Finished exercise 6

Coryab/implement problem 2

.gitignore

@@ -41,4 +41,6 @@
 
 # C++ specifics
 src/*
+!src/Makefile
 !src/*.cpp
+!src/*.py

latex/assignment_1.tex

@@ -74,6 +74,9 @@
 %%
 %% Don't ask me why, I don't know.
 
+% custom stuff
+\graphicspath{{./images/}}
+
 \begin{document}
 
 \title{Project 1}      % self-explanatory

latex/problems/problem2.tex

@@ -1,3 +1,11 @@
 \section*{Problem 2}
 
-% Write which .cpp/.hpp/.py (using a link?) files are relevant for this and show the plot generated.
+The code for generating the points and plotting them can be found under.
+
+Point generator code: https://github.uio.no/FYS3150-G2-203/Project-1/blob/main/src/analyticPlot.cpp
+
+Plotting code: https://github.uio.no/FYS3150-G2-2023/Project-1/blob/main/src/analyticPlot.py
+
+Here is the plot of the analytical solution for $u(x)$.
+
+\includegraphics[scale=.5]{analytical_solution.pdf}

latex/problems/problem4.tex

@@ -2,7 +2,7 @@
 
 % Show that each iteration of the discretized version naturally creates a matrix equation.
 
-The value of $u(x_{0})$ and $u(x_{1})$ is known, using the discretized equation we can approximate the value of $f(x_{i}) = f_{i}$. This will result in a set of equations   
+The value of $u(x_{0})$ and $u(x_{1})$ is known, using the discretized equation we solve for $\vec{v}$. This will result in a set of equations 
 \begin{align*}
     - v_{0} + 2 v_{1} - v_{2} &= h^{2} \cdot f_{1} \\
     - v_{1} + 2 v_{2} - v_{3} &= h^{2} \cdot f_{2} \\
@@ -10,7 +10,7 @@ The value of $u(x_{0})$ and $u(x_{1})$ is known, using the discretized equation
     - v_{m-2} + 2 v_{m-1} - v_{m} &= h^{2} \cdot f_{m-1} \\
 \end{align*}
 
-Rearranging the first and last equation, moving terms of known boundary values to the RHS
+where $v_{i} = v(x_{i})$ and $f_{i} = f(x_{i})$. Rearranging the first and last equation, moving terms of known boundary values to the RHS
 \begin{align*}
     2 v_{1} - v_{2} &= h^{2} \cdot f_{1} + v_{0} \\
     - v_{1} + 2 v_{2} - v_{3} &= h^{2} \cdot f_{2} \\
@@ -40,5 +40,5 @@ We now have a number of linear eqations, corresponding to the number of unknown
       \right.
     \right]
     \end{align*}
-    where $g_{i} = h^{2} f_{i}$. An augmented matrix can be represented as $\boldsymbol{A} \vec{x} = \vec{b}$. In this case $\boldsymbol{A}$ is the coefficient matrix with a tridiagonal signature $(-1, 2, -1)$ and dimension $n \cross n$, where $n=m-2$.
+    Since the boundary values are equal to $0$ the RHS can be renamed $g_{i} = h^{2} f_{i}$ for all $i$. An augmented matrix can be represented as $\boldsymbol{A} \vec{x} = \vec{b}$. In this case $\boldsymbol{A}$ is the coefficient matrix with a tridiagonal signature $(-1, 2, -1)$ and dimension $n \cross n$, where $n=m-2$.
 

latex/problems/problem6.tex

@@ -1,9 +1,46 @@
 \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*}
+    \vec{a} &= [a_{2}, a_{3}, ..., a_{n-1}, a_{n}] \\
+    \vec{b} &= [b_{1}, b_{2}, 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 followed by a backward sweep to obtain $\vec{v}$
+\begin{algorithm}[H]
+    \caption{General algorithm}\label{algo:general}
+    \begin{algorithmic}
+        \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}
+        \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
+        \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}
+
 
 \subsection*{b)}
-
+% Figure out FLOPs
-% Figure it out
+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 $8(n-1)+1$.

src/Makefile

@@ -1,10 +1,15 @@
 CC=g++
 
-all: simpleFile
+.PHONY: clean
+
+all: simpleFile analyticPlot
 
 simpleFile: simpleFile.o
 	$(CC) -o $@ $^
 
+analyticPlot: analyticPlot.o
+	$(CC) -o $@ $^
+
 %.o: %.cpp
 	$(CC) -c $< -o $@
 

src/analyticPlot.cpp

@@ -6,17 +6,36 @@
 #include <fstream>
 #include <iomanip>
 
+#define RANGE 1000
+#define FILENAME "analytical_solution.txt"
+
 double u(double x);
 void generate_range(std::vector<double> &vec, double start, double stop, int n);
+void write_analytical_solution(std::string filename, int n);
 
 int main() {
-    int n = 1000;
+    write_analytical_solution(FILENAME, RANGE);
 
+    return 0;
+};
+
+double u(double x) {
+    return 1 - (1 - exp(-10))*x - exp(-10*x);
+};
+
+void generate_range(std::vector<double> &vec, double start, double stop, int n) {
+    double step = (stop - start) / n;
+
+    for (int i = 0; i <= vec.size(); i++) {
+        vec[i] = i * step;
+    }
+}
+
+void write_analytical_solution(std::string filename, int n) {
     std::vector<double> x(n), y(n);
     generate_range(x, 0.0, 1.0, n);
 
     // Set up output file and strem
-    std::string filename = "datapoints.txt";
     std::ofstream outfile;
     outfile.open(filename);
 
@@ -32,21 +51,5 @@ int main() {
             << std::endl;
     }
     outfile.close();
-
-    return 0;
-};
-
-double u(double x) {
-    double result;
-
-    result = 1 - (1 - exp(-10))*x - exp(-10*x);
-    return result;
-};
-
-void generate_range(std::vector<double> &vec, double start, double stop, int n) {
-    double step = (stop - start) / n;
-
-    for (int i = 0; i <= vec.size(); i++) {
-        vec[i] = i * step;
-    }
 }
+

src/analyticPlot.py

@@ -1,17 +1,19 @@
 import numpy as np 
 import matplotlib.pyplot as plt
 
+def main():
+    FILENAME = "analytical_solution.pdf"
     x = []
-y = []
     v = []
-with open('testdata.txt') as f:
-    for line in f:
-        a, b, c = line.strip().split()
-        x.append(float(a))
-        # y.append(float(b))
-        v.append(float(c))
 
-fig, ax = plt.subplots()
+    with open('analytical_solution.txt') as f:
-ax.plot(x, v)
+        for line in f:
-plt.show()
+            a, b = line.strip().split()
-# plt.savefig("main.png")
+            x.append(float(a))
+            v.append(float(b))
+
+    plt.plot(x, v)
+    plt.savefig(FILENAME)
+
+if __name__ == "__main__":
+    main()