Add ex 13 solution

exercises/ex13/solutions/competition/TAs.py

+p=print
+a=b=''
+i=26
+while i:l=chr(123-i);a+=l;b+=l.upper()+l;i-=1
+c='-'*23
+p(b[::2]+a);p(b);p(c)
+while b:p('| %-20s|'%' '.join(b[:10]));b=b[10:]
+p(c)
+d='prelpamilpoveene'
+for e in a:
+ if e in'aglo':x=e+d[i::4];b+=', %s begins with '%x+e;e=e+': ',x;i+=1
+ p([*e])
+p(b[2:])
\ No newline at end of file

exercises/ex13/solutions/harmonic_chain/.gitignore

+results_py.*

exercises/ex13/solutions/harmonic_chain/harmonic_chain.py

+#!/usr/bin/env python
+# -*- coding: utf-8 -*-
+#
+# Programming Techniques for Scientific Simulations, HS 2020
+# Exercise 13.1
+
+from __future__ import (absolute_import, division, print_function,
+                        unicode_literals)
+
+import h5py
+
+import numpy as np
+
+
+def main(K=1.0, m=1.0, N=4):
+    # Set up H (only upper triangle).
+    H = 2 * np.eye(N) - np.eye(N, k=1)
+    # Compute eigenvalues and eigenvectors.
+    eigval, eigvec = np.linalg.eigh(H, UPLO='U')
+    print("EW ", eigval)
+    # Compute frequencies from eigenvalues.
+    omega = np.sqrt(K / m * eigval)
+    # Exact frequencies.
+    exact_omega = np.sqrt(
+        K / m * (2 - 2 * np.cos(
+            np.pi * np.arange(1, N + 1) / (N + 1)
+            ))
+        )
+
+    # Print a preview of the results to screen.
+    eigvec = np.transpose(eigvec)  # We want to print eigenvectors on rows.
+    np.set_printoptions(threshold=100)
+    print(
+        "\nHamiltonian (upper triangle):", H,
+        "\nEigenfrequencies omega:", omega,
+        "\nEigenvectors:", eigvec,
+        sep='\n',
+        end='\n\n'
+        )
+
+    # Print the full results to a file.
+    np.savetxt(
+        "results_py.data",
+        np.c_[exact_omega, omega, eigvec],
+        header="exact_omega  omega  eigenvector",
+        fmt=str('%.15g')
+        )
+
+    # Save in HDF5 format.
+    with h5py.File('results_py.h5', 'w') as f:
+        f['eigenvectors'] = eigvec
+        f['omega'] = omega
+        f['exact_omega'] = exact_omega
+
+
+if __name__ == "__main__":
+    main()

exercises/ex13/solutions/plot/.gitignore

+*.pdf

exercises/ex13/solutions/plot/plot.py

+#!/usr/bin/env python
+# -*- coding: utf-8 -*-
+#
+# Programming Techniques for Scientific Simulations, HS 2020
+# Exercise 13.2
+
+from __future__ import (division, print_function)
+
+import numpy as np
+import scipy.stats as st
+import scipy.linalg as la
+import matplotlib.pyplot as plt
+
+
+def get_samples(n=500):
+    return np.random.rand(n, 2) * 2 - 1
+
+
+def get_r(samples):
+    return la.norm(samples, axis=1)
+
+
+def calculate_pi(samples):
+    num_inside = np.sum(get_r(samples) < 1.)
+    num_total = len(samples)
+    return 4 * num_inside / num_total
+
+
+def create_scatter_plot(samples):
+    fig, ax = plt.subplots()
+    ax.set_aspect('equal')
+    ax.set_xticks([-1, 0, 1])
+    ax.set_yticks([-1, 0, 1])
+    ax.set_xlabel(r'$x$')
+    ax.set_ylabel(r'$y$', rotation='horizontal')
+    ax.set_title('Points used to calculate Pi')
+
+    x, y = samples.T
+    ax.scatter(x, y, c=get_r(samples) < 1., cmap='bwr')
+
+    phi = np.linspace(0, 2 * np.pi, 200)
+    ax.plot(np.cos(phi), np.sin(phi), 'k')
+
+    fig.savefig('scatter_plot.pdf', bbox_inches='tight')
+
+
+def create_contour_plot(samples):
+    density_func = st.gaussian_kde(samples.T, bw_method=0.2)
+    X, Y = np.mgrid[-1.1:1.1:200j, -1.1:1.1:200j]
+    positions = np.stack([X.flat, Y.flat])
+    Z = np.reshape(density_func(positions), X.shape)
+
+    fig, ax = plt.subplots()
+    ax.set_aspect('equal')
+
+    ax.set_xticks([-1, 0, 1])
+    ax.set_yticks([-1, 0, 1])
+    ax.set_xlabel(r'$x$')
+    ax.set_ylabel(r'$y$', rotation='horizontal')
+    ax.set_title('Interpolated Point Density')
+
+    heatmap = ax.contourf(X, Y, Z)
+    ax.contour(X, Y, Z, linewidths=0.3, colors='k')
+    fig.colorbar(heatmap)
+
+    fig.savefig('contour_plot.pdf', bbox_inches='tight')
+
+
+def main():
+    np.random.seed(42)
+    samples = get_samples()
+    pi_estimate = calculate_pi(samples)
+    print('Pi ~= {}'.format(pi_estimate))
+
+    create_scatter_plot(samples)
+    create_contour_plot(samples)
+
+
+if __name__ == '__main__':
+    main()