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diff --git a/tutorials/module_3/notebook_3/numerical_differentiation.ipynb b/tutorials/module_3/notebook_3/numerical_differentiation.ipynb new file mode 100644 index 0000000..0892554 --- /dev/null +++ b/tutorials/module_3/notebook_3/numerical_differentiation.ipynb @@ -0,0 +1,140 @@ +{ + "cells": [ + { + "cell_type": "markdown", + "id": "d66e2b4b-047b-4670-897d-7877d27a1597", + "metadata": {}, + "source": [ + "# Numerical Differentiation\n", + "\n", + "Finding a derivative of tabular data can be done using a finite\n", + "difference. Here we essentially pick two points on a function or a set\n", + "of data points and calculate the slope from there. Let’s imagine a\n", + "domain $x$ as a vector such that $\\vec{x}$ =\n", + "$\\pmatrix{x_0, x_1, x_2, ...}$. Then we can use the following methods to\n", + "approximate derivatives\n", + "\n", + "## Forward Difference\n", + "\n", + "Uses the point at which we want to find the derivative and a point\n", + "forwards on the line. $$\n", + "f'(x_i) = \\frac{f(x_{i+1})-f(x_i)}{x_{i+1}-x_i}\n", + "$$ *Hint: Consider what happens at the last point.*" + ] + }, + { + "cell_type": "code", + "execution_count": 2, + "id": "d2fe3ad1-8af0-499a-8954-41a88a49b834", + "metadata": {}, + "outputs": [ + { + "ename": "IndexError", + "evalue": "index 99 is out of bounds for axis 0 with size 99", + "output_type": "error", + "traceback": [ + "\u001b[0;31m---------------------------------------------------------------------------\u001b[0m", + "\u001b[0;31mIndexError\u001b[0m Traceback (most recent call last)", + "Cell \u001b[0;32mIn[2], line 10\u001b[0m\n\u001b[1;32m 6\u001b[0m y \u001b[38;5;241m=\u001b[39m \u001b[38;5;241m34\u001b[39m \u001b[38;5;241m*\u001b[39m np\u001b[38;5;241m.\u001b[39mexp(\u001b[38;5;241m3\u001b[39m \u001b[38;5;241m*\u001b[39m x)\n\u001b[1;32m 8\u001b[0m dydx \u001b[38;5;241m=\u001b[39m (y[\u001b[38;5;241m1\u001b[39m:] \u001b[38;5;241m-\u001b[39m y[:\u001b[38;5;241m-\u001b[39m\u001b[38;5;241m1\u001b[39m]) \u001b[38;5;241m/\u001b[39m (x[\u001b[38;5;241m1\u001b[39m:] \u001b[38;5;241m-\u001b[39m x[:\u001b[38;5;241m-\u001b[39m\u001b[38;5;241m1\u001b[39m])\n\u001b[0;32m---> 10\u001b[0m dydx[\u001b[38;5;28mlen\u001b[39m(y)]\u001b[38;5;241m=\u001b[39mdydx[\u001b[38;5;28mlen\u001b[39m(dydx)]\n\u001b[1;32m 12\u001b[0m \u001b[38;5;66;03m# Plot the function\u001b[39;00m\n\u001b[1;32m 13\u001b[0m plt\u001b[38;5;241m.\u001b[39mplot(x, y, label\u001b[38;5;241m=\u001b[39m\u001b[38;5;124mr\u001b[39m\u001b[38;5;124m'\u001b[39m\u001b[38;5;124m$y(x)$\u001b[39m\u001b[38;5;124m'\u001b[39m)\n", + "\u001b[0;31mIndexError\u001b[0m: index 99 is out of bounds for axis 0 with size 99" + ] + } + ], + "source": [ + "import numpy as np\n", + "import matplotlib.pyplot as plt\n", + "\n", + "# Initiate vectors\n", + "x = np.linspace(0, 2, 100) \n", + "y = 34 * np.exp(3 * x)\n", + "\n", + "dydx = (y[1:] - y[:-1]) / (x[1:] - x[:-1])\n", + "\n", + "dydx[len(y)]=dydx[len(dydx)]\n", + "\n", + "# Plot the function\n", + "plt.plot(x, y, label=r'$y(x)$')\n", + "plt.plot(x, dydx, label=r'$\\frac{dy}{dx}$')\n", + "plt.xlabel('x')\n", + "plt.ylabel('y')\n", + "plt.title('Plot of $34e^{3x}$')\n", + "plt.grid(True)\n", + "plt.legend()\n", + "plt.show()" + ] + }, + { + "cell_type": "markdown", + "id": "a54ad04f-05e5-4589-a633-d6dea2a5acfc", + "metadata": {}, + "source": [ + "## Backwards Difference\n", + "\n", + "Uses the point at which we want to find $$\n", + "f'(x_i) = \\frac{f(x_{i})-f(x_{i-1})}{x_i - x_{i-1}}\n", + "$$" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "415d0b3c-a70d-48ef-93ad-6e827127b222", + "metadata": {}, + "outputs": [], + "source": [ + "dydx = (y[1:] - y[:-1]) / (x[1:] - x[:-1])\n", + "\n", + "# Plot the function\n", + "plt.plot(x, y, label=r'$y(x)$')\n", + "plt.plot(x, dydx, label=b'$/frac{dy}{dx}$')\n", + "plt.xlabel('x')\n", + "plt.ylabel('y')\n", + "plt.title('Plot of $34e^{3x}$')\n", + "plt.grid(True)\n", + "plt.legend()\n", + "plt.show()" + ] + }, + { + "cell_type": "markdown", + "id": "21c87595-add2-4f7a-8d10-2da3b4284e02", + "metadata": {}, + "source": [ + "## Central Difference\n", + "\n", + "$$\n", + "f'(x_i) = \\frac{f(x_{i+1})-f(x_{i-1})}{x_{i+1}-x_{i-1}}\n", + "$$" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "6a67e63c-9516-415d-87d6-97d9d986ba33", + "metadata": {}, + "outputs": [], + "source": [] + } + ], + "metadata": { + "kernelspec": { + "display_name": "Python 