From 598519d94dca50d2e7d576009741145143fed347 Mon Sep 17 00:00:00 2001 From: "Ryan C. Cooper" Date: Tue, 4 Apr 2017 12:56:30 -0400 Subject: [PATCH 1/2] added hw6 --- HW6/.~lock.Primary_Energy_monthly.csv# | 1 + HW6/README.md | 86 +++++ HW6/README.pdf | Bin 0 -> 168643 bytes HW6/boussinesq_lookup.m | 22 ++ HW6/cost_logistic.m | 27 ++ HW6/problem_1_data.m | 15 + lecture_18/lecture_18.ipynb | 90 +++--- lecture_18/octave-workspace | Bin 7470 -> 2001 bytes lecture_19/lecture 19.ipynb | 426 ++++++++++++++++++++++--- lecture_19/octave-workspace | Bin 0 -> 10815 bytes 10 files changed, 588 insertions(+), 79 deletions(-) create mode 100644 HW6/.~lock.Primary_Energy_monthly.csv# create mode 100644 HW6/README.md create mode 100644 HW6/README.pdf create mode 100644 HW6/boussinesq_lookup.m create mode 100644 HW6/cost_logistic.m create mode 100644 HW6/problem_1_data.m create mode 100644 lecture_19/octave-workspace diff --git a/HW6/.~lock.Primary_Energy_monthly.csv# b/HW6/.~lock.Primary_Energy_monthly.csv# new file mode 100644 index 0000000..eef216f --- /dev/null +++ b/HW6/.~lock.Primary_Energy_monthly.csv# @@ -0,0 +1 @@ +,ryan,fermi,31.03.2017 16:47,file:///home/ryan/.config/libreoffice/4; \ No newline at end of file diff --git a/HW6/README.md b/HW6/README.md new file mode 100644 index 0000000..1f3c86c --- /dev/null +++ b/HW6/README.md @@ -0,0 +1,86 @@ +# Homework #6 +## due 4/14 by 11:59pm + + +0. Create a new github repository called 'curve_fitting'. + + a. Add rcc02007 and pez16103 as collaborators. + + b. Clone the repository to your computer. + + +1. Create a least-squares function called `least_squares.m` that accepts a Z-matrix and +dependent variable y as input and returns the vector of best-fit constants, a, the +best-fit function evaluated at each point $f(x_{i})$, and the coefficient of +determination, r2. + +```matlab +[a,fx,r2]=least_squares(Z,y); +``` + + Test your function on the sets of data in script `problem_1_data.m` and show that the + following functions are the best fit lines: + + a. y=0.3745+0.98644x+0.84564/x + + b. y=-11.4887+7.143817x-1.04121 x^2+0.046676 x^3 + + c. y=4.0046e^(-1.5x)+2.9213e^(-0.3x)+1.5647e^(-0.05x) + + +2. Use the Temperature and failure data from the Challenger O-rings in lecture_18 +(challenger_oring.csv). Your independent variable is temerature and your dependent +variable is failure (1=fail, 0=pass). Create a function called `cost_logistic.m` that +takes the vector `a`, and independent variable `x` and dependent variable `y`. Use the +function, $\sigma(t)=\frac{1}{1+e^{-t}}$ where $t=a_{0}+a_{1}x$. Use the cost function, + + $J(a_{0},a_{1})=\sum_{i=1}^{n}\left[-y_{i}\log(\sigma(t_{i}))-(1-y_{i})\log((1-\sigma(t_{i})))\right]$ + + and gradient + + $\frac{\partial J}{\partial a_{i}}= + 1/m\sum_{k=1}^{N}\left(\sigma(t_{k})-y_{k}\right)t_{k}$ + + a. edit `cost_logistic.m` so that the output is `[J,grad]` or [cost, gradient] + + b. use the following code to solve for a0 and a1 + +```matlab +% Set options for fminunc +options = optimset('GradObj', 'on', 'MaxIter', 400); +% Run fminunc to obtain the optimal theta +% This function will return theta and the cost +[theta, cost] = ... +fminunc(@(a)(costFunction(a, x, y)), initial_a, options); +``` + + c. plot the data and the best-fit logistic regression model + +```matlab +plot(x,y, x, sigma(a(1)+a(2)*x)) +``` + +3. The vertical stress under a corner of a rectangular area subjected to a uniform load of +intensity $q$ is given by the solution of the Boussinesq's equation: + + $\sigma_{z} = + \frac{q}{4\pi}\left(\frac{2mn\sqrt{m^{2}+n^{2}+1}}{m^{2}+n^{2}+1+m^{2}n^{2}}\frac{m^{2}+n^{2}+2}{m^{2}+n^{2}+1}+sin^{-1}\left(\frac{2mn\sqrt{m^{2}+n^{2}+1}}{m^{2}+n^{2}+1+m^{2}n^{2}}\right)\right)$ + + Typically, this equation is solved as a table of values where: + + $\sigma_{z}=q f(m,n)$ + + where $f(m,n)$ is the influence value, q is the uniform load, m=a/z, n=b/z, a and b are + width and length of the rectangular area and z is the depth below the area. + + a. Finish the function `boussinesq_lookup.m` so that when you enter a force, q, + dimensions of rectangular area a, b, and depth, z, it uses a third-order polynomial + interpolation of the four closest values of m to determine the stress in the vertical + direction, sigma_z=$\sigma_{z}$. Use a $0^{th}$-order, polynomial interpolation for + the value of n (i.e. round to the closest value of n). + + b. Copy the `boussinesq_lookup.m` to a file called `boussinesq_spline.m` and use a + cubic spline to interpolate in two dimensions, both m and n, that returns sigma_z. + + + diff --git a/HW6/README.pdf b/HW6/README.pdf new file mode 100644 index 0000000000000000000000000000000000000000..3cc89911b274f1e5fcca0d59039ce6695c387b84 GIT binary patch literal 168643 zcma&NL$EMh(4={7+qP}nwr$(CZQI6sZQHhOYret1>WSz+tE|;oRaHEhQKSkYVzi8O ztWc!$%OmShjLZZK1olQ&P&_Vgh{SNIo>L(cp`fDake746G!!PS+wu(3v8@%MB1JyKoIk;^C< zqT=J`$GYTGhDoY&zVTt@#`ssZdHhBmG*kU~Ga+65c#6D@Gdml=R~CslxR5S-@kJe^?fL%VB*|I2nV6pHx1pHY zXC9?^5WnbgS@puz<)X5Nsjwr{Ktmykgs9|%dA8h1#aq0Y?E^2qIl1r_QKG)pS#09m zW&7k6g|9)F0t0D~ESr*nQYT<;CWa{dBBLm00rlAwc#{j`nNT@cG5&!wYcNi$Z0AwT6Ki@Ad(ysiI5EXH5yEc{pMBRNvsqq5h+L3(#EcNMX5{o zwAJSLY-V>eHbQ27bvQs@6yWH2vXPTs-7Qk*g>WZ_w62Q2al$38<#iY+EmhV$tT32B zH*8Zn?kYC&6ZbX;O<}AWi zj`w>ZVq0Wib8*n5)N| zv-DFrb=aK-YUx z@-XHD^+2_DsQA?7A1c@E_z8khvVQn5Qp$k`A(rOHclqE)gWQON!OE{uZs&-xtxbMM 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The calculated solutions are in the fmn data + % m=fmn(:,1) + % in column 2, fmn(:,2), n=1.2 + % in column 3, fmn(:,2), n=1.4 + % in column 4, fmn(:,2), n=1.6 + + fmn= [0.1,0.02926,0.03007,0.03058 + 0.2,0.05733,0.05894,0.05994 + 0.3,0.08323,0.08561,0.08709 + 0.4,0.10631,0.10941,0.11135 + 0.5,0.12626,0.13003,0.13241 + 0.6,0.14309,0.14749,0.15027 + 0.7,0.15703,0.16199,0.16515 + 0.8,0.16843,0.17389,0.17739]; + + m=a/z; + n=b/z; + + %... +end diff --git a/HW6/cost_logistic.m b/HW6/cost_logistic.m new file mode 100644 index 0000000..169718f --- /dev/null +++ b/HW6/cost_logistic.m @@ -0,0 +1,27 @@ +function [J, grad] = cost_logistic(a, x, y) +% cost_logistic Compute cost and gradient for logistic regression +% J = cost_logistic(theta, X, y) computes the cost of using theta as the +% parameter for logistic regression and the gradient of the cost +% w.r.t. to the parameters. + +% Initialize some useful values +N = length(y); % number of training examples + +% You need to return the following variables correctly +J = 0; +grad = zeros(size(theta)); + +% ====================== YOUR CODE HERE ====================== +% Instructions: Compute the cost of a particular choice of a. +% Compute the partial derivatives and set grad to the partial +% derivatives of the cost w.r.t. each parameter in theta +% +% Note: grad should have the same dimensions as theta +% + + + +% ============================================================= + +end + diff --git a/HW6/problem_1_data.m b/HW6/problem_1_data.m new file mode 100644 index 0000000..6af2956 --- /dev/null +++ b/HW6/problem_1_data.m @@ -0,0 +1,15 @@ + +% part a +xa=[1 2 3 4 5]'; +yb=[2.2 2.8 3.6 4.5 5.5]'; + +% part b + +xb=[3 4 5 7 8 9 11 12]'; +yb=[1.6 3.6 4.4 3.4 2.2 2.8 3.8 4.6]'; + +% part c + +xc=[0.5 1 2 3 4 5 6 7 9]; +yc=[6 4.4 3.2 2.7 2.2 1.9 1.7 1.4 1.1]; + diff --git a/lecture_18/lecture_18.ipynb b/lecture_18/lecture_18.ipynb index