The [[conjugate gradient method]] is a [[numerical analysis|numerical algorithm]] that solves a [[system of linear equations]]
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:<math>A x= b.\,</math>
If the [[matrix]] 'A' is [[ill-conditioned]], i.e. it has a large [[condition number]] <math>\kappa(A)</math>, it is often useful to use a [[preconditioner|preconditioning matrix]] <math>P^{-1}</math> that is chosen such that <math>P^{-1} \approx A^{-1}</math> and solve the system
