## Version 1.2.0

12th April 2013

Recent Changes

• Single
• Double

### HSL_MI31 Symmetric positive-deﬁnite system: conjugate gradient method, stopping according to the A-norm of the error

The package uses the preconditioned conjugate gradient method to solve the $n×n$ symmetric positive-deﬁnite linear system

$Au=b,$

and implements several stopping criteria based on lower and upper bounds of the A-norm of the error. If $M={U}^{T}U$ is the preconditioning matrix, the routine actually solves the preconditioned system

$\stackrel{̄}{A}Y=\stackrel{̄}{b},$

with $\stackrel{̄}{A}={U}^{-T}A{U}^{-1}$ and $\stackrel{̄}{b}={U}^{-T}b$ and recovers the solution $u={U}^{-1}y$.

Reverse communication is used. Control is returned to the user for preconditioning operations and the products of $A$ with a vector $z$.