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Merge pull request #11 from FYS3150-G2-2023/janitaws/finish-text-document

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Janitaws/finish text document
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Cory Balaton 2023-09-25 14:21:56 +02:00 committed by GitHub Enterprise
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37
README.md
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@ -6,39 +6,32 @@
## Compile and run C++ programs ## Compile and run C++ programs
Compiling and linking is done using Make, make sure you are in the root folder. To create executable files, run shell commands Compiling and linking is done using Make, make sure you are in the root directory.
```shell There are two alternative ways to compile the code. The first alternative is to change
cd src/ directory to `src/` and compile
make
```
To run `script-name`
```shell
./main
```
There are two ways of compiling the code.
The first way is to go into the src directory and compile
```shell ```shell
cd src && make cd src && make
``` ```
or you could use make in the project directory The second alternative does not involve changing directory to `src/`, use
make in the projects root directory
```shell ```shell
make code make code
``` ```
and it will compile everything with you needing to move into **src**. You can run any of the compiled programs by changing directory to `src/`, then
You can run any of the compiled programs by going into the src directory
and then doing
```shell ```shell
./program-name ./test_suite
./main
```
Remove object files and executables from `src/`
```shell
make clean
``` ```
@ -64,7 +57,7 @@ If you want to generate the documentation you can do
make docs make docs
``` ```
in the project root, and the documentation will be made into the **docs** in the project root, and the documentation will be made into the `docs/`
directory. directory.
## Generate project document ## Generate project document
@ -75,4 +68,4 @@ If you want to recompile the Pdf file, you can do
make latex make latex
``` ```
and a Pdf file will be produced inside the **latex** directory. and a Pdf file will be produced inside the `latex/` directory.

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98 9.5049504950e-01 2.1797196485e-02 4.3068226072e-02 -6.3299628759e-02 2.1797195856e-02 4.3068226575e-02 -6.3299628182e-02
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101 9.8019801980e-01 8.7484791764e-03 1.7463117510e-02 -2.6110190428e-02 8.7484808507e-03 1.7463115529e-02 -2.6110188805e-02
102 9.9009900990e-01 4.3763588619e-03 8.7484790494e-03 -1.3112138188e-02 4.3763573469e-03 8.7484808507e-03 -1.3112140764e-02
103 1.0000000000e+00 0.0000000000e+00 0.0000000000e+00 0.0000000000e+00 0.0000000000e+00 0.0000000000e+00 0.0000000000e+00

18
latex/output/eigenvector_6.csv
View File

@ -1,9 +1,9 @@
x,Vector 1,Vector 2,Vector 3 x,Vector 1,Vector 2,Vector 3,Analytic 1,Analytic 2,Analytic 3
0.0000000000e+00,0.0000000000e+00,0.0000000000e+00,0.0000000000e+00 0.0000000000e+00,0.0000000000e+00,0.0000000000e+00,0.0000000000e+00,0.0000000000e+00,0.0000000000e+00,0.0000000000e+00
1.4285714286e-01,2.3192061397e-01,-4.1790650593e-01,-5.2112088916e-01 1.4285714286e-01,2.3192061397e-01,-4.1790650593e-01,-5.2112088916e-01,2.3192061392e-01,-4.1790650594e-01,-5.2112088917e-01
2.8571428571e-01,4.1790650598e-01,-5.2112088916e-01,-2.3192061388e-01 2.8571428571e-01,4.1790650598e-01,-5.2112088916e-01,-2.3192061388e-01,4.1790650594e-01,-5.2112088917e-01,-2.3192061392e-01
4.2857142857e-01,5.2112088920e-01,-2.3192061385e-01,4.1790650595e-01 4.2857142857e-01,5.2112088920e-01,-2.3192061385e-01,4.1790650595e-01,5.2112088917e-01,-2.3192061392e-01,4.1790650594e-01
5.7142857143e-01,5.2112088915e-01,2.3192061400e-01,4.1790650592e-01 5.7142857143e-01,5.2112088915e-01,2.3192061400e-01,4.1790650592e-01,5.2112088917e-01,2.3192061392e-01,4.1790650594e-01
7.1428571429e-01,4.1790650588e-01,5.2112088921e-01,-2.3192061394e-01 7.1428571429e-01,4.1790650588e-01,5.2112088921e-01,-2.3192061394e-01,4.1790650594e-01,5.2112088917e-01,-2.3192061392e-01
8.5714285714e-01,2.3192061389e-01,4.1790650591e-01,-5.2112088921e-01 8.5714285714e-01,2.3192061389e-01,4.1790650591e-01,-5.2112088921e-01,2.3192061392e-01,4.1790650594e-01,-5.2112088917e-01
1.0000000000e+00,0.0000000000e+00,0.0000000000e+00,0.0000000000e+00 1.0000000000e+00,0.0000000000e+00,0.0000000000e+00,0.0000000000e+00,0.0000000000e+00,0.0000000000e+00,0.0000000000e+00

