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| { | |
| "cells": [ | |
| { | |
| "cell_type": "code", | |
| "execution_count": 1, | |
| "metadata": { | |
| "collapsed": true | |
| }, | |
| "outputs": [], | |
| "source": [ | |
| "%plot --format svg" | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "execution_count": 2, | |
| "metadata": { | |
| "collapsed": true | |
| }, | |
| "outputs": [], | |
| "source": [ | |
| "setdefaults" | |
| ] | |
| }, | |
| { | |
| "cell_type": "markdown", | |
| "metadata": {}, | |
| "source": [ | |
| "## My question from last class \n", | |
| "\n", | |
| "\n", | |
| "\n", | |
| "\n", | |
| "\n", | |
| "\n", | |
| "\n", | |
| "\n", | |
| "## Your questions from last class\n", | |
| "\n", | |
| "1. Do we have to submit a link for HW #4 somewhere or is uploading to Github sufficient?\n", | |
| " \n", | |
| " -no, your submission from HW 3 is sufficient\n", | |
| "\n", | |
| "2. How do I get the formulas/formatting in markdown files to show up on github?\n", | |
| " \n", | |
| " -no luck for markdown equations in github, this is an ongoing request\n", | |
| " \n", | |
| "3. Confused about the p=1 norm part and ||A||_1\n", | |
| "\n", | |
| "4. When's the exam?\n", | |
| " \n", | |
| " -next week (3/9)\n", | |
| "\n", | |
| "5. What do you recommend doing to get better at figuring out the homeworks?\n", | |
| "\n", | |
| " -time and experimenting (try going through the lecture examples, verify my work)\n", | |
| " \n", | |
| "6. Could we have an hw or extra credit with a video lecture to learn some simple python?\n", | |
| " \n", | |
| " -Sounds great! how simple? \n", | |
| " \n", | |
| " -[Installing Python and Jupyter Notebook (via Anaconda) - https://www.continuum.io/downloads](https://www.continuum.io/downloads)\n", | |
| " \n", | |
| " -[Running Matlab kernel in Jupyter - https://anneurai.net/2015/11/12/matlab-based-ipython-notebooks/](https://anneurai.net/2015/11/12/matlab-based-ipython-notebooks/)\n", | |
| " \n", | |
| " -[Running Octave kernel in Jupyter - https://anaconda.org/pypi/octave_kernel](https://anaconda.org/pypi/octave_kernel)\n", | |
| " \n", | |
| " " | |
| ] | |
| }, | |
| { | |
| "cell_type": "markdown", | |
| "metadata": {}, | |
| "source": [ | |
| "# Markdown examples\n", | |
| "\n", | |
| "` \" ' ` `\n", | |
| "\n", | |
| "```matlab\n", | |
| "x=linspace(0,1);\n", | |
| "y=x.^2;\n", | |
| "plot(x,y)\n", | |
| "for i = 1:10\n", | |
| " fprintf('markdown is pretty')\n", | |
| "end\n", | |
| "```" | |
| ] | |
| }, | |
| { | |
| "cell_type": "markdown", | |
| "metadata": {}, | |
| "source": [ | |
| "## Condition of a matrix \n", | |
| "### *just checked in to see what condition my condition was in*\n", | |
| "### Matrix norms\n", | |
| "\n", | |
| "The Euclidean norm of a vector is measure of the magnitude (in 3D this would be: $|x|=\\sqrt{x_{1}^{2}+x_{2}^{2}+x_{3}^{2}}$) in general the equation is:\n", | |
| "\n", | |
| "$||x||_{e}=\\sqrt{\\sum_{i=1}^{n}x_{i}^{2}}$\n", | |
| "\n", | |
| "For a matrix, A, the same norm is called the Frobenius norm:\n", | |
| "\n", | |
| "$||A||_{f}=\\sqrt{\\sum_{i=1}^{n}\\sum_{j=1}^{m}A_{i,j}^{2}}$\n", | |
| "\n", | |
| "In general we can calculate any $p$-norm where\n", | |
| "\n", | |
| "$||A||_{p}=\\sqrt{\\sum_{i=1}^{n}\\sum_{i=1}^{m}A_{i,j}^{p}}$\n", | |
| "\n", | |
| "so the p=1, 1-norm is \n", | |
| "\n", | |
| "$||A||_{1}=\\sqrt{\\sum_{i=1}^{n}\\sum_{i=1}^{m}A_{i,j}^{1}}=\\sum_{i=1}^{n}\\sum_{i=1}^{m}|A_{i,j}|$\n", | |
| "\n", | |
| "$||A||_{\\infty}=\\sqrt{\\sum_{i=1}^{n}\\sum_{i=1}^{m}A_{i,j}^{\\infty}}=\\max_{1\\le i \\le n}\\sum_{j=1}^{m}|A_{i,j}|$\n", | |
| "\n", | |
| "### Condition of Matrix\n", | |
| "\n", | |
| "The matrix condition is the product of \n", | |
| "\n", | |
| "$Cond(A) = ||A||\\cdot||A^{-1}||$ \n", | |
| "\n", | |
| "So each norm will have a different condition number, but the limit is $Cond(A)\\ge 1$\n", | |
| "\n", | |
| "An estimate of the rounding error is based on the condition of A:\n", | |
| "\n", | |
| "$\\frac{||\\Delta x||}{x} \\le Cond(A) \\frac{||\\Delta A||}{||A||}$\n", | |
| "\n", | |
| "So if the coefficients of A have accuracy to $10^{-t}\n", | |
| "\n", | |
| "and the condition of A, $Cond(A)=10^{c}$\n", | |
| "\n", | |
| "then the solution for x can have rounding errors up to $10^{c-t}$\n" | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "execution_count": 7, | |
| "metadata": { | |
| "collapsed": false | |
| }, | |
| "outputs": [ | |
| { | |
| "name": "stdout", | |
| "output_type": "stream", | |
| "text": [ | |
| "A =\n", | |
| "\n", | |
| " 1.00000 0.50000 0.33333\n", | |
| " 0.50000 0.33333 0.25000\n", | |
| " 0.33333 0.25000 0.20000\n", | |
| "\n", | |
| "L =\n", | |
| "\n", | |
| " 1.00000 0.00000 0.00000\n", | |
| " 0.50000 1.00000 0.00000\n", | |
| " 0.33333 1.00000 1.00000\n", | |
| "\n", | |
| "U =\n", | |
| "\n", | |
| " 1.00000 0.50000 0.33333\n", | |
| " 0.00000 0.08333 0.08333\n", | |
| " 0.00000 -0.00000 0.00556\n", | |
| "\n" | |
| ] | |
| } | |
| ], | |
| "source": [ | |
| "A=[1,1/2,1/3;1/2,1/3,1/4;1/3,1/4,1/5]\n", | |
| "[L,U]=LU_naive(A)" | |
| ] | |
| }, | |
| { | |
| "cell_type": "markdown", | |
| "metadata": { | |
| "collapsed": true | |
| }, | |
| "source": [ | |
| "Then, $A^{-1}=(LU)^{-1}=U^{-1}L^{-1}$\n", | |
| "\n", | |
| "$Ld_{1}=\\left[\\begin{array}{c}\n", | |
| "1 \\\\\n", | |
| "0 \\\\\n", | |
| "0 \\end{array}\\right]$, $Ux_{1}=d_{1}$ ..." | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "execution_count": 8, | |
| "metadata": { | |
| "collapsed": false | |
| }, | |
| "outputs": [ | |
| { | |
| "name": "stdout", | |
| "output_type": "stream", | |
| "text": [ | |
| "invA =\n", | |
| "\n", | |
| " 9.0000 -36.0000 30.0000\n", | |
| " -36.0000 192.0000 -180.0000\n", | |
| " 30.0000 -180.0000 180.0000\n", | |
| "\n", | |
| "ans =\n", | |
| "\n", | |
| " 1.0000e+00 3.5527e-15 2.9976e-15\n", | |
| " -1.3249e-14 1.0000e+00 -9.1038e-15\n", | |
| " 8.5117e-15 7.1054e-15 1.0000e+00\n", | |
| "\n" | |
| ] | |
| } | |
| ], | |
| "source": [ | |
| "invA=zeros(3,3);\n", | |
| "d1=L\\[1;0;0];\n", | |
| "d2=L\\[0;1;0];\n", | |
| "d3=L\\[0;0;1];\n", | |
| "invA(:,1)=U\\d1; % shortcut invA(:,1)=A\\[1;0;0]\n", | |
| "invA(:,2)=U\\d2;\n", | |
| "invA(:,3)=U\\d3\n", | |
| "invA*A" | |
| ] | |
| }, | |
| { | |
| "cell_type": "markdown", | |
| "metadata": {}, | |
| "source": [ | |
| "Find the condition of A, $cond(A)$" | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "execution_count": 9, | |
| "metadata": { | |
| "collapsed": false | |
| }, | |
| "outputs": [ | |
| { | |
| "name": "stdout", | |
| "output_type": "stream", | |
| "text": [ | |
| "normf_A = 1.4136\n", | |
| "normf_invA = 372.21\n", | |
| "cond_f_A = 526.16\n", | |
| "ans = 1.4136\n", | |
| "norm1_A = 1.8333\n", | |
| "norm1_invA = 30.000\n", | |
| "ans = 1.8333\n", | |
| "cond_1_A = 55.000\n", | |
| "norminf_A = 1.8333\n", | |
| "norminf_invA = 30.000\n", | |
| "ans = 1.8333\n", | |
| "cond_inf_A = 55.000\n" | |
| ] | |
| } | |
| ], | |
| "source": [ | |
| "% Frobenius norm\n", | |
| "normf_A = sqrt(sum(sum(A.^2)))\n", | |
| "normf_invA = sqrt(sum(sum(invA.^2)))\n", | |
| "\n", | |
| "cond_f_A = normf_A*normf_invA\n", | |
| "\n", | |
| "norm(A,'fro')\n", | |
| "\n", | |
| "% p=1, column sum norm\n", | |
| "norm1_A = max(sum(A,2))\n", | |
| "norm1_invA = max(sum(invA,2))\n", | |
| "norm(A,1)\n", | |
| "\n", | |
| "cond_1_A=norm1_A*norm1_invA\n", | |
| "\n", | |
| "% p=inf, row sum norm\n", | |
| "norminf_A = max(sum(A,1))\n", | |
| "norminf_invA = max(sum(invA,1))\n", | |
| "norm(A,inf)\n", | |
| "\n", | |
| "cond_inf_A=norminf_A*norminf_invA\n" | |
| ] | |
| }, | |
| { | |
| "cell_type": "markdown", | |
| "metadata": { | |
| "collapsed": true | |
| }, | |
| "source": [ | |
| "Consider the problem again from the intro to Linear Algebra, 4 masses are connected in series to 4 springs with spring constants $K_{i}$. What does a high condition number mean for this problem? \n", | |
| "\n", | |
| "\n", | |
| "\n", | |
| "The masses haves the following amounts, 1, 2, 3, and 4 kg for masses 1-4. Using a FBD for each mass:\n", | |
| "\n", | |
| "$m_{1}g+k_{2}(x_{2}-x_{1})-k_{1}x_{1}=0$\n", | |
| "\n", | |
| "$m_{2}g+k_{3}(x_{3}-x_{2})-k_{2}(x_{2}-x_{1})=0$\n", | |
| "\n", | |
| "$m_{3}g+k_{4}(x_{4}-x_{3})-k_{3}(x_{3}-x_{2})=0$\n", | |
| "\n", | |
| "$m_{4}g-k_{4}(x_{4}-x_{3})=0$\n", | |
| "\n", | |
| "in matrix form:\n", | |
| "\n", | |
| "$\\left[ \\begin{array}{cccc}\n", | |
| "k_{1}+k_{2} & -k_{2} & 0 & 0 \\\\\n", | |
| "-k_{2} & k_{2}+k_{3} & -k_{3} & 0 \\\\\n", | |
| "0 & -k_{3} & k_{3}+k_{4} & -k_{4} \\\\\n", | |
| "0 & 0 & -k_{4} & k_{4} \\end{array} \\right]\n", | |
| "\\left[ \\begin{array}{c}\n", | |
| "x_{1} \\\\\n", | |
| "x_{2} \\\\\n", | |
| "x_{3} \\\\\n", | |
| "x_{4} \\end{array} \\right]=\n", | |
| "\\left[ \\begin{array}{c}\n", | |
| "m_{1}g \\\\\n", | |
| "m_{2}g \\\\\n", | |
| "m_{3}g \\\\\n", | |
| "m_{4}g \\end{array} \\right]$" | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "execution_count": 10, | |
| "metadata": { | |
| "collapsed": false | |
| }, | |
| "outputs": [ | |
| { | |
| "name": "stdout", | |
| "output_type": "stream", | |
| "text": [ | |
| "K =\n", | |
| "\n", | |
| " 100010 -100000 0 0\n", | |
| " -100000 100010 -10 0\n", | |
| " 0 -10 11 -1\n", | |
| " 0 0 -1 1\n", | |
| "\n", | |
| "y =\n", | |
| "\n", | |
| " 9.8100\n", | |
| " 19.6200\n", | |
| " 29.4300\n", | |
| " 39.2400\n", | |
| "\n" | |
| ] | |
| } | |
| ], | |
| "source": [ | |
| "k1=10; % N/m\n", | |
| "k2=100000;\n", | |
| "k3=10;\n", | |
| "k4=1;\n", | |
| "m1=1; % kg\n", | |
| "m2=2;\n", | |
| "m3=3;\n", | |
| "m4=4;\n", | |
| "g=9.81; % m/s^2\n", | |
| "K=[k1+k2 -k2 0 0; -k2 k2+k3 -k3 0; 0 -k3 k3+k4 -k4; 0 0 -k4 k4]\n", | |
| "y=[m1*g;m2*g;m3*g;m4*g]" | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "execution_count": 11, | |
| "metadata": { | |
| "collapsed": false | |
| }, | |
| "outputs": [ | |
| { | |
| "name": "stdout", | |
| "output_type": "stream", | |
| "text": [ | |
| "ans = 3.2004e+05\n", | |
| "ans = 3.2004e+05\n", | |
| "ans = 2.5925e+05\n", | |
| "ans = 2.5293e+05\n" | |
| ] | |
| } | |
| ], | |
| "source": [ | |
| "cond(K,inf)\n", | |
| "cond(K,1)\n", | |
| "cond(K,'fro')\n", | |
| "cond(K,2)" | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "execution_count": 26, | |
| "metadata": { | |
| "collapsed": false | |
| }, | |
| "outputs": [ | |
| { | |
| "name": "stdout", | |
| "output_type": "stream", | |
| "text": [ | |
