## Version 1.1.0

• Single
• Double

### MA61 Sparse symmetric positive-deﬁnite system: incomplete factorization

To solve a symmetric, sparse and positive deﬁnite set of linear equations $Ax=b$ i.e.

$\sum _{j=1}^{n}{a}_{ij}{x}_{j}={b}_{i},i=1,2,...,n.$

The solution is found by a preconditioned conjugate gradient technique, where the preconditioning is done by incomplete factorization.

(a) MA61A performs the incomplete factorization based on an ${LDL}^{T}$ decomposition. New entries which have small numerical values compared to the corresponding diagonal entries are dropped, and the diagonal entries are modiﬁed to ensure positive deﬁniteness. This results in a preconditioning matrix $C$ held in ${LDL}^{T}$ form.
(b) MA61B performs the iteration procedure using the preconditioned coeﬃcient matrix (i.e. ${AC}^{-1}$ ) as the iteration matrix for the conjugate gradient algorithm.