bf7731c..f1fd1b3 100644 --- a/lecture_18/lecture_18.ipynb +++ b/lecture_18/lecture_18.ipynb @@ -2,7 +2,7 @@ "cells": [ { "cell_type": "code", - "execution_count": 146, + "execution_count": 2, "metadata": { "collapsed": true }, @@ -13,7 +13,7 @@ }, { "cell_type": "code", - "execution_count": 107, + "execution_count": 3, "metadata": { "collapsed": true }, @@ -1959,7 +1959,7 @@ }, { "cell_type": "code", - "execution_count": 162, + "execution_count": 8, "metadata": { "collapsed": false }, @@ -1969,7 +1969,7 @@ "output_type": "stream", "text": [ "ln(2)=0.693147\n", - "ln(2)~0.366349\n" + "ln(2)~0.693147\n" ] }, { @@ -2014,113 +2014,117 @@ "\n", "\n", "\t\n", - "\t\t\n", + "\t\t\n", "\t\n", "\n", "\n", "\n", "\n", - "\t\t\n", + "\t\t\n", "\t\t-2\n", "\t\n", "\n", "\n", - "\t\t\n", - "\t\t0\n", + "\t\t\n", + "\t\t-1.5\n", "\t\n", "\n", "\n", - "\t\t\n", - "\t\t2\n", + "\t\t\n", + "\t\t-1\n", "\t\n", "\n", "\n", - "\t\t\n", - "\t\t4\n", + "\t\t\n", + "\t\t-0.5\n", "\t\n", "\n", "\n", - "\t\t\n", - "\t\t6\n", + "\t\t\n", + "\t\t0\n", "\t\n", "\n", "\n", - "\t\t\n", - "\t\t8\n", + "\t\t\n", + "\t\t0.5\n", "\t\n", "\n", "\n", - "\t\t\n", - "\t\t10\n", + "\t\t\n", + "\t\t1\n", "\t\n", "\n", "\n", - "\t\t\n", - "\t\t0\n", + "\t\t\n", + "\t\t1.5\n", "\t\n", "\n", "\n", - "\t\t\n", - "\t\t10\n", + "\t\t\n", + "\t\t2\n", "\t\n", "\n", "\n", - "\t\t\n", - "\t\t20\n", + "\t\t\n", + "\t\t0\n", "\t\n", "\n", "\n", - "\t\t\n", - "\t\t30\n", + "\t\t\n", + "\t\t1\n", "\t\n", "\n", "\n", - "\t\t\n", - "\t\t40\n", + "\t\t\n", + "\t\t2\n", "\t\n", "\n", "\n", - "\t\t\n", - "\t\t50\n", + "\t\t\n", + "\t\t3\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t4\n", "\t\n", "\n", "\n", "\t\t\n", - "\t\t60\n", + "\t\t5\n", "\t\n", "\n", "\n", "\n", "\n", - "\t\n", + "\t\n", "\n", "\n", "\tgnuplot_plot_1a\n", "\n", "\n", "\n", - "\t\n", + "\t\n", "\t\n", "\tgnuplot_plot_2a\n", "\n", "\t \n", - "\t\n", + "\t\n", "\n", "\t\n", "\tgnuplot_plot_3a\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", "\n", "\t\n", 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"\n", + "plot(x,y,'o',x,a(1)*x.^2+a(2)*exp(-2*x)+a(3)*sinh(x/2))" + ] + }, { "cell_type": "markdown", "metadata": {}, @@ -51,6 +328,8 @@ "source": [ "![q4](q4.png)\n", "\n", + "# Answer: It depends\n", + "\n", "#### Other:\n", "\n", "Twice the amount of points needed\n", @@ -101,7 +380,7 @@ "cell_type": "markdown", "metadata": {}, "source": [ - "# Splines (Brief introduction before next section)\n", + "# Splines (Brief introduction before integrals)\n", "\n", "Following interpolation discussion, instead of estimating 9 data points with an eighth-order polynomial, it makes more sense to fit sections of the curve to lower-order polynomials:\n", "\n", @@ -315,7 +594,7 @@ }, { "cell_type": "code", - "execution_count": 9, + "execution_count": 75, "metadata": { "collapsed": false }, @@ -532,12 +811,12 @@ "source": [ "### Example: Accelerate then hold velocity\n", "\n", - "Here the time is given as vector t in seconds and the velocity is in mph. " + "Test driving a car, the accelerator is pressed, then released, then pressed again for 20-second intervals, until speed is 120 mph. Here the time is given as vector t in seconds and the velocity is in mph. " ] }, { "cell_type": "code", - "execution_count": 10, + "execution_count": 82, "metadata": { "collapsed": false }, @@ -683,12 +962,12 @@ "\n", "\n", "\n", - "\t\n", + "\t\n", "\tdata\n", "\n", "\n", "\n", - "\t\n", + "\t\n", "\t\tdata\n", "\t\n", "\n", @@ -696,7 +975,6 @@ "\t\n", "\t\n", "\t\n", - "\t\n", "\t\n", "\t\n", "\t\n", @@ -706,34 +984,45 @@ "\t\n", "\n", "\t\n", - "\tlinear\n", + "\tremoved data