| "e =\n", | |
| "\n", | |
| " 7.9078e-01\n", | |
| " 3.5881e+00\n", | |
| " 1.7621e+01\n", | |
| " 2.0001e+05\n", | |
| "\n", | |
| "ans = 2.5293e+05\n" | |
| ] | |
| } | |
| ], | |
| "source": [ | |
| "e=eig(K)\n", | |
| "max(e)/min(e)" | |
| ] | |
| }, | |
| { | |
| "cell_type": "markdown", | |
| "metadata": {}, | |
| "source": [ | |
| "## P=2 norm is ratio of biggest eigenvalue to smallest eigenvalue!\n", | |
| "\n", | |
| "no need to calculate the inv(K)" | |
| ] | |
| }, | |
| { | |
| "cell_type": "markdown", | |
| "metadata": {}, | |
| "source": [ | |
| "# Iterative Methods\n", | |
| "\n", | |
| "## Gauss-Seidel method\n", | |
| "\n", | |
| "If we have an intial guess for each value of a vector $x$ that we are trying to solve, then it is easy enough to solve for one component given the others. \n", | |
| "\n", | |
| "Take a 3$\\times$3 matrix \n", | |
| "\n", | |
| "$Ax=b$\n", | |
| "\n", | |
| "$\\left[ \\begin{array}{ccc}\n", | |
| "3 & -0.1 & -0.2 \\\\\n", | |
| "0.1 & 7 & -0.3 \\\\\n", | |
| "0.3 & -0.2 & 10 \\end{array} \\right]\n", | |
| "\\left[ \\begin{array}{c}\n", | |
| "x_{1} \\\\\n", | |
| "x_{2} \\\\\n", | |
| "x_{3} \\end{array} \\right]=\n", | |
| "\\left[ \\begin{array}{c}\n", | |
| "7.85 \\\\\n", | |
| "-19.3 \\\\\n", | |
| "71.4\\end{array} \\right]$\n", | |
| "\n", | |
| "$x_{1}=\\frac{7.85+0.1x_{2}+0.2x_{3}}{3}$\n", | |
| "\n", | |
| "$x_{2}=\\frac{-19.3-0.1x_{1}+0.3x_{3}}{7}$\n", | |
| "\n", | |
| "$x_{3}=\\frac{71.4+0.1x_{1}+0.2x_{2}}{10}$" | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "execution_count": 12, | |
| "metadata": { | |
| "collapsed": false | |
| }, | |
| "outputs": [ | |
| { | |
| "name": "stdout", | |
| "output_type": "stream", | |
| "text": [ | |
| "A =\n", | |
| "\n", | |
| " 3.00000 -0.10000 -0.20000\n", | |
| " 0.10000 7.00000 -0.30000\n", | |
| " 0.30000 -0.20000 10.00000\n", | |
| "\n", | |
| "b =\n", | |
| "\n", | |
| " 7.8500\n", | |
| " -19.3000\n", | |
| " 71.4000\n", | |
| "\n", | |
| "x =\n", | |
| "\n", | |
| " 3.0000\n", | |
| " -2.5000\n", | |
| " 7.0000\n", | |
| "\n" | |
| ] | |
| } | |
| ], | |
| "source": [ | |
| "A=[3 -0.1 -0.2;0.1 7 -0.3;0.3 -0.2 10]\n", | |
| "b=[7.85;-19.3;71.4]\n", | |
| "\n", | |
| "x=A\\b" | |
| ] | |
| }, | |
| { | |
| "cell_type": "markdown", | |
| "metadata": { | |
| "collapsed": true | |
| }, | |
| "source": [ | |
| "### Gauss-Seidel Iterative approach\n", | |
| "\n", | |
| "As a first guess, we can use $x_{1}=x_{2}=x_{3}=0$\n", | |
| "\n", | |
| "$x_{1}=\\frac{7.85+0.1(0)+0.3(0)}{3}=2.6167$\n", | |
| "\n", | |
| "$x_{2}=\\frac{-19.3-0.1(2.6167)+0.3(0)}{7}=-2.7945$\n", | |
| "\n", | |
| "$x_{3}=\\frac{71.4+0.1(2.6167)+0.2(-2.7945)}{10}=7.0056$\n", | |
| "\n", | |
| "Then, we update the guess:\n", | |
| "\n", | |
| "$x_{1}=\\frac{7.85+0.1(-2.7945)+0.3(7.0056)}{3}=2.9906$\n", | |
| "\n", | |
| "$x_{2}=\\frac{-19.3-0.1(2.9906)+0.3(7.0056)}{7}=-2.4996$\n", | |
| "\n", | |
| "$x_{3}=\\frac{71.4+0.1(2.9906)+0.2(-2.4966)}{10}=7.00029$\n", | |
| "\n", | |
| "The results are conveerging to the solution we found with `\\` of $x_{1}=3,~x_{2}=-2.5,~x_{3}=7$\n", | |
| "\n", | |
| "We could also use an iterative method that solves for all of the x-components in one step:\n", | |
| "\n", | |
| "### Jacobi method\n", | |
| "\n", | |
| "$x_{1}^{i}=\\frac{7.85+0.1x_{2}^{i-1}+0.3x_{3}^{i-1}}{3}$\n", | |
| "\n", | |
| "$x_{2}^{i}=\\frac{-19.3-0.1x_{1}^{i-1}+0.3x_{3}^{i-1}}{7}$\n", | |
| "\n", | |
| "$x_{3}^{i}=\\frac{71.4+0.1x_{1}^{i-1}+0.2x_{2}^{i-1}}{10}$\n", | |
| "\n", | |
| "Here the solution is a matrix multiplication and vector addition\n", | |
| "\n", | |
| "$\\left[ \\begin{array}{c}\n", | |
| "x_{1}^{i} \\\\\n", | |
| "x_{2}^{i} \\\\\n", | |
| "x_{3}^{i} \\end{array} \\right]=\n", | |
| "\\left[ \\begin{array}{c}\n", | |
| "7.85/3 \\\\\n", | |
| "-19.3/7 \\\\\n", | |
| "71.4/10\\end{array} \\right]-\n", | |
| "\\left[ \\begin{array}{ccc}\n", | |
| "0 & 0.1/3 & 0.2/3 \\\\\n", | |
| "0.1/7 & 0 & -0.3/7 \\\\\n", | |
| "0.3/10 & -0.2/10 & 0 \\end{array} \\right]\n", | |
| "\\left[ \\begin{array}{c}\n", | |
| "x_{1}^{i-1} \\\\\n", | |
| "x_{2}^{i-1} \\\\\n", | |
| "x_{3}^{i-1} \\end{array} \\right]$\n", | |
| "\n", | |
| "|x_{j}|Jacobi method |vs| Gauss-Seidel |\n", | |
| "|--------|------------------------------|---|-------------------------------|\n", | |
| "|$x_{1}^{i}=$ | $\\frac{7.85+0.1x_{2}^{i-1}+0.3x_{3}^{i-1}}{3}$ | | $\\frac{7.85+0.1x_{2}^{i-1}+0.3x_{3}^{i-1}}{3}$|\n", | |
| "|$x_{2}^{i}=$ | $\\frac{-19.3-0.1x_{1}^{i-1}+0.3x_{3}^{i-1}}{7}$ | | $\\frac{-19.3-0.1x_{1}^{i}+0.3x_{3}^{i-1}}{7}$ |\n", | |
| "|$x_{3}^{i}=$ | $\\frac{71.4+0.1x_{1}^{i-1}+0.2x_{2}^{i-1}}{10}$ | | $\\frac{71.4+0.1x_{1}^{i}+0.2x_{2}^{i}}{10}$|" | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "execution_count": 14, | |
| "metadata": { | |
| "collapsed": false | |
| }, | |
| "outputs": [ | |
| { | |
| "name": "stdout", | |
| "output_type": "stream", | |
| "text": [ | |
| "ba =\n", | |
| "\n", | |
| " 2.6167\n", | |
| " -2.7571\n", | |
| " 7.1400\n", | |
| "\n", | |
| "sA =\n", | |
| "\n", | |
| " 0.00000 -0.10000 -0.20000\n", | |
| " 0.10000 0.00000 -0.30000\n", | |
| " 0.30000 -0.20000 0.00000\n", | |
| "\n", | |
| "sA =\n", | |
| "\n", | |
| " 0.000000 -0.033333 -0.066667\n", | |
| " 0.014286 0.000000 -0.042857\n", | |
| " 0.030000 -0.020000 0.000000\n", | |
| "\n", | |
| "x1 =\n", | |
| "\n", | |
| " 2.6167\n", | |
| " -2.7571\n", | |
| " 7.1400\n", | |
| "\n", | |
| "x2 =\n", | |
| "\n", | |
| " 3.0008\n", | |
| " -2.4885\n", | |
| " 7.0064\n", | |
| "\n", | |
| "x3 =\n", | |
| "\n", | |
| " 3.0008\n", | |
| " -2.4997\n", | |
| " 7.0002\n", | |
| "\n", | |
| "solution is converging to [3,-2.5,7]]\n" | |
| ] | |
| } | |
| ], | |
| "source": [ | |
| "ba=b./diag(A) % or ba=b./[A(1,1);A(2,2);A(3,3)]\n", | |
| "sA=A-diag(diag(A)) % A with zeros on diagonal\n", | |
| "sA(1,:)=sA(1,:)/A(1,1);\n", | |
| "sA(2,:)=sA(2,:)/A(2,2);\n", | |
| "sA(3,:)=sA(3,:)/A(3,3)\n", | |
| "x0=[0;0;0];\n", | |
| "x1=ba-sA*x0\n", | |
| "x2=ba-sA*x1\n", | |
| "x3=ba-sA*x2\n", | |
| "fprintf('solution is converging to [3,-2.5,7]]\\n')" | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "execution_count": 16, | |
| "metadata": { | |
| "collapsed": false | |
| }, | |
| "outputs": [ | |
| { | |
| "name": "stdout", | |
| "output_type": "stream", | |
| "text": [ | |
| "ans =\n", | |
| "\n", | |
| " 3\n", | |
| " 7\n", | |
| " 10\n", | |
| "\n", | |
| "ans =\n", | |
| "\n", | |
| "Diagonal Matrix\n", | |
| "\n", | |
| " 3 0 0\n", | |
| " 0 7 0\n", | |
| " 0 0 10\n", | |
| "\n" | |
| ] | |
| } | |
| ], | |
| "source": [ | |
| "diag(A)\n", | |
| "diag(diag(A))" | |
| ] | |
| }, | |
| { | |
| "cell_type": "markdown", | |
| "metadata": {}, | |
| "source": [ | |
| "This method works if problem is diagonally dominant, \n", | |
| "\n", | |
| "$|a_{ii}|>\\sum_{j=1,j\\ne i}^{n}|a_{ij}|$\n", | |
| "\n", | |
| "If this condition is true, then Jacobi or Gauss-Seidel should converge\n", | |
| "\n" | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "execution_count": 15, | |
| "metadata": { | |
| "collapsed": false | |
| }, | |
| "outputs": [ | |
| { | |
| "name": "stdout", | |
| "output_type": "stream", | |
| "text": [ | |
| "A =\n", | |
| "\n", | |
| " 0.10000 1.00000 3.00000\n", | |
| " 1.00000 0.20000 3.00000\n", | |
| " 5.00000 2.00000 0.30000\n", | |
| "\n", | |
| "b =\n", | |
| "\n", | |
| " 12\n", | |
| " 2\n", | |
| " 4\n", | |
| "\n", | |
| "ans =\n", | |
| "\n", | |
| " -2.9393\n", | |
| " 9.1933\n", | |
| " 1.0336\n", | |
| "\n" | |
| ] | |
| } | |
| ], | |
| "source": [ | |
| "A=[0.1,1,3;1,0.2,3;5,2,0.3]\n", | |
| "b=[12;2;4]\n", | |
| "A\\b" | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "execution_count": 16, | |
| "metadata": { | |
| "collapsed": false | |
| }, | |
| "outputs": [ | |
| { | |
| "name": "stdout", | |
| "output_type": "stream", | |
| "text": [ | |
| "ba =\n", | |
| "\n", | |
| " 120.000\n", | |
| " 10.000\n", | |
| " 13.333\n", | |
| "\n", | |
| "sA =\n", | |
| "\n", | |
| " 0 1 3\n", | |
| " 1 0 3\n", | |
| " 5 2 0\n", | |
| "\n", | |
| "sA =\n", | |
| "\n", | |
| " 0.00000 10.00000 30.00000\n", | |
| " 5.00000 0.00000 15.00000\n", | |
| " 16.66667 6.66667 0.00000\n", | |
| "\n", | |
| "x1 =\n", | |
| "\n", | |
| " 120.000\n", | |
| " 10.000\n", | |
| " 13.333\n", | |
| "\n", | |
| "x2 =\n", | |
| "\n", | |
| " -380.00\n", | |
| " -790.00\n", | |
| " -2053.33\n", | |
| "\n", | |
| "x3 =\n", | |
| "\n", | |
| " 6.9620e+04\n", | |
| " 3.2710e+04\n", | |
| " 1.1613e+04\n", | |
| "\n", | |
| "solution is not converging to [-2.93,9.19,1.03]\n" | |
| ] | |
| } | |
| ], | |
| "source": [ | |
| "ba=b./diag(A) % or ba=b./[A(1,1);A(2,2);A(3,3)]\n", | |
| "sA=A-diag(diag(A)) % A with zeros on diagonal\n", | |
| "sA(1,:)=sA(1,:)/A(1,1);\n", | |
| "sA(2,:)=sA(2,:)/A(2,2);\n", | |
| "sA(3,:)=sA(3,:)/A(3,3)\n", | |
| "x0=[0;0;0];\n", | |
| "x1=ba-sA*x0\n", | |
| "x2=ba-sA*x1\n", | |
| "x3=ba-sA*x2\n", | |
| "fprintf('solution is not converging to [-2.93,9.19,1.03]\\n')" | |
| ] | |
| }, | |
| { | |
| "cell_type": "markdown", | |
| "metadata": {}, | |
| "source": [ | |
| "## Gauss-Seidel with Relaxation\n", | |
| "\n", | |
| "In order to force the solution to converge faster, we can introduce a relaxation term $\\lambda$. \n", | |
| "\n", | |
| "where the new x values are weighted between the old and new:\n", | |
| "\n", | |
| "$x^{i}=\\lambda x^{i}+(1-\\lambda)x^{i-1}$\n", | |
| "\n", | |
| "after solving for x, lambda weights the current approximation with the previous approximation for the updated x\n" | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "execution_count": 17, | |
| "metadata": { | |
| "collapsed": false | |
| }, | |
| "outputs": [ | |
| { | |
| "name": "stdout", | |
| "output_type": "stream", | |
| "text": [ | |
| "A =\n", | |
| "\n", | |
| " 3.00000 -0.10000 -0.20000\n", | |
| " 0.10000 7.00000 -0.30000\n", | |
| " 0.30000 -0.20000 10.00000\n", | |
| "\n", | |
| "b =\n", | |
| "\n", | |
| " 7.8500\n", | |
| " -19.3000\n", | |
| " 71.4000\n", | |
| "\n" | |
| ] | |
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| "\t</g>\n", | |
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| "</g>\n", | |
| "<g color=\"black\" fill=\"none\" stroke=\"black\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\" stroke-width=\"0.50\">\n", | |
| "</g>\n", | |
| "<g color=\"black\" fill=\"none\" stroke=\"currentColor\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\" stroke-width=\"0.50\">\n", | |
