From 3519f1bfbda115e72a58482e7476c05bb3087b78 Mon Sep 17 00:00:00 2001 From: "Ryan C. Cooper" Date: Tue, 7 Feb 2017 20:46:45 -0500 Subject: [PATCH 01/10] added HW3 --- HW3/README.html | 60 ++ HW3/README.md | 74 +++ lecture_07/.newtraph.m.swp | Bin 0 -> 12288 bytes lecture_07/lecture_07.ipynb | 544 +++++++++++++++--- lecture_07/lecture_07.md | 160 ++++-- lecture_07/lecture_07.pdf | Bin 61249 -> 79416 bytes .../lecture_07_files/lecture_07_15_1.svg | 131 +++++ .../lecture_07_files/lecture_07_16_1.svg | 151 +++++ .../lecture_07_files/lecture_07_24_1.svg | 59 +- .../lecture_07_files/lecture_07_26_1.svg | 182 ++++++ 10 files changed, 1225 insertions(+), 136 deletions(-) create mode 100644 HW3/README.html create mode 100644 HW3/README.md create mode 100644 lecture_07/.newtraph.m.swp create mode 100644 lecture_07/lecture_07_files/lecture_07_15_1.svg create mode 100644 lecture_07/lecture_07_files/lecture_07_16_1.svg create mode 100644 lecture_07/lecture_07_files/lecture_07_26_1.svg diff --git a/HW3/README.html b/HW3/README.html new file mode 100644 index 0000000..950e6d3 --- /dev/null +++ b/HW3/README.html @@ -0,0 +1,60 @@ + + + + + + + + + + +

Homework #3

+

due 2/15/17 by 11:59pm

+
    +

  1. Create a new github repository called ‘roots_and_optimization’.

    +
      +

    1. Add rcc02007 and pez16103 as collaborators.

      2. +

    2. Clone the repository to your computer.

      3. +

    3. Copy your projectile.m function into the ‘roots_and_optimization’ folder. Disable the plotting routine for the solvers

      4. +

    4. Use the four solvers falsepos.m, incsearch.m, newtraph.m and mod_secant.m to solve for the angle needed to reach h=1.72, with an initial speed of 1.5 m/s.

      5. +

    5. The newtraph.m function needs a derivative, calculate the derivative of your function with respect to theta, dprojectile_dtheta.m. This function should output d(h)/d(theta).

      6. +

    6. In your README.md file, you will document the following under the heading # Homework #3:

      +
        +
      1. Compare the number of iterations that each function needed to reach an accuracy of 0.00001%. Include a table in your README.md with:
        2. +
      +
      | solver | initial guess(es) | ea | number of iterations|
+| --- | --- | --- | --- |
+|falsepos   |  |  |  |
+|incsearch  |  |  |  |
+|newtraph   |  |  |  |
+|mod_secant |  |  |  |
      +
        +
      1. Compare the convergence of the 4 methods. Plot the approximate error vs the number of iterations that the solver has calculated. Save the plot as convergence.png and display the plot in your README.md with:
        2. +
      +

      ![Plot of convergence for four numerical solvers.](convergence.png)

      +
        +
      1. In the README.md provide a description of the files used to create the table and the convergence plot.
        2. +
      7. +
    2. +

  2. The Newton-Raphson method and the modified secant method do not always converge to a solution. One simple example is the function f(x) = x*exp(-x^2). The root is at 0, but using the numerical solvers, newtraph.m and mod_secant.m, there are certain initial guesses that do not converge.

    +
      +

    1. Calculate the first 5 iterations for the Newton-Raphson method with an initial guess of x_i=2.

      2. +

    2. Add the results to a table in the README.md with:

      3. +
    +
    ### divergence of Newton-Raphson method
+
+| iteration | x_i | approx error |
+| --- | --- | --- |
+| 0 | 2 | n/a |
+| 1 |   |     |
+| 2 |   |     |
+| 3 |   |     |
+| 4 |   |     |
+| 5 |   |     |
    +
      +
    1. Repeat steps a-b for an initial guess of 0.2. (But change the heading from ‘divergence’ to ‘convergence’)
      2. +
    3. +

  3. Commit your changes to your repository. Sync your local repository with github. Then copy and paste the “clone URL” into the following Google Form Homework #3

