Merge pull request #11 from FYS3150-G2-2023/janitaws/finish-text-document
Janitaws/finish text document
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37
README.md
37
README.md
@ -6,39 +6,32 @@
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## Compile and run C++ programs
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Compiling and linking is done using Make, make sure you are in the root folder. To create executable files, run shell commands
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Compiling and linking is done using Make, make sure you are in the root directory.
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```shell
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cd src/
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make
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```
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To run `script-name`
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```shell
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./main
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```
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There are two ways of compiling the code.
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The first way is to go into the src directory and compile
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There are two alternative ways to compile the code. The first alternative is to change
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directory to `src/` and compile
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```shell
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cd src && make
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```
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or you could use make in the project directory
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The second alternative does not involve changing directory to `src/`, use
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make in the projects root directory
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```shell
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make code
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```
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and it will compile everything with you needing to move into **src**.
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You can run any of the compiled programs by going into the src directory
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and then doing
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You can run any of the compiled programs by changing directory to `src/`, then
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```shell
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./program-name
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./test_suite
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./main
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```
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Remove object files and executables from `src/`
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```shell
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make clean
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```
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@ -64,7 +57,7 @@ If you want to generate the documentation you can do
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make docs
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```
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in the project root, and the documentation will be made into the **docs**
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in the project root, and the documentation will be made into the `docs/`
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directory.
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## Generate project document
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@ -75,4 +68,4 @@ If you want to recompile the Pdf file, you can do
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make latex
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```
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and a Pdf file will be produced inside the **latex** directory.
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and a Pdf file will be produced inside the `latex/` directory.
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latex/images/eigenvector_10.pdf
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latex/images/eigenvector_10.pdf
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latex/output/eigenvector_10.csv
Normal file
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latex/output/eigenvector_10.csv
Normal file
@ -0,0 +1,13 @@
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|
||||
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+00,0.0000000000e+00,0.0000000000e+00,0.0000000000e+00
|
||||
|
||||
|
@ -12,5 +12,5 @@ Scaling will result in a dimensionless variable $\hat{x} = \frac{1}{L}$.
|
||||
\end{align*}
|
||||
Now we insert the expression into the original equation
|
||||
\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*}
|
||||
@ -2,7 +2,7 @@
|
||||
|
||||
\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}.
|
||||
|
||||
\subsection*{b)}
|
||||
|
||||
@ -1 +1,20 @@
|
||||
\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 $N \cross N$.
|
||||
@ -1 +1,21 @@
|
||||
\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
40
src/main.cpp
@ -87,6 +87,7 @@ void write_eigenvec(int N)
|
||||
|
||||
// Create tridiagonal matrix
|
||||
arma::mat A = create_symmetric_tridiagonal(N, a, d);
|
||||
arma::mat analytic = arma::mat(N, N);
|
||||
arma::vec eigval;
|
||||
arma::mat eigvec;
|
||||
|
||||
@ -96,13 +97,36 @@ void write_eigenvec(int N)
|
||||
// Solve using Jacobi rotation method
|
||||
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;
|
||||
// Create file based on matrix size, and write header line to file
|
||||
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
|
||||
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) << std::endl;
|
||||
@ -113,10 +137,16 @@ void write_eigenvec(int N)
|
||||
ofile << scientific_format(x, 16)<< ","
|
||||
<< scientific_format(eigvec(i,0), 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
|
||||
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) << std::endl;
|
||||
@ -126,9 +156,9 @@ void write_eigenvec(int N)
|
||||
|
||||
int main()
|
||||
{
|
||||
write_transformation_tridiag(100);
|
||||
write_transformation_dense(100);
|
||||
write_eigenvec(6);
|
||||
write_transformation_tridiag(100);
|
||||
write_transformation_dense(100);
|
||||
write_eigenvec(10);
|
||||
write_eigenvec(100);
|
||||
return 0;
|
||||
}
|
||||
|
||||
19
src/plot.py
19
src/plot.py
@ -24,29 +24,28 @@ def plot_transformations(save: bool=False) -> None:
|
||||
fig.savefig("../latex/images/transform.pdf")
|
||||
|
||||
|
||||
|
||||
def plot_eigenvectors(N: int, save: bool=False) -> None:
|
||||
# Load data based on matrix size
|
||||
path = f"../latex/output/eigenvector_{N}.csv"
|
||||
eigvec = pd.read_csv(path, header=0)
|
||||
|
||||
fig, ax = plt.subplots()
|
||||
ax.plot(eigvec['x'], eigvec['Vector 1'], label='Vector 1')
|
||||
ax.plot(eigvec['x'], eigvec['Vector 2'], label='Vector 2')
|
||||
ax.plot(eigvec['x'], eigvec['Vector 3'], label='Vector 3')
|
||||
ax.plot(eigvec['x'], eigvec['Vector 1'], label=r'$\vec{v}_{1}$')
|
||||
ax.plot(eigvec['x'], eigvec['Vector 2'], label=r'$\vec{v}_{2}$')
|
||||
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_ylabel(r'Value of element $v_{i}$')
|
||||
ax.legend()
|
||||
ax.set_ylabel(r'Element $v_{i}$')
|
||||
ax.legend(loc='upper left')
|
||||
|
||||
# Save to file
|
||||
if save is True:
|
||||
fig.savefig(f"../latex/images/eigenvector_{N}.pdf")
|
||||
|
||||
|
||||
|
||||
|
||||
if __name__ == '__main__':
|
||||
plot_transformations(True)
|
||||
plot_eigenvectors(6, True)
|
||||
plot_eigenvectors(10, True)
|
||||
plot_eigenvectors(100, True)
|
||||
# plt.show()
|
||||
Loading…
Reference in New Issue
Block a user