STABILITY OF THE DIAGONAL PIVOTING METHOD WITH PARTIAL PIVOTING

LAPACK and LINPACK both solve symmetric indefinite linear systems usin g the diagonal pivoting method with the partial pivoting strategy of B unch and Kaufman [Math. Comp., 31 (1977), pp. 163-119]. No proof of th e stability of this method has appeared in the literature. It is tempt ing to argue that the diagonal pivoting method is stable for a given p ivoting strategy if the growth factor is small. We show that this argu ment is false in general and give a sufficient condition for stability . This condition is not satisfied by the partial pivoting strategy bec ause the multipliers are unbounded. Nevertheless, using a more specifi c approach we are able to prove the stability of partial pivoting, the reby filling a gap in the body of theory supporting LAPACK and LINPACK .