Merge branch 'main' into 7-solve-problem-7

Coryab/change latex structure

latex/assignment_1.tex

@@ -84,116 +84,22 @@
     
 \textit{https://github.uio.no/FYS3150-G2-2023/Project-1}
     
-\section*{Problem 1}
+\input{problems/problem1}
 
-\begin{align*}
+\input{problems/problem2}
-    u(x) &= \int \int \frac{d^2 u}{dx^2} dx^2\\
-         &= \int \int -100 e^{-10x} dx^2 \\
-         &= \int \frac{-100 e^{-10x}}{-10} + c_1 dx \\
-         &= \int 10 e^{-10x} + c_1 dx \\
-         &= \frac{10 e^{-10x}}{-10} + c_1 x + c_2 \\
-         &= -e^{-10x} + c_1 x + c_2
-\end{align*}
 
-Using the boundary conditions, we can find $c_1$ and $c_2$ as shown below:
+\input{problems/problem3}
 
-\begin{align*}
+\input{problems/problem4}
-    u(0) &= 0 \\
-    -e^{-10 \cdot 0} + c_1 \cdot 0 + c_2 &= 0 \\
-    -1 + c_2 &= 0 \\
-    c_2 &= 1
-\end{align*}
 
-\begin{align*}
+\input{problems/problem5}
-    u(1) &= 0 \\
-    -e^{-10 \cdot 1} + c_1 \cdot 1 + c_2 &= 0 \\
-    -e^{-10} + c_1 + c_2 &= 0 \\
-    c_1 &= e^{-10} - c_2\\
-    c_1 &= e^{-10} - 1\\
-\end{align*}
 
-Using the values that we found for $c_1$ and $c_2$, we get
+\input{problems/problem6}
 
-\begin{align*}
+\input{problems/problem7}
-    u(x) &= -e^{-10x} + (e^{-10} - 1) x + 1 \\
-         &= 1 - (1 - e^{-10}) - e^{-10x}
-\end{align*}
 
-\section*{Problem 2}
+\input{problems/problem8}
 
-% Write which .cpp/.hpp/.py (using a link?) files are relevant for this and show the plot generated.
+\input{problems/problem9}
-
-\section*{Problem 3}
-
-To derive the discretized version of the Poisson equation, we first need
-the taylor expansion for $u(x)$ around $x + h$ and $x - h$.
-
-\begin{align*}
-    u(x+h) &= u(x) + u'(x) h + \frac{1}{2} u''(x) h^2 + \frac{1}{6} u'''(x) h^3 + \mathcal{O}(h^4)
-\end{align*}
-
-\begin{align*}
-    u(x-h) &= u(x) - u'(x) h + \frac{1}{2} u''(x) h^2 - \frac{1}{6} u'''(x) h^3 + \mathcal{O}(h^4)
-\end{align*}
-
-If we add the equations above, we get this new equation:
-
-\begin{align*}
-    u(x+h) + u(x-h) &= 2 u(x) + u''(x) h^2 + \mathcal{O}(h^4) \\
-    u(x+h) - 2 u(x) + u(x-h) + \mathcal{O}(h^4) &= u''(x) h^2 \\
-    u''(x) &= \frac{u(x+h) - 2 u(x) + u(x-h)}{h^2} + \mathcal{O}(h^2) \\
-    u_i''(x) &= \frac{u_{i+1} - 2 u_i + u_{i-1}}{h^2} + \mathcal{O}(h^2) \\
-\end{align*}
-
-We can then replace $\frac{d^2u}{dx^2}$ with the RHS (right-hand side) of the equation:
-
-\begin{align*}
-    - \frac{d^2u}{dx^2} &= 100 e^{-10x} \\
-    \frac{ - u_{i+1} + 2 u_i - u_{i-1}}{h^2} + \mathcal{O}(h^2) &= 100 e^{-10x} \\
-\end{align*}
-
-And lastly, we leave out $\mathcal{O}(h^2)$ and change $u_i$ to $v_i$ to 
-differentiate between the exact solution and the approximate solution, 
-and get the discretized version of the equation:
-
-\begin{align*}
-align*    \frac{ - u_{i+1} + 2 u_i - u_{i-1}}{h^2} &= 100 e^{-10x} \\
-\end{align*}
-
-\section*{Problem 4}
-
-% Show that each iteration of the discretized version naturally creates a matrix equation.
-
-\section*{Problem 5}
-
-\subsection*{a)}
-
-\subsection*{b)}
-
-\section*{Problem 6}
-
-\subsection*{a)}
-
-% Use Gaussian elimination, and then use backwards substitution to solve the equation
-
-\subsection*{b)}
-
-% Figure it out
-
-\section*{Problem 7}
-
-% Link to relevant files on gh and possibly add some comments
-
-\section*{Problem 8}
-
-%link to relvant files and show plots
-
-\section*{Problem 9}
-
-% Show the algorithm, then calculate FLOPs, then link to relevant files
-
-\section*{Problem 10}
-
-% Time and show result, and link to relevant files
 
