Implementing the Retraction Algorithm for factoring symmetric banded matrices

Linda Kaufman, William Paterson University, Wayne, N.J.
 
Problems arising when modeling structures subject to vibrations or in designing
optical fibers wrapped around a spool give rise to eigenvalue problems and
linear systems with symmetric, banded matrices.  We present various variations
of the retraction algorithm which factors a matrix into a product of matrices
while preserving symmetry, the bandwidth, and element growth even when the
original matrix is indefinite.  The factorization may be used to solve a system
or compute the inertia of a matrix.  It is based on the algorithm of Bunch and
Kaufman, 1977, for symmetric nonbanded systems, which grew out of the first
paper that Gene Golub asked Linda Kaufman to referee while she was his graduate
student research assistant in 1973.  Theoretically the operation count is
about 1/2 that of nonsymmetric banded Gaussian elimination and 2/3 the space.
Experimental results will be presented comparing the various versions of the
retraction algorithm with LAPACK's nonsymmetric band routines and the Snap-back
code of Irony and Toledo from Tel Aviv University