***Ph.D. Dissertation Abstract
Andrew Allen Anda, Ph.D.

    A suite of self-scaling fast circular plane rotation algorithms is
developed which obviates the monitoring and periodic rescaling necessitated
by the standard set of fast plane rotation algorithms. Self-Scaling fast
rotations dynamically preserve the normed scaling of the diagonal factor
matrix whereas standard fast rotations exhibit divergent scaling. Variations
on standard fast rotation matrices are developed and algorithms which
implement them are offered. Self-Scaling fast rotations are shown to be
essentially as accurate as slow rotations and at least as efficient as
standard fast rotations. Computational experimental results utilizing the
Cray-2 illustrate the effectively stable scaling exhibited by self-scaling
fast rotations. Jacobi-class algorithms with one-sided alterations are
developed for the algebraic eigenvalue decomposition using self-scaling fast
plane rotations and one-sided modifications. The new algorithms are shown to
be both accurate and efficient on both vector and parallel architectures.
The utility is described of applying fast plane rotations towards the
rank-one update and downdate of least squares factorizations. The
equivalence is illuminated of LINPACK, hyperbolic rotation, and fast
negatively weighted plane rotation downdating. Algorithms are presented
which apply self-scaling fast plane rotations to the QR factorization for
stiff least squares problems. Both fast and standard Givens rotation-based
algorithms are shown to produce accurate results regardless of row sorting
even with extremely heavily row weighted matrices. Such matrices emanate,
e.g., from equality constrained least squares problems solved via the
weighting method. The necessity of column sorting is emphasized. Numerical
tests expose the Householder QR factorization algorithm to be sensitive to
row sorting and it yields less accurate results for greater weights.
Additionally, the modified Gram-Schmidt algorithm is shown to be sensitive
to row sorting to a notably significant but lesser degree. Self-Scaling fast
plane rotation algorithms, having competitive computational complexities,
must therefore be the method of choice for the QR factorization of stiff
matrices. Timing results on the Cray 2, [XY]M/P, and C90, of rotations both
in and out of a matrix factorization context are presented. Architectural
features that can best exploit the advantageous features of the new fast
rotations are subsequently discussed.

***[end of Ph.D. Dissertation Abstract text]