1 x Vector 1 Vector 2 Vector 3 Analytic 1 Analytic 2 Analytic 3
2 0.0000000000e+00 0.0000000000e+00 0.0000000000e+00 0.0000000000e+00 0.0000000000e+00 0.0000000000e+00 0.0000000000e+00
3 1.4285714286e-01 2.3192061397e-01 -4.1790650593e-01 -5.2112088916e-01 2.3192061392e-01 -4.1790650594e-01 -5.2112088917e-01
4 2.8571428571e-01 4.1790650598e-01 -5.2112088916e-01 -2.3192061388e-01 4.1790650594e-01 -5.2112088917e-01 -2.3192061392e-01
5 4.2857142857e-01 5.2112088920e-01 -2.3192061385e-01 4.1790650595e-01 5.2112088917e-01 -2.3192061392e-01 4.1790650594e-01
6 5.7142857143e-01 5.2112088915e-01 2.3192061400e-01 4.1790650592e-01 5.2112088917e-01 2.3192061392e-01 4.1790650594e-01
7 7.1428571429e-01 4.1790650588e-01 5.2112088921e-01 -2.3192061394e-01 4.1790650594e-01 5.2112088917e-01 -2.3192061392e-01
8 8.5714285714e-01 2.3192061389e-01 4.1790650591e-01 -5.2112088921e-01 2.3192061392e-01 4.1790650594e-01 -5.2112088917e-01
9 1.0000000000e+00 0.0000000000e+00 0.0000000000e+00 0.0000000000e+00 0.0000000000e+00 0.0000000000e+00 0.0000000000e+00

2
latex/problems/problem-1.tex
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@ -12,5 +12,5 @@ Scaling will result in a dimensionless variable $\hat{x} = \frac{1}{L}$.
\end{align*} \end{align*}
Now we insert the expression into the original equation Now we insert the expression into the original equation
\begin{align*} \begin{align*}
\frac{d u(\hat{x})}{d\hat{x}^{2}} &= - \frac{F L^{2}}{\gamma} u(\hat{x}) \\ \frac{d u(\hat{x})}{d\hat{x}^{2}} &= - \frac{F L^{2}}{\gamma} u(\hat{x}). \\
\end{align*} \end{align*}

2
latex/problems/problem-3.tex
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@ -2,7 +2,7 @@
\subsection*{a)} \subsection*{a)}
The function for found the largest off-diagonal can be found in The function to find the largest off-diagonal can be found in
\textbf{matrix.hpp} and \textbf{matrix.cpp}. \textbf{matrix.hpp} and \textbf{matrix.cpp}.
\subsection*{b)} \subsection*{b)}

19
latex/problems/problem-5.tex
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@ -1 +1,20 @@
\section*{Problem 5} \section*{Problem 5}
\subsection*{a)}
We used the Jacobi's rotation method to solve $\boldsymbol{A} \vec{v} = \lambda \vec{v}$, for $\boldsymbol{A}_{(N \cross N)}$ with $N \in [5, 100]$,
and increased the matrix size by $3$ rows and columns for every new matrix generated. The number of similarity transformations performed for a tridiagonal matrix
of is presented in Figure \ref{fig:transform}. We chose to run the program using dense matrices of same size as the tridiagonal matrices, to compare the scaling data.
What we see is that the number of similarity transformations necessary to solve the system is proportional to the matrix size.
\begin{figure}[h]
\centering
\includegraphics[width=0.8\textwidth]{images/transform.pdf}
\caption{Similarity transformations performed as a function of matrix size (N), data is presented in a logarithmic scale.}
\label{fig:transform}
\end{figure}
\subsection*{b)}
For both the tridiagonal and dense matrices we are checking off-diagonal elements above the main diagonal, since these are symmetric matrices.
The max value is found at index $(k,l)$ and for every rotation of the matrix, we update the remaining elements along row $k$ and $l$. This can lead to an increased
value of off-diagonal elements, that previously were close to zero, and extra rotations has to be performed due to these elements. Which suggest that the
number of similarity transformations perfomed on a matrix does not depend on its initial number of non-zero elements, making the Jacobi's rotation algorithm as
computationally expensive for both dense and tridiagonal matrices of size $N \cross N$.

