From 53bf5d8739f2fed6b20fce92afdf9046098e3e12 Mon Sep 17 00:00:00 2001 From: "Ryan C. Cooper" Date: Wed, 21 Feb 2018 15:01:06 -0500 Subject: [PATCH] added intro to matlab ode-solvers --- Matlab refresher.ipynb | 34 + Untitled.ipynb | 6 + intro_to_numerical_integration.ipynb | 1392 ++++++++++++++++++++++++++ octave-workspace | Bin 629036 -> 153 bytes 4 files changed, 1432 insertions(+) create mode 100644 Matlab refresher.ipynb create mode 100644 Untitled.ipynb create mode 100644 intro_to_numerical_integration.ipynb diff --git a/Matlab refresher.ipynb b/Matlab refresher.ipynb new file mode 100644 index 0000000..8cefa3e --- /dev/null +++ b/Matlab refresher.ipynb @@ -0,0 +1,34 @@ +{ + "cells": [ + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "collapsed": true + }, + "outputs": [], + "source": [] + } + ], + "metadata": { + "kernelspec": { + "display_name": "Octave", + "language": "octave", + "name": "octave" + }, + "language_info": { + "file_extension": ".m", + "help_links": [ + { + "text": "MetaKernel Magics", + "url": "https://github.com/calysto/metakernel/blob/master/metakernel/magics/README.md" + } + ], + "mimetype": "text/x-octave", + "name": "octave", + "version": "0.19.14" + } + }, + "nbformat": 4, + "nbformat_minor": 2 +} diff --git a/Untitled.ipynb b/Untitled.ipynb new file mode 100644 index 0000000..2fd6442 --- /dev/null +++ b/Untitled.ipynb @@ -0,0 +1,6 @@ +{ + "cells": [], + "metadata": {}, + "nbformat": 4, + "nbformat_minor": 2 +} diff --git a/intro_to_numerical_integration.ipynb b/intro_to_numerical_integration.ipynb new file mode 100644 index 0000000..41b8e77 --- /dev/null +++ b/intro_to_numerical_integration.ipynb @@ -0,0 +1,1392 @@ +{ + "cells": [ + { + "cell_type": "code", + "execution_count": 9, + "metadata": { + "collapsed": true, + "slideshow": { + "slide_type": "skip" + } + }, + "outputs": [], + "source": [ + "setdefaults" + ] + }, + { + "cell_type": "code", + "execution_count": 33, + "metadata": { + "collapsed": true, + "slideshow": { + "slide_type": "skip" + } + }, + "outputs": [], + "source": [ + "%plot --format svg" + ] + }, + { + "cell_type": "code", + "execution_count": 34, + "metadata": { + "collapsed": true, + "slideshow": { + "slide_type": "skip" + } + }, + "outputs": [], + "source": [ + "pkg load odepkg" + ] + }, + { + "cell_type": "markdown", + "metadata": { + "slideshow": { + "slide_type": "slide" + } + }, + "source": [ + "## Intro to functions and m-files" + ] + }, + { + "cell_type": "markdown", + "metadata": { + "slideshow": { + "slide_type": "slide" + } + }, + "source": [ + "## Building our own functions." + ] + }, + { + "cell_type": "markdown", + "metadata": { + "collapsed": true, + "slideshow": { + "slide_type": "fragment" + } + }, + "source": [ + "Functions are saved in memory (or better yet) in a folder in your path or current directory\n", + "\n", + "Example of storing a function in memory\n", + "\n", + "$f(x,y) = (xy^{3}-x^{3}y)$" + ] + }, + { + "cell_type": "code", + "execution_count": 11, + "metadata": { + "collapsed": false, + "slideshow": { + "slide_type": "fragment" + } + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "f =\n", + "\n", + "@(x, y) (x .* y .^ 3 - x .^ 3 .* y)\n", + "\n", + "ans = 6\n" + ] + } + ], + "source": [ + "f= @(x,y) (x.*y.^3-x.^3.