3 (ipykernel)", + "language": "python", + "name": "python3" + }, + "language_info": { + "codemirror_mode": { + "name": "ipython", + "version": 3 + }, + "file_extension": ".py", + "mimetype": "text/x-python", + "name": "python", + "nbconvert_exporter": "python", + "pygments_lexer": "ipython3", + "version": "3.13.2" + } + }, + "nbformat": 4, + "nbformat_minor": 5 +} diff --git a/tutorials/module_3/notebook_3/numerical_integration.ipynb b/tutorials/module_3/notebook_3/numerical_integration.ipynb new file mode 100644 index 0000000..1638ad5 --- /dev/null +++ b/tutorials/module_3/notebook_3/numerical_integration.ipynb @@ -0,0 +1,27 @@ +{ + "cells": [ + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Midpoint Method\n", + "\n", + "## Trapezoidal Method\n", + "\n", + "## Romberg Integration\n", + "\n", + "## Gaussian Integration\n", + "\n", + "## Simpson’s Rule\n", + "\n", + "### Simpsons 1/3\n", + "\n", + "### Simpsons 3/8" + ], + "id": "81ed366c-8dc9-4d6c-9a5e-36a28f7811b5" + } + ], + "nbformat": 4, + "nbformat_minor": 5, + "metadata": {} +} diff --git a/tutorials/module_3/notebook_3/numerical_methods.ipynb b/tutorials/module_3/notebook_3/numerical_methods.ipynb new file mode 100644 index 0000000..e5bf7df --- /dev/null +++ b/tutorials/module_3/notebook_3/numerical_methods.ipynb @@ -0,0 +1,52 @@ +{ + "cells": [ + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# Numerical Methods\n", + "\n", + "Engineering\n", + "\n", + "## What is a numerical method?\n", + "\n", + "Numerical methods are techniques that transform mathematical problems\n", + "into forms that can be solved using arithmetic and logical operations.\n", + "Because digital computers excel at these computations, numerical methods\n", + "are often referred to as computer mathematics.\n", + "\n", + "## Numerical Differentiation\n", + "\n", + "Forwards difference Backwards difference Central Difference method [Read\n", + "More](https://pythonnumericalmethods.studentorg.berkeley.edu/notebooks/chapter20.00-Numerical-Differentiation.html)\n", + "\n", + "## Roots and Optimization\n", + "\n", + "Incremental Search Bisection Modified Secant Newton-Raphson\n", + "\n", + "## System of Equations\n", + "\n", + "Guassian Method LU Decomposition\n", + "\n", + "## Numerical Integration\n", + "\n", + "Midpoint Trapezoidal Romberg Gaussian Simpson’s Rule\n", + "\n", + "[Read\n", + "More](https://pythonnumericalmethods.studentorg.berkeley.edu/notebooks/chapter21.00-Numerical-Integration.html)\n", + "\n", + "## Numerical Solutions of Ordinary Differential Equations\n", + "\n", + "Euler’s Method - Forward - Backwards\n", + "\n", + "Runge-Kutte\n", + "\n", + "[ReadMore](https://pythonnumericalmethods.studentorg.berkeley.edu/notebooks/chapter22.00-ODE-Initial-Value-Problems.html)" + ], + "id": "508fb71a-8f30-4cb3-88f8-28b45a334596" + } + ], + "nbformat": 4, + "nbformat_minor": 5, + "metadata": {} +} diff --git a/tutorials/module_3/notebook_3/ode.ipynb b/tutorials/module_3/notebook_3/ode.ipynb new file mode 100644 index 0000000..a343f4f --- /dev/null +++ b/tutorials/module_3/notebook_3/ode.ipynb @@ -0,0 +1,23 @@ +{ + "cells": [ + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# Numerical Solutions of Ordinary Differential Equations\n", + "\n", + "## Euler’s Method\n", + "\n", + "### Forwards Eulers\n", + "\n", + "### Backwards Eulers\n", + "\n", + "## Runge-Kutta" + ], + "id": "4c1220ef-f6ce-43a7-9017-6c22fc8fd4af" + } + ], + "nbformat": 4, + "nbformat_minor": 5, + "metadata": {} +} diff --git a/tutorials/module_3/notebook_3/roots_optimization.ipynb b/tutorials/module_3/notebook_3/roots_optimization.ipynb new file mode 100644 index 0000000..3e764f2 --- /dev/null +++ b/tutorials/module_3/notebook_3/roots_optimization.ipynb @@ -0,0 +1,25 @@ +{ + "cells": [ + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# Root Finding Methods\n", + "\n", + "Root Finding Methods or non-linear equation solvers.\n", + "\n", + "## Incremental Search\n", + "\n", + "## Bisection\n", + "\n", + "## Modified Secant\n", + "\n", + "## Newton-Raphson" + ], + "id": "9483ba01-3002-4544-9286-e134b3d2538b" + } + ], + "nbformat": 4, + "nbformat_minor": 5, + "metadata": {} +} |