| "</g>\n", | |
| "</g>\n", | |
| "</svg>" | |
| ], | |
| "text/plain": [ | |
| "<IPython.core.display.SVG object>" | |
| ] | |
| }, | |
| "metadata": {}, | |
| "output_type": "display_data" | |
| } | |
| ], | |
| "source": [ | |
| "% rearrange A and b\n", | |
| "A=[3 -0.1 -0.2;0.1 7 -0.3;0.3 -0.2 10]\n", | |
| "b=[7.85;-19.3;71.4]\n", | |
| "\n", | |
| "iters=zeros(100,1);\n", | |
| "for i=1:100\n", | |
| " lambda=2/100*i;\n", | |
| " [x,ea,iters(i)]=Jacobi_rel(A,b,lambda);\n", | |
| "end\n", | |
| "plot([1:100]*2/100,iters) " | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "execution_count": 107, | |
| "metadata": { | |
| "collapsed": false | |
| }, | |
| "outputs": [ | |
| { | |
| "name": "stdout", | |
| "output_type": "stream", | |
| "text": [ | |
| "l = 0.99158\r\n" | |
| ] | |
| } | |
| ], | |
| "source": [ | |
| "l=fminbnd(@(l) lambda_fcn(A,b,l),0.5,1.5)" | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "execution_count": 108, | |
| "metadata": { | |
| "collapsed": false | |
| }, | |
| "outputs": [ | |
| { | |
| "name": "stdout", | |
| "output_type": "stream", | |
| "text": [ | |
| "ans =\n", | |
| "\n", | |
| " 3.0000\n", | |
| " -2.5000\n", | |
| " 7.0000\n", | |
| "\n" | |
| ] | |
| } | |
| ], | |
| "source": [ | |
| "A\\b" | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "execution_count": 109, | |
| "metadata": { | |
| "collapsed": false | |
| }, | |
| "outputs": [ | |
| { | |
| "name": "stdout", | |
| "output_type": "stream", | |
| "text": [ | |
| "x =\n", | |
| "\n", | |
| " 3.0000\n", | |
| " -2.5000\n", | |
| " 7.0000\n", | |
| "\n", | |
| "ea =\n", | |
| "\n", | |
| " 1.8289e-07\n", | |
| " 2.1984e-08\n", | |
| " 2.3864e-08\n", | |
| "\n", | |
| "iter = 8\n", | |
| "x =\n", | |
| "\n", | |
| " 3.0000\n", | |
| " -2.5000\n", | |
| " 7.0000\n", | |
| "\n", | |
| "ea =\n", | |
| "\n", | |
| " 1.9130e-08\n", | |
| " 7.6449e-08\n", | |
| " 3.3378e-08\n", | |
| "\n", | |
| "iter = 8\n" | |
| ] | |
| } | |
| ], | |
| "source": [ | |
| "[x,ea,iter]=Jacobi_rel(A,b,l,0.000001)\n", | |
| "[x,ea,iter]=Jacobi_rel(A,b,1,0.000001)\n" | |
| ] | |
| }, | |
| { | |
| "cell_type": "markdown", | |
| "metadata": {}, | |
| "source": [ | |
| "## Nonlinear Systems\n", | |
| "\n", | |
| "Consider two simultaneous nonlinear equations with two unknowns:\n", | |
| "\n", | |
| "$x_{1}^{2}+x_{1}x_{2}=10$\n", | |
| "\n", | |
| "$x_{2}+3x_{1}x_{2}^{2}=57$\n", | |
| "\n", | |
| "Graphically, we are looking for the solution:\n" | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "execution_count": 19, | |
| "metadata": { | |
| "collapsed": false | |
| }, | |
| "outputs": [ | |
| { | |
| "data": { | |
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| "\t <feFlood flood-color=\"white\" flood-opacity=\"1\" result=\"bgnd\"/>\n", | |
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| "<g color=\"black\" fill=\"none\" stroke=\"currentColor\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\" stroke-width=\"0.50\">\n", | |
| "\t<path d=\"M55.3,344.4 L67.8,344.4 M535.0,344.4 L522.5,344.4 \" stroke=\"black\"/>\t<g fill=\"rgb(0,0,0)\" font-family=\"{}\" font-size=\"16.00\" stroke=\"none\" text-anchor=\"end\" transform=\"translate(47.0,350.4)\">\n", | |
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| "\t<path d=\"M55.3,262.5 L67.8,262.5 M535.0,262.5 L522.5,262.5 \" stroke=\"black\"/>\t<g fill=\"rgb(0,0,0)\" font-family=\"{}\" font-size=\"16.00\" stroke=\"none\" text-anchor=\"end\" transform=\"translate(47.0,268.5)\">\n", | |
| "\t\t<text><tspan font-family=\"{}\">5</tspan></text>\n", | |
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| "</g>\n", | |
| "<g color=\"black\" fill=\"none\" stroke=\"currentColor\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\" stroke-width=\"0.50\">\n", | |
| "\t<path d=\"M55.3,180.5 L67.8,180.5 M535.0,180.5 L522.5,180.5 \" stroke=\"black\"/>\t<g fill=\"rgb(0,0,0)\" font-family=\"{}\" font-size=\"16.00\" stroke=\"none\" text-anchor=\"end\" transform=\"translate(47.0,186.5)\">\n", | |
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| "\t\t<text><tspan font-family=\"{}\">0</tspan></text>\n", | |
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| "\t\t<text><tspan font-family=\"{}\">1</tspan></text>\n", | |
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| "\t<g id=\"gnuplot_plot_3a\"><title>gnuplot_plot_3a</title>\n", | |
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| "source": [ | |
| "x11=linspace(0.5,3);\n", | |
| "x12=(10-x11.^2)./x11;\n", | |
| "\n", | |
| "x22=linspace(2,8);\n", | |
| "x21=(57-x22).*x22.^-2/3;\n", | |
| "\n", | |
| "plot(x11,x12,x21,x22)\n", | |
| "% Solution at x_1=2, x_2=3\n", | |
| "hold on;\n", | |
| "plot(2,3,'o')\n", | |
| "xlabel('x_1')\n", | |
| "ylabel('x_2')" | |
| ] | |
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| { | |
| "cell_type": "markdown", | |
| "metadata": {}, | |
| "source": [ | |
| "## Newton-Raphson part II\n", | |
| "\n", | |
| "Remember the first order approximation for the next point in a function is:\n", | |
| "\n", | |
| "$f(x_{i+1})=f(x_{i})+(x_{i+1}-x_{i})f'(x_{i})$\n", | |
| "\n", | |
| "then, $f(x_{i+1})=0$ so we are left with:\n", | |
| "\n", | |
| "$x_{i+1}=x_{i}-\\frac{f(x_{i})}{f'(x_{i})}$\n", | |
| "\n", | |
| "We can use the same formula, but now we have multiple dimensions so we need to determine the Jacobian\n", | |
| "\n", | |
| "$[J]=\\left[ \\begin{array}{cccc}\n", | |
| "\\frac{\\partial f_{1,i}}{\\partial x_{1}} & \\frac{\\partial f_{1,i}}{\\partial x_{2}} & \n", | |
| "\\cdots & \\frac{\\partial f_{1,i}}{\\partial x_{n}} \\\\\n", | |
| "\\frac{\\partial f_{2,i}}{\\partial x_{1}} & \\frac{\\partial f_{2,i}}{\\partial x_{2}} & \n", | |
| "\\cdots & \\frac{\\partial f_{2,i}}{\\partial x_{n}} \\\\\n", | |
| "\\vdots & \\vdots & & \\vdots \\\\\n", | |
| "\\frac{\\partial f_{n,i}}{\\partial x_{1}} & \\frac{\\partial f_{n,i}}{\\partial x_{2}} & \n", | |
| "\\cdots & \\frac{\\partial f_{n,i}}{\\partial x_{n}} \\\\\n", | |
| "\\end{array} \\right]$\n", | |
| "\n", | |
| "$\\left[ \\begin{array}{c}\n", | |
| "f_{1,i+1} \\\\\n", | |
| "f_{2,i+1} \\\\\n", | |
| "\\vdots \\\\\n", | |
| "f_{n,i+1}\\end{array} \\right]=\n", | |
| "\\left[ \\begin{array}{c}\n", | |
| "f_{1,i} \\\\\n", | |
| "f_{2,i} \\\\\n", | |
| "\\vdots \\\\\n", | |
| "f_{n,i}\\end{array} \\right]+\n", | |
| "\\left[ \\begin{array}{cccc}\n", | |
| "\\frac{\\partial f_{1,i}}{\\partial x_{1}} & \\frac{\\partial f_{1,i}}{\\partial x_{2}} & \n", | |
| "\\cdots & \\frac{\\partial f_{1,i}}{\\partial x_{n}} \\\\\n", | |
| "\\frac{\\partial f_{2,i}}{\\partial x_{1}} & \\frac{\\partial f_{2,i}}{\\partial x_{2}} & \n", | |
| "\\cdots & \\frac{\\partial f_{2,i}}{\\partial x_{n}} \\\\\n", | |
| "\\vdots & \\vdots & & \\vdots \\\\\n", | |
| "\\frac{\\partial f_{n,i}}{\\partial x_{1}} & \\frac{\\partial f_{n,i}}{\\partial x_{2}} & \n", | |
| "\\cdots & \\frac{\\partial f_{n,i}}{\\partial x_{n}} \\\\\n", | |
| "\\end{array} \\right]\n", | |
| "\\left( \\left[ \\begin{array}{c}\n", | |
| "x_{i+1} \\\\\n", | |
| "x_{i+1} \\\\\n", | |
| "\\vdots \\\\\n", | |
| "x_{i+1}\\end{array} \\right]-\n", | |
| "\\left[ \\begin{array}{c}\n", | |
| "x_{1,i} \\\\\n", | |
| "x_{2,i} \\\\\n", | |
| "\\vdots \\\\\n", | |
| "x_{n,i}\\end{array} \\right]\\right)$\n", | |
| "\n", | |
| "### Solution is again in the form Ax=b\n", | |
| "\n", | |
| "$[J]([x_{i+1}]-[x_{i}])=-[f]$\n", | |
| "\n", | |
| "so\n", | |
| "\n", | |
| "$[x_{i+1}]= [x_{i}]-[J]^{-1}[f]$\n", | |
| "\n", | |
| "## Example of Jacobian calculation\n", | |
| "\n", | |
| "### Nonlinear springs supporting two masses in series\n", | |
| "\n", | |
| "Two springs are connected to two masses, with $m_1$=1 kg and $m_{2}$=2 kg. The springs are identical, but they have nonlinear spring constants, of $k_1$=100 N/m and $k_2$=-10 N/m\n", | |
| "\n", | |
| "We want to solve for the final position of the masses ($x_1$ and $x_2$)\n", | |
| "\n", | |
| "$m_{1}g+k_{1}(x_{2}-x_{1})+k_{2}(x_{2}-x_{1})^{2}+k_{1}x_{1}+k_{2}x_{1}^{2}=0$\n", | |
| "\n", | |
| "$m_{2}g-k_{1}(x_{2}-x_{1})-k_{2}(x_2-x_1)^{2}=0$\n", | |
| "\n", | |
| "$J(1,1)=\\frac{\\partial f_{1}}{\\partial x_{1}}=-k_{1}-2k_{2}(x_{2}-x_{1})+k_{1}+2k_{2}x_{1}$\n", | |
| "\n", | |
| "$J(1,2)=\\frac{\\partial f_1}{\\partial x_{2}}=k_{1}+2k_{2}(x_{2}-x_{1})$\n", | |
| "\n", | |
| "$J(2,1)=\\frac{\\partial f_2}{\\partial x_{1}}=k_{1}+2k_{2}(x_{2}-x_{1})$\n", | |
| "\n", | |
| "$J(2,2)=\\frac{\\partial f_2}{\\partial x_{2}}=-k_{1}-2k_{2}(x_{2}-x_{1})$\n", | |
| "\n" | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "execution_count": null, | |
| "metadata": { | |
| "collapsed": true | |
| }, | |
| "outputs": [], | |
| "source": [ | |
| "m1=1; % kg \n", | |
| "m2=2; % kg\n", | |
| "k1=100; % N/m\n", | |
| "k2=-10; % N/m^2" | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "execution_count": 20, | |
| "metadata": { | |
| "collapsed": false | |
| }, | |
| "outputs": [], | |
| "source": [ | |
| "function [f,J]=mass_spring(x)\n", | |
| " % Function to calculate function values f1 and f2 as well as Jacobian \n", | |
| " % for 2 masses and 2 identical nonlinear springs\n", | |
| " m1=1; % kg \n", | |
| " m2=2; % kg\n", | |
| " k1=100; % N/m\n", | |
| " k2=-10; % N/m^2\n", | |
| " g=9.81; % m/s^2\n", | |
| " x1=x(1);\n", | |
| " x2=x(2);\n", | |
| " J=[-k1-2*k2*(x2-x1)-k1-2*k2*x1,k1+2*k2*(x2-x1);\n", | |
| " k1+2*k2*(x2-x1),-k1-2*k2*(x2-x1)];\n", | |
| " f=[m1*g+k1*(x2-x1)+k2*(x2-x1).^2-k1*x1-k2*x1^2;\n", | |
| " m2*g-k1*(x2-x1)-k2*(x2-x1).^2];\n", | |
| "end\n", | |
| " " | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "execution_count": 21, | |
| "metadata": { | |
| "collapsed": false | |
| }, | |
| "outputs": [ | |
| { | |
| "name": "stdout", | |
| "output_type": "stream", | |
| "text": [ | |
| "f =\n", | |
| "\n", | |
| " -190.19\n", | |
| " 129.62\n", | |
| "\n", | |
| "J =\n", | |
| "\n", | |
| " -200 120\n", | |
| " 120 -120\n", | |
| "\n" | |
| ] | |
| } | |
| ], | |
| "source": [ | |
| "[f,J]=mass_spring([1,0])" | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "execution_count": 22, | |
| "metadata": { | |
| "collapsed": false | |
| }, | |
| "outputs": [ | |
| { | |
| "name": "stdout", | |
| "output_type": "stream", | |
| "text": [ | |
| "x1 =\n", | |
| "\n", | |
| " -1.5142\n", | |
| " -1.4341\n", | |
| "\n", | |
| "ea =\n", | |
| "\n", | |
| " 2.9812\n", | |
| " 2.3946\n", | |
| "\n", | |
| "x2 =\n", | |
| "\n", | |
| " 0.049894\n", | |
| " 0.248638\n", | |
| "\n", | |
| "ea =\n", | |
