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‎00_intro_numerical_methods-extra-examples.ipynb

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{
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"cells": [
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{
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"cell_type": "markdown",
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"source": [
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"<table>\n",
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" <tr align=left><td><img align=left src=\"./images/CC-BY.png\">\n",
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" <td>Text provided under a Creative Commons Attribution license, CC-BY. All code is made available under the FSF-approved MIT license. (c) Kyle T. Mandli</td>\n",
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"</table>"
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{
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"cell_type": "code",
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"execution_count": null,
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"metadata": {
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}
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},
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"outputs": [],
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"source": [
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"from __future__ import print_function\n",
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"\n",
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"%matplotlib inline\n",
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"import numpy\n",
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"import matplotlib.pyplot as plt"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {
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}
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},
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"source": [
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"Note to lecturers: This notebook is designed to work best as a classic Jupyter Notebook with nbextensions \n",
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"* hide_input: to hide selected python cells particularly for just plotting\n",
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"* RISE: Interactive js slide presentations"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {
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"slideshow": {
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}
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},
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"source": [
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"## Why should I care?"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {
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"slideshow": {
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}
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},
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"source": [
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"*Because I have to be here for a requirement...*"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {
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"slideshow": {
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"slide_type": "skip"
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}
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},
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"source": [
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"*or I might actually need to solve something important...like can I ever retire?*"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {
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"slideshow": {
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}
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},
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"source": [
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"### An example: The Retirement Problem\n",
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"\n",
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"$$A = \\frac{P}{r} \\left((1+r)^n - 1 \\right)$$\n",
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"\n",
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"$A$ is the total amount after n payments\n",
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"\n",
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"$P$ is the incremental payment\n",
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"\n",
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"$r$ is the interest rate per payment period\n",
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"\n",
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"$n$ is the number of payments\n",
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"\n",
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"\n",
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"\n",
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"Note that these can all be functions of $r$, $n$, and time"
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]
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},
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{
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"cell_type": "markdown",
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"slide_type": "subslide"
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}
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},
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"source": [
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"What is the minimum interest rate $r$ required to accrue \\$1M over 20 years saving \\\\$1000 per month?\n",
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"\n",
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"$$\n",
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"A = \\frac{P}{r} \\left((1+r)^n - 1 \\right)\n",
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"$$"
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"metadata": {
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},
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"outputs": [],
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"source": [
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"def A(P, r, n):\n",
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" return P / r * ((1 + r)**n - 1)\n",
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"\n",
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"P = 1000.\n",
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"years = numpy.linspace(0,20,241)\n",
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"n = 12*years\n",
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"target = 1.e6"
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]
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{
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"outputs": [],
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"source": [
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"# find the root using scipy's brentq method\n",
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"from scipy.optimize import brentq\n",
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"n_max = n[-1]\n",
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"f = lambda r : target - A(P,r/12., n_max)\n",
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"r_target = brentq(f,0.1,0.15)"
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]
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"fig = plt.figure(figsize=(8,6))\n",
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"fig.set_figwidth(fig.get_figwidth() * 2)\n",
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"\n",
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"axes = fig.add_subplot(1, 2, 1)\n",
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"for r in [0.02, 0.05, 0.08, 0.1, 0.12, 0.15]:\n",
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" axes.plot(years, A(P, r/12., n),label='r = {}'.format(r))\n",
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"axes.plot(years, numpy.ones(years.shape) * target, 'k--',linewidth=2,label=\"Target\")\n",
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"axes.legend(loc='best')\n",
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"axes.set_xlabel(\"Years\",fontsize=18)\n",
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"axes.set_ylabel(\"Annuity Value (Dollars)\",fontsize=18)\n",
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"axes.set_title(\"The Forward Problem $A(P,r,n)$\",fontsize=20)\n",
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"axes.grid()\n",
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"\n",
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"axes = fig.add_subplot(1, 2, 2)\n",
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"r = numpy.linspace(1.e-6,0.15,100)\n",
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"target = 1.e6\n",
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"axes.plot(A(P,r/12.,n_max),r,linewidth=2)\n",
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"axes.plot(numpy.ones(r.shape) * target, r,'k--',linewidth=2)\n",
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"axes.scatter(A(P,r_target/12,n_max),r_target,s=100,c='r')\n",
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"axes.set_xlabel('Annuity Value (Dollars)',fontsize=18)\n",
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"axes.set_title('The Inverse Problem $r(A,P,n)$, $r\\geq$ {:.3f}'.format(r_target),fontsize=20)\n",
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"axes.grid()\n",
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"plt.show()"
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]
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},
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{
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"cell_type": "markdown",
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"slide_type": "fragment"
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}
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},
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"source": [
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"This is a rootfinding problem for a function of a single variable"
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]
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},
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{
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"source": [
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"### Another Example: Boat race (numerical quadrature of arc length)\n",
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"Given a river with coordinates $(x,f(x))$ find the total length actually rowed over a given interval $x\\in [0,X]$\n",
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"\n",
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"Example: a sinusoidal river $$f(x) = A \\sin x$$"
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]
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},
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{
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}
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},
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"outputs": [],
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"source": [
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"A=.5\n",
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"fig = plt.figure(figsize=(8,6))\n",
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"x = numpy.linspace(0,10,100)\n",
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"plt.plot(x,A*numpy.sin(x),'r',linewidth=2,)\n",
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"plt.xlabel('distance (km)')\n",
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"plt.grid()\n",
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"plt.title('The Boat Race: $f(x)=A\\sin(x)')\n",
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"plt.show()"
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]
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},
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{
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"cell_type": "markdown",
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"source": [
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"We need to calculate the function $f(x)$'s arc-length from $[0, X]$ e.g.\n",
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"\n",
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"\\begin{align}\n",
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" L(X) &= \\int_0^{X} \\sqrt{1 + |f'(x)|^2} dx\\\\\n",
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" &= \\int_0^{X} \\sqrt{1 + A^2\\cos^2(x)} dx\\\\\n",
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"\\end{align}\n",
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"\n",
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"In general the value of this integral cannot be solved analytically (the answer is actually an elliptic integral). We usually need to approximate the integral using a numerical quadrature."
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]
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"source": [
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"## Why is this exciting?\n",
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"\n",
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"Computers have revolutionized our ability to explore...\n",
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"\n",
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"[Top 10 Algorithms of the 20th Century](http://www.siam.org/pdf/news/637.pdf?t=1&cn=ZmxleGlibGVfcmVjcw%3D%3D&refsrc=email&iid=658bdab6af614c83a8df1f8e02035eae&uid=755271476&nid=244+285282312)\n",
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"\n",
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"...and there is more to come!"
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]
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