We develop an algorithmic framework for reducing the bandwidth of symmetric
matrices via orthogonal similarity transformations. This framework include
s the reduction of full matrices to banded or tridiagonal form and the redu
ction of banded matrices to narrower banded or tridiagonal form, possibly i
n multiple steps. Our framework leads to algorithms that require fewer floa
ting-point: operations than do standard algorithms, if only the eigenvalues
are required. In addition, it allows for space-time tradeoffs and enables
or increases the use of blocked transformations.