20
latex/problems/problem-6.tex
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@ -1 +1,21 @@
\section*{Problem 6} \section*{Problem 6}
\subsection*{a)}
The plot in Figure \ref{fig:eigenvector_10} is showing the discretization of $\hat{x}$ with $n=10$.
The eigenvectors and corresponding analytical eigenvectors have a complete overlap suggesting the implementation of the algorithm is correct.
We have included the boundary points for each vector to show a complete solution.
\begin{figure}[h]
\centering
\includegraphics[width=0.8\textwidth]{images/eigenvector_10.pdf}
\caption{The plot is showing the elements of eigenvector $\vec{v}_{1}, \vec{v}_{2}, \vec{v}_{3}$, corresponding to the three lowest eigenvalues of matrix $\boldsymbol{A} (10 \cross 10)$, against the position $\hat{x}$. The analytical eigenvectors $\vec{v}^{(1)}, \vec{v}^{(2)}, \vec{v}^{(3)}$ are also included in the plot.}
\label{fig:eigenvector_10}
\end{figure}
\subsection*{b)}
For the discretization with $n=100$ the solution is visually close to a continuous curve, with a complete overlap of the analytical eigenvectors, presented in Figure \ref{fig:eigenvector_100}.
\begin{figure}
\centering
\includegraphics[width=0.8\textwidth]{images/eigenvector_100.pdf}
\caption{The plot is showing the elements of eigenvector $\vec{v}_{1}, \vec{v}_{2}, \vec{v}_{3}$, corresponding to the three lowest eigenvalues of matrix $\boldsymbol{A} (100 \cross 100)$, against the position $\hat{x}$. The analytical eigenvectors $\vec{v}^{(1)}, \vec{v}^{(2)}, \vec{v}^{(3)}$ are also included in the plot.}
\label{fig:eigenvector_100}
\end{figure}

40
src/main.cpp
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@ -87,6 +87,7 @@ void write_eigenvec(int N)
// Create tridiagonal matrix // Create tridiagonal matrix
arma::mat A = create_symmetric_tridiagonal(N, a, d); arma::mat A = create_symmetric_tridiagonal(N, a, d);
arma::mat analytic = arma::mat(N, N);
arma::vec eigval; arma::vec eigval;
arma::mat eigvec; arma::mat eigvec;
@ -96,13 +97,36 @@ void write_eigenvec(int N)
// Solve using Jacobi rotation method // Solve using Jacobi rotation method
jacobi_eigensolver(A, 10e-14, eigval, eigvec, 100000, iters, converged); jacobi_eigensolver(A, 10e-14, eigval, eigvec, 100000, iters, converged);
// Build analytic eigenvectors
arma::vec v, analytic_vec = arma::vec(N);
for (int i=0; i < N; i++) {
v = eigvec.col(i);
for (int j=0; j < N; j++) {
analytic_vec(j) = std::sin(((j+1.)*(i+1.)*M_PI) / (N+1.));
}
analytic_vec = arma::normalise(analytic_vec);
// Flip the sign of the analytic vector if they are different
if (analytic_vec(0)*v(0) < 0.) {
analytic_vec *= -1;
}
analytic.col(i) = analytic_vec;
}
std::ofstream ofile; std::ofstream ofile;
// Create file based on matrix size, and write header line to file // Create file based on matrix size, and write header line to file
ofile.open("../latex/output/eigenvector_" + std::to_string(N) + ".csv"); ofile.open("../latex/output/eigenvector_" + std::to_string(N) + ".csv");
ofile << "x,Vector 1,Vector 2,Vector 3" << std::endl; ofile << "x,"
<< "Vector 1,Vector 2,Vector 3,"
<< "Analytic 1,Analytic 2,Analytic 3" << std::endl;
// Add boundary value for x=0 // Add boundary value for x=0
ofile << scientific_format(0., 16) << "," ofile << scientific_format(0., 16) << ","
<< scientific_format(0., 16) << ","
<< scientific_format(0., 16) << ","
<< scientific_format(0., 16) << ","
<< scientific_format(0., 16) << "," << scientific_format(0., 16) << ","
<< scientific_format(0., 16) << "," << scientific_format(0., 16) << ","
<< scientific_format(0., 16) << std::endl; << scientific_format(0., 16) << std::endl;
@ -113,10 +137,16 @@ void write_eigenvec(int N)
ofile << scientific_format(x, 16)<< "," ofile << scientific_format(x, 16)<< ","
<< scientific_format(eigvec(i,0), 16) << "," << scientific_format(eigvec(i,0), 16) << ","
<< scientific_format(eigvec(i,1), 16) << "," << scientific_format(eigvec(i,1), 16) << ","
<< scientific_format(eigvec(i,2), 16) << std::endl; << scientific_format(eigvec(i,2), 16) << ","
<< scientific_format(analytic(i,0), 16) << ","
<< scientific_format(analytic(i,1), 16) << ","
<< scientific_format(analytic(i,2), 16) << std::endl;
} }
// Add boundary value for x=1 // Add boundary value for x=1
ofile << scientific_format(1., 16) << "," ofile << scientific_format(1., 16) << ","
<< scientific_format(0., 16) << ","
<< scientific_format(0., 16) << ","
<< scientific_format(0., 16) << ","
<< scientific_format(0., 16) << "," << scientific_format(0., 16) << ","
<< scientific_format(0., 16) << "," << scientific_format(0., 16) << ","
<< scientific_format(0., 16) << std::endl; << scientific_format(0., 16) << std::endl;
@ -126,9 +156,9 @@ void write_eigenvec(int N)
int main() int main()
{ {
write_transformation_tridiag(100); write_transformation_tridiag(100);
write_transformation_dense(100); write_transformation_dense(100);
write_eigenvec(6); write_eigenvec(10);
write_eigenvec(100); write_eigenvec(100);
return 0; return 0;
} }