| "\n", | |
| " 31.3492\n", | |
| " 6.7678\n", | |
| "\n", | |
| "x3 =\n", | |
| "\n", | |
| " 0.29701\n", | |
| " 0.49722\n", | |
| "\n", | |
| "ea =\n", | |
| "\n", | |
| " 0.83201\n", | |
| " 0.49995\n", | |
| "\n", | |
| "x =\n", | |
| "\n", | |
| " 0.29701\n", | |
| " 0.49722\n", | |
| "\n", | |
| "ea =\n", | |
| "\n", | |
| " 0.021392\n", | |
| " 0.012890\n", | |
| "\n", | |
| "ea =\n", | |
| "\n", | |
| " 1.4786e-05\n", | |
| " 8.9091e-06\n", | |
| "\n", | |
| "ea =\n", | |
| "\n", | |
| " 7.0642e-12\n", | |
| " 4.2565e-12\n", | |
| "\n" | |
| ] | |
| } | |
| ], | |
| "source": [ | |
| "x0=[3;2];\n", | |
| "[f0,J0]=mass_spring(x0);\n", | |
| "x1=x0-J0\\f0\n", | |
| "ea=(x1-x0)./x1\n", | |
| "[f1,J1]=mass_spring(x1);\n", | |
| "x2=x1-J1\\f1\n", | |
| "ea=(x2-x1)./x2\n", | |
| "[f2,J2]=mass_spring(x2);\n", | |
| "x3=x2-J2\\f2\n", | |
| "ea=(x3-x2)./x3\n", | |
| "x=x3\n", | |
| "for i=1:3\n", | |
| " xold=x;\n", | |
| " [f,J]=mass_spring(x);\n", | |
| " x=x-J\\f;\n", | |
| " ea=(x-xold)./x\n", | |
| "end" | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "execution_count": 23, | |
| "metadata": { | |
| "collapsed": false | |
| }, | |
| "outputs": [ | |
| { | |
| "name": "stdout", | |
| "output_type": "stream", | |
| "text": [ | |
| "x =\n", | |
| "\n", | |
| " 0.30351\n", | |
| " 0.50372\n", | |
| "\n", | |
| "X0 =\n", | |
| "\n", | |
| " 0.30351\n", | |
| " 0.50372\n", | |
| "\n" | |
| ] | |
| } | |
| ], | |
| "source": [ | |
| "x\n", | |
| "X0=fsolve(@(x) mass_spring(x),[3;5])" | |
| ] | |
| }, | |
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| "cell_type": "code", | |
| "execution_count": 26, | |
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| "\t</g>\n", | |
| "\t<g id=\"gnuplot_plot_2a\"><title>gnuplot_plot_2a</title>\n", | |
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| "</g>\n", | |
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| "</g>\n", | |
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| "<g color=\"black\" fill=\"none\" stroke=\"currentColor\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\" stroke-width=\"0.50\">\n", | |
| "\t<path d=\"M223.8,372.3 L74.3,286.4 \" stroke=\"black\"/></g>\n", | |
| "<g color=\"black\" fill=\"none\" stroke=\"currentColor\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\" stroke-width=\"0.50\">\n", | |
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| "<g color=\"black\" fill=\"none\" stroke=\"currentColor\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\" stroke-width=\"0.50\">\n", | |
| "\t<path d=\"M418.5,306.4 L327.2,253.9 \" stroke=\"black\"/></g>\n", | |
| "<g color=\"black\" fill=\"none\" stroke=\"currentColor\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\" stroke-width=\"0.50\">\n", | |
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| "<g color=\"black\" fill=\"none\" stroke=\"currentColor\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\" stroke-width=\"0.50\">\n", | |
| "\t<path d=\"M269.0,33.1 L269.0,188.1 \" stroke=\"black\"/></g>\n", | |
| "<g color=\"black\" fill=\"none\" stroke=\"currentColor\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\" stroke-width=\"0.50\">\n", | |
| "\t<path d=\"M418.5,119.0 L418.5,306.4 \" stroke=\"black\"/></g>\n", | |
| "<g color=\"black\" fill=\"none\" stroke=\"currentColor\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\" stroke-width=\"0.50\">\n", | |
| "\t<path d=\"M223.8,339.8 L223.8,372.3 \" stroke=\"black\"/></g>\n", | |
| "<g color=\"black\" fill=\"none\" stroke=\"currentColor\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\" stroke-width=\"0.50\">\n", | |
| "\t<path d=\"M74.3,99.0 L269.0,33.1 \" stroke=\"black\"/></g>\n", | |
| "<g color=\"black\" fill=\"none\" stroke=\"currentColor\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\" stroke-width=\"0.50\">\n", | |
| "\t<path d=\"M418.5,119.0 L269.0,33.1 \" stroke=\"black\"/></g>\n", | |
| "<g color=\"gray\" fill=\"none\" stroke=\"currentColor\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\" stroke-width=\"0.50\">\n", | |
| "\t<path class=\"gridline\" d=\"M223.8,372.3 L74.3,286.4 \" stroke=\"gray\" stroke-dasharray=\"2,4\"/></g>\n", | |
| "<g color=\"black\" fill=\"none\" stroke=\"currentColor\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\" stroke-width=\"0.50\">\n", | |
| "\t<path d=\"M232.5,377.3 L223.8,372.3 \" stroke=\"black\"/></g>\n", | |
| "<g color=\"black\" fill=\"none\" stroke=\"currentColor\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\" stroke-width=\"0.50\">\n", | |
| "\t<path d=\"M74.3,286.4 L65.6,281.4 \" stroke=\"black\"/>\t<g fill=\"rgb(0,0,0)\" font-family=\"{}\" font-size=\"16.00\" stroke=\"none\" text-anchor=\"middle\" transform=\"translate(238.2,390.4)\">\n", | |
| "\t\t<text><tspan font-family=\"{}\">0</tspan></text>\n", | |
| "\t</g>\n", | |
| "</g>\n", | |
| "<g color=\"black\" fill=\"none\" stroke=\"currentColor\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\" stroke-width=\"0.50\">\n", | |
| "</g>\n", | |
| "<g color=\"gray\" fill=\"none\" stroke=\"currentColor\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\" stroke-width=\"0.50\">\n", | |
| "\t<path class=\"gridline\" d=\"M262.7,359.1 L226.3,338.1 \" stroke=\"gray\" stroke-dasharray=\"2,4\"/></g>\n", | |
| "<g color=\"black\" fill=\"none\" stroke=\"currentColor\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\" stroke-width=\"0.50\">\n", | |
| "\t<path d=\"M271.4,364.1 L262.7,359.1 \" stroke=\"black\"/>\t<g fill=\"rgb(0,0,0)\" font-family=\"{}\" font-size=\"16.00\" stroke=\"none\" text-anchor=\"middle\" transform=\"translate(277.1,377.2)\">\n", | |
| "\t\t<text><tspan font-family=\"{}\">2</tspan></text>\n", | |
| "\t</g>\n", | |
| "</g>\n", | |
| "<g color=\"black\" fill=\"none\" stroke=\"currentColor\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\" stroke-width=\"0.50\">\n", | |
| "</g>\n", | |
| "<g color=\"gray\" fill=\"none\" stroke=\"currentColor\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\" stroke-width=\"0.50\">\n", | |
| "\t<path class=\"gridline\" d=\"M301.6,345.9 L253.8,318.5 \" stroke=\"gray\" stroke-dasharray=\"2,4\"/></g>\n", | |
| "<g color=\"black\" fill=\"none\" stroke=\"currentColor\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\" stroke-width=\"0.50\">\n", | |
| "\t<path d=\"M310.3,350.9 L301.6,345.9 \" stroke=\"black\"/>\t<g fill=\"rgb(0,0,0)\" font-family=\"{}\" font-size=\"16.00\" stroke=\"none\" text-anchor=\"middle\" transform=\"translate(316.1,364.1)\">\n", | |
| "\t\t<text><tspan font-family=\"{}\">4</tspan></text>\n", | |
| "\t</g>\n", | |
| "</g>\n", | |
| "<g color=\"black\" fill=\"none\" stroke=\"currentColor\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\" stroke-width=\"0.50\">\n", | |
| "</g>\n", | |
| "<g color=\"gray\" fill=\"none\" stroke=\"currentColor\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\" stroke-width=\"0.50\">\n", | |
| "\t<path class=\"gridline\" d=\"M340.6,332.7 L279.6,297.7 \" stroke=\"gray\" stroke-dasharray=\"2,4\"/></g>\n", | |
| "<g color=\"black\" fill=\"none\" stroke=\"currentColor\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\" stroke-width=\"0.50\">\n", | |
| "\t<path d=\"M349.3,337.7 L340.6,332.7 \" stroke=\"black\"/>\t<g fill=\"rgb(0,0,0)\" font-family=\"{}\" font-size=\"16.00\" stroke=\"none\" text-anchor=\"middle\" transform=\"translate(355.0,350.9)\">\n", | |
| "\t\t<text><tspan font-family=\"{}\">6</tspan></text>\n", | |
| "\t</g>\n", | |
| "</g>\n", | |
| "<g color=\"black\" fill=\"none\" stroke=\"currentColor\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\" stroke-width=\"0.50\">\n", | |
| "</g>\n", | |
| "<g color=\"gray\" fill=\"none\" stroke=\"currentColor\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\" stroke-width=\"0.50\">\n", | |
| "\t<path class=\"gridline\" d=\"M379.6,319.6 L304.1,276.2 \" stroke=\"gray\" stroke-dasharray=\"2,4\"/></g>\n", | |
| "<g color=\"black\" fill=\"none\" stroke=\"currentColor\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\" stroke-width=\"0.50\">\n", | |
| "\t<path d=\"M388.3,324.6 L379.6,319.6 \" stroke=\"black\"/>\t<g fill=\"rgb(0,0,0)\" font-family=\"{}\" font-size=\"16.00\" stroke=\"none\" text-anchor=\"middle\" transform=\"translate(394.0,337.7)\">\n", | |
| "\t\t<text><tspan font-family=\"{}\">8</tspan></text>\n", | |
| "\t</g>\n", | |
| "</g>\n", | |
| "<g color=\"black\" fill=\"none\" stroke=\"currentColor\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\" stroke-width=\"0.50\">\n", | |
| "</g>\n", | |
| "<g color=\"gray\" fill=\"none\" stroke=\"currentColor\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\" stroke-width=\"0.50\">\n", | |
| "\t<path class=\"gridline\" d=\"M418.5,306.4 L327.2,253.9 \" stroke=\"gray\" stroke-dasharray=\"2,4\"/></g>\n", | |
| "<g color=\"black\" fill=\"none\" stroke=\"currentColor\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\" stroke-width=\"0.50\">\n", | |
| "\t<path d=\"M427.2,311.4 L418.5,306.4 \" stroke=\"black\"/>\t<g fill=\"rgb(0,0,0)\" font-family=\"{}\" font-size=\"16.00\" stroke=\"none\" text-anchor=\"middle\" transform=\"translate(433.0,324.5)\">\n", | |
| "\t\t<text><tspan font-family=\"{}\">10</tspan></text>\n", | |
| "\t</g>\n", | |
| "</g>\n", | |
| "<g color=\"black\" fill=\"none\" stroke=\"currentColor\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\" stroke-width=\"0.50\">\n", | |
| "\t<g fill=\"rgb(0,0,0)\" font-family=\"{}\" font-size=\"16.00\" stroke=\"none\" text-anchor=\"middle\" transform=\"translate(367.2,371.8)\">\n", | |
| "\t\t<text><tspan font-family=\"{}\">x</tspan><tspan dy=\"4.80px\" font-family=\"{}\" font-size=\"12.8\">1</tspan><tspan dy=\"-4.80\" font-size=\"16.0\"/></text>\n", | |
| "\t</g>\n", | |
| "</g>\n", | |
| "<g color=\"black\" fill=\"none\" stroke=\"currentColor\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\" stroke-width=\"0.50\">\n", | |
| "</g>\n", | |
| "<g color=\"gray\" fill=\"none\" stroke=\"currentColor\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\" stroke-width=\"0.50\">\n", | |
| "\t<path class=\"gridline\" d=\"M223.8,372.3 L418.5,306.4 \" stroke=\"gray\" stroke-dasharray=\"2,4\"/></g>\n", | |
| "<g color=\"black\" fill=\"none\" stroke=\"currentColor\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\" stroke-width=\"0.50\">\n", | |
| "\t<path d=\"M213.8,375.7 L223.8,372.3 \" stroke=\"black\"/></g>\n", | |
| "<g color=\"black\" fill=\"none\" stroke=\"currentColor\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\" stroke-width=\"0.50\">\n", | |
