Overview

The ILLIAC's memory is sufficient to accommodate a system of 39 equations when used with Routine 51.

The additional length of Routine 100 restricts to 37 the number of equations that it can handle. With 37 equations the operation time of Routine 100 is about 4 minutes per iteration.

- JAMES N. SNYDER, On the Improvement of the Solutions to a Set of Simultaneous Linear Equations Using the ILLIAC (1955)

In a short mantissa computing environment the presence of an iterative improvement routine can significantly widen the class of solvable Ax = b problems.

- GENE H. GOLUB and CHARLES F. VAN LOAN,

Matrix Computations (1996)

Most problems involve inexact input data and obtaining a highly accurate solution to an imprecise problem may not be justified.

- J. J. DONGARRA, J. R. BUNCH, C. B. MOLER, and G. W. STEWART,

LINPACK Users' Guide (1979)

It is shown that even a single iteration of iterative refinement in single precision is enough to make Gaussian elimination stable in a very strong sense.

- ROBERT D. SKEEL, Iterative Refinement Implies Numerical Stability for Gaussian Elimination (1980)

Overview

Iterative refinement is an established technique for improving a computed solution to a linear system Ax = b. The process consists of three steps:

1. Computer r = b ? .

2. Solve Ad = r.

3. Update .

   (Repeat from step 1 if necessary, with replaced by y.)

If there were no rounding errors in the computation of