*y)\n", + "f(1,2)" + ] + }, + { + "cell_type": "code", + "execution_count": 12, + "metadata": { + "collapsed": false, + "slideshow": { + "slide_type": "subslide" + } + }, + "outputs": [ + { + "data": { + "image/svg+xml": [ + "\n", + "\n", + "Gnuplot\n", + "Produced by GNUPLOT 5.0 patchlevel 3 \n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t \n", + "\t \n", + "\t\n", + "\t\n", + "\t \n", + "\t \n", + "\t\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\tgnuplot_plot_1a\n", + "\n", + "\n", + "\n", + "\t\n", + "\n", + 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"\t\t\n", + "\t\t1\n", + "\t\n", + "\n", + "\n", + "\n", + "\n", + "\t\n", + "\n", + "\t\n", + "\n", + "\t\n", + "\n", + "\t\t\n", + "\t\t-0.4\n", + "\t\n", + "\n", + "\n", + "\n", + "\n", + "\t\n", + "\n", + "\t\n", + "\n", + "\t\n", + "\n", + "\t\n", + "\n", + "\t\t\n", + "\t\t-0.3\n", + "\t\n", + "\n", + "\n", + "\n", + "\n", + "\t\n", + "\n", + "\t\n", + "\n", + "\t\n", + "\n", + "\t\t\n", + "\t\t-0.2\n", + "\t\n", + "\n", + "\n", + "\n", + "\n", + "\t\n", + "\n", + "\t\n", + "\n", + "\t\n", + "\n", + "\t\n", + "\n", + "\t\n", + "\n", + "\t\t\n", + "\t\t-0.1\n", + "\t\n", + "\n", + "\n", + "\n", + "\n", + "\t\n", + "\n", + "\t\n", + "\n", + "\t\n", + "\n", + "\t\t\n", + "\t\t0\n", + "\t\n", + "\n", + "\n", + "\n", + "\n", + "\t\n", + "\n", + "\t\n", + "\n", + "\t\n", + "\n", + "\t\n", + "\n", + "\t\n", + "\n", + "\t\n", + "\n", + "\t\t\n", + "\t\t0.1\n", + "\t\n", + "\n", + "\n", + "\n", + "\n", + "\t\n", + "\n", + "\t\n", + "\n", + "\t\n", + "\n", + "\t\n", + "\n", + "\t\n", + "\n", + "\t\n", + "\n", + "\t\t\n", + "\t\t0.2\n", + "\t\n", + "\n", + "\n", + "\n", + "\n", + "\t\n", + "\n", + "\t\n", + "\n", + "\t\n", + "\n", + "\t\n", + "\n", + "\t\n", + "\n", + "\t\t\n", + "\t\t0.3\n", + "\t\n", + "\n", + "\n", + "\n", + "\n", + "\t\n", + "\n", + "\t\n", + "\n", + "\t\n", + "\n", + "\t\t\n", + "\t\t0.4\n", + "\t\n", + "\n", + "\n", + "\n", + "\n", + "\t\n", + "\n", + "" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "x=linspace(-1,1,10); % 10 data points from -1 to 1\n", + "y=linspace(-1,1,10); % 10 data points from -1 to 1\n", + "[X,Y]=meshgrid(x,y); % 100 data points from -1 to 1 on the x- and y-axes\n", + "Z=f(X,Y); % Z=f(X,Y) evaluated at the 100 data points\n", + "\n", + "mesh(X,Y,Z)" + ] + }, + { + "cell_type": "markdown", + "metadata": { + "slideshow": { + "slide_type": "slide" + } + }, + "source": [ + "Here we will save a function called `my_ode` as `my_ode.m` For a single-DOF forced spring-mass-damper:\n", + "\n", + "$\\ddot{x}+2\\dot{x}+9x=\\cos(t/2)$" + ] + }, + { + "cell_type": "code", + "execution_count": 38, + "metadata": { + "collapsed": true, + "slideshow": { + "slide_type": "fragment" + } + }, + "outputs": [], + "source": [ + "function dr = my_ode(t,r)\n", + " % Help documentation for \"my_ode.m\"\n", + " % input is time, t (s) and r=[position (m); velocity (m/s)] and time\n", + " % output is dr=[velocity (m/s); acceleration (m/s/s)] at time, t\n", + " % the ODE is defined by:\n", + " %\n", + " % a = -2*v-9*x+cos(t/2)\n", + " \n", + " dr=zeros(size(r));\n", + " dr(1)=r(2);\n", + " dr(2)=-2*r(2)-9*r(1)+cos(2*t);\n", + "\n", + "end" + ] + }, + { + "cell_type": "code", + "execution_count": 39, + "metadata": { + "collapsed": false, + "slideshow": { + "slide_type": "subslide" + } + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "'my_ode' is a command-line function\n", + "\n", + " Help documentation for \"my_ode.m\"\n", + " input is time, t (s) and r=[position (m); velocity (m/s)] and time\n", + " output is dr=[velocity (m/s); acceleration (m/s/s)] at time, t\n", + " the ODE is defined by:\n", + "\n", + " a = -2*v-9*x+cos(t/2)\n", + "\n", + "\n", + "Additional help for built-in functions and operators is\n", + "available in the online version of the manual. Use the command\n", + "'doc ' to search the manual index.