point\n", "\n", - "\t\n", + "\t\n", + "\t\tremoved data point\n", + "\t\n", + "\n", + "\n", + "\t\n", + "\t\n", + "\n", + "\t\n", + "\tlinear\n", + "\n", + "\t\n", "\t\tlinear\n", "\t\n", "\n", "\n", - "\t\n", + "\t\n", "\t\n", - "\tcubic spline\n", + "\tcubic spline\n", "\n", - "\t\n", + "\t\n", "\t\tcubic spline\n", "\t\n", "\n", "\n", - "\t\n", + "\t\n", "\t\n", - "\tpiecewise cubic\n", + "\tpiecewise cubic\n", "\n", - "\t\n", + "\t\n", "\t\tpiecewise cubic\n", "\t\n", "\n", "\n", - "\t\n", + "\t\n", "\t\n", - "\n", + "\n", "\n", "\n", "\n", @@ -753,22 +1042,22 @@ } ], "source": [ - "t=[0 20 40 56 68 80 84 96 104 110]';\n", - "v=[0 20 20 38 80 80 100 100 125 125]';\n", + "t=[0 20 40 68 80 84 96 104 110]';\n", + "v=[0 20 20 80 80 100 100 125 125]';\n", "tt=linspace(0,110)';\n", "v_lin=interp1(t,v,tt);\n", "v_spl=interp1(t,v,tt,'spline');\n", "v_cub=interp1(t,v,tt,'cubic');\n", "\n", - "plot(t,v,'o',tt,v_lin,tt,v_spl,tt,v_cub)\n", + "plot(t,v,'o',56,38,'s',tt,v_lin,tt,v_spl,tt,v_cub)\n", "xlabel('t (s)')\n", "ylabel('v (mph)')\n", - "legend('data','linear','cubic spline','piecewise cubic','Location','NorthWest')" + "legend('data','removed data point','linear','cubic spline','piecewise cubic','Location','NorthWest')" ] }, { "cell_type": "code", - "execution_count": 14, + "execution_count": 83, "metadata": { "collapsed": false }, @@ -1206,7 +1495,7 @@ }, { "cell_type": "code", - "execution_count": 48, + "execution_count": 86, "metadata": { "collapsed": false }, @@ -1215,21 +1504,21 @@ "name": "stdout", "output_type": "stream", "text": [ - "For 5 steps\n", - "trapezoid approximation of integral is 0.79 \n", + "For 70 steps\n", + "trapezoid approximation of integral is 0.99 \n", " actual integral is 1.00\n" ] } ], "source": [ - "N=5;\n", + "N=70;\n", "I_trap=trap(@(x) sin(x),0,pi/2,N);\n", "fprintf('For %i steps\\ntrapezoid approximation of integral is %1.2f \\n actual integral is %1.2f',N,I_trap,I_act)" ] }, { "cell_type": "code", - "execution_count": 43, + "execution_count": 92, "metadata": { "collapsed": false }, @@ -1365,11 +1654,76 @@ "\t\n", "\tgnuplot_plot_2a\n", "\n", - "\t \n", + "\t\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", + "\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", + "\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", + "\t\n", "\t\n", "\n", "\t\n", @@ -1393,8 +1747,8 @@ } ], "source": [ - "\n", - "plot(x,sin(x),linspace(0,pi/2,N),sin(linspace(0,pi/2,N)),'o')" + "x=linspace(0,pi);\n", + "plot(x,sin(x),linspace(0,pi/2,N),sin(linspace(0,pi/2,N)),'-o')" ] }, { @@ -1425,7 +1779,7 @@ }, { "cell_type": "code", - "execution_count": 68, + "execution_count": 93, "metadata": { "collapsed": false }, diff --git a/lecture_19/octave-workspace b/lecture_19/octave-workspace new file mode 100644 index 0000000000000000000000000000000000000000..4bd89dd249beee3bc2d071dccd02246ca43bf138 GIT binary patch literal 10815 zcmd5?2{_f;+TT=)CWS;qDn*gl=DqiN31ukrJY_5m=CL}_s8dpDq>0i&8ih2j-5jMs zsfdouGpW$TxA$Jl>D+VfeeQj}@A;nR?x(E&Yj}s>`}@8B6|DUteWC-@=BrsTMlcu* 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{ + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# Questions from last class\n", + "\n", + "The interp2 function uses splines to interpolate between data points. What are three options for interpolation:\n", + "\n", + "- cubic spline\n", + "\n", + "- piecewise cubic spline\n", + "\n", + "- linear spline\n", + "\n", + "- quadratic spline\n", + "\n", + "- fourth-order spline\n", + "\n", + "![q1](q1.png)\n", + "\n", + "Numerical integration is a general application of the Newton-Cotes formulas. What is the first order approximation of the Newton-Cotes formula? *\n", + "\n", + "- trapezoidal rule\n", + "\n", + "- Simpson's 1/3 rule\n", + "\n", + "- Simpson's 3/8 rule\n", + "\n", + "- linear approximation of integral\n", + "\n", + "- constant approximation of integral (sum(f(x)*dx))\n", + "\n", + "![q2](q2.png)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# Questions from you\n", + "\n", + "- Is spring here to stay?