The preconditioned solver

Summing up the ideas above, we start from fitting goals  

$$\begin{array} {lll} \bold 0 &\approx& \bold F \bold m - \bold d \\ \bold 0 &\approx& \bold A \bold m\end{array}$$ (8)

and we change variables from $\bold m$ to $\bold p$ using $\bold m = \bold A^{-1} \bold p$ 

$$\begin{array} {llllcl} \bold 0 &\approx & \bold F \bold m - ... ...d 0 &\approx & \bold A \bold m &=& \bold I & \bold p\end{array}$$ (9)

Preconditioning means iteratively fitting by adjusting the $\bold p$ variables and then finding the model by using $\bold m = \bold A^{-1} \bold p$.

A new reusable preconditioned solver is the module prec_solver . The variable x in prec_solver refers to $\bold m$.Likewise the modeling operator $\bold F$ is called oper and the smoothing operator $\bold A^{-1}$ is called prec. Details of the code are only slightly different from the regularized solver reg_solver .