| "\t<path d=\"M418.5,306.4 L428.5,303.0 \" stroke=\"black\"/>\t<g fill=\"rgb(0,0,0)\" font-family=\"{}\" font-size=\"16.00\" stroke=\"none\" text-anchor=\"middle\" transform=\"translate(207.2,386.5)\">\n", | |
| "\t\t<text><tspan font-family=\"{}\">0</tspan></text>\n", | |
| "\t</g>\n", | |
| "</g>\n", | |
| "<g color=\"black\" fill=\"none\" stroke=\"currentColor\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\" stroke-width=\"0.50\">\n", | |
| "</g>\n", | |
| "<g color=\"gray\" fill=\"none\" stroke=\"currentColor\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\" stroke-width=\"0.50\">\n", | |
| "\t<path class=\"gridline\" d=\"M193.9,355.1 L388.6,289.2 \" stroke=\"gray\" stroke-dasharray=\"2,4\"/></g>\n", | |
| "<g color=\"black\" fill=\"none\" stroke=\"currentColor\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\" stroke-width=\"0.50\">\n", | |
| "\t<path d=\"M183.9,358.5 L193.9,355.1 \" stroke=\"black\"/></g>\n", | |
| "<g color=\"black\" fill=\"none\" stroke=\"currentColor\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\" stroke-width=\"0.50\">\n", | |
| "\t<path d=\"M388.6,289.2 L398.6,285.8 \" stroke=\"black\"/>\t<g fill=\"rgb(0,0,0)\" font-family=\"{}\" font-size=\"16.00\" stroke=\"none\" text-anchor=\"middle\" transform=\"translate(177.3,369.3)\">\n", | |
| "\t\t<text><tspan font-family=\"{}\">2</tspan></text>\n", | |
| "\t</g>\n", | |
| "</g>\n", | |
| "<g color=\"black\" fill=\"none\" stroke=\"currentColor\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\" stroke-width=\"0.50\">\n", | |
| "</g>\n", | |
| "<g color=\"gray\" fill=\"none\" stroke=\"currentColor\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\" stroke-width=\"0.50\">\n", | |
| "\t<path class=\"gridline\" d=\"M277.6,299.5 L358.7,272.0 \" stroke=\"gray\" stroke-dasharray=\"2,4\"/></g>\n", | |
| "<g color=\"gray\" fill=\"none\" stroke=\"currentColor\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\" stroke-width=\"0.50\">\n", | |
| "\t<path class=\"gridline\" d=\"M164.0,337.9 L186.9,330.2 \" stroke=\"gray\" stroke-dasharray=\"2,4\"/></g>\n", | |
| "<g color=\"black\" fill=\"none\" stroke=\"currentColor\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\" stroke-width=\"0.50\">\n", | |
| "\t<path d=\"M154.0,341.3 L164.0,337.9 \" stroke=\"black\"/></g>\n", | |
| "<g color=\"black\" fill=\"none\" stroke=\"currentColor\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\" stroke-width=\"0.50\">\n", | |
| "\t<path d=\"M358.7,272.0 L368.7,268.7 \" stroke=\"black\"/>\t<g fill=\"rgb(0,0,0)\" font-family=\"{}\" font-size=\"16.00\" stroke=\"none\" text-anchor=\"middle\" transform=\"translate(147.4,352.1)\">\n", | |
| "\t\t<text><tspan font-family=\"{}\">4</tspan></text>\n", | |
| "\t</g>\n", | |
| "</g>\n", | |
| "<g color=\"black\" fill=\"none\" stroke=\"currentColor\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\" stroke-width=\"0.50\">\n", | |
| "</g>\n", | |
| "<g color=\"gray\" fill=\"none\" stroke=\"currentColor\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\" stroke-width=\"0.50\">\n", | |
| "\t<path class=\"gridline\" d=\"M325.0,256.2 L328.8,254.9 \" stroke=\"gray\" stroke-dasharray=\"2,4\"/></g>\n", | |
| "<g color=\"gray\" fill=\"none\" stroke=\"currentColor\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\" stroke-width=\"0.50\">\n", | |
| "\t<path class=\"gridline\" d=\"M134.1,320.8 L152.6,314.5 \" stroke=\"gray\" stroke-dasharray=\"2,4\"/></g>\n", | |
| "<g color=\"black\" fill=\"none\" stroke=\"currentColor\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\" stroke-width=\"0.50\">\n", | |
| "\t<path d=\"M124.1,324.1 L134.1,320.8 \" stroke=\"black\"/></g>\n", | |
| "<g color=\"black\" fill=\"none\" stroke=\"currentColor\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\" stroke-width=\"0.50\">\n", | |
| "\t<path d=\"M328.8,254.9 L338.8,251.5 \" stroke=\"black\"/>\t<g fill=\"rgb(0,0,0)\" font-family=\"{}\" font-size=\"16.00\" stroke=\"none\" text-anchor=\"middle\" transform=\"translate(117.5,335.0)\">\n", | |
| "\t\t<text><tspan font-family=\"{}\">6</tspan></text>\n", | |
| "\t</g>\n", | |
| "</g>\n", | |
| "<g color=\"black\" fill=\"none\" stroke=\"currentColor\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\" stroke-width=\"0.50\">\n", | |
| "</g>\n", | |
| "<g color=\"gray\" fill=\"none\" stroke=\"currentColor\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\" stroke-width=\"0.50\">\n", | |
| "\t<path class=\"gridline\" d=\"M104.2,303.6 L124.5,296.7 \" stroke=\"gray\" stroke-dasharray=\"2,4\"/></g>\n", | |
| "<g color=\"black\" fill=\"none\" stroke=\"currentColor\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\" stroke-width=\"0.50\">\n", | |
| "\t<path d=\"M94.2,307.0 L104.2,303.6 \" stroke=\"black\"/>\t<g fill=\"rgb(0,0,0)\" font-family=\"{}\" font-size=\"16.00\" stroke=\"none\" text-anchor=\"middle\" transform=\"translate(87.6,317.8)\">\n", | |
| "\t\t<text><tspan font-family=\"{}\">8</tspan></text>\n", | |
| "\t</g>\n", | |
| "</g>\n", | |
| "<g color=\"black\" fill=\"none\" stroke=\"currentColor\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\" stroke-width=\"0.50\">\n", | |
| "</g>\n", | |
| "<g color=\"gray\" fill=\"none\" stroke=\"currentColor\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\" stroke-width=\"0.50\">\n", | |
| "\t<path class=\"gridline\" d=\"M74.3,286.4 L100.1,277.7 \" stroke=\"gray\" stroke-dasharray=\"2,4\"/></g>\n", | |
| "<g color=\"black\" fill=\"none\" stroke=\"currentColor\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\" stroke-width=\"0.50\">\n", | |
| "\t<path d=\"M64.3,289.8 L74.3,286.4 \" stroke=\"black\"/>\t<g fill=\"rgb(0,0,0)\" font-family=\"{}\" font-size=\"16.00\" stroke=\"none\" text-anchor=\"middle\" transform=\"translate(57.7,300.6)\">\n", | |
| "\t\t<text><tspan font-family=\"{}\">10</tspan></text>\n", | |
| "\t</g>\n", | |
| "</g>\n", | |
| "<g color=\"black\" fill=\"none\" stroke=\"currentColor\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\" stroke-width=\"0.50\">\n", | |
| "\t<g fill=\"rgb(0,0,0)\" font-family=\"{}\" font-size=\"16.00\" stroke=\"none\" text-anchor=\"middle\" transform=\"translate(90.3,355.2)\">\n", | |
| "\t\t<text><tspan font-family=\"{}\">x</tspan><tspan dy=\"4.80px\" font-family=\"{}\" font-size=\"12.8\">2</tspan><tspan dy=\"-4.80\" font-size=\"16.0\"/></text>\n", | |
| "\t</g>\n", | |
| "</g>\n", | |
| "<g color=\"black\" fill=\"none\" stroke=\"currentColor\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\" stroke-width=\"0.50\">\n", | |
| "</g>\n", | |
| "<g color=\"gray\" fill=\"none\" stroke=\"currentColor\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\" stroke-width=\"0.50\">\n", | |
| "\t<path class=\"gridline\" d=\"M74.3,286.4 L100.1,277.7 \" stroke=\"gray\" stroke-dasharray=\"2,4\"/></g>\n", | |
| "<g color=\"gray\" fill=\"none\" stroke=\"currentColor\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\" stroke-width=\"0.50\">\n", | |
| "\t<path class=\"gridline\" d=\"M418.5,306.4 L327.2,253.9 \" stroke=\"gray\" stroke-dasharray=\"2,4\"/></g>\n", | |
| "<g color=\"black\" fill=\"none\" stroke=\"currentColor\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\" stroke-width=\"0.50\">\n", | |
| "\t<path d=\"M61.8,286.4 L74.3,286.4 \" stroke=\"black\"/>\t<g fill=\"rgb(0,0,0)\" font-family=\"{}\" font-size=\"16.00\" stroke=\"none\" text-anchor=\"end\" transform=\"translate(43.7,292.4)\">\n", | |
| "\t\t<text><tspan font-family=\"{}\">-500</tspan></text>\n", | |
| "\t</g>\n", | |
| "</g>\n", | |
| "<g color=\"black\" fill=\"none\" stroke=\"currentColor\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\" stroke-width=\"0.50\">\n", | |
| "</g>\n", | |
| "<g color=\"black\" fill=\"none\" stroke=\"currentColor\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\" stroke-width=\"0.50\">\n", | |
| "\t<path d=\"M431.0,306.4 L418.5,306.4 \" stroke=\"black\"/></g>\n", | |
| "<g color=\"gray\" fill=\"none\" stroke=\"currentColor\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\" stroke-width=\"0.50\">\n", | |
| "\t<path class=\"gridline\" d=\"M78.3,253.8 L265.0,190.7 \" stroke=\"gray\" stroke-dasharray=\"2,4\"/></g>\n", | |
| "<g color=\"gray\" fill=\"none\" stroke=\"currentColor\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\" stroke-width=\"0.50\">\n", | |
| "\t<path class=\"gridline\" d=\"M74.3,255.2 L75.2,254.8 \" stroke=\"gray\" stroke-dasharray=\"2,4\"/></g>\n", | |
| "<g color=\"gray\" fill=\"none\" stroke=\"currentColor\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\" stroke-width=\"0.50\">\n", | |
| "\t<path class=\"gridline\" d=\"M418.5,275.1 L346.7,233.9 \" stroke=\"gray\" stroke-dasharray=\"2,4\"/></g>\n", | |
| "<g color=\"black\" fill=\"none\" stroke=\"currentColor\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\" stroke-width=\"0.50\">\n", | |
| "\t<path d=\"M61.8,255.2 L74.3,255.2 \" stroke=\"black\"/>\t<g fill=\"rgb(0,0,0)\" font-family=\"{}\" font-size=\"16.00\" stroke=\"none\" text-anchor=\"end\" transform=\"translate(43.7,261.2)\">\n", | |
| "\t\t<text><tspan font-family=\"{}\">0</tspan></text>\n", | |
| "\t</g>\n", | |
| "</g>\n", | |
| "<g color=\"black\" fill=\"none\" stroke=\"currentColor\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\" stroke-width=\"0.50\">\n", | |
| "</g>\n", | |
| "<g color=\"black\" fill=\"none\" stroke=\"currentColor\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\" stroke-width=\"0.50\">\n", | |
| "\t<path d=\"M431.0,275.1 L418.5,275.1 \" stroke=\"black\"/></g>\n", | |
| "<g color=\"gray\" fill=\"none\" stroke=\"currentColor\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\" stroke-width=\"0.50\">\n", | |
| "\t<path class=\"gridline\" d=\"M74.3,223.9 L269.0,158.1 \" stroke=\"gray\" stroke-dasharray=\"2,4\"/></g>\n", | |
| "<g color=\"gray\" fill=\"none\" stroke=\"currentColor\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\" stroke-width=\"0.50\">\n", | |
| "\t<path class=\"gridline\" d=\"M322.0,188.5 L269.0,158.1 \" stroke=\"gray\" stroke-dasharray=\"2,4\"/></g>\n", | |
| "<g color=\"gray\" fill=\"none\" stroke=\"currentColor\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\" stroke-width=\"0.50\">\n", | |
| "\t<path class=\"gridline\" d=\"M418.5,243.9 L365.4,213.4 \" stroke=\"gray\" stroke-dasharray=\"2,4\"/></g>\n", | |
| "<g color=\"black\" fill=\"none\" stroke=\"currentColor\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\" stroke-width=\"0.50\">\n", | |
