Update docs

Author: lorenzotomada

docs/Documentation.ipynb

@@ -754,8 +754,32 @@
     "\n",
     "Now we do the same for runtime vs matrix size and number of processes.\n",
     "\n",
-    "<!-- TODO: Add image -->\n",
-    "<!-- ![Time profiling](images/plot_time.png) -->"
+    "![Time profiling](images/plot_time.png)\n",
+    "\n",
+    "Based on this plot, we would be tempted to say that not only the execution time is much bigger that it is for `numpy` and `scipy`, but it might also seem that our method does not scale with respect to the number of processes.\n",
+    "However, running a single time the file `shell/time_profile.sh`, we notice that this is likely a problem related to how `time.time()` saves the results.\n",
+    "\n",
+    "Running, for instance,\n",
+    "```bash\n",
+    "$ shell/time_profile.sh 1 1500 # 1 refers to n_procs, 1000 to the matrix size\n",
+    "```\n",
+    "we get the following results:\n",
+    "```mermaid\n",
+    "Some results\n",
+    "```\n",
+    "Re-running with `n_procs=2`, we obtain\n",
+    "```mermaid\n",
+    "Even more results\n",
+    "```\n",
+    "Finally, for `n_procs=4`, we obtain\n",
+    "```mermaid\n",
+    "Final results\n",
+    "```\n",
+    "The previous results suggest that the method scales well with the number of processes, and that the performance (while worse than `numpy` and `scipy`) is such that the comparison goes much better than it seemed to do earlier.\n",
+    "We believe that the reason for such a behavior is related to the execution of multiple scripts, which can have an impact on execution times as measured with `time.time()`.\n",
+    "\n",
+    "Notice that we parallelized everything that could be parallelized (except for the secular solver, which usually takes no more than $5\\%$ of the total time): the bottleneck is given by the Lanczos method, which cannot be parallelized.\n",
+    "If the Lanczos method is not needed (that is, if the matrix $A$ of which we want to compute the eigenvalues and eigenvectors is already tridiagonal), then the execution time of our solver becomes comparable to the one of `numpy` and `scipy`."
    ]
   },
   {
@@ -817,6 +841,15 @@
     "        ), \"Error. The eigenvalues were not computed correctly.\""
    ]
   },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "# Final remarks\n",
+    "We hope this file, along with the `README.md`, provides sufficient information to understand the work done in our project.\n",
+    "If anything remains unclear, feel free to reach out: we’ll be happy to answer any questions you may have."
+   ]
+  },
   {
    "cell_type": "markdown",
    "metadata": {

shell/time_profile.sh

@@ -15,4 +15,4 @@ sed -i "s/^dim: .*/dim: $2/" $CONFIG_FILE
 mv "${CONFIG_FOLDER}/config.bak" "${CONFIG_FILE}"
 
 echo "Running with $1 process(es) with a matrix of size $2"
-mpirun -np "$1" python "$python_script"
+mpirun -np "$1" python "$python_script"