# ABS algorithm

### From Glossary

This is a family of algorithms to solve a linear system of equations, , with variables such that . Its distinguishing property is that the k-th iterate, , is a solution to the first equations. While algorithms in this class date back to 1934, the family was recognized and formally developed by J. Abaffy, C.G. Broyden and E. Spedicato, so the algorithm family name bears their initials. The ABS algorithm is given by the following.

Notation: is the i-th row of .

- Initialize: choose , so that is it nonsingular, set
- Set search direction for any that does not satisfy . Compute residuals: .
- Iterate: , where is the step size: .
- Test for convergance (update residuals, if used in test): Is solution within tolerance? If so, stop; else, continue.
- Update: , where is a rank 1 update of the form for such that .
- Increment: let and go to step 1.

The ABS algorithm extends to nonlinear equations and to linear diophantine equations.