diff --git a/latex/assignment_1.pdf b/latex/assignment_1.pdf index a1e7e76..1f17a59 100644 Binary files a/latex/assignment_1.pdf and b/latex/assignment_1.pdf differ diff --git a/latex/problems/problem6.tex b/latex/problems/problem6.tex index 535e69d..18c9b60 100644 --- a/latex/problems/problem6.tex +++ b/latex/problems/problem6.tex @@ -1,7 +1,6 @@ \section*{Problem 6} \subsection*{a)} - % Use Gaussian elimination, and then use backwards substitution to solve the equation Renaming the sub-, main-, and supdiagonal of matrix $\boldsymbol{A}$ \begin{align*} @@ -10,40 +9,39 @@ Renaming the sub-, main-, and supdiagonal of matrix $\boldsymbol{A}$ \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 +Following Thomas algorithm for gaussian elimination, we first perform a forward sweep followed by a backward sweep to obtain $\vec{v}$ \begin{algorithm}[H] - \caption{Foreward sweep}\label{algo:foreward} + \caption{General algorithm}\label{algo:general} \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 $b -= d*(*sup_diag)(i-1);$ - $(*g_vec)(i) -= d*(*g_vec)(i-1);$ + \Procedure{Forward sweep}{$\vec{a}$, $\vec{b}$, $\vec{c}$} + \State $n \leftarrow$ length of $\vec{b}$ + \State $\vec{\hat{b}}$, $\vec{\hat{g}} \leftarrow$ vectors of length $n$. + \State $\hat{b}_{1} \leftarrow b_{1}$ \Comment{Handle first element in main diagonal outside loop} + \State $\hat{g}_{1} \leftarrow g_{1}$ + \For{$i = 2, 3, ..., n$} + \State $d \leftarrow \frac{a_{i}}{\hat{b}_{i-1}}$ \Comment{Calculating common expression} + \State $\hat{b}_{i} \leftarrow b_{i} - d \cdot c_{i-1}$ + \State $\hat{g}_{i} \leftarrow g_{i} - d \cdot \hat{g}_{i-1}$ \EndFor - \While{Some condition} - \State Do something more here - \EndWhile - \State Maybe even some more math here, e.g $\int_0^1 f(x) \dd x$ + \Return $\vec{\hat{b}}$, $\vec{\hat{g}}$ + \EndProcedure + + \Procedure{Backward sweep}{$\vec{\hat{b}}$, $\vec{\hat{g}}$} + \State $n \leftarrow$ length of $\vec{\hat{b}}$ + \State $\vec{v} \leftarrow$ vector of length $n$. + \State $v_{n} \leftarrow \frac{\hat{g}_{n}}{\hat{b}_{n}}$ + \For{$i = n-1, n-2, ..., 1$} + \State $v_{i} \leftarrow \frac{\hat{g}_{i} - c_{i} \cdot v_{i+1}}{\hat{b}_{i}}$ + \EndFor + \Return $\vec{v}$ + \EndProcedure \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)} - -% Figure it out +% Figure out FLOPs +Counting the number of FLOPs for the general algorithm by looking at one procedure at a time. +For every iteration of i in forward sweep we have 1 division, 2 multiplications, and 2 subtractions, resulting in $5(n-1)$ FLOPs. +For backward sweep we have 1 division, and for every iteration of i we have 1 subtraction, 1 multiplication, and 1 division, resulting in $3(n-1)+1$ FLOPs. +Total FLOPs for the general algorithm is $8(n-1)+1$. diff --git a/latex/problems/problem9.tex b/latex/problems/problem9.tex index e5bd862..5bbd5ea 100644 --- a/latex/problems/problem9.tex +++ b/latex/problems/problem9.tex @@ -1,3 +1,55 @@ \section*{Problem 9} -% Show the algorithm, then calculate FLOPs, then link to relevant files +\subsection*{a)} +% Specialize algorithm +The special algorithm does not require the values of all $a_{i}$, $b_{i}$, $c_{i}$. +We find the values of $\hat{b}_{i}$ from simplifying the general case +\begin{align*} + \hat{b}_{i} &= b_{i} - \frac{a_{i} \cdot c_{i-1}}{\hat{b}_{i-1}} \\ + \hat{b}_{i} &= 2 - \frac{1}{\hat{b}_{i-1}} +\end{align*} +Calculating the first values to see a pattern +\begin{align*} + \hat{b}_{1} &= 2 \\ + \hat{b}_{2} &= 2 - \frac{1}{2} = \frac{3}{2} \\ + \hat{b}_{3} &= 2 - \frac{1}{\frac{3}{2}} = \frac{4}{3} \\ + \hat{b}_{4} &= 2 - \frac{1}{\frac{4}{3}} = \frac{5}{4} \\ + \vdots & \\ + \hat{b}_{i} &= \frac{i+1}{i} && \text{for $i = 1, 2, ..., n$} +\end{align*} + + +\begin{algorithm}[H] + \caption{Special algorithm}\label{algo:special} + \begin{algorithmic} + \Procedure{Forward sweep}{$\vec{b}$} + \State $n \leftarrow$ length of $\vec{b}$ + \State $\vec{\hat{b}}$, $\vec{\hat{g}} \leftarrow$ vectors of length $n$. + \State $\hat{b}_{1} \leftarrow 2$ \Comment{Handle first element in main diagonal outside loop} + \State $\hat{g}_{1} \leftarrow g_{1}$ + \For{$i = 2, 3, ..., n$} + \State $\hat{b}_{i} \leftarrow \frac{i+1}{i}$ + \State $\hat{g}_{i} \leftarrow g_{i} + \frac{\hat{g}_{i-1}}{\hat{b}_{i-1}}$ + \EndFor + \Return $\vec{\hat{b}}$, $\vec{\hat{g}}$ + \EndProcedure + + \Procedure{Backward sweep}{$\vec{\hat{b}}$, $\vec{\hat{g}}$} + \State $n \leftarrow$ length of $\vec{\hat{b}}$ + \State $\vec{v} \leftarrow$ vector of length $n$. + \State $v_{n} \leftarrow \frac{\hat{g}_{n}}{\hat{b}_{n}}$ + \For{$i = n-1, n-2, ..., 1$} + \State $v_{i} \leftarrow \frac{\hat{g}_{i} + v_{i+1}}{\hat{b}_{i}}$ + \EndFor + \Return $\vec{v}$ + \EndProcedure + \end{algorithmic} +\end{algorithm} + + +\subsection*{b)} +% Find FLOPs +For every iteration of i in forward sweep we have 2 divisions, and 2 additions, resulting in $4(n-1)$ FLOPs. +For backward sweep we have 1 division, and for every iteration of i we have 1 addition, and 1 division, resulting in $2(n-1)+1$ FLOPs. +Total FLOPs for the special algorithm is $6(n-1)+1$. + diff --git a/src/problem2.cpp b/src/problem2.cpp deleted file mode 100644 index e69de29..0000000