    4. +
+ + diff --git a/HW3/README.md b/HW3/README.md new file mode 100644 index 0000000..494e0d5 --- /dev/null +++ b/HW3/README.md @@ -0,0 +1,74 @@ +# Homework #3 +## due 2/15/17 by 11:59pm + + +1. Create a new github repository called 'roots_and_optimization'. + + a. Add rcc02007 and pez16103 as collaborators. + + b. Clone the repository to your computer. + + c. Copy your `projectile.m` function into the 'roots_and_optimization' folder. + *Disable the plotting routine for the solvers* + + d. Use the four solvers `falsepos.m`, `incsearch.m`, `newtraph.m` and `mod_secant.m` + to solve for the angle needed to reach h=1.72 m, with an initial speed of 1.5 m/s. + + e. The `newtraph.m` function needs a derivative, calculate the derivative of your + function with respect to theta, `dprojectile_dtheta.m`. This function should + output d(h)/d(theta). + + + f. In your `README.md` file, document the following under the heading `# + Homework #3`: + + i. Compare the number of iterations that each function needed to reach an + accuracy of 0.00001%. Include a table in your README.md with: + + ``` + | solver | initial guess(es) | ea | number of iterations| + | --- | --- | --- | --- | + |falsepos | | | | + |incsearch | | | | + |newtraph | | | | + |mod_secant | | | | + ``` + + ii. Compare the convergence of the 4 methods. Plot the approximate error vs the + number of iterations that the solver has calculated. Save the plot as + `convergence.png` and display the plot in your `README.md` with: + + `![Plot of convergence for four numerical solvers.](convergence.png)` + + iii. In the `README.md` provide a description of the files used to create the + table and the convergence plot. + +2. The Newton-Raphson method and the modified secant method do not always converge to a +solution. One simple example is the function f(x) = x*exp(-x^2). The root is at 0, but +using the numerical solvers, `newtraph.m` and `mod_secant.m`, there are certain initial +guesses that do not converge. + + a. Calculate the first 5 iterations for the Newton-Raphson method with an initial + guess of x_i=2. + + b. Add the results to a table in the `README.md` with: + + ``` + ### divergence of Newton-Raphson method + + | iteration | x_i | approx error | + | --- | --- | --- | + | 0 | 2 | n/a | + | 1 | | | + | 2 | | | + | 3 | | | + | 4 | | | + | 5 | | | + ``` + + c. Repeat steps a-b for an initial guess of 0.2. (But change the heading from + 'divergence' to 'convergence') + +3. Commit your changes to your repository. Sync your local repository with github. Then +copy and paste the "clone URL" into the following Google Form [Homework +#3](https://goo.gl/forms/UJBGwp0fQcSxImkq2) diff --git a/lecture_07/.newtraph.m.swp b/lecture_07/.newtraph.m.swp new file mode 100644 index 0000000000000000000000000000000000000000..9759dd203c57d4a4571f3ffdbff4740cc9c0af64 GIT binary patch literal 12288 zcmeI2zi-<{6vxjEGPO>ObSOFmcuf#QOc{}qEf}!P4Ef=pLsFzcf}llXoWzq&n7m_; zq%8-5Gk0m`4Ba~iom-$&ySM*GhIY^Qj*@MtgPSqMBkWwMo-9P03x9qnggubjJwELkD_dlzIl+GD0!FvZs40}LtLV`!qjhxe&T@Y+ zhzjL~t%v=%w;GL1V=J|_oz+pUMs;J=R=gT%wO?CV?sSX)kc?fJ027!auqc-Mz1Tgx zeCad#_`;((J5QMa6JP>NfC(@GCcp%k02BB>6R67t@d6EgFm3YUbX=Glr&Au7025#W zOn?b60Vco%m;e)C0!)AjFoE}wfJ}w>72o>5{0PP4`~U3!|1al-_yKwVeF`l>f1eZL z59l>?2yH^=p|jANMInBNeu8$Po6slFB6J4&2l4-ceuI97UP9kPPob|Nm;XM*7ZYFt zOn?b60Vco%m;e)C0`Cn0YfMcEP3*c*Iu%6On3SZAgj7Y(pHo&VOE(fKp-@;&oT@vFpl7 zS5*a}s*<=H$8m4YaXD1h8r!)lYs!_ZYFegDmrYH`*&dE2tG;V8tJ13ogDh9*s88J# zLo{s0zGGQuMpxe6f-6gFCRrhCg%~-^dy0HI>RbuD-*l0{J|}mNho-5Ke$AW9URd_S zZnx`}Ax$%v0G_3^vQR)>QB$duh6kRlv-;OO6ujcbqA5t`x!IRPpmkW=GO3fQv21d; zM+R|Q4K_wXsj6JF|El=dy#+b#z%Os6Q5p){ZdBC*@Jozn8ihJOsj}H#O8_&s{zuii z$qCOBHop)p3G`XfeVjTs`ubNmcwQMzK(J#{stt`Zjd2H<-KL)=0E%ziJXA7teSe$G zPOW!bj?kAFPAv9DyN$vl+<{9v^In_J>yzYW(^b literal 0 HcmV?d00001 diff --git a/lecture_07/lecture_07.ipynb b/lecture_07/lecture_07.ipynb index d7ff495..08d29c4 100644 --- a/lecture_07/lecture_07.ipynb +++ b/lecture_07/lecture_07.ipynb @@ -2,7 +2,7 @@ "cells": [ { "cell_type": "code", - "execution_count": 1, + "execution_count": 13, "metadata": { "collapsed": true }, @@ -40,12 +40,12 @@ "\n", "$x_{i+1}=x_{i}-\\frac{f(x_{i})}{f'(x_{i})}$\n", "\n", - "Use Newton-Raphson to find solution when $e^{x}=x$" + "Use Newton-Raphson to find solution when $e^{-x}=x$" ] }, { "cell_type": "code", - "execution_count": 20, + "execution_count": 2, "metadata": { "collapsed": false }, @@ -54,7 +54,8 @@ "name": "stdout", "output_type": "stream", "text": [ - "error_approx = 1\r\n" + "x_r = 0.50000\n", + "error_approx = 1\n" ] } ], @@ -63,13 +64,14 @@ "df= @(x) -exp(-x)-1;\n", "\n", "x_i= 0;\n", + "x_r = x_i-f(x_i)/df(x_i)\n", "error_approx = abs((x_r-x_i)/x_r)\n", - "x_r=x_i;\n" + "x_i=x_r;\n" ] }, { "cell_type": "code", - "execution_count": 21, + "execution_count": 3, "metadata": { "collapsed": false }, @@ -78,8 +80,8 @@ "name": "stdout", "output_type": "stream", "text": [ - "x_r = 0.50000\n", - "error_approx = 1\n" + "x_r = 0.56631\n", + "error_approx = 0.11709\n" ] } ], @@ -91,7 +93,7 @@ }, { "cell_type": "code", - "execution_count": 22, + "execution_count": 4, "metadata": { "collapsed": false }, @@ -100,8 +102,8 @@ "name": "stdout", "output_type": "stream", "text": [ - "x_r = 0.56631\n", - "error_approx = 0.11709\n" + "x_r = 0.56714\n", + "error_approx = 0.0014673\n" ] } ], @@ -113,7 +115,7 @@ }, { "cell_type": "code", - "execution_count": 23, + "execution_count": 5, "metadata": { "collapsed": false }, @@ -123,7 +125,7 @@ "output_type": "stream", "text": [ "x_r = 0.56714\n", - "error_approx = 0.0014673\n" + "error_approx = 2.2106e-07\n" ] } ], @@ -149,7 +151,7 @@ }, { "cell_type": "code", - "execution_count": 7, + "execution_count": 6, "metadata": { "collapsed": true }, @@ -176,12 +178,14 @@ "name": "stdout", "output_type": "stream", "text": [ - "ans = 142.74\r\n" + "root = 142.74\n", + "ea = 8.0930e-06\n", + "iter = 48\n" ] } ], "source": [ - "newtraph(f_m,df_m,140,0.00001)" + "[root,ea,iter]=newtraph(f_m,df_m,140,0.00001)" ] }, { @@ -217,7 +221,7 @@ }, { "cell_type": "code", - "execution_count": 9, + "execution_count": 11, "metadata": { "collapsed": false }, @@ -226,17 +230,24 @@ "name": "stdout", "output_type": "stream", "text": [ - "ans = 142.74\r\n" + "root = 142.74\n", + "ea = 3.0615e-07\n", + "iter = 7\n" ] } ], "source": [ - "mod_secant(f_m,1e-6,50,0.00001)" + "[root,ea,iter]=mod_secant(f_m,1,50,0.00001)" ] }, + { + "cell_type": "raw", + "metadata": {}, + "source": [] + }, { "cell_type": "code", - "execution_count": 10, + "execution_count": 15, "metadata": { "collapsed": false }, @@ -245,23 +256,159 @@ "name": "stdout", "output_type": "stream", "text": [ - "error: 'plot_bool' undefined near line 12 column 6\n", - "error: called from\n", - " car_payments at line 12 column 3\n", - " mod_secant at line 22 column 8\n", - "error: 'Amt' undefined near line 1 column 14\n", - "error: evaluating argument list element number 1\n" + "ans = 1.1185e+04\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\t10000\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t15000\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t20000\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t25000\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t30000\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", + "\t\t\n", + "\t\t5\n", + "\t\n", + "\n", + "\n", + "\n", + "\n", + "\t\n", + "\n", + "\t\n", + "\t\tprinciple amount left ($)\n", + "\t\n", + "\n", + "\n", + "\t\n", + "\t\ttime (years)\n", + "\t\n", + "\n", + "\n", + "\n", + "\tgnuplot_plot_1a\n", + "\n", + "\n", + "\n", + "\t\n", + "\t\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" } ], "source": [ - "Amt_numerical=mod_secant(@(A) car_payments(A,30000,0.05,5),1e-6,50,0.001)\n", - "car_payments(Amt,30000,0.05,5)" + "car_payments(400,30000,0.05,5,1)" ] }, { "cell_type": "code", - "execution_count": 11, + "execution_count": 17, "metadata": { "collapsed": false }, @@ -270,12 +417,195 @@ "name": "stdout", "output_type": "stream", "text": [ - "error: 'Amt' undefined near line 1 column 1\r\n" + "Amt_numerical = 5467.0\n", + "ans = 3.9755e-04\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\t0\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t100000\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t200000\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t300000\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t400000\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t500000\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t600000\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t700000\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", + "\t\t\n", + "\t\t25\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t30\n", + "\t\n", + "\n", + "\n", + "\n", + "\n", + "\t\n", + "\n", + "\t\n", + "\t\tprinciple amount left ($)\n", + "\t\n", + "\n", + "\n", + "\t\n", + "\t\ttime (years)\n", + "\t\n", + "\n", + "\n", + "\n", + "\tgnuplot_plot_1a\n", + "\n", + "\n", + "\n", + "\t\n", + "\t\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "Amt_numerical=mod_secant(@(A) car_payments(A,700000,0.0875,30,0),1e-6,50,0.001)\n", + "car_payments(Amt_numerical,700000,0.0875,30,1)" + ] + }, + { + "cell_type": "code", + "execution_count": 18, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "ans = 1.9681e+06\r\n" ] } ], "source": [ - "Amt*12*5" + "Amt_numerical*12*30" ] }, { @@ -287,7 +617,7 @@ }, { "cell_type": "code", - "execution_count": 12, + "execution_count": 19, "metadata": { "collapsed": false }, @@ -398,7 +728,7 @@ }, { "cell_type": "code", - "execution_count": 14, + "execution_count": 20, "metadata": { "collapsed": false }, @@ -428,7 +758,7 @@ }, { "cell_type": "code", - "execution_count": 15, + "execution_count": 23, "metadata": { "collapsed": false }, @@ -611,7 +941,7 @@ "\t\n", "\n", "\n", - "\t\n", + "\t\n", "\t\n", "\tmod-secant\n", "\n", @@ -667,19 +997,20 @@ "ea_fp=zeros(1,N); % appr error false point method\n", "ea_bs=zeros(1,N); % appr error bisect method\n", "for i=1:length(iterations)\n", - " [root_nr,ea_nr(i),iter_nr]=newtraph(f_m,df_m,200,0,iterations(i));\n", + " [root_nr,ea_nr(i),iter_nr]=newtraph(f_m,df_m,300,0,iterations(i));\n", " [root_ms,ea_ms(i),iter_ms]=mod_secant(f_m,1e-6,300,0,iterations(i));\n", " [root_fp,ea_fp(i),iter_fp]=falsepos(f_m,1,300,0,iterations(i));\n", " [root_bs,ea_bs(i),iter_bs]=bisect(f_m,1,300,0,iterations(i));\n", "end\n", - " \n", + "\n", + "setdefaults\n", "semilogy(iterations,abs(ea_nr),iterations,abs(ea_ms),iterations,abs(ea_fp),iterations,abs(ea_bs))\n", "legend('newton-raphson','mod-secant','false point','bisection')" ] }, { "cell_type": "code", - "execution_count": 16, + "execution_count": 22, "metadata": { "collapsed": false }, @@ -688,34 +1019,30 @@ "name": "stdout", "output_type": "stream", "text": [ - "ea_ms =\n", - "\n", - " Columns 1 through 6:\n", + "ea_nr =\n", "\n", - " 2.3382e+03 1.9171e-14 1.9171e-14 1.9171e-14 1.9171e-14 1.9171e-14\n", + " Columns 1 through 8:\n", "\n", - " Columns 7 through 12:\n", + " 6.36591 0.06436 0.00052 0.00000 0.00000 0.00000 0.00000 0.00000\n", "\n", - " 1.9171e-14 1.9171e-14 1.9171e-14 1.9171e-14 1.9171e-14 1.9171e-14\n", + " Columns 9 through 16:\n", "\n", - " Columns 13 through 18:\n", + " 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000\n", "\n", - " 1.9171e-14 1.9171e-14 1.9171e-14 1.9171e-14 1.9171e-14 1.9171e-14\n", + " Columns 17 through 20:\n", "\n", - " Columns 19 and 20:\n", - "\n", - " 1.9171e-14 1.9171e-14\n", + " 0.00000 0.00000 0.00000 0.00000\n", "\n" ] } ], "source": [ - "ea_ms" + "ea_nr" ] }, { "cell_type": "code", - "execution_count": 17, + "execution_count": 26, "metadata": { "collapsed": false }, @@ -953,7 +1280,7 @@ }, { "cell_type": "code", - "execution_count": 18, + "execution_count": 27, "metadata": { "collapsed": false }, @@ -962,30 +1289,26 @@ "name": "stdout", "output_type": "stream", "text": [ - "ea_bs =\n", + "ea_nr =\n", "\n", - " Columns 1 through 6:\n", + " Columns 1 through 7:\n", "\n", - " 9.5357e+03 -4.7554e-01 -2.1114e-01 6.0163e-02 -2.4387e-03 6.1052e-04\n", + " 99.03195 11.11111 11.11111 11.11111 11.11111 11.11111 11.11111\n", "\n", - " Columns 7 through 12:\n", + " Columns 8 through 14:\n", "\n", - " 2.2891e-04 -9.5367e-06 2.3842e-06 8.9407e-07 -2.2352e-07 9.3132e-09\n", + " 11.11111 11.11111 11.11111 11.11109 11.11052 11.10624 10.99684\n", "\n", - " Columns 13 through 18:\n", + " Columns 15 through 20:\n", "\n", - " -2.3283e-09 -8.7311e-10 3.6380e-11 -9.0949e-12 -3.4106e-12 8.5265e-13\n", - "\n", - " Columns 19 and 20:\n", - "\n", - " -3.5527e-14 8.8818e-15\n", + " 8.76956 2.12993 0.00000 0.00000 0.00000 0.00000\n", "\n", "ans = 16.208\n" ] } ], "source": [ - "ea_bs\n", + "ea_nr\n", "newtraph(f,df,0.5,0,12)" ] }, @@ -1008,6 +1331,99 @@ "df(300)" ] }, + { + "cell_type": "code", + "execution_count": 28, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "f =\n", + "\n", + "@(x) tan (x) - (x - 1) .^ 2\n", + "\n", + "ans = 0.37375\n" + ] + } + ], + "source": [ + "% our class function\n", + "f= @(x) tan(x)-(x-1).^2\n", + "mod_secant(f,1e-3,1)" + ] + }, + { + "cell_type": "code", + "execution_count": 29, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "ans = -3.5577e-13\r\n" + ] + } + ], + "source": [ + "f(ans)" + ] + }, + { + "cell_type": "code", + "execution_count": 30, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "ans = 0.39218\n", + "ans = 0.39219\n" + ] + } + ], + "source": [ + "tan(0.37375)\n", + "(0.37375-1)^2" + ] + }, + { + "cell_type": "code", + "execution_count": 31, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "ans =\n", + "\n", + " Columns 1 through 8:\n", + "\n", + " -1.0000 1.5574 -3.1850 -4.1425 -7.8422 -19.3805 -25.2910 -35.1286\n", + "\n", + " Columns 9 through 11:\n", + "\n", + " -55.7997 -64.4523 -80.3516\n", + "\n" + ] + } + ], + "source": [ + "f([0:10])" + ] + }, { "cell_type": "code", "execution_count": null, diff --git a/lecture_07/lecture_07.md b/lecture_07/lecture_07.md index ffdc48d..3aacefb 100644 --- a/lecture_07/lecture_07.md +++ b/lecture_07/lecture_07.md @@ -18,7 +18,7 @@ $f'(x_{i})=\frac{f(x_{i})-0}{x_{i}-x_{i+1}}$ $x_{i+1}=x_{i}-\frac{f(x_{i})}{f'(x_{i})}$ -Use Newton-Raphson to find solution when $e^{x}=x$ +Use Newton-Raphson to find solution when $e^{-x}=x$ ```octave @@ -26,12 +26,14 @@ f= @(x) exp(-x)-x; df= @(x) -exp(-x)-1; x_i= 0; +x_r = x_i-f(x_i)/df(x_i) error_approx = abs((x_r-x_i)/x_r) -x_r=x_i; +x_i=x_r; ``` - error_approx = 1 + x_r = 0.50000 + error_approx = 1 @@ -41,8 +43,8 @@ error_approx = abs((x_r-x_i)/x_r) x_i=x_r; ``` - x_r = 0.50000 - error_approx = 1 + x_r = 0.56631 + error_approx = 0.11709 @@ -52,8 +54,8 @@ error_approx = abs((x_r-x_i)/x_r) x_i=x_r; ``` - x_r = 0.56631 - error_approx = 0.11709 + x_r = 0.56714 + error_approx = 0.0014673 @@ -64,7 +66,7 @@ x_i=x_r; ``` x_r = 0.56714 - error_approx = 0.0014673 + error_approx = 2.2106e-07 In the bungee jumper example, we created a function f(m) that when f(m)=0, then the mass had been chosen such that at t=4 s, the velocity is 36 m/s. @@ -90,10 +92,12 @@ df_m = @(m) 1/2*sqrt(g./m/c_d).