 \end{document}

latex/problems/problem1.tex

@@ -0,0 +1,35 @@
+\section*{Problem 1}
+
+% Do the double integral
+\begin{align*}
+    u(x) &= \int \int \frac{d^2 u}{dx^2} dx^2\\
+         &= \int \int -100 e^{-10x} dx^2 \\
+         &= \int \frac{-100 e^{-10x}}{-10} + c_1 dx \\
+         &= \int 10 e^{-10x} + c_1 dx \\
+         &= \frac{10 e^{-10x}}{-10} + c_1 x + c_2 \\
+         &= -e^{-10x} + c_1 x + c_2
+\end{align*}
+
+Using the boundary conditions, we can find $c_1$ and $c_2$ as shown below:
+
+\begin{align*}
+    u(0) &= 0 \\
+    -e^{-10 \cdot 0} + c_1 \cdot 0 + c_2 &= 0 \\
+    -1 + c_2 &= 0 \\
+    c_2 &= 1
+\end{align*}
+
+\begin{align*}
+    u(1) &= 0 \\
+    -e^{-10 \cdot 1} + c_1 \cdot 1 + c_2 &= 0 \\
+    -e^{-10} + c_1 + c_2 &= 0 \\
+    c_1 &= e^{-10} - c_2\\
+    c_1 &= e^{-10} - 1\\
+\end{align*}
+
+Using the values that we found for $c_1$ and $c_2$, we get
+
+\begin{align*}
+    u(x) &= -e^{-10x} + (e^{-10} - 1) x + 1 \\
+         &= 1 - (1 - e^{-10}) - e^{-10x}
+\end{align*}

latex/problems/problem10.tex

@@ -0,0 +1,3 @@
+\section*{Problem 10}
+
+% Time and show result, and link to relevant files

latex/problems/problem2.tex

@@ -0,0 +1,3 @@
+\section*{Problem 2}
+
+% Write which .cpp/.hpp/.py (using a link?) files are relevant for this and show the plot generated.

latex/problems/problem3.tex

@@ -0,0 +1,4 @@
+
+\section*{Problem 3}
+
+% Show how it's derived and where we found the derivation.

latex/problems/problem4.tex

@@ -0,0 +1,3 @@
+\section*{Problem 4}
+
+% Show that each iteration of the discretized version naturally creates a matrix equation.

latex/problems/problem5.tex

@@ -0,0 +1,6 @@
+
+\section*{Problem 5}
+
+\subsection*{a)}
+
+\subsection*{b)}

latex/problems/problem6.tex

@@ -0,0 +1,9 @@
+\section*{Problem 6}
+
+\subsection*{a)}
+
+% Use Gaussian elimination, and then use backwards substitution to solve the equation
+
+\subsection*{b)}
+
+% Figure it out

latex/problems/problem7.tex

@@ -0,0 +1,3 @@
+\section*{Problem 7}
+
+% Link to relevant files on gh and possibly add some comments

latex/problems/problem8.tex

@@ -0,0 +1,3 @@
+\section*{Problem 8}
+
+%link to relvant files and show plots

latex/problems/problem9.tex

@@ -0,0 +1,3 @@
+\section*{Problem 9}
+
+% Show the algorithm, then calculate FLOPs, then link to relevant files

src/analyticPlot.cpp

@@ -0,0 +1,52 @@
+#include <iostream>
+#include <cmath>
+#include <vector>
+#include <string>
+#include <numeric>
+#include <fstream>
+#include <iomanip>
+
+double u(double x);
+void generate_range(std::vector<double> &vec, double start, double stop, int n);
+
+int main() {
+    int n = 1000;
+
+    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);
+
+    // Parameters for formatting 
+    int width = 12;
+    int prec = 4;
+
+    // Calculate u(x) and write to file
+    for (int i = 0; i <= x.size(); i++) {
+        y[i] = u(x[i]);
+        outfile << std::setw(width) << std::setprecision(prec) << std::scientific << x[i] 
+            << std::setw(width) << std::setprecision(prec) << std::scientific << y[i] 
+            << 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

@@ -0,0 +1,17 @@
+import numpy as np 
+import matplotlib.pyplot as plt
+
+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()
+ax.plot(x, v)
+plt.show()
+# plt.savefig("main.png")

src/main.cpp

@@ -0,0 +1,24 @@
+#include "GeneralAlgorithm.hpp"
+#include <armadillo>
+#include <iostream>
+
+double f(double x) {
+    return 100. * std::exp(-10.*x);
+}
+
+double a_sol(double x) {
+    return 1. - (1. - std::exp(-10)) * x - std::exp(-10*x);
+}
+
+int main() {
+    arma::mat A = arma::eye(3,3);
+
+    GeneralAlgorithm ga(3, &A, f, a_sol, 0., 1.);
+
+    ga.solve();
+    std::cout << "Time: " << ga.time(5) << std::endl;
+    ga.error();
+
+    return 0;
+
+}

