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I have to code Jacobian Free version of GMRES with scaling and reordering algorithms separately. But I have serious problems about the convergence of inner gmres iterations and I have doubts on my formulation about jacobian-vector product for scaled equations since its bookkeeping gets harder.

Please notice me if the following is wrong; (For simplicity lets forget reordering now)

For row(S1) and column(S2) scalings of right preconditioned jacobian-vector product;

(1) (S1*A*S2) * (S1*M*S2)^-1 * (S1*v) = S1 * [F(Q+eps*(S1*M*S2)^-1 * (S1*v)) - F(Q)]/eps

where eps is perturbation epsilon, v is search direction, F(Q) is the function, M is right preconditioner matrix and A is jacobian matrix. Notation is similar to Yousef Saad's notation. I'm pretty sure about my formulation and application on code is right. But I'm not sure if I have to scale my residual vector. If answer is yes, how? Can someone share good references,

Thank you all.

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# A Scaling And Reordering of Jacobian Free GMRES

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