\n", + " \n", + " - [Punsxatawney Phil](http://www.groundhog.org/)\n", + "\n", + "- The time is now.\n", + "\n", + "- Final Project Sheet?\n", + " \n", + " - coming this evening/tomorrow\n", + "\n", + "- can you provide some sort of hw answer key?\n", + " \n", + " - we can go through some of the HW \n", + "\n", + "- What's the most 1337 thing you've ever done?\n", + "\n", + " - sorry, I'm n00b to this\n", + "\n", + "- Can we do out more examples by hand (doc cam or drawing on computer notepad) instead of with pre-written code?\n", + "\n", + " - forthcoming\n", + "\n", + "- Favorite movie?\n", + " \n", + " - Big Lebowski\n", + " " + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# Integrals in practice\n", + "\n", + "### Example: Compare toughness of two steels\n", + "\n", + "![Stress-strain plot of steel](steel_psi.jpg)\n", + "\n", + "Use the plot shown to determine the toughness of stainless steel and the toughness of structural steel.\n", + "\n" + ] + }, + { + "cell_type": "code", + "execution_count": 49, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "toughness of structural steel is 10.2 psi\n", + "toughness of stainless steel is 18.6 psi\n" + ] + } + ], + "source": [ + "fe_c=load('structural_steel_psi.jpg.dat');\n", + "fe_cr =load('stainless_steel_psi.jpg.dat');\n", + "\n", + "fe_c_toughness=trapz(fe_c(:,1),fe_c(:,2));\n", + "fe_cr_toughness=trapz(fe_cr(:,1),fe_cr(:,2));\n", + "\n", + "fprintf('toughness of structural steel is %1.1f psi\\n',fe_c_toughness)\n", + "fprintf('toughness of stainless steel is %1.1f psi',fe_cr_toughness)\n" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# Gauss Quadrature (for functions)\n", + "\n", + "Evaluating an integral, we assumed a polynomial form for each Newton-Cotes approximation.\n", + "\n", + "If we can evaluate the function at any point, it makes more sense to choose points more wisely rather than just using endpoints\n", + "\n", + "![trapezoidal example](trap_example.png)\n", + "\n", + "Let us set up two unknown constants, $c_{0}$ and $x_{0}$ and determine a *wise* place to evaluate f(x) such that \n", + "\n", + "$I=c_{0}f(x_{0})$\n", + "\n", + "and I is exact for polynomial of n=0, 1\n", + "\n", + "$\\int_{a}^{b}1dx=b-a=c_{0}$\n", + "\n", + "$\\int_{a}^{b}xdx=\\frac{b^2-a^2}{2}=c_{0}x_{0}$\n", + "\n", + "so $c_{0}=b-a$ and $x_{0}=\\frac{b+a}{2}$\n", + "\n", + "$I=\\int_{a}^{b}f(x)dx \\approx (b-a)f\\left(\\frac{b+a}{2}\\right)$\n", + "\n" + ] + }, + { + "cell_type": "code", + "execution_count": 147, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "f1 =\n", + "\n", + "@(x) x + 1\n", + "\n", + "f2 =\n", + "\n", + "@(x) 1 / 2 * x .^ 2 + x + 1\n", + "\n", + "f3 =\n", + "\n", + "@(x) 1 / 6 * x .^ 3 + 1 / 2 * x .^ 2 + x\n", + "\n" + ] + }, + { + "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-1000\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t-500\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t0\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t500\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t1000\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t1500\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t-20\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t-15\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t-10\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t-5\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t0\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t5\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t10\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t15\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t20\n", + "\t\n", + "\n", + "\n", + "\n", + "\n", + "\t\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": [ + "f1=@(x) x+1\n", + "f2=@(x) 1/2*x.