19
src/plot.py
View File

@ -24,29 +24,28 @@ def plot_transformations(save: bool=False) -> None:
fig.savefig("../latex/images/transform.pdf") fig.savefig("../latex/images/transform.pdf")
def plot_eigenvectors(N: int, save: bool=False) -> None: def plot_eigenvectors(N: int, save: bool=False) -> None:
# Load data based on matrix size # Load data based on matrix size
path = f"../latex/output/eigenvector_{N}.csv" path = f"../latex/output/eigenvector_{N}.csv"
eigvec = pd.read_csv(path, header=0) eigvec = pd.read_csv(path, header=0)
fig, ax = plt.subplots() fig, ax = plt.subplots()
ax.plot(eigvec['x'], eigvec['Vector 1'], label='Vector 1') ax.plot(eigvec['x'], eigvec['Vector 1'], label=r'$\vec{v}_{1}$')
ax.plot(eigvec['x'], eigvec['Vector 2'], label='Vector 2') ax.plot(eigvec['x'], eigvec['Vector 2'], label=r'$\vec{v}_{2}$')
ax.plot(eigvec['x'], eigvec['Vector 3'], label='Vector 3') ax.plot(eigvec['x'], eigvec['Vector 3'], label=r'$\vec{v}_{3}$')
ax.plot(eigvec['x'], eigvec['Analytic 1'], '--', label=r'$\vec{v}^{(1)}$')
ax.plot(eigvec['x'], eigvec['Analytic 2'], '--', label=r'$\vec{v}^{(2)}$')
ax.plot(eigvec['x'], eigvec['Analytic 3'], '--', label=r'$\vec{v}^{(3)}$')
ax.set_xlabel(r'Element $\hat{x}_{i}$') ax.set_xlabel(r'Element $\hat{x}_{i}$')
ax.set_ylabel(r'Value of element $v_{i}$') ax.set_ylabel(r'Element $v_{i}$')
ax.legend() ax.legend(loc='upper left')
# Save to file # Save to file
if save is True: if save is True:
fig.savefig(f"../latex/images/eigenvector_{N}.pdf") fig.savefig(f"../latex/images/eigenvector_{N}.pdf")
if __name__ == '__main__': if __name__ == '__main__':
plot_transformations(True) plot_transformations(True)
plot_eigenvectors(6, True) plot_eigenvectors(10, True)
plot_eigenvectors(100, True) plot_eigenvectors(100, True)
# plt.show()