 
Summary: Beiträge zur Numerischen Mathematik
12 (1984), 719
On improving approximate triangular factorizations
iteratively
GÖTZALEFELD and JON G. ROKNE
Summary. Newton's methoq. applied iteratively to the improvement of an approximate
triangular factorization of a matrix is discussed in detail. Particular consideration is given to
the effect of rounding errors on the convergence of the iteration. It is shown that on a computer
employing fixed length floatingpoint arithmetic Newton's method converges with an arbi
trary starti:m.g value after 2nl steps to the same value as that obtained by Gaussian elimi
nation. Finally, a new method is proposed for the iterative improvement of bounds for the
elements of the triangular factorization where the effects of the rounding errors are also
considered.
1. Intr:oduction
It is frequently necessary to solve a system of linear equations Ax = b for a variety
of righthand sides b. Since this is often done by factoring A as (1 + L*) U* and
then solving the resulting simpler sets of equations, it is important to calculate L*
and U* as accurately as possible.
\Vith this in mind J. W. SCHMIDT[3] recently proposed to apply Newton's method
in order to correct an approximate factorization of a nonsingular matrix A. SCH:MIDT
