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b_{3}, ..., b_{n-1}, b_{n}] \\ + \vec{c} &= [c_{1}, c_{2}, c_{3}, ..., c_{n-1}] \\ +\end{align*} + +Following Thomas algorithm for gaussian elimination, we first perform a forward sweep +\begin{algorithm}[H] + \caption{Foreward sweep}\label{algo:foreward} + \begin{algorithmic} + \State Create new vector \vec{\hat{b}} of length n. + \State \hat{b}[0] &= b[0] \Comment{Handle first element in main diagonal outside loop} + \For{$i = 1, ..., n-1$} + \State d = \frac{a[i-1]}{b[i-1]} + \State b -= d*(*sup_diag)(i-1); + (*g_vec)(i) -= d*(*g_vec)(i-1); + \EndFor + \While{Some condition} + \State Do something more here + \EndWhile + \State Maybe even some more math here, e.g $\int_0^1 f(x) \dd x$ + \end{algorithmic} +\end{algorithm} +Here the index i does not refer to the element in the vector, ie. $b_{i}$, but to the index in the vector where the element is found ie. $b[i] = b_{i+1}$. + +\begin{align*} + \hat{b}_{1} &= b_{1} \\ + \hat{b}_{2} &= b_{2} - \frac{a_{2}}{\hat{b}_{1}} \cdot c_{1} \\ + \vdots & \\ + \hat{b}_{i} &= b_{i} - \frac{a_{i}}{\hat{b}_{i-1}} \cdot c_{i-1} \\ + \vdots & \\ +\end{align*} + + +% as g_hat also make use of the ratio (ai/bi-1_hat) it only needs to be calculated once + +% for n steps we have n+1 points, that is values of x to evaluate u(x) at, when boundary values are known +% we have to perform (n+1)-2 = n-1 calculations (equal number of col in matrix). Dealing +% with a quadratic matrix number of col = number of rows \subsection*{b)}