## Version 1.3.0

26th March 2013

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

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

$\bar{\mathbf{A}} \bar{\mathbf{x}} = \bar{\mathbf{b}},$

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