| "\t<path d=\"M61.8,223.9 L74.3,223.9 \" stroke=\"black\"/>\t<g fill=\"rgb(0,0,0)\" font-family=\"{}\" font-size=\"16.00\" stroke=\"none\" text-anchor=\"end\" transform=\"translate(43.7,229.9)\">\n", | |
| "\t\t<text><tspan font-family=\"{}\">500</tspan></text>\n", | |
| "\t</g>\n", | |
| "</g>\n", | |
| "<g color=\"black\" fill=\"none\" stroke=\"currentColor\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\" stroke-width=\"0.50\">\n", | |
| "</g>\n", | |
| "<g color=\"black\" fill=\"none\" stroke=\"currentColor\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\" stroke-width=\"0.50\">\n", | |
| "\t<path d=\"M431.0,243.9 L418.5,243.9 \" stroke=\"black\"/></g>\n", | |
| "<g color=\"gray\" fill=\"none\" stroke=\"currentColor\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\" stroke-width=\"0.50\">\n", | |
| "\t<path class=\"gridline\" d=\"M74.3,192.8 L269.0,126.9 \" stroke=\"gray\" stroke-dasharray=\"2,4\"/></g>\n", | |
| "<g color=\"gray\" fill=\"none\" stroke=\"currentColor\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\" stroke-width=\"0.50\">\n", | |
| "\t<path class=\"gridline\" d=\"M360.1,179.2 L269.0,126.9 \" stroke=\"gray\" stroke-dasharray=\"2,4\"/></g>\n", | |
| "<g color=\"gray\" fill=\"none\" stroke=\"currentColor\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\" stroke-width=\"0.50\">\n", | |
| "\t<path class=\"gridline\" d=\"M418.5,212.6 L383.4,192.6 \" stroke=\"gray\" stroke-dasharray=\"2,4\"/></g>\n", | |
| "<g color=\"black\" fill=\"none\" stroke=\"currentColor\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\" stroke-width=\"0.50\">\n", | |
| "\t<path d=\"M61.8,192.8 L74.3,192.8 \" stroke=\"black\"/>\t<g fill=\"rgb(0,0,0)\" font-family=\"{}\" font-size=\"16.00\" stroke=\"none\" text-anchor=\"end\" transform=\"translate(43.7,198.8)\">\n", | |
| "\t\t<text><tspan font-family=\"{}\">1000</tspan></text>\n", | |
| "\t</g>\n", | |
| "</g>\n", | |
| "<g color=\"black\" fill=\"none\" stroke=\"currentColor\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\" stroke-width=\"0.50\">\n", | |
| "</g>\n", | |
| "<g color=\"black\" fill=\"none\" stroke=\"currentColor\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\" stroke-width=\"0.50\">\n", | |
| "\t<path d=\"M431.0,212.6 L418.5,212.6 \" stroke=\"black\"/></g>\n", | |
| "<g color=\"gray\" fill=\"none\" stroke=\"currentColor\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\" stroke-width=\"0.50\">\n", | |
| "\t<path class=\"gridline\" d=\"M74.3,161.5 L269.0,95.6 \" stroke=\"gray\" stroke-dasharray=\"2,4\"/></g>\n", | |
| "<g color=\"gray\" fill=\"none\" stroke=\"currentColor\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\" stroke-width=\"0.50\">\n", | |
| "\t<path class=\"gridline\" d=\"M390.9,165.6 L269.0,95.6 \" stroke=\"gray\" stroke-dasharray=\"2,4\"/></g>\n", | |
| "<g color=\"gray\" fill=\"none\" stroke=\"currentColor\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\" stroke-width=\"0.50\">\n", | |
| "\t<path class=\"gridline\" d=\"M418.5,181.5 L400.9,171.4 \" stroke=\"gray\" stroke-dasharray=\"2,4\"/></g>\n", | |
| "<g color=\"black\" fill=\"none\" stroke=\"currentColor\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\" stroke-width=\"0.50\">\n", | |
| "\t<path d=\"M61.8,161.5 L74.3,161.5 \" stroke=\"black\"/>\t<g fill=\"rgb(0,0,0)\" font-family=\"{}\" font-size=\"16.00\" stroke=\"none\" text-anchor=\"end\" transform=\"translate(43.7,167.5)\">\n", | |
| "\t\t<text><tspan font-family=\"{}\">1500</tspan></text>\n", | |
| "\t</g>\n", | |
| "</g>\n", | |
| "<g color=\"black\" fill=\"none\" stroke=\"currentColor\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\" stroke-width=\"0.50\">\n", | |
| "</g>\n", | |
| "<g color=\"black\" fill=\"none\" stroke=\"currentColor\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\" stroke-width=\"0.50\">\n", | |
| "\t<path d=\"M431.0,181.5 L418.5,181.5 \" stroke=\"black\"/></g>\n", | |
| "<g color=\"gray\" fill=\"none\" stroke=\"currentColor\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\" stroke-width=\"0.50\">\n", | |
| "\t<path class=\"gridline\" d=\"M74.3,130.3 L269.0,64.4 \" stroke=\"gray\" stroke-dasharray=\"2,4\"/></g>\n", | |
| "<g color=\"gray\" fill=\"none\" stroke=\"currentColor\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\" stroke-width=\"0.50\">\n", | |
| "\t<path class=\"gridline\" d=\"M417.5,149.7 L269.0,64.4 \" stroke=\"gray\" stroke-dasharray=\"2,4\"/></g>\n", | |
| "<g color=\"gray\" fill=\"none\" stroke=\"currentColor\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\" stroke-width=\"0.50\">\n", | |
| "\t<path class=\"gridline\" d=\"M418.5,150.2 L417.9,149.9 \" stroke=\"gray\" stroke-dasharray=\"2,4\"/></g>\n", | |
| "<g color=\"black\" fill=\"none\" stroke=\"currentColor\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\" stroke-width=\"0.50\">\n", | |
| "\t<path d=\"M61.8,130.3 L74.3,130.3 \" stroke=\"black\"/>\t<g fill=\"rgb(0,0,0)\" font-family=\"{}\" font-size=\"16.00\" stroke=\"none\" text-anchor=\"end\" transform=\"translate(43.7,136.3)\">\n", | |
| "\t\t<text><tspan font-family=\"{}\">2000</tspan></text>\n", | |
| "\t</g>\n", | |
| "</g>\n", | |
| "<g color=\"black\" fill=\"none\" stroke=\"currentColor\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\" stroke-width=\"0.50\">\n", | |
| "</g>\n", | |
| "<g color=\"black\" fill=\"none\" stroke=\"currentColor\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\" stroke-width=\"0.50\">\n", | |
| "\t<path d=\"M431.0,150.2 L418.5,150.2 \" stroke=\"black\"/></g>\n", | |
| "<g color=\"gray\" fill=\"none\" stroke=\"currentColor\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\" stroke-width=\"0.50\">\n", | |
| "\t<path class=\"gridline\" d=\"M74.3,99.0 L269.0,33.1 \" stroke=\"gray\" stroke-dasharray=\"2,4\"/></g>\n", | |
| "<g color=\"gray\" fill=\"none\" stroke=\"currentColor\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\" stroke-width=\"0.50\">\n", | |
| "\t<path class=\"gridline\" d=\"M418.5,119.0 L269.0,33.1 \" stroke=\"gray\" stroke-dasharray=\"2,4\"/></g>\n", | |
| "<g color=\"black\" fill=\"none\" stroke=\"currentColor\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\" stroke-width=\"0.50\">\n", | |
| "\t<path d=\"M61.8,99.0 L74.3,99.0 \" stroke=\"black\"/>\t<g fill=\"rgb(0,0,0)\" font-family=\"{}\" font-size=\"16.00\" stroke=\"none\" text-anchor=\"end\" transform=\"translate(43.7,105.0)\">\n", | |
| "\t\t<text><tspan font-family=\"{}\">2500</tspan></text>\n", | |
| "\t</g>\n", | |
| "</g>\n", | |
| "<g color=\"black\" fill=\"none\" stroke=\"currentColor\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\" stroke-width=\"0.50\">\n", | |
| "</g>\n", | |
| "<g color=\"black\" fill=\"none\" stroke=\"currentColor\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\" stroke-width=\"0.50\">\n", | |
| "\t<path d=\"M431.0,119.0 L418.5,119.0 \" stroke=\"black\"/></g>\n", | |
| "<g color=\"black\" fill=\"none\" stroke=\"black\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\" stroke-width=\"1.00\">\n", | |
| "</g>\n", | |
| "<g color=\"black\" fill=\"none\" stroke=\"currentColor\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\" stroke-width=\"1.00\">\n", | |
| "\t<g shape-rendering=\"crispEdges\" stroke=\"none\">\n", | |
| "\t\t<polygon fill=\"rgb(255, 255, 255)\" points=\"463.4,373.8 489.2,373.8 489.2,31.7 463.4,31.7 \"/>\n", | |
| "\t</g>\n", | |
| "</g>\n", | |
| "<g color=\"black\" fill=\"none\" stroke=\"rgb(255, 255, 255)\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\" stroke-width=\"0.50\">\n", | |
| "</g>\n", | |
| "<g color=\"black\" fill=\"none\" stroke=\"currentColor\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\" stroke-width=\"0.50\">\n", | |
| "\t<path d=\"M463.4,31.6 L463.4,373.8 L489.3,373.8 L489.3,31.6 L463.4,31.6 Z \" stroke=\"black\"/></g>\n", | |
| "<g color=\"black\" fill=\"none\" stroke=\"currentColor\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\" stroke-width=\"0.50\">\n", | |
| "</g>\n", | |
| "\t<g id=\"gnuplot_plot_1b\"><title>gnuplot_plot_1b</title>\n", | |
| "<g color=\"white\" fill=\"none\" stroke=\"black\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\" stroke-width=\"1.00\">\n", | |
| "</g>\n", | |
| "<g color=\"black\" fill=\"none\" stroke=\"currentColor\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\" stroke-width=\"1.00\">\n", | |
| "<image height=\"342.2\" preserveAspectRatio=\"none\" width=\"25.9\" x=\"463.4\" xlink:href=\"gp_image_01.png\" y=\"31.6\"/>;\n", | |
| "</g>\n", | |
| "\t</g>\n", | |
| "<g color=\"white\" fill=\"none\" stroke=\"rgb(148, 0, 211)\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\" stroke-width=\"0.50\">\n", | |
| "</g>\n", | |
| "<g color=\"black\" fill=\"none\" stroke=\"currentColor\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\" stroke-width=\"0.50\">\n", | |
| "\t<path d=\"M489.3,338.9 L476.8,338.9 M463.4,338.9 L475.9,338.9 \" stroke=\"black\"/>\t<g fill=\"rgb(0,0,0)\" font-family=\"{}\" font-size=\"16.00\" stroke=\"none\" text-anchor=\"start\" transform=\"translate(497.6,344.9)\">\n", | |
| "\t\t<text><tspan font-family=\"{}\">0</tspan></text>\n", | |
| "\t</g>\n", | |
| "</g>\n", | |
| "<g color=\"black\" fill=\"none\" stroke=\"currentColor\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\" stroke-width=\"0.50\">\n", | |
| "\t<path d=\"M489.3,262.8 L476.8,262.8 M463.4,262.8 L475.9,262.8 \" stroke=\"black\"/>\t<g fill=\"rgb(0,0,0)\" font-family=\"{}\" font-size=\"16.00\" stroke=\"none\" text-anchor=\"start\" transform=\"translate(497.6,268.8)\">\n", | |
| "\t\t<text><tspan font-family=\"{}\">500</tspan></text>\n", | |
| "\t</g>\n", | |
| "</g>\n", | |
| "<g color=\"black\" fill=\"none\" stroke=\"currentColor\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\" stroke-width=\"0.50\">\n", | |
| "\t<path d=\"M489.3,186.7 L476.8,186.7 M463.4,186.7 L475.9,186.7 \" stroke=\"black\"/>\t<g fill=\"rgb(0,0,0)\" font-family=\"{}\" font-size=\"16.00\" stroke=\"none\" text-anchor=\"start\" transform=\"translate(497.6,192.7)\">\n", | |
| "\t\t<text><tspan font-family=\"{}\">1000</tspan></text>\n", | |
| "\t</g>\n", | |
| "</g>\n", | |
| "<g color=\"black\" fill=\"none\" stroke=\"currentColor\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\" stroke-width=\"0.50\">\n", | |
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| "\t<path d=\"M74.3,99.0 L74.3,286.4 \" stroke=\"black\"/></g>\n", | |
