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scratchpad4

-              v8
+              v9
 We can also write the equation as
 $\beta = \alpha_0 T'_0 + \displaystyle \sum_{\pm j} \alpha_{\pm j} T'_{\pm \hat{j}} + \sum_{\pm i, \pm j,|i| \ne |j|} \alpha_{\pm i, \pm j} T'_{\pm \hat{i} \pm \hat{j}}$
+$-T_0 - \alpha_0 T^{\lambda+1}_0 - \displaystyle \sum_{\pm j} \alpha_{\pm j} T^{\lambda+1}_{\pm \hat{j}} - \sum_{\pm i, \pm j,|i| \ne |j|} \alpha_{\pm i, \pm j} T^{\lambda+1}_{\pm \hat{i} \pm \hat{j}} = -T'_0 + \alpha_0 T^{\lambda}_0T'_0 + \displaystyle \sum_{\pm j} \alpha_{\pm j} T^{\lambda}_{\pm \hat{j}} T'_{\pm \hat{j}} + \sum_{\pm i, \pm j,|i| \ne |j|} \alpha_{\pm i, \pm j} T^{\lambda}_{\pm \hat{i} \pm \hat{j}} T'_{\pm \hat{i} \pm \hat{j}}$
+and then work out the coefficients for the matrix equation
+where
 $\alpha_0 = -\frac{1}{\Delta t} -\frac{2E_{jj}}{\Delta x^2}T_0^\lambda$