Band reduction algorithms revisited

In this paper we explain some of the changes that have been incorporated in the latest version of the LAPACK subroutine for reducing a symmetric bande d matrix to tridiagonal form. These modifications improve the performance f or larger-bandwidth problems and reduce the number of operations when accum ulating the transformations onto the identity matrix, by taking advantage o f the structure of the initial matrix. We show that similar modifications c an be made to the LAPACK subroutines for reducing a symmetric positive defi nite generalized eigenvalue problem to a standard symmetric banded eigenval ue problem and for reducing a general banded matrix to bidiagonal form to f acilitate the computation of the singular values of the matrix.