^2+x+1\n", + "f3=@(x) 1/6*x.^3+1/2*x.^2+x\n", + "plot(linspace(-18,18),f3(linspace(-18,18)))" + ] + }, + { + "cell_type": "code", + "execution_count": 148, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "integral of f1 from 2 to 3 = 3.500000\n", + "integral of f1 from 2 to 3 ~ 3.500000\n", + "integral of f2 from 2 to 3 = 6.666667\n", + "integral of f2 from 2 to 3 ~ 6.625000\n" + ] + } + ], + "source": [ + "fprintf('integral of f1 from 2 to 3 = %f',f2(3)-f2(2))\n", + "fprintf('integral of f1 from 2 to 3 ~ %f',(3-2)*f1(3/2+2/2))\n", + "\n", + "fprintf('integral of f2 from 2 to 3 = %f',f3(3)-f3(2))\n", + "fprintf('integral of f2 from 2 to 3 ~ %f',(3-2)*f2(3/2+2/2))\n" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "This process is called **Gauss Quadrature**. Usually, the bounds are fixed at -1 and 1 instead of a and b\n", + "\n", + "$I=c_{0}f(x_{0})$\n", + "\n", + "and I is exact for polynomial of n=0, 1\n", + "\n", + "$\\int_{-1}^{1}1dx=b-a=c_{0}$\n", + "\n", + "$\\int_{-1}^{1}xdx=\\frac{1^2-(-1)^2}{2}=c_{0}x_{0}$\n", + "\n", + "so $c_{0}=2$ and $x_{0}=0$\n", + "\n", + "$I=\\int_{-1}^{1}f(x)dx \\approx 2f\\left(0\\right)$\n", + "\n", + "Now, integrals can be performed with a change of variable\n", + "\n", + "a=2\n", + "\n", + "b=3\n", + "\n", + "x= 2 to 3\n", + "\n", + "or $x_{d}=$ -1 to 1\n", + "\n", + "$x=a_{1}+a_{2}x_{d}$\n", + "\n", + "at $x_{d}=-1$, x=a\n", + "\n", + "at $x_{d}=1$, x=b\n", + "\n", + "so \n", + "\n", + "$x=\\frac{(b+a) +(b-a)x_{d}}{2}$\n", + "\n", + "$dx=\\frac{b-a}{2}dx_{d}$\n", + "\n", + "$\\int_{2}^{3}x+1dx=\\int_{-1}^{1}\\left(\\frac{(2+3) +(3-2)x_{d}}{2}\n", + "+1\\right)\n", + "\\frac{3-2}{2}dx_{d}$\n", + "\n", + "$\\int_{2}^{3}x+1dx=\\int_{-1}^{1}\\left(\\frac{5 +x_{d}}{2}\n", + "+1\\right)\n", + "\\frac{3-2}{2}dx_{d}$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$\\int_{2}^{3}x+1dx=\\int_{-1}^{1}\\left(\\frac{7}{4}+\\frac{1}{4}x_{d}\\right)dx_{d}=2\\frac{7}{4}=3.5$" + ] + }, + { + "cell_type": "code", + "execution_count": 15, + "metadata": { + "collapsed": true + }, + "outputs": [], + "source": [ + "function I=gauss_1pt(func,a,b)\n", + " % Gauss quadrature using single point\n", + " % exact for n<1 polynomials\n", + " c0=2;\n", + " xd=0;\n", + " dx=(b-a)/2;\n", + " x=(b+a)/2+(b-a)/2*xd;\n", + " I=func(x).*dx*c0;\n", + "end" + ] + }, + { + "cell_type": "code", + "execution_count": 13, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "ans = 3.5000\r\n" + ] + } + ], + "source": [ + "gauss_1pt(f1,2,3)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## General Gauss weights and points\n", + "\n", + "![Gauss quadrature table](gauss_weights.png)\n", + "\n", + "### If you need to evaluate an integral, to increase accuracy, increase number of Gauss points\n", + "\n", + "### Adaptive Quadrature\n", + "\n", + "Matlab/Octave built-in functions use two types of adaptive quadrature to increase accuracy of integrals of functions. \n", + "\n", + "1. `quad`: Simpson quadrature good for nonsmooth functions\n", + "\n", + "2. `quadl`: Lobatto quadrature good for smooth functions\n", + "\n", + "```matlab\n", + "q = quad(fun, a, b, tol, trace, p1, p2, …)\n", + "fun : function to be integrates\n", + "a, b: