Breakdowns in the Implementation of the L Anczos Method for Solving Linear Systems

The LL anczos method for solving systems of linear equations is based on formal orthogonal polynomi-als. Its implementation is realized via some recurrence relationships between polynomials of a family of orthogonal polynomials or between those of two adjacent families of orthogonal polynomials. A division by zero can occur in such recurrence relations, thus causing a breakdown in the algorithm which has to be stopped. In this paper, two types of breakdowns are discussed. The true breakdowns which are due to the non{existence of some polynomials and the ghost breakdowns which are due to the recurrence relationship used. Among all the recurrence relationships which can be used and all the algorithms for implementing the LL anczos method which came out from them, the only reliable algorithm is LL anczos/Orthodir which can only suuer from true breakdowns. It is showed how to avoid true breakdowns in this algorithm. Other algorithms are also discussed and the case of near{breakdown is treated. The same treatment applies to other methods related to LL anczos'.