Simple expressions for project 1

For the special matrix we can can actually precalculate the updated matrix elements $\tilde{d}_i$. The non-diagonal elements $e_i$ are unchanged. For our particular matrix in project 1 we have $$\tilde{d}_i= 2 - \frac{1}{\tilde{d}_{i-1}}=\frac{i+1}{i},$$ and the new righthand side $\tilde{f}_i$ given by $$\tilde{f}_i= f_i + \frac{(i-1)\tilde{f}_{i-1}}{i}.$$ Recall that $\tilde{d}_1=2$ and $\tilde{f}_1=f_1$. These arrays can be set up before computing $u$.

The backward substitution gives then the final solution $$u_{i-1}= \frac{i-1}{i}\left(\tilde{f}_{i-1}+u_i\right),$$ with $u_n=\tilde{f}_{n}/\tilde{b}_{n}$.

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