integration bounds\n", + "tol: desired absolute tolerance (default: 10-6)\n", + "trace: flag to display details or not\n", + "p1, p2, …: extra parameters for fun\n", + "quadl has the same arguments\n", + "```\n" + ] + }, + { + "cell_type": "code", + "execution_count": 149, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "ans = 6.6667\n", + "ans = 6.6667\n", + "ans = 6.6667\n" + ] + } + ], + "source": [ + "% integral of quadratic\n", + "quad(f2,2,3)\n", + "quadl(f2,2,3)\n", + "f3(3)-f3(2)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# Numerical Differentiation\n", + "\n", + "Expanding the Taylor Series:\n", + "\n", + "$f(x_{i+1})=f(x_{i})+f'(x_{i})h+\\frac{f''(x_{i})}{2!}h^2+\\cdots$" + ] + }, + { + "cell_type": "code", + "execution_count": 82, + "metadata": { + "collapsed": false + }, + "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-1.5\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t-1\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t-0.5\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t0\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t0.5\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t1\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t1.5\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t-4\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t-3\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t-2\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t-1\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t0\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t1\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t2\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t3\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t4\n", + "\t\n", + "\n", + "\n", + "\n", + "\n", + "\t\n", + "\n", + "\t\n", + "\t\tLow noise in sin wave\n", + "\t\n", + "\n", + "\n", + "\n", + "\tgnuplot_plot_1a\n", + "\n", + "\n", + "\n", + "\t\n", + "\t\n", + "\tgnuplot_plot_2a\n", + "\n", + "\t\n", + "\t\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "x=linspace(-pi,pi);\n", + "y_smooth=sin(x);\n", + "y_noise =y_smooth+rand(size(x))*0.1-0.05;\n", + "plot(x,y_smooth,x,y_noise)\n", + "title('Low noise in sin wave')" + ] + }, + { + "cell_type": "code", + "execution_count": 98, + "metadata": { + "collapsed": false + }, + "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-1.5\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t-1\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t-0.5\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t0\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t0.5\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t1\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t1.5\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t-4\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t-3\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t-2\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t-1\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t0\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t1\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t2\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t3\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t4\n", + "\t\n", + "\n", + "\n", + "\n", + "\n", + "\t\n", + "\n", + "\t\n", + "\t\tNoise Amplified with derivative\n", + "\t\n", + "\n", + "\n", + "\n", + "\tgnuplot_plot_1a\n", + "\n", + "\n", + "\n", + "\t\n", + "\t\n", + "\tgnuplot_plot_2a\n", + "\n", + "\t\n", + "\t\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "% Central Difference