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In this paper Penalty and front-fixing methods for the numerical solution of American option problems a front fixing method based on Newton is described for an American put option is described. I am trying to implement this method. Can someone confirm that the following is the Jacobian matrix $J$. $$\left( \begin{array}{ccccccc} \gamma^n_1 & & &&&&-\beta^n_1-\alpha^n_1(1+\Delta x)+(\gamma^n_1)'p^n_2+(\beta^n_1)'(E-\bar{S}^n)\\ \alpha^n_2 &\gamma^n_2 & &&&&-\beta^n_2(1+\Delta x)+(\gamma^n_2)'p^n_3+(\beta^n_2)'(E-(1+\Delta x)\bar{S}^n)\\ \beta^n_3 & \alpha^n_3 & \gamma^n_3& &&&(\beta^n_3)'p^n_2+(\gamma^n_3)'p^n_4\\ &\ddots&\ddots&\ddots&&&\vdots\\ &&\ddots&\ddots&\ddots&&\vdots\\ &&& \beta^n_{M-1} & \alpha^n_{M-1} & \gamma^n_{M-1}&(\beta^n_{M-1})'p^n_{M-2}+(\gamma^n_{M-1})'p^n_{M} \\ &&&&\beta^n_{M} &\alpha^n_{M}&(\beta^n_{M})'p^n_{M-1}\\ \end{array} \right)$$

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