src/simpleFile.cpp

@@ -0,0 +1,162 @@
+#include <armadillo>
+#include <cmath>
+#include <ctime>
+#include <fstream>
+#include <iomanip>
+#include <ios>
+#include <string>
+
+#define TIMING_ITERATIONS 5
+
+arma::vec* general_algorithm(
+    arma::vec* sub_diag,
+    arma::vec* main_diag,
+    arma::vec* sup_diag,
+    arma::vec* g_vec
+)
+{
+    int n = g_vec->n_elem;
+    double d;
+
+    for (int i = 1; i < n; i++) {
+        d = (*sub_diag)(i-1) / (*main_diag)(i-1);
+        (*main_diag)(i) -= d*(*sup_diag)(i-1);
+        (*g_vec)(i) -= d*(*g_vec)(i-1);
+    }
+
+    (*g_vec)(n-1) /= (*main_diag)(n-1);
+
+    for (int i = n-2; i >= 0; i--) {
+        (*g_vec)(i) = ((*g_vec)(i) - (*sup_diag)(i) * (*g_vec)(i+1)) / (*main_diag)(i);
+    }
+    return g_vec;
+}
+
+
+arma::vec* special_algorithm(
+    double sub_sig,
+    double main_sig,
+    double sup_sig,
+    arma::vec* g_vec
+)
+{
+    int n = g_vec->n_elem;
+    arma::vec diag = arma::vec(n);
+
+    for (int i = 1; i < n; i++) {
+        // Calculate values for main diagonal based on indices
+        diag(i-1) = (double)(i+1) / i;
+        (*g_vec)(i) += (*g_vec)(i-1) / diag(i-1);
+    }
+    // The last element in main diagonal has value (i+1)/i = (n+1)/n
+    (*g_vec)(n-1) /= (double)(n+1) / (n);
+
+    for (int i = n-2; i >= 0; i--) {
+        (*g_vec)(i) = ((*g_vec)(i) + (*g_vec)(i+1))/ diag(i);
+    }
+
+    return g_vec;
+}
+
+void error(
+    std::string filename,
+    arma::vec* x_vec,
+    arma::vec* v_vec,
+    arma::vec* a_vec
+)
+{
+    std::ofstream ofile;
+    ofile.open(filename);
+
+    if (!ofile.is_open()) {
+        exit(1);
+    }
+
+    for (int i=0; i < a_vec->n_elem; i++) {
+        double sub = (*a_vec)(i) - (*v_vec)(i);
+        ofile << std::setprecision(8) << std::scientific << (*x_vec)(i)
+            << std::setprecision(8) << std::scientific << std::log10(std::abs(sub))
+            << std::setprecision(8) << std::scientific << std::log10(std::abs(sub/(*a_vec)(i)))
+            << std::endl;
+    }
+    
+    ofile.close();
+
+}
+
+double f(double x) {
+    return 100*std::exp(-10*x);
+}
+
+void build_array(
+    int n_steps, 
+    arma::vec* sub_diag, 
+    arma::vec* main_diag, 
+    arma::vec* sup_diag,
+    arma::vec* g_vec
+) 
+{
+    sub_diag->resize(n_steps-2);
+    main_diag->resize(n_steps-1);
+    sup_diag->resize(n_steps-2);
+
+    sub_diag->fill(-1);
+    main_diag->fill(2);
+    sup_diag->fill(-1);
+
+    g_vec->resize(n_steps-1);
+
+    double step_size = 1./ (double) n_steps;
+    for (int i=0; i < n_steps-1; i++) {
+        (*g_vec)(i) = f((i+1)*step_size);
+    }
+
+}
+
+void timing() {
+    arma::vec sub_diag, main_diag, sup_diag, g_vec;
+    int n_steps;
+
+    std::ofstream ofile;
+    ofile.open("timing.txt");
+
+    // Timing
+    for (int i=1; i <= 8; i++) {
+        n_steps = std::pow(10, i);
+        clock_t g_1, g_2, s_1, s_2;
+        double g_res = 0, s_res = 0;
+      
+        for (int j=0; j < TIMING_ITERATIONS; j++) {
+            build_array(n_steps, &sub_diag, &main_diag, &sup_diag, &g_vec);
+
+            g_1 = clock();
+
+            general_algorithm(&sub_diag, &main_diag, &sup_diag, &g_vec);
+
+            g_2 = clock();
+
+            g_res += (double) (g_2 - g_1) / CLOCKS_PER_SEC;
+            build_array(n_steps, &sub_diag, &main_diag, &sup_diag, &g_vec);
+
+            s_1 = clock();
+
+            special_algorithm(-1., 2., -1., &g_vec);
+
+            s_2 = clock();
+
+            s_res += (double) (s_2 - s_1) / CLOCKS_PER_SEC;
+
+        }
+        ofile 
+            << n_steps << ","
+            << g_res / (double) TIMING_ITERATIONS << ","
+            << s_res / (double) TIMING_ITERATIONS << std::endl;
+    }
+
+    ofile.close();
+}
+
+int main() 
+{
+    timing();
+}