derivative\n", + "\n", + "dy_smooth=zeros(size(x));\n", + "dy_smooth([1,end])=NaN;\n", + "dy_smooth(2:end-1)=(y_smooth(3:end)-y_smooth(1:end-2))/2/(x(2)-x(1));\n", + "\n", + "dy_noise=zeros(size(x));\n", + "dy_noise([1,end])=NaN;\n", + "dy_noise(2:end-1)=(y_noise(3:end)-y_noise(1:end-2))/2/(x(2)-x(1));\n", + "\n", + "plot(x,dy_smooth,x,dy_noise)\n", + "title('Noise Amplified with derivative')" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "### Reduce noise\n", + "\n", + "Options:\n", + "\n", + "1. Fit a function and take derivative\n", + " \n", + " a. splines won't help much\n", + " \n", + " b. best fit curve (better)\n", + " \n", + "2. Smooth data (does not matter if you smooth before/after derivative)" + ] + }, + { + "cell_type": "code", + "execution_count": 99, + "metadata": { + "collapsed": false + }, + "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-1.5\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t-1\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t-0.5\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t0\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t0.5\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t1\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t1.5\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t-4\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t-3\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t-2\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t-1\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t0\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t1\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t2\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t3\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t4\n", + "\t\n", + "\n", + "\n", + "\n", + "\n", + "\t\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\t\n", + "\tderiv of spline\n", + "\n", + "\n", + "\n", + "\t\n", + "\t\tderiv of spline\n", + "\t\n", + "\n", + "\n", + "\t\n", + "\t\n", + "\tno change\n", + "\n", + "\t\n", + "\t\tno change\n", + "\t\n", + "\n", + "\n", + "\t\n", + "\t\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "y_spline_i1=interp1(x,y_noise,x+0.1);\n", + "y_spline_in1=interp1(x,y_noise,x-0.1);\n", + "dy_spline=(y_spline_i1-y_spline_in1)/0.2;\n", + "plot(x,dy_spline,x,dy_noise)\n", + "legend('deriv of spline','no change')" + ] + }, + { + "cell_type": "code", + "execution_count": 100, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "a = 0.99838\r\n" + ] + }, + { + "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-1.5\n", + "\t\n", + "\n", + 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"\n", + "\n", + "\n", + "" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "Z=[sin(x')];\n", + "a=Z\\y_noise'\n", + "plot(x,a*sin(x),x,y_noise)" + ] + }, + { + "cell_type": "code", + "execution_count": 103, + "metadata": { + "collapsed": false + }, + "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-1\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t-0.5\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t0\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + 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"metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "a =\n", + "\n", + " 1 2 3 4 5\n", + "\n", + "pa =\n", + "\n", + " 2 1 1 2 3 4 5 5 4\n", + "\n", + "ans =\n", + "\n", + " 1 9\n", + "\n" + ] + } + ], + "source": [ + "a=[1,2,3,4,5]\n", + "pa=[a(4/2:-1:1) a a(end:-1:end-4/2+1)]\n", + "size(pa)" + ] + }, + { + "cell_type": "code", + "execution_count": 137, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "ans =\n", + "\n", + " 1 100\n", + "\n" + ] + }, + { + "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", + 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