Refactor code for improved readability and consistency across multiple Jupyter notebooks

- Added missing commas in various print statements and function calls for better syntax.
- Reformatted code to enhance clarity, including breaking long lines and aligning parameters.
- Updated function signatures to use float type for sigma parameters instead of int for better precision.
- Cleaned up comments and documentation strings for clarity and consistency.
- Ensured consistent formatting in plotting functions and data handling.

M1/Numerical Optimisation/ComputerSession1.ipynb

@@ -682,7 +682,7 @@
     "        [\n",
     "            (F(x + delta * e(i, d)) - F(x - delta * e(i, d))) / (2 * delta)\n",
     "            for i in range(d)\n",
-    "        ]\n",
+    "        ],\n",
     "    )\n",
     "\n",
     "\n",

M1/Numerical Optimisation/ComputerSession2.ipynb

@@ -384,7 +384,7 @@
     "optimal_point_newton, iterations_newton = newton_method(initial_guess_newton)\n",
     "print(f\"Optimal point (Newton): {optimal_point_newton}\")\n",
     "print(\n",
-    "    f\"Objective function value at optimal point (Newton): {objective_function(optimal_point_newton)}\"\n",
+    "    f\"Objective function value at optimal point (Newton): {objective_function(optimal_point_newton)}\",\n",
     ")\n",
     "print(f\"Number of iterations (Newton): {iterations_newton}\")\n",
     "\n",
@@ -395,7 +395,7 @@
     "optimal_point_dichotomy, iterations_dichotomy = dichotomy_method(aL, aR)\n",
     "print(f\"Optimal point (Dichotomy): {optimal_point_dichotomy}\")\n",
     "print(\n",
-    "    f\"Objective function value at optimal point (Dichotomy): {objective_function(optimal_point_dichotomy)}\"\n",
+    "    f\"Objective function value at optimal point (Dichotomy): {objective_function(optimal_point_dichotomy)}\",\n",
     ")\n",
     "print(f\"Number of iterations (Dichotomy): {iterations_dichotomy}\")"
    ]

M1/Numerical Optimisation/ComputerSession3.ipynb

@@ -46,7 +46,7 @@
     "def generate_thetas(n):\n",
     "    random_steps = np.random.random(n)\n",
     "    return np.concatenate(\n",
-    "        ([0], np.cumsum(random_steps / np.sum(random_steps) * (2 * np.pi)))\n",
+    "        ([0], np.cumsum(random_steps / np.sum(random_steps) * (2 * np.pi))),\n",
     "    )\n",
     "\n",
     "\n",