| "<g color=\"black\" fill=\"none\" stroke=\"currentColor\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\" stroke-width=\"0.50\">\n", | |
| "\t<path d=\"M269.0,33.1 L269.0,69.9 \" stroke=\"black\"/></g>\n", | |
| "<g color=\"black\" fill=\"none\" stroke=\"currentColor\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\" stroke-width=\"0.50\">\n", | |
| "\t<path d=\"M418.5,119.0 L418.5,306.4 \" stroke=\"black\"/></g>\n", | |
| "<g color=\"black\" fill=\"none\" stroke=\"currentColor\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\" stroke-width=\"0.50\">\n", | |
| "\t<path d=\"M223.8,221.6 L223.8,372.3 \" stroke=\"black\"/></g>\n", | |
| "<g color=\"black\" fill=\"none\" stroke=\"currentColor\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\" stroke-width=\"0.50\">\n", | |
| "\t<path d=\"M74.3,99.0 L269.0,33.1 \" stroke=\"black\"/></g>\n", | |
| "<g color=\"black\" fill=\"none\" stroke=\"currentColor\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\" stroke-width=\"0.50\">\n", | |
| "\t<path d=\"M418.5,119.0 L269.0,33.1 \" stroke=\"black\"/></g>\n", | |
| "<g color=\"gray\" fill=\"none\" stroke=\"currentColor\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\" stroke-width=\"0.50\">\n", | |
| "\t<path class=\"gridline\" d=\"M223.8,372.3 L74.3,286.4 \" stroke=\"gray\" stroke-dasharray=\"2,4\"/></g>\n", | |
| "<g color=\"black\" fill=\"none\" stroke=\"currentColor\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\" stroke-width=\"0.50\">\n", | |
| "\t<path d=\"M232.5,377.3 L223.8,372.3 \" stroke=\"black\"/></g>\n", | |
| "<g color=\"black\" fill=\"none\" stroke=\"currentColor\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\" stroke-width=\"0.50\">\n", | |
| "\t<path d=\"M74.3,286.4 L65.6,281.4 \" stroke=\"black\"/>\t<g fill=\"rgb(0,0,0)\" font-family=\"{}\" font-size=\"16.00\" stroke=\"none\" text-anchor=\"middle\" transform=\"translate(238.2,390.4)\">\n", | |
| "\t\t<text><tspan font-family=\"{}\">0</tspan></text>\n", | |
| "\t</g>\n", | |
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| "<g color=\"black\" fill=\"none\" stroke=\"currentColor\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\" stroke-width=\"0.50\">\n", | |
| "</g>\n", | |
| "<g color=\"gray\" fill=\"none\" stroke=\"currentColor\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\" stroke-width=\"0.50\">\n", | |
| "\t<path class=\"gridline\" d=\"M262.7,359.1 L113.2,273.2 \" stroke=\"gray\" stroke-dasharray=\"2,4\"/></g>\n", | |
| "<g color=\"black\" fill=\"none\" stroke=\"currentColor\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\" stroke-width=\"0.50\">\n", | |
| "\t<path d=\"M271.4,364.1 L262.7,359.1 \" stroke=\"black\"/></g>\n", | |
| "<g color=\"black\" fill=\"none\" stroke=\"currentColor\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\" stroke-width=\"0.50\">\n", | |
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| "\t\t<text><tspan font-family=\"{}\">2</tspan></text>\n", | |
| "\t</g>\n", | |
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| "<g color=\"black\" fill=\"none\" stroke=\"currentColor\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\" stroke-width=\"0.50\">\n", | |
| "</g>\n", | |
| "<g color=\"gray\" fill=\"none\" stroke=\"currentColor\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\" stroke-width=\"0.50\">\n", | |
| "\t<path class=\"gridline\" d=\"M301.6,345.9 L152.2,260.0 \" stroke=\"gray\" stroke-dasharray=\"2,4\"/></g>\n", | |
| "<g color=\"black\" fill=\"none\" stroke=\"currentColor\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\" stroke-width=\"0.50\">\n", | |
| "\t<path d=\"M310.3,350.9 L301.6,345.9 \" stroke=\"black\"/></g>\n", | |
| "<g color=\"black\" fill=\"none\" stroke=\"currentColor\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\" stroke-width=\"0.50\">\n", | |
| "\t<path d=\"M152.2,260.0 L143.5,255.0 \" stroke=\"black\"/>\t<g fill=\"rgb(0,0,0)\" font-family=\"{}\" font-size=\"16.00\" stroke=\"none\" text-anchor=\"middle\" transform=\"translate(316.1,364.1)\">\n", | |
| "\t\t<text><tspan font-family=\"{}\">4</tspan></text>\n", | |
| "\t</g>\n", | |
| "</g>\n", | |
| "<g color=\"black\" fill=\"none\" stroke=\"currentColor\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\" stroke-width=\"0.50\">\n", | |
| "</g>\n", | |
| "<g color=\"gray\" fill=\"none\" stroke=\"currentColor\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\" stroke-width=\"0.50\">\n", | |
| "\t<path class=\"gridline\" d=\"M340.6,332.7 L191.2,246.9 \" stroke=\"gray\" stroke-dasharray=\"2,4\"/></g>\n", | |
| "<g color=\"black\" fill=\"none\" stroke=\"currentColor\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\" stroke-width=\"0.50\">\n", | |
| "\t<path d=\"M349.3,337.7 L340.6,332.7 \" stroke=\"black\"/></g>\n", | |
| "<g color=\"black\" fill=\"none\" stroke=\"currentColor\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\" stroke-width=\"0.50\">\n", | |
| "\t<path d=\"M191.2,246.9 L182.5,241.9 \" stroke=\"black\"/>\t<g fill=\"rgb(0,0,0)\" font-family=\"{}\" font-size=\"16.00\" stroke=\"none\" text-anchor=\"middle\" transform=\"translate(355.0,350.9)\">\n", | |
| "\t\t<text><tspan font-family=\"{}\">6</tspan></text>\n", | |
| "\t</g>\n", | |
| "</g>\n", | |
| "<g color=\"black\" fill=\"none\" stroke=\"currentColor\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\" stroke-width=\"0.50\">\n", | |
| "</g>\n", | |
| "<g color=\"gray\" fill=\"none\" stroke=\"currentColor\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\" stroke-width=\"0.50\">\n", | |
| "\t<path class=\"gridline\" d=\"M379.6,319.6 L230.1,233.7 \" stroke=\"gray\" stroke-dasharray=\"2,4\"/></g>\n", | |
| "<g color=\"black\" fill=\"none\" stroke=\"currentColor\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\" stroke-width=\"0.50\">\n", | |
| "\t<path d=\"M388.3,324.6 L379.6,319.6 \" stroke=\"black\"/></g>\n", | |
| "<g color=\"black\" fill=\"none\" stroke=\"currentColor\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\" stroke-width=\"0.50\">\n", | |
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| "\t\t<text><tspan font-family=\"{}\">8</tspan></text>\n", | |
| "\t</g>\n", | |
| "</g>\n", | |
| "<g color=\"black\" fill=\"none\" stroke=\"currentColor\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\" stroke-width=\"0.50\">\n", | |
| "</g>\n", | |
| "<g color=\"gray\" fill=\"none\" stroke=\"currentColor\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\" stroke-width=\"0.50\">\n", | |
| "\t<path class=\"gridline\" d=\"M418.5,306.4 L413.4,303.4 \" stroke=\"gray\" stroke-dasharray=\"2,4\"/></g>\n", | |
| "<g color=\"black\" fill=\"none\" stroke=\"currentColor\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\" stroke-width=\"0.50\">\n", | |
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| "\t\t<text><tspan font-family=\"{}\">10</tspan></text>\n", | |
| "\t</g>\n", | |
| "</g>\n", | |
| "<g color=\"black\" fill=\"none\" stroke=\"currentColor\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\" stroke-width=\"0.50\">\n", | |
| "\t<g fill=\"rgb(0,0,0)\" font-family=\"{}\" font-size=\"16.00\" stroke=\"none\" text-anchor=\"middle\" transform=\"translate(367.2,371.8)\">\n", | |
| "\t\t<text><tspan font-family=\"{}\">x</tspan><tspan dy=\"4.80px\" font-family=\"{}\" font-size=\"12.8\">1</tspan><tspan dy=\"-4.80\" font-size=\"16.0\"/></text>\n", | |
| "\t</g>\n", | |
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| "<g color=\"black\" fill=\"none\" stroke=\"currentColor\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\" stroke-width=\"0.50\">\n", | |
| "</g>\n", | |
| "<g color=\"gray\" fill=\"none\" stroke=\"currentColor\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\" stroke-width=\"0.50\">\n", | |
| "\t<path class=\"gridline\" d=\"M223.8,372.3 L418.5,306.4 \" stroke=\"gray\" stroke-dasharray=\"2,4\"/></g>\n", | |
| "<g color=\"black\" fill=\"none\" stroke=\"currentColor\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\" stroke-width=\"0.50\">\n", | |
| "\t<path d=\"M213.8,375.7 L223.8,372.3 \" stroke=\"black\"/></g>\n", | |
| "<g color=\"black\" fill=\"none\" stroke=\"currentColor\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\" stroke-width=\"0.50\">\n", | |
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| "\t\t<text><tspan font-family=\"{}\">0</tspan></text>\n", | |
| "\t</g>\n", | |
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| "<g color=\"black\" fill=\"none\" stroke=\"currentColor\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\" stroke-width=\"0.50\">\n", | |
| "</g>\n", | |
| "<g color=\"gray\" fill=\"none\" stroke=\"currentColor\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\" stroke-width=\"0.50\">\n", | |
| "\t<path class=\"gridline\" d=\"M193.9,355.1 L384.0,290.8 \" stroke=\"gray\" stroke-dasharray=\"2,4\"/></g>\n", | |
| "<g color=\"black\" fill=\"none\" stroke=\"currentColor\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\" stroke-width=\"0.50\">\n", | |
| "\t<path d=\"M183.9,358.5 L193.9,355.1 \" stroke=\"black\"/>\t<g fill=\"rgb(0,0,0)\" font-family=\"{}\" font-size=\"16.00\" stroke=\"none\" text-anchor=\"middle\" transform=\"translate(177.3,369.3)\">\n", | |
| "\t\t<text><tspan font-family=\"{}\">2</tspan></text>\n", | |
| "\t</g>\n", | |
| "</g>\n", | |
| "<g color=\"black\" fill=\"none\" stroke=\"currentColor\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\" stroke-width=\"0.50\">\n", | |
| "</g>\n", | |
| "<g color=\"gray\" fill=\"none\" stroke=\"currentColor\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\" stroke-width=\"0.50\">\n", | |
| "\t<path class=\"gridline\" d=\"M164.0,337.9 L348.6,275.5 \" stroke=\"gray\" stroke-dasharray=\"2,4\"/></g>\n", | |
| "<g color=\"black\" fill=\"none\" stroke=\"currentColor\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\" stroke-width=\"0.50\">\n", | |
| "\t<path d=\"M154.0,341.3 L164.0,337.9 \" stroke=\"black\"/>\t<g fill=\"rgb(0,0,0)\" font-family=\"{}\" font-size=\"16.00\" stroke=\"none\" text-anchor=\"middle\" transform=\"translate(147.4,352.1)\">\n", | |
| "\t\t<text><tspan font-family=\"{}\">4</tspan></text>\n", | |
| "\t</g>\n", | |
| "</g>\n", | |
| "<g color=\"black\" fill=\"none\" stroke=\"currentColor\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\" stroke-width=\"0.50\">\n", | |
| "</g>\n", | |
| "<g color=\"gray\" fill=\"none\" stroke=\"currentColor\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\" stroke-width=\"0.50\">\n", | |
| "\t<path class=\"gridline\" d=\"M134.1,320.8 L313.1,260.2 \" stroke=\"gray\" stroke-dasharray=\"2,4\"/></g>\n", | |
