## Version 1.3.0

26th March 2013

Recent Changes

• Single
• Double

### MI21 Symmetric positive-deﬁnite system: conjugate gradient method

This routine uses the Conjugate Gradient method to solve the $n×n$ symmetric positive-deﬁnite linear system $Ax=b$, optionally using preconditioning. If $P{P}^{T}$ is the preconditioning matrix, the routine actually solves the preconditioned system

$\stackrel{̄}{A}\stackrel{̄}{x}=\stackrel{̄}{b},$

with $\stackrel{̄}{A}=PA{P}^{T}$ and $\stackrel{̄}{b}=Pb$ and recovers the solution $x={P}^{T}\stackrel{̄}{x}$. Reverse communication is used for preconditioning operations and matrix-vector products of the form $Az$.