*tanh(sqrt(g*c_d./m)*t)-g/2./m*sech(sqrt(g*c_d./m ```octave -newtraph(f_m,df_m,140,0.00001) +[root,ea,iter]=newtraph(f_m,df_m,140,0.00001) ``` - ans = 142.74 + root = 142.74 + ea = 8.0930e-06 + iter = 48 ## Secant Methods @@ -119,32 +123,46 @@ $x_{i+1}=x_{i}-\frac{f(x_{i})(\delta x_{i})}{f(x_{i}+\delta x_{i})-f(x_{i})}$ ```octave -mod_secant(f_m,1e-6,50,0.00001) +[root,ea,iter]=mod_secant(f_m,1,50,0.00001) ``` - ans = 142.74 + root = 142.74 + ea = 3.0615e-07 + iter = 7 ```octave -Amt_numerical=mod_secant(@(A) car_payments(A,30000,0.05,5),1e-6,50,0.001) -car_payments(Amt,30000,0.05,5) +car_payments(400,30000,0.05,5,1) ``` - error: 'plot_bool' undefined near line 12 column 6 - error: called from - car_payments at line 12 column 3 - mod_secant at line 22 column 8 - error: 'Amt' undefined near line 1 column 14 - error: evaluating argument list element number 1 + ans = 1.1185e+04 + + + +![svg](lecture_07_files/lecture_07_15_1.svg) ```octave -Amt*12*5 +Amt_numerical=mod_secant(@(A) car_payments(A,700000,0.0875,30,0),1e-6,50,0.001) +car_payments(Amt_numerical,700000,0.0875,30,1) ``` - error: 'Amt' undefined near line 1 column 1 + Amt_numerical = 5467.0 + ans = 3.9755e-04 + + + +![svg](lecture_07_files/lecture_07_16_1.svg) + + + +```octave +Amt_numerical*12*30 +``` + + ans = 1.9681e+06 Amortization calculation makes the same calculation for the monthly payment amount, A, paying off the principle amount, P, over n pay periods with monthly interest rate, r. @@ -257,12 +275,13 @@ ea_ms=zeros(1,N); % appr error Modified Secant ea_fp=zeros(1,N); % appr error false point method ea_bs=zeros(1,N); % appr error bisect method for i=1:length(iterations) - [root_nr,ea_nr(i),iter_nr]=newtraph(f_m,df_m,200,0,iterations(i)); + [root_nr,ea_nr(i),iter_nr]=newtraph(f_m,df_m,300,0,iterations(i)); [root_ms,ea_ms(i),iter_ms]=mod_secant(f_m,1e-6,300,0,iterations(i)); [root_fp,ea_fp(i),iter_fp]=falsepos(f_m,1,300,0,iterations(i)); [root_bs,ea_bs(i),iter_bs]=bisect(f_m,1,300,0,iterations(i)); end - + +setdefaults semilogy(iterations,abs(ea_nr),iterations,abs(ea_ms),iterations,abs(ea_fp),iterations,abs(ea_bs)) legend('newton-raphson','mod-secant','false point','bisection') ``` @@ -281,31 +300,27 @@ legend('newton-raphson','mod-secant','false point','bisection') -![svg](lecture_07_files/lecture_07_22_1.svg) +![svg](lecture_07_files/lecture_07_24_1.svg) ```octave -ea_ms +ea_nr ``` - ea_ms = + ea_nr = - Columns 1 through 6: + Columns 1 through 8: - 2.3382e+03 1.9171e-14 1.9171e-14 1.9171e-14 1.9171e-14 1.9171e-14 + 6.36591 0.06436 0.00052 0.00000 0.00000 0.00000 0.00000 0.00000 - Columns 7 through 12: + Columns 9 through 16: - 1.9171e-14 1.9171e-14 1.9171e-14 1.9171e-14 1.9171e-14 1.9171e-14 + 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 - Columns 13 through 18: + Columns 17 through 20: - 1.9171e-14 1.9171e-14 1.9171e-14 1.9171e-14 1.9171e-14 1.9171e-14 - - Columns 19 and 20: - - 1.9171e-14 1.9171e-14 + 0.00000 0.00000 0.00000 0.00000 @@ -344,32 +359,28 @@ legend('newton-raphson','mod-secant','false point','bisection') -![svg](lecture_07_files/lecture_07_24_1.svg) +![svg](lecture_07_files/lecture_07_26_1.svg) ```octave -ea_bs +ea_nr newtraph(f,df,0.5,0,12) ``` - ea_bs = - - Columns 1 through 6: + ea_nr = - 9.5357e+03 -4.7554e-01 -2.1114e-01 6.0163e-02 -2.4387e-03 6.1052e-04 + Columns 1 through 7: - Columns 7 through 12: + 99.03195 11.11111 11.11111 11.11111 11.11111 11.11111 11.11111 - 2.2891e-04 -9.5367e-06 2.3842e-06 8.9407e-07 -2.2352e-07 9.3132e-09 + Columns 8 through 14: - Columns 13 through 18: + 11.11111 11.11111 11.11111 11.11109 11.11052 11.10624 10.99684 - -2.3283e-09 -8.7311e-10 3.6380e-11 -9.0949e-12 -3.4106e-12 8.5265e-13 + Columns 15 through 20: - Columns 19 and 20: - - -3.5527e-14 8.8818e-15 + 8.76956 2.12993 0.00000 0.00000 0.00000 0.00000 ans = 16.208 @@ -383,6 +394,55 @@ df(300) +```octave +% our class function +f= @(x) tan(x)-(x-1).^2 +mod_secant(f,1e-3,1) +``` + + f = + + @(x) tan (x) - (x - 1) .^ 2 + + ans = 0.37375 + + + +```octave +f(ans) +``` + + ans = -3.5577e-13 + + + +```octave +tan(0.37375) +(0.37375-1)^2 +``` + + ans = 0.39218 + ans = 0.39219 + + + +```octave +f([0:10]) +``` + + ans = + + Columns 1 through 8: + + -1.0000 1.5574 -3.1850 -4.1425 -7.8422 -19.3805 -25.2910 -35.1286 + + Columns 9 through 11: + + -55.7997 -64.4523 -80.3516 + + + + ```octave ``` diff --git a/lecture_07/lecture_07.pdf b/lecture_07/lecture_07.pdf index 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b/lecture_07/lecture_07_files/lecture_07_15_1.svg new file mode 100644 index 0000000..903c445 --- /dev/null +++ b/lecture_07/lecture_07_files/lecture_07_15_1.svg @@ -0,0 +1,131 @@ + + +Gnuplot +Produced by GNUPLOT 5.0 patchlevel 3 + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + 10000 + + + + + 15000 + + + + + 20000 + + + + + 25000 + + + + + 30000 + + + + + 0 + + + + + 1 + + + + + 2 + + + + + 3 + + + + + 4 + + + + + 5 + + + + + + + + + principle amount left ($) + + + + + time (years) + + + + + gnuplot_plot_1a + + + + + + + + + + + + + + + \ No newline at end of file diff --git a/lecture_07/lecture_07_files/lecture_07_16_1.svg b/lecture_07/lecture_07_files/lecture_07_16_1.svg new file mode 100644 index 0000000..a3a2125 --- /dev/null +++ b/lecture_07/lecture_07_files/lecture_07_16_1.svg @@ -0,0 +1,151 @@ + + +Gnuplot +Produced by GNUPLOT 5.0 patchlevel 3 + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + 0 + + + + + 100000 + + + + + 200000 + + + + + 300000 + + + + + 400000 + + + + + 500000 + + + + + 600000 + + + + + 700000 + + + + + 0 + + + + + 5 + + + + + 10 + + + + + 15 + + + + + 20 + + + + + 25 + + + + + 30 + + + + + + + + + principle amount left ($) + + + + + time (years) + + + + + gnuplot_plot_1a + + + + + + + + + + + + + + + \ No newline at end of file diff --git a/lecture_07/lecture_07_files/lecture_07_24_1.svg b/lecture_07/lecture_07_files/lecture_07_24_1.svg index a8b614c..b511b10 100644 --- a/lecture_07/lecture_07_files/lecture_07_24_1.svg +++ b/lecture_07/lecture_07_files/lecture_07_24_1.svg @@ -43,47 +43,47 @@ - + 10-14 - + 10-12 - + 10-10 - + 10-8 - + 10-6 - + 10-4 - + 10-2 - + 100 - + 102 @@ -98,28 +98,43 @@ - - 10 + + 50 + + + + + 100 - - 20 + + 150 - - 30 + + 200 - - 40 + + 250 + + + + + 300 + + + + + 350 - 50 + 400 @@ -141,7 +156,7 @@ - + mod-secant @@ -150,7 +165,7 @@ - + false point @@ -159,7 +174,7 @@ - + bisection @@ -168,7 +183,7 @@ - + diff --git a/lecture_07/lecture_07_files/lecture_07_26_1.svg b/lecture_07/lecture_07_files/lecture_07_26_1.svg new file mode 100644 index 0000000..a8b614c --- /dev/null +++ b/lecture_07/lecture_07_files/lecture_07_26_1.svg @@ -0,0 +1,182 @@ + + +Gnuplot +Produced by GNUPLOT 5.0 patchlevel 3 + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + 10-14 + + + + + 10-12 + + + + + 10-10 + + + + + 10-8 + + + + + 10-6 + + + + + 10-4 + + + + + 10-2 + + + + + 100 + + + + + 102 + + + + + 104 + + + + + 0 + + + + + 10 + + + + + 20 + + + + + 30 + + + + + 40 + + + + + 50 + + + + + + + + + + + + + newton-raphson + + + + + newton-raphson + + + + + + mod-secant + + + mod-secant + + + + + + false point + + + false point + + + + + + bisection + + + bisection + + + + + + + + + + + + + + + \ No newline at end of file From 6e109623ad724ad4a78871fcba90ae3fef495c3a Mon Sep 17 00:00:00 2001 From: "Ryan C. Cooper" Date: Tue, 7 Feb 2017 20:47:05 -0500 Subject: [PATCH 02/10] added HW3 --- HW3/README.html | 60 ------------------------------------------------- 1 file changed, 60 deletions(-) delete mode 100644 HW3/README.html diff --git a/HW3/README.html b/HW3/README.html deleted file mode 100644 index 950e6d3..0000000 --- a/HW3/README.html +++ /dev/null @@ -1,60 +0,0 @@ - - - - - - - - - - -