| "<g color=\"black\" fill=\"none\" stroke=\"currentColor\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\" stroke-width=\"0.50\">\n", | |
| "\t<path d=\"M124.1,324.1 L134.1,320.8 \" stroke=\"black\"/>\t<g fill=\"rgb(0,0,0)\" font-family=\"{}\" font-size=\"16.00\" stroke=\"none\" text-anchor=\"middle\" transform=\"translate(117.5,335.0)\">\n", | |
| "\t\t<text><tspan font-family=\"{}\">6</tspan></text>\n", | |
| "\t</g>\n", | |
| "</g>\n", | |
| "<g color=\"black\" fill=\"none\" stroke=\"currentColor\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\" stroke-width=\"0.50\">\n", | |
| "</g>\n", | |
| "<g color=\"gray\" fill=\"none\" stroke=\"currentColor\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\" stroke-width=\"0.50\">\n", | |
| "\t<path class=\"gridline\" d=\"M104.2,303.6 L277.7,244.9 \" stroke=\"gray\" stroke-dasharray=\"2,4\"/></g>\n", | |
| "<g color=\"black\" fill=\"none\" stroke=\"currentColor\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\" stroke-width=\"0.50\">\n", | |
| "\t<path d=\"M94.2,307.0 L104.2,303.6 \" stroke=\"black\"/>\t<g fill=\"rgb(0,0,0)\" font-family=\"{}\" font-size=\"16.00\" stroke=\"none\" text-anchor=\"middle\" transform=\"translate(87.6,317.8)\">\n", | |
| "\t\t<text><tspan font-family=\"{}\">8</tspan></text>\n", | |
| "\t</g>\n", | |
| "</g>\n", | |
| "<g color=\"black\" fill=\"none\" stroke=\"currentColor\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\" stroke-width=\"0.50\">\n", | |
| "</g>\n", | |
| "<g color=\"gray\" fill=\"none\" stroke=\"currentColor\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\" stroke-width=\"0.50\">\n", | |
| "\t<path class=\"gridline\" d=\"M74.3,286.4 L242.3,229.6 \" stroke=\"gray\" stroke-dasharray=\"2,4\"/></g>\n", | |
| "<g color=\"black\" fill=\"none\" stroke=\"currentColor\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\" stroke-width=\"0.50\">\n", | |
| "\t<path d=\"M64.3,289.8 L74.3,286.4 \" stroke=\"black\"/>\t<g fill=\"rgb(0,0,0)\" font-family=\"{}\" font-size=\"16.00\" stroke=\"none\" text-anchor=\"middle\" transform=\"translate(57.7,300.6)\">\n", | |
| "\t\t<text><tspan font-family=\"{}\">10</tspan></text>\n", | |
| "\t</g>\n", | |
| "</g>\n", | |
| "<g color=\"black\" fill=\"none\" stroke=\"currentColor\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\" stroke-width=\"0.50\">\n", | |
| "\t<g fill=\"rgb(0,0,0)\" font-family=\"{}\" font-size=\"16.00\" stroke=\"none\" text-anchor=\"middle\" transform=\"translate(90.3,355.2)\">\n", | |
| "\t\t<text><tspan font-family=\"{}\">x</tspan><tspan dy=\"4.80px\" font-family=\"{}\" font-size=\"12.8\">2</tspan><tspan dy=\"-4.80\" font-size=\"16.0\"/></text>\n", | |
| "\t</g>\n", | |
| "</g>\n", | |
| "<g color=\"black\" fill=\"none\" stroke=\"currentColor\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\" stroke-width=\"0.50\">\n", | |
| "</g>\n", | |
| "<g color=\"gray\" fill=\"none\" stroke=\"currentColor\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\" stroke-width=\"0.50\">\n", | |
| "\t<path class=\"gridline\" d=\"M74.3,286.4 L242.3,229.6 \" stroke=\"gray\" stroke-dasharray=\"2,4\"/></g>\n", | |
| "<g color=\"gray\" fill=\"none\" stroke=\"currentColor\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\" stroke-width=\"0.50\">\n", | |
| "\t<path class=\"gridline\" d=\"M418.5,306.4 L413.4,303.4 \" stroke=\"gray\" stroke-dasharray=\"2,4\"/></g>\n", | |
| "<g color=\"black\" fill=\"none\" stroke=\"currentColor\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\" stroke-width=\"0.50\">\n", | |
| "\t<path d=\"M61.8,286.4 L74.3,286.4 \" stroke=\"black\"/>\t<g fill=\"rgb(0,0,0)\" font-family=\"{}\" font-size=\"16.00\" stroke=\"none\" text-anchor=\"end\" transform=\"translate(43.7,292.4)\">\n", | |
| "\t\t<text><tspan font-family=\"{}\">-2000</tspan></text>\n", | |
| "\t</g>\n", | |
| "</g>\n", | |
| "<g color=\"black\" fill=\"none\" stroke=\"currentColor\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\" stroke-width=\"0.50\">\n", | |
| "</g>\n", | |
| "<g color=\"black\" fill=\"none\" stroke=\"currentColor\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\" stroke-width=\"0.50\">\n", | |
| "\t<path d=\"M431.0,306.4 L418.5,306.4 \" stroke=\"black\"/></g>\n", | |
| "<g color=\"gray\" fill=\"none\" stroke=\"currentColor\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\" stroke-width=\"0.50\">\n", | |
| "\t<path class=\"gridline\" d=\"M74.3,248.9 L206.7,204.1 \" stroke=\"gray\" stroke-dasharray=\"2,4\"/></g>\n", | |
| "<g color=\"gray\" fill=\"none\" stroke=\"currentColor\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\" stroke-width=\"0.50\">\n", | |
| "\t<path class=\"gridline\" d=\"M418.5,268.9 L392.6,254.0 \" stroke=\"gray\" stroke-dasharray=\"2,4\"/></g>\n", | |
| "<g color=\"black\" fill=\"none\" stroke=\"currentColor\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\" stroke-width=\"0.50\">\n", | |
| "\t<path d=\"M61.8,248.9 L74.3,248.9 \" stroke=\"black\"/>\t<g fill=\"rgb(0,0,0)\" font-family=\"{}\" font-size=\"16.00\" stroke=\"none\" text-anchor=\"end\" transform=\"translate(43.7,254.9)\">\n", | |
| "\t\t<text><tspan font-family=\"{}\">-1500</tspan></text>\n", | |
| "\t</g>\n", | |
| "</g>\n", | |
| "<g color=\"black\" fill=\"none\" stroke=\"currentColor\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\" stroke-width=\"0.50\">\n", | |
| "</g>\n", | |
| "<g color=\"black\" fill=\"none\" stroke=\"currentColor\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\" stroke-width=\"0.50\">\n", | |
| "\t<path d=\"M431.0,268.9 L418.5,268.9 \" stroke=\"black\"/></g>\n", | |
| "<g color=\"gray\" fill=\"none\" stroke=\"currentColor\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\" stroke-width=\"0.50\">\n", | |
| "\t<path class=\"gridline\" d=\"M74.3,211.4 L175.2,177.4 \" stroke=\"gray\" stroke-dasharray=\"2,4\"/></g>\n", | |
| "<g color=\"gray\" fill=\"none\" stroke=\"currentColor\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\" stroke-width=\"0.50\">\n", | |
| "\t<path class=\"gridline\" d=\"M418.5,231.4 L362.0,199.0 \" stroke=\"gray\" stroke-dasharray=\"2,4\"/></g>\n", | |
| "<g color=\"black\" fill=\"none\" stroke=\"currentColor\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\" stroke-width=\"0.50\">\n", | |
| "\t<path d=\"M61.8,211.4 L74.3,211.4 \" stroke=\"black\"/>\t<g fill=\"rgb(0,0,0)\" font-family=\"{}\" font-size=\"16.00\" stroke=\"none\" text-anchor=\"end\" transform=\"translate(43.7,217.4)\">\n", | |
| "\t\t<text><tspan font-family=\"{}\">-1000</tspan></text>\n", | |
| "\t</g>\n", | |
| "</g>\n", | |
| "<g color=\"black\" fill=\"none\" stroke=\"currentColor\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\" stroke-width=\"0.50\">\n", | |
| "</g>\n", | |
| "<g color=\"black\" fill=\"none\" stroke=\"currentColor\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\" stroke-width=\"0.50\">\n", | |
| "\t<path d=\"M431.0,231.4 L418.5,231.4 \" stroke=\"black\"/></g>\n", | |
| "<g color=\"gray\" fill=\"none\" stroke=\"currentColor\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\" stroke-width=\"0.50\">\n", | |
| "\t<path class=\"gridline\" d=\"M74.3,174.0 L136.1,153.1 \" stroke=\"gray\" stroke-dasharray=\"2,4\"/></g>\n", | |
| "<g color=\"gray\" fill=\"none\" stroke=\"currentColor\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\" stroke-width=\"0.50\">\n", | |
| "\t<path class=\"gridline\" d=\"M418.5,194.0 L324.6,140.0 \" stroke=\"gray\" stroke-dasharray=\"2,4\"/></g>\n", | |
| "<g color=\"black\" fill=\"none\" stroke=\"currentColor\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\" stroke-width=\"0.50\">\n", | |
| "\t<path d=\"M61.8,174.0 L74.3,174.0 \" stroke=\"black\"/>\t<g fill=\"rgb(0,0,0)\" font-family=\"{}\" font-size=\"16.00\" stroke=\"none\" text-anchor=\"end\" transform=\"translate(43.7,180.0)\">\n", | |
| "\t\t<text><tspan font-family=\"{}\">-500</tspan></text>\n", | |
| "\t</g>\n", | |
| "</g>\n", | |
| "<g color=\"black\" fill=\"none\" stroke=\"currentColor\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\" stroke-width=\"0.50\">\n", | |
| "</g>\n", | |
| "<g color=\"black\" fill=\"none\" stroke=\"currentColor\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\" stroke-width=\"0.50\">\n", | |
| "\t<path d=\"M431.0,194.0 L418.5,194.0 \" stroke=\"black\"/></g>\n", | |
| "<g color=\"gray\" fill=\"none\" stroke=\"currentColor\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\" stroke-width=\"0.50\">\n", | |
| "\t<path class=\"gridline\" d=\"M74.3,136.5 L76.0,135.9 \" stroke=\"gray\" stroke-dasharray=\"2,4\"/></g>\n", | |
| "<g color=\"gray\" fill=\"none\" stroke=\"currentColor\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\" stroke-width=\"0.50\">\n", | |
| "\t<path class=\"gridline\" d=\"M418.5,156.5 L270.4,71.4 \" stroke=\"gray\" stroke-dasharray=\"2,4\"/></g>\n", | |
| "<g color=\"black\" fill=\"none\" stroke=\"currentColor\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\" stroke-width=\"0.50\">\n", | |
| "\t<path d=\"M61.8,136.5 L74.3,136.5 \" stroke=\"black\"/>\t<g fill=\"rgb(0,0,0)\" font-family=\"{}\" font-size=\"16.00\" stroke=\"none\" text-anchor=\"end\" transform=\"translate(43.7,142.5)\">\n", | |
| "\t\t<text><tspan font-family=\"{}\">0</tspan></text>\n", | |
| "\t</g>\n", | |
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| "source": [ | |
| "[X,Y]=meshgrid(linspace(0,10,20),linspace(0,10,20));\n", | |
| "[N,M]=size(X);\n", | |
| "F=zeros(size(X));\n", | |
| "for i=1:N\n", | |
| " for j=1:M\n", | |
| " [f,~]=mass_spring([X(i,j),Y(i,j)]);\n", | |
| " F1(i,j)=f(1);\n", | |
| " F2(i,j)=f(2);\n", | |
| " end\n", | |
| "end\n", | |
| "mesh(X,Y,F1)\n", | |
| "xlabel('x_1')\n", | |
| "ylabel('x_2')\n", | |
| "colorbar()\n", | |
| "figure()\n", | |
| "mesh(X,Y,F2)\n", | |
| "xlabel('x_1')\n", | |
| "ylabel('x_2')\n", | |
| "colorbar()" | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "execution_count": null, | |
| "metadata": { | |
| "collapsed": true | |
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| "source": [] | |
| } | |
| ], | |
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| "name": "octave" | |
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| "language_info": { | |
| "file_extension": ".m", | |
| "help_links": [ | |
| { | |
| "text": "MetaKernel Magics", | |
| "url": "https://github.com/calysto/metakernel/blob/master/metakernel/magics/README.md" | |
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