\n", + "\n", + "Help and information about Octave is also available on the WWW\n", + "at http://www.octave.org and via the help@octave.org\n", + "mailing list.\n" + ] + } + ], + "source": [ + "help my_ode" + ] + }, + { + "cell_type": "markdown", + "metadata": { + "slideshow": { + "slide_type": "slide" + } + }, + "source": [ + "## Euler approximation\n", + "\n", + "$r_{i+1}=r_{i}+\\frac{dr_{i}}{dt}\\Delta t$" + ] + }, + { + "cell_type": "code", + "execution_count": 40, + "metadata": { + "collapsed": false, + "slideshow": { + "slide_type": "fragment" + } + }, + "outputs": [], + "source": [ + "dt=0.1;\n", + "t=[0:dt:6*pi/2];\n", + "x0=0.1;\n", + "v0=0;\n", + "r=zeros(length(t),2);\n", + "r(1,:)=[x0;v0];\n", + "for i=2:length(t)\n", + " dr=my_ode(t(i-1),r(i-1,:));\n", + " r(i,:)=r(i-1,:)+dr*dt;\n", + "end\n", + " " + ] + }, + { + "cell_type": "code", + "execution_count": 41, + "metadata": { + "collapsed": false, + "slideshow": { + "slide_type": "subslide" + } + }, + "outputs": [ + { + "data": { + "image/svg+xml": [ + "\n", + "\n", + "Gnuplot\n", + "Produced by GNUPLOT 5.0 patchlevel 3 \n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t \n", + "\t \n", + "\t\n", + "\t\n", + "\t \n", + "\t \n", + "\t\n", + "\n", + "\n", + "\n", + "\n", + "\t\n", + "\t\t\n", + "\t\n", + "\n", + "\n", + "\n", + "\n", + "\t\t\n", + "\t\t-0.2\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t-0.15\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t-0.1\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t-0.05\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t0\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t0.05\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t0.1\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t0.15\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t0.2\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t0\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t2\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t4\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t6\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t8\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t10\n", + "\t\n", + "\n", + "\n", + "\n", + "\n", + "\t\n", + "\n", + "\t\n", + "\t\tposition (m)\n", + "\t\n", + "\n", + "\n", + "\t\n", + "\t\ttime (s)\n", + "\t\n", + "\n", + "\n", + "\n", + "\tgnuplot_plot_1a\n", + "\n", + "\n", + "\n", + "\t\n", + "\t\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "plot(t,r(:,1))\n", + "xlabel('time (s)')\n", + "ylabel('position (m)')" + ] + }, + { + "cell_type": "markdown", + "metadata": { + "collapsed": false, + "slideshow": { + "slide_type": "subslide" + } + }, + "source": [ + "## Runge-Kutta\n", + "\n", + "\n", + "Using `ode23` and `ode45` have the same syntax\n", + "\n", + "Numerical integration algorithms for differential equations:" + ] + }, + { + "cell_type": "code", + "execution_count": 49, + "metadata": { + "collapsed": false, + "slideshow": { + "slide_type": "subslide" + } + }, + "outputs": [ + { + "data": { + "image/svg+xml": [ + "\n", + "\n", + "Gnuplot\n", + "Produced by GNUPLOT 5.0 patchlevel 3 \n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t \n", + "\t \n", + "\t\n", + "\t\n", + "\t \n", + "\t \n", + "\t\n", + "\n", + "\n", + "\n", + "\n", + "\t\n", + "\t\t\n", + "\t\n", + "\n", + "\n", + "\n", + "\n", + "\t\t\n", + "\t\t-0.2\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t-0.1\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t0\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + 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