Homework #3

-

due 2/15/17 by 11:59pm

- - - From 889acf40251fbf3602f3714991430b0b91864a1c Mon Sep 17 00:00:00 2001 From: "Ryan C. Cooper" Date: Mon, 13 Feb 2017 16:54:49 -0500 Subject: [PATCH 03/10] update HW3 --- HW3/README.md | 6 +- lecture_07/.lecture_07.md.swo | Bin 0 -> 16384 bytes lecture_07/.lecture_07.md.swp | Bin 0 -> 16384 bytes "lecture_08/\020\267Y\001" | 0 .../lecture_08-checkpoint.ipynb | 6 + lecture_08/Auchain_model.gif | Bin 0 -> 17748 bytes lecture_08/au_chain.jpg | Bin 0 -> 234971 bytes lecture_08/goldenratio.png | Bin 0 -> 94208 bytes lecture_08/lecture_08.ipynb | 1941 +++++++++++++++++ lecture_08/lennard_jones.m | 4 + lecture_08/octave-workspace | Bin 0 -> 4894 bytes "lecture_08/\34040\001" | 0 12 files changed, 1954 insertions(+), 3 deletions(-) create mode 100644 lecture_07/.lecture_07.md.swo create mode 100644 lecture_07/.lecture_07.md.swp create mode 100644 "lecture_08/\020\267Y\001" create mode 100644 lecture_08/.ipynb_checkpoints/lecture_08-checkpoint.ipynb create mode 100644 lecture_08/Auchain_model.gif create mode 100644 lecture_08/au_chain.jpg create mode 100644 lecture_08/goldenratio.png create mode 100644 lecture_08/lecture_08.ipynb create mode 100644 lecture_08/lennard_jones.m create mode 100644 lecture_08/octave-workspace create mode 100644 "lecture_08/\34040\001" diff --git a/HW3/README.md b/HW3/README.md index 494e0d5..eb93c41 100644 --- a/HW3/README.md +++ b/HW3/README.md @@ -12,7 +12,7 @@ *Disable the plotting routine for the solvers* d. Use the four solvers `falsepos.m`, `incsearch.m`, `newtraph.m` and `mod_secant.m` - to solve for the angle needed to reach h=1.72 m, with an initial speed of 1.5 m/s. + to solve for the angle needed to reach h=1.72 m, with an initial speed of 15 m/s. e. The `newtraph.m` function needs a derivative, calculate the derivative of your function with respect to theta, `dprojectile_dtheta.m`. This function should @@ -29,7 +29,7 @@ | solver | initial guess(es) | ea | number of iterations| | --- | --- | --- | --- | |falsepos | | | | - |incsearch | | | | + |bisect | | | | |newtraph | | | | |mod_secant | | | | ``` @@ -49,7 +49,7 @@ using the numerical solvers, `newtraph.m` and `mod_secant.m`, there are certain guesses that do not converge. a. Calculate the first 5 iterations for the Newton-Raphson method with an initial - guess of x_i=2. + guess of x_i=2 for f(x)=x*exp(-x^2). b. 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maximum based on given constraints. Engineers and scientists typically use energy balance equations to find the conditions of minimum energy, but this value may never go to 0 and the actual value of energy may not be of interest. " + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "collapsed": true + }, + "outputs": [], + "source": [] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "collapsed": true + }, + "outputs": [], + "source": [] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "The Lennard-Jones potential is commonly used to model interatomic bonding. \n", + "\n", + "$E_{LJ}(x)=4\\epsilon \\left(\\left(\\frac{\\sigma}{x}\\right)^{12}-\\left(\\frac{\\sigma}{x}\\right)^{6}\\right)$\n", + "\n", + "Considering a 1-D gold chain, we can calculate the bond length, $x_{b}$, with no force applied to the chain and even for tension, F. This will allow us to calculate the nonlinear spring constant of a 1-D gold chain. \n", + "\n", + "![TEM image of Gold chain](au_chain.jpg)\n", + "\n", + "Computational Tools to Study and Predict the Long-Term Stability of Nanowires.\n", + "By Martin E. Zoloff Michoff, Patricio Vélez, Sergio A. Dassie and Ezequiel P. M. Leiva \n", + "\n", + "![Model of Gold chain, from molecular dynamics simulation](Auchain_model.gif)\n", + "\n", + "[Single atom gold chain mechanics](http://www.uam.es/personal_pdi/ciencias/agrait/)\n", + "\n", + "### First, let's find the minimum energy $\\min(E_{LJ}(x))$\n", + "\n", + "## Brute force" + ] + }, + { + "cell_type": "code", + "execution_count": 3, + "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-0.04\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t-0.02\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t0\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t0.02\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t0.04\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t0.06\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t0.08\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t2.5\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t3\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t3.5\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t4\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t4.5\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t5\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t5.5\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t6\n", + "\t\n", + "\n", + "\n", + "\n", + "\n", + "\t\n", + "\n", + "\n", + "\tgnuplot_plot_1a\n", + "\n", + "\n", + "\n", + "\t\n", + "\t\n", + "\tgnuplot_plot_2a\n", + "\n", + "\t \n", + "\t\n", + "\n", + "\t\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "setdefaults\n", + "epsilon = 0.039; % kcal/mol\n", + "sigma = 2.934; % Angstrom\n", + "x=linspace(2.8,6,200); % bond length in Angstrom\n", + "\n", + "Ex = lennard_jones(x,sigma,epsilon);\n", + "\n", + "[Emin,imin]=min(Ex);\n", + "\n", + "plot(x,Ex,x(imin),Emin,'o')" + ] + }, + { + "cell_type": "code", + "execution_count": 4, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "ans = 3.2824\n", + "ans = 3.2985\n", + "ans = 3.3146\n" + ] + } + ], + "source": [ + "x(imin-1)\n", + "x(imin)\n", + "x(imin+1)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Golden Search Algorithm\n", + "\n", + "We can't just look for a sign change for the problem (unless we can take a derivative) so we need a new approach to determine whether we have a maximum between the two bounds.\n", + "\n", + "Rather than using the midpoint of initial bounds, here the problem is more difficult. We need to compare the values of 4 function evaluations. The golden search uses the golden ratio to determine two interior points. \n", + "\n", + "![golden ratio](goldenratio.png)\n", + "\n", + "Start with bounds of 2.5 and 6 Angstrom. " + ] + }, + { + "cell_type": "code", + "execution_count": 5, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "current_min = -0.019959\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-0.04\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t-0.02\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t0\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t0.02\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t0.04\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t0.06\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t0.08\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t2.5\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t3\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t3.5\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t4\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t4.5\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t5\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t5.5\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t6\n", + "\t\n", + "\n", + "\n", + "\n", + "\n", + "\t\n", + "\n", + "\n", + "\tgnuplot_plot_1a\n", + "\n", + "\n", + "\n", + "\t\n", + "\t\n", + "\tgnuplot_plot_2a\n", + "\n", + "\t \n", + "\t\n", + "\n", + "\t\n", + "\tgnuplot_plot_3a\n", + "\n", + "\t \n", + "\t\n", + "\n", + "\t\n", + "\tgnuplot_plot_4a\n", + "\n", + "\t \n", + "\t\n", + "\n", + "\t\n", + "\tgnuplot_plot_5a\n", + "\n", + "\t \n", + "\t\n", + "\n", + "\t\n", + "\tgnuplot_plot_6a\n", + "\n", + "\t\n", + "\t\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "% define Au atomic potential\n", + "epsilon = 0.039; % kcal/mol\n", + "sigma = 2.934; % Angstrom\n", + "Au_x= @(x) lennard_jones(x,sigma,epsilon);\n", + "\n", + "% calculate golden ratio\n", + "phi = 1/2+sqrt(5)/2;\n", + "% set initial limits\n", + "x_l=2.8; \n", + "x_u=6; \n", + "\n", + "% Iteration #1\n", + "d=(phi-1)*(x_u-x_l);\n", + "\n", + "x1=x_l+d; % define point 1\n", + "x2=x_u-d; % define point 2\n", + "\n", + "\n", + "% evaluate Au_x(x1) and Au_x(x2)\n", + "\n", + "f1=Au_x(x1);\n", + "f2=Au_x(x2);\n", + "plot(x,Au_x(x),x_l,Au_x(x_l),'ro',x2,f2,'rs',x1,f1,'gs',x_u,Au_x(x_u),'go')\n", + "hold on;\n", + "\n", + "if f2\n", + "\n", + "Gnuplot\n", + "Produced by GNUPLOT 5.0 patchlevel 3 \n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t \n", + "\t \n", + "\t\n", + "\t\n", + "\t \n", + "\t \n", + "\t\n", + "\n", + "\n", + "\n", + "\n", + "\t\n", + "\t\t\n", + "\t\n", + "\n", + "\n", + "\n", + "\n", + "\t\t\n", + "\t\t-0.04\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t-0.02\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t0\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t0.02\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t0.04\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t0.06\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t0.08\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t2.5\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t3\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t3.5\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t4\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t4.5\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t5\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t5.5\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t6\n", + "\t\n", + "\n", + "\n", + "\n", + "\n", + "\t\n", + "\n", + "\n", + "\tgnuplot_plot_1a\n", + "\n", + "\n", + "\n", + "\t\n", + "\t\n", + "\tgnuplot_plot_2a\n", + "\n", + "\t \n", + "\t\n", + "\n", + "\t\n", + "\tgnuplot_plot_3a\n", + "\n", + "\t \n", + "\t\n", + "\n", + "\t\n", + "\tgnuplot_plot_4a\n", + "\n", + "\t \n", + "\t\n", + "\n", + "\t\n", + "\tgnuplot_plot_5a\n", + "\n", + "\t \n", + "\t\n", + "\n", + "\t\n", + "\tgnuplot_plot_6a\n", + "\n", + "\t\n", + "\t\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "% Iteration #2\n", + "d=(phi-1)*(x_u-x_l);\n", + "\n", + "x1=x_l+d; % define point 1\n", + "x2=x_u-d; % define point 2\n", + "\n", + "% evaluate Au_x(x1) and Au_x(x2)\n", + "\n", + "f1=Au_x(x1);\n", + "f2=Au_x(x2);\n", + "plot(x,Au_x(x),x_l,Au_x(x_l),'ro',x2,f2,'rs',x1,f1,'gs',x_u,Au_x(x_u),'go')\n", + "hold on;\n", + "\n", + "if f2\n", + "\n", + "Gnuplot\n", + "Produced by GNUPLOT 5.0 patchlevel 3 \n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t \n", + "\t \n", + "\t\n", + "\t\n", + "\t \n", + "\t \n", + "\t\n", + "\n", + "\n", + "\n", + "\n", + "\t\n", + "\t\t\n", + "\t\n", + "\n", + "\n", + "\n", + "\n", + "\t\t\n", + "\t\t-0.04\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t-0.02\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t0\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t0.02\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t0.04\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t0.06\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t0.08\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t2.5\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t3\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t3.5\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t4\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t4.5\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t5\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t5.5\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t6\n", + "\t\n", + "\n", + "\n", + "\n", + "\n", + "\t\n", + "\n", + "\n", + "\tgnuplot_plot_1a\n", + "\n", + "\n", + "\n", + "\t\n", + "\t\n", + "\tgnuplot_plot_2a\n", + "\n", + "\t \n", + "\t\n", + "\n", + "\t\n", + "\tgnuplot_plot_3a\n", + "\n", + "\t \n", + "\t\n", + "\n", + "\t\n", + "\tgnuplot_plot_4a\n", + "\n", + "\t \n", + "\t\n", + "\n", + "\t\n", + "\tgnuplot_plot_5a\n", + "\n", + "\t \n", + "\t\n", + "\n", + "\t\n", + "\tgnuplot_plot_6a\n", + "\n", + "\t\n", + "\t\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "% Iteration #3\n", + "d=(phi-1)*(x_u-x_l);\n", + "\n", + "x1=x_l+d; % define point 1\n", + "x2=x_u-d; % define point 2\n", + "\n", + "% evaluate Au_x(x1) and Au_x(x2)\n", + "\n", + "f1=Au_x(x1);\n", + "f2=Au_x(x2);\n", + "plot(x,Au_x(x),x_l,Au_x(x_l),'ro',x2,f2,'rs',x1,f1,'gs',x_u,Au_x(x_u),'go')\n", + "hold on;\n", + "\n", + "if f2\n", + "\n", + "Gnuplot\n", + "Produced by GNUPLOT 5.0 patchlevel 3 \n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t \n", + "\t \n", + "\t\n", + "\t\n", + "\t \n", + "\t \n", + "\t\n", + "\n", + "\n", + "\n", + "\n", + "\t\n", + "\t\t\n", + "\t\n", + "\n", + "\n", + "\n", + "\n", + "\t\t\n", + "\t\t-0.04\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t-0.02\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t0\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t0.02\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t0.04\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t0.06\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t0.08\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t2.5\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t3\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t3.5\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t4\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t4.5\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t5\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t5.5\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t6\n", + "\t\n", + "\n", + "\n", + "\n", + "\n", + "\t\n", + "\n", + "\n", + "\tgnuplot_plot_1a\n", + "\n", + "\n", + "\n", + "\t\n", + "\t\n", + "\tgnuplot_plot_2a\n", + "\n", + "\t \n", + "\t\n", + "\n", + "\t\n", + "\tgnuplot_plot_3a\n", + "\n", + "\t \n", + "\t\n", + "\n", + "\t\n", + "\tgnuplot_plot_4a\n", + "\n", + "\t \n", + "\t\n", + "\n", + "\t\n", + "\tgnuplot_plot_5a\n", + "\n", + "\t \n", + "\t\n", + "\n", + "\t\n", + "\tgnuplot_plot_6a\n", + "\n", + "\t\n", + "\t\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "% Iteration #3\n", + "d=(phi-1)*(x_u-x_l);\n", + "\n", + "x1=x_l+d; % define point 1\n", + "x2=x_u-d; % define point 2\n", + "\n", + "% evaluate Au_x(x1) and Au_x(x2)\n", + "\n", + "f1=Au_x(x1);\n", + "f2=Au_x(x2);\n", + "plot(x,Au_x(x),x_l,Au_x(x_l),'ro',x2,f2,'rs',x1,f1,'gs',x_u,Au_x(x_u),'go')\n", + "hold on;\n", + "\n", + "if f2\n", + "\n", + "Gnuplot\n", + "Produced by GNUPLOT 5.0 patchlevel 3 \n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t \n", + "\t \n", + "\t\n", + "\t\n", + "\t \n", + "\t \n", + "\t\n", + "\n", + "\n", + "\n", + "\n", + "\t\n", + "\t\t\n", + "\t\n", + "\n", + "\n", + "\n", + "\n", + "\t\t\n", + "\t\t-0.06\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t-0.04\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t-0.02\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t0\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t0.02\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t0.04\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t0.06\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t0.08\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t2.5\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t3\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t3.5\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t4\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t4.5\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t5\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t5.5\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t6\n", + "\t\n", + "\n", + "\n", + "\n", + "\n", + "\t\n", + "\n", + "\n", + "\tgnuplot_plot_1a\n", + "\n", + "\n", + "\n", + "\t\n", + "\t\n", + "\tgnuplot_plot_2a\n", + "\n", + "\t\n", + "\t\n", + "\tgnuplot_plot_3a\n", + "\n", + "\t \n", + "\t\n", + "\t\n", + "\t\n", + "\n", + "\t\n", + "\tgnuplot_plot_4a\n", + "\n", + "\t \n", + "\t\n", + "\n", + "\t\n", + "\tgnuplot_plot_5a\n", + "\n", + "\t\n", + "\t\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "% define Au atomic potential\n", + "epsilon = 0.039; % kcal/mol\n", + "sigma = 2.934; % Angstrom\n", + "Au_x= @(x) lennard_jones(x,sigma,epsilon);\n", + "\n", + "% set initial limits\n", + "x_l=2.8; \n", + "x_u=3.5; \n", + "\n", + "% Iteration #1\n", + "x1=x_l;\n", + "x2=mean([x_l,x_u]);\n", + "x3=x_u;\n", + "\n", + "% evaluate Au_x(x1), Au_x(x2) and Au_x(x3)\n", + " \n", + "f1=Au_x(x1);\n", + "f2=Au_x(x2);\n", + "f3=Au_x(x3);\n", + "p = polyfit([x1,x2,x3],[f1,f2,f3],2);\n", + "x_fit = linspace(x1,x3,20);\n", + "y_fit = polyval(p,x_fit);\n", + "\n", + "plot(x,Au_x(x),x_fit,y_fit,[x1,x2,x3],[f1,f2,f3],'o')\n", + "hold on\n", + "if f2x2\n", + " plot(x4,f4,'*',[x1,x2],[f1,f2])\n", + " x1=x2;\n", + " f1=f2;\n", + " else\n", + " plot(x4,f4,'*',[x3,x2],[f3,f2])\n", + " x3=x2;\n", + " f3=f2;\n", + " end\n", + " x2=x4; f2=f4;\n", + "else\n", + " error('no minimum in bracket')\n", + "end\n", + "hold off" + ] + }, + { + "cell_type": "code", + "execution_count": 32, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "f3 = -0.038878\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-0.04\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t-0.02\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t0\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t0.02\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t0.04\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t0.06\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t0.08\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t2.5\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t3\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t3.5\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t4\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t4.5\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t5\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t5.5\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t6\n", + "\t\n", + "\n", + "\n", + "\n", + "\n", + "\t\n", + "\n", + "\n", + "\tgnuplot_plot_1a\n", + "\n", + "\n", + "\n", + "\t\n", + "\t\n", + "\tgnuplot_plot_2a\n", + "\n", + "\t\n", + "\t\n", + "\tgnuplot_plot_3a\n", + "\n", + "\t \n", + "\t\n", + "\t\n", + "\t\n", + "\n", + "\t\n", + "\tgnuplot_plot_4a\n", + "\n", + "\t \n", + "\t\n", + "\n", + "\t\n", + "\tgnuplot_plot_5a\n", + "\n", + "\t\n", + "\t\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "p = polyfit([x1,x2,x3],[f1,f2,f3],2);\n", + "x_fit = linspace(x1,x3,20);\n", + "y_fit = polyval(p,x_fit);\n", + "\n", + "plot(x,Au_x(x),x_fit,y_fit,[x1,x2,x3],[f1,f2,f3],'o')\n", + "hold on\n", + "if f2x2\n", + " plot(x4,f4,'*',[x1,x2],[f1,f2])\n", + " x1=x2;\n", + " f1=f2;\n", + " else\n", + " plot(x4,f4,'*',[x3,x2],[f3,f2])\n", + " x3=x2;\n", + " f3=f2;\n", + " end\n", + " x2=x4; f2=f4;\n", + "else\n", + " error('no minimum in bracket')\n", + "end\n", + "hold off\n" + ] + }, + { + "cell_type": "code", + "execution_count": 33, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "f3 = -0.038878\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-0.04\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t-0.02\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t0\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t0.02\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t0.04\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t0.06\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t0.08\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t2.5\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t3\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t3.5\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t4\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t4.5\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t5\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t5.5\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t6\n", + "\t\n", + "\n", + "\n", + "\n", + "\n", + "\t\n", + "\n", + "\n", + "\tgnuplot_plot_1a\n", + "\n", + "\n", + "\n", + "\t\n", + "\t\n", + "\tgnuplot_plot_2a\n", + "\n", + "\t\n", + "\t\n", + "\tgnuplot_plot_3a\n", + "\n", + "\t \n", + "\t\n", + "\t\n", + "\t\n", + "\n", + "\t\n", + "\tgnuplot_plot_4a\n", + "\n", + "\t \n", + "\t\n", + "\n", + "\t\n", + "\tgnuplot_plot_5a\n", + "\n", + "\t\n", + "\t\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "p = polyfit([x1,x2,x3],[f1,f2,f3],2);\n", + "x_fit = linspace(x1,x3,20);\n", + "y_fit = polyval(p,x_fit);\n", + "\n", + "plot(x,Au_x(x),x_fit,y_fit,[x1,x2,x3],[f1,f2,f3],'o')\n", + "hold on\n", + "if f2x2\n", + " plot(x4,f4,'*',[x1,x2],[f1,f2])\n", + " x1=x2;\n", + " f1=f2;\n", + " else\n", + " plot(x4,f4,'*',[x3,x2],[f3,f2])\n", + " x3=x2;\n", + " f3=f2;\n", + " end\n", + " x2=x4; f2=f4;\n", + "else\n", + " error('no minimum in bracket')\n", + "end\n", + "hold off\n" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Parabolic interpolation does not converge in many scenarios even though it it a bracketing method. Instead, functions like `fminbnd` in Matlab and Octave use a combination of the two (Golden Ratio and Parabolic)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Using the solutions to minimization for the nonlinear spring constant\n", + "\n", + "Now, we have two routines to find minimums of a univariate function (Golden Ratio and Parabolic). Let's use these to solve for the minimum energy given a range of applied forces to the single atom gold chain\n", + "\n", + "$E_{total}(\\Delta x) = E_{LJ}(x_{min}+\\Delta x) - F \\cdot \\Delta x$" + ] + }, + { + "cell_type": "code", + "execution_count": 37, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "xmin = 3.2933\r\n" + ] + } + ], + "source": [ + "xmin = fminbnd(Au_x,2.8,6)\n", + "\n", + "Etotal = @(dx,F) Au_x(xmin+dx)-F.*dx;\n" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "collapsed": true + }, + "outputs": [], + "source": [] + } + ], + "metadata": { + "kernelspec": { + "display_name": "Octave", + "language": "octave", + "name": "octave" + }, + "language_info": { + "file_extension": ".m", + "help_links": [ + { + "text": "MetaKernel Magics", + "url": "https://github.com/calysto/metakernel/blob/master/metakernel/magics/README.md" + } + ], + "mimetype": "text/x-octave", + "name": "octave", + "version": "0.19.14" + } + }, + "nbformat": 4, + "nbformat_minor": 2 +} diff --git a/lecture_08/lennard_jones.m b/lecture_08/lennard_jones.m new file mode 100644 index 0000000..d18a6ad --- /dev/null +++ b/lecture_08/lennard_jones.m @@ -0,0 +1,4 @@ +function E_LJ =lennard_jones(x,sigma,epsilon) + E_LJ = 4*epsilon*((sigma./x).^12-(sigma./x).^6); +end + diff --git a/lecture_08/octave-workspace b/lecture_08/octave-workspace new file mode 100644 index 0000000000000000000000000000000000000000..f505b0a7d58e9e91f9e09b3ff78567b472aeda59 GIT binary patch literal 4894 zcmdT|cUV)|5|5&Ugcec(5s|f^SWvM885=4Jt_@L;>L}ATMu!UlW~~6UY%g z@gyBjXMw<8_^mUvF)u!s0>P(wVbpKEAAw?g8kHx@}{U@#HFr38a zhlNS3yzKo%?&5EQgM-6}f3?JqI^R#}!QIiN{N--afu#}0`6~?-l((!o!tYr7BFO!K zgipT|y5~z?4u5gN;?UuZX?)*NIxm~cWBCR5?+;g{h4LlY4^A!GY0bYP9Bnd*>4A*~ z3SQZaa1_1XFnpxPUc4B+Yi5gqS{JM$!19=I!6exLx~ zR}Z!(Vgb&$b1knVya1_@`ZW% zm^{pQho~bDxgRc^o8_E`g{67xzcd`g!{(`n`>YP)!9FuLj|aK<(8pqOh(#{GdzN@% zV_gnr@seXlFVDe2+p8*{s=-@T->qpL~ipR7U+^Gd=to1_8{%_NMRvv7J}aw1M?u3=gABw*HE_HV5l6R?)5 zwJGOPJeC-$y>Ximj}^?Xosb%bW#u;cJh?c`?4a+yvr`WV;?^As*PYe?ZYc8 zOv4-+W3Vf3`8MIU82r6|+SKU{d-3O%YULFBy{MItWf0dIjScVnUowQzI2V1ytUsdg zw0n10^xi1cPsmHNQ;))3>(&IN6z{=|iKjSf{5|NQKUaM3(QZt7R{Nw{up8UW_4YT@ zcH@&?10(GtkyvirxZBe-65SU}*>JTr0*ySYF0o@HaNg_iK`Vm@q!{&@e!_*QIpL_zae0$=FAQsMmTa=08-{jz zsR?!kq4-$PbS_*a6n9>)vOX2M6MblnTN=7U(DJVEU7u?RhBr??($lm9D}0uU&ROq3 z-HNe^3+sb1a^$8|PwQYTthIDAY6`-qgN9}~EkGDaNVgW*6=Ui&6RT#C7os zVmz1~NzKm)L=EK+c>|LJu~_AdekKNBXQAZ?7s~*=bYtVKO`ZPu@j$hk!#;n^KKzar zJkcMw#q^Cyt?|Rkz;qjls~=iTmt4r9`=MdcW4FMQzUcHa=*2`!Upz`ttC4;aVXh62 z`Km~S_h+1MeY8P@F4-wZ#KT3H)Fo=iQXiCy9v3cg@WDXc`2rrt2bFZ|-*avXv1^ai zdw-}9U%v87ik~gSz>Kp?-`0EMnUdz!32VGjZdO`m_jLijEUNllo-e?&TR3KT!3$MY z+YOZFdtrK;u}$N7Pwe-uY@2NCiC4})o%iah2ky&wDF4aK17F?fv)cS z9iN^YuKlXZ9lNW{S#Ni_V~ziq1TM!NL%2&Q8;jkLeRtZJoJDTvRprs#^xPFs7*HeB z16(m8P04>A%M~qz3c@*ME?E0G)lp%+3%0nl1t0q4jNJtxn!~c3(bc!8fep?WP`x(m zajO$9`WFR2N@38z1 zA>^L2;&W{<9RIYZ5+i z+@JtY*w@q7`71!>VBO|L=?ZYT{hpRVg#yGqdfce}Q~`XeUPzSt6kyA-X6LUOG+310 zrq{86270^RcRbrl169wDCu&1z@Y%B8ry`dIFBef}7go}sdW!s)v_={n<0)N?q|hNn zrU)H^-tiXMFQ$Xt)6pY-cBI3KYd_sE*-eLOCnoKidYBH}%<1M@HFOvt)orE2Yr(PbA4lZtRlFM+$@nER|JbL8|RceiZHw0i4ai{ zv}(3@c&jRbT&eJcje!!hX9xJ0uT_Fyqe2L%mEfFDMDOHOCCIbhdqJaI38I!K>{NK9 z1cI4q(ECXVHjK`a`<<%{Mhf{g59ca_?uWxM*Q}I*-f(PnS&%Yx)t0IjWGTbb)1*a| z0c8eJhC+hNYd{&|{ceu3)>Z*O+g}>Y_$si~>`}pdI~7AafAlc)fpZ2QP- zD#V^|OFp)!KtofHuw{fQT)*{Y*8TCSaQy7I&sCc-Ea`Mqj}I z^VTuPcUdt&W@a}A7~UW6Eev8n5#c?pF$@@WS%0Q&76TH>PNY6AX29^G8K2BAG9Vzg z-17P@1`vy#GkU=Q8GLUUaBrva4*dZJER{Kq8dSI~G11gk1AUvCyyR(WaB%&~VKlxP zFjwAo2w$THdra0;_u8sKAEBkUPz}f^_&r<=nzWy`ZA(^z^=uL`HMl_|0at?sgLcZ* zSJmLe#}>jf)Ihi0N-EFbC}SDBFm)lOmIB&se(F( z36G1#8eyhPFw6fV__HMwD#`G1Wx|ZuVJ5!?GU0IeZo(CqpdLUS_ArA9aUS~|Rvl%6 zY}@lp@FiATP{)MdSFxf?pEF^LEFhS0XO0H3Boh`*$st^x1?6MMC?{&MK#wO8$WLWK zj?#Dn8WyO^0AvB_b)GgX_)MBxz=G|~29{gHSn%=*=}0X2S+*bxs;{08zH*8M^9YpZ zU17no5;Q*bD+{!xi}NNnvmhnoIwh}%1@Z}`2e2V1^47I@78~A)Eyj-=#|AqA>FI2E z=(wTn!%{Y^Afm@(BO5MSkP*#>X-i)!Z3$q5WIoBs2JZBhUsfDoL(Djr8C6HvAaj>< zY!J!V(Q1C~_eg zIiO{&Ey!u&K+@WRUs&B7kXt^EFdY{HjE)dT3>Vr7F9_1*g3ZK|jomZ3@GEhexh>(s zQs!y(ruAGnN1YSA$%zY72F}&r_2WWb&q8C-Tw z;)LQt)8)Ex*-yA2J4V~MP)ml#7cLmd9GV9wiRch(@IW_avr^Y29;B0QwSWh7S>W?v zr#oq29>|U-4<0y?<_O`zQ^FdT;&`x%|NiW(Tpm;!k_z)+CaFdx4+_Tw)@R=3fd;4F zNUf0vvas*ufh;Le)Ma0>;-Tqzmh7Q@Cl5`}7HdzsJGy`W1{M!Z&sjDREeA4xkcXz{ zKWCGd6k=N!cR!ghvJCn6?8+tx)IUX8H5tb-OljaWb`|%X6%9hvpHS;OXppAr*jE@z zgX8j@?UBhexIPg4!tM}}Nj}QoHLjq+r;ZfVtfPT)i*{-63mQy#UMzXoNduE#=cJVm zr-Nnf)jbIuItb7r&_j<7(Py8#tyoBhyb_l!<1Oe=aj0k2aC;qZ>Hh!L@yAt-Y=EL9yNLO(7Y!s#LH`pbordoP zRg*?bsNq@(+GQRs&)6eDu-+utaaMv=4sVaWXptbL#LerHvJ}G?+%D5Jlwy@4FQnH} ziupP#Z>|fFVw2b*Yf1{yvK>2j7fTU7n$>PaDeCYv$CutC_;%hKxu=1kPZVr_(k4Ya l&WCkHoy54QE%x>h>s~B7d7WsheBb@VB*w$NRkg@c?O#Po{_p?* literal 0 HcmV?d00001 diff --git "a/lecture_08/\34040\001" "b/lecture_08/\34040\001" new file mode 100644 index 0000000..e69de29 From 4b1d857f609a380cb94af481215f4ad7f27da4c1 Mon Sep 17 00:00:00 2001 From: "Ryan C. Cooper" Date: Mon, 13 Feb 2017 16:58:14 -0500 Subject: [PATCH 04/10] update HW3 --- HW3/README.md | 3 +-- 1 file changed, 1 insertion(+), 2 deletions(-) diff --git a/HW3/README.md b/HW3/README.md index eb93c41..891eb1d 100644 --- a/HW3/README.md +++ b/HW3/README.md @@ -70,5 +70,4 @@ guesses that do not converge. 'divergence' to 'convergence') 3. Commit your changes to your repository. Sync your local repository with github. Then -copy and paste the "clone URL" into the following Google Form [Homework -#3](https://goo.gl/forms/UJBGwp0fQcSxImkq2) +copy and paste the "clone URL" into the following Google Form [Homework 3](https://goo.gl/forms/UJBGwp0fQcSxImkq2) From 593b1de7d5b91589adbe5d2b6706479e798fed4e Mon Sep 17 00:00:00 2001 From: Ryan C Cooper Date: Mon, 13 Feb 2017 17:01:25 -0500 Subject: [PATCH 05/10] updated HW3 1d --- HW3/README.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/HW3/README.md b/HW3/README.md index 891eb1d..45214a4 100644 --- a/HW3/README.md +++ b/HW3/README.md @@ -11,7 +11,7 @@ c. Copy your `projectile.m` function into the 'roots_and_optimization' folder. *Disable the plotting routine for the solvers* - d. Use the four solvers `falsepos.m`, `incsearch.m`, `newtraph.m` and `mod_secant.m` + d. Use the four solvers `falsepos.m`, `bisect.m`, `newtraph.m` and `mod_secant.m` to solve for the angle needed to reach h=1.72 m, with an initial speed of 15 m/s. e. The `newtraph.m` function needs a derivative, calculate the derivative of your From 59658f7e3b0880cbaa0d79038195e7ba8af476af Mon Sep 17 00:00:00 2001 From: Ryan C Cooper Date: Mon, 13 Feb 2017 17:06:10 -0500 Subject: [PATCH 06/10] =?UTF-8?q?Delete=20=10=EF=BF=BDY=01?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- "lecture_08/\020\267Y\001" | 0 1 file changed, 0 insertions(+), 0 deletions(-) delete mode 100644 "lecture_08/\020\267Y\001" diff --git "a/lecture_08/\020\267Y\001" "b/lecture_08/\020\267Y\001" deleted file mode 100644 index e69de29..0000000 From e5f170b8bf33104074821192dd93176526ac7fd3 Mon Sep 17 00:00:00 2001 From: Ryan C Cooper Date: Mon, 13 Feb 2017 17:06:15 -0500 Subject: [PATCH 07/10] =?UTF-8?q?Delete=20=EF=BF=BD40=01?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- "lecture_08/\34040\001" | 0 1 file changed, 0 insertions(+), 0 deletions(-) delete mode 100644 "lecture_08/\34040\001" diff --git "a/lecture_08/\34040\001" "b/lecture_08/\34040\001" deleted file mode 100644 index e69de29..0000000 From 5128cfdd16e951bae9c50931c11a4a10062adbc6 Mon Sep 17 00:00:00 2001 From: "Ryan C. 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Engineers and scientists typically use energy balance equations to find the conditions of minimum energy, but this value may never go to 0 and the actual value of energy may not be of interest. " ] }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": true - }, - "outputs": [], - "source": [] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": true - }, - "outputs": [], - "source": [] - }, { "cell_type": "markdown", "metadata": {}, @@ -64,7 +46,7 @@ }, { "cell_type": "code", - "execution_count": 3, + "execution_count": 2, "metadata": { "collapsed": false }, @@ -243,7 +225,7 @@ }, { "cell_type": "code", - "execution_count": 4, + "execution_count": 3, "metadata": { "collapsed": false }, @@ -281,7 +263,7 @@ }, { "cell_type": "code", - "execution_count": 5, + "execution_count": 4, "metadata": { "collapsed": false }, @@ -515,7 +497,7 @@ }, { "cell_type": "code", - "execution_count": 6, + "execution_count": 5, "metadata": { "collapsed": false }, @@ -738,7 +720,7 @@ }, { "cell_type": "code", - "execution_count": 10, + "execution_count": 6, "metadata": { "collapsed": false }, @@ -748,7 +730,7 @@ "output_type": "stream", "text": [ "current_min = -0.038904\n", - "error_app = 0\n" + "error_app = 0.13359\n" ] }, { @@ -888,30 +870,30 @@ "\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", "\n", "\t\n", "\tgnuplot_plot_4a\n", "\n", "\t \n", - "\t\n", + "\t\n", "\n", "\t\n", "\tgnuplot_plot_5a\n", "\n", "\t \n", - "\t\n", + "\t\n", "\n", "\t\n", "\tgnuplot_plot_6a\n", "\n", - "\t\n", + "\t\n", "\t\n", "\n", "\n", @@ -961,7 +943,7 @@ }, { "cell_type": "code", - "execution_count": 13, + "execution_count": 7, "metadata": { "collapsed": false }, @@ -970,7 +952,7 @@ "name": "stdout", "output_type": "stream", "text": [ - "current_min = -0.039000\n", + "current_min = -0.038904\n", "error_app = 0\n" ] }, @@ -1111,30 +1093,30 @@ "\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", "\n", "\t\n", "\tgnuplot_plot_4a\n", "\n", "\t \n", - "\t\n", + "\t\n", "\n", "\t\n", "\tgnuplot_plot_5a\n", "\n", "\t \n", - "\t\n", + "\t\n", "\n", "\t\n", "\tgnuplot_plot_6a\n", "\n", - "\t\n", + "\t\n", "\t\n", "\n", "\n", @@ -1188,7 +1170,7 @@ "collapsed": true }, "source": [ - "## Parabolc Interpolation