e., a replacement of an extradiagonal element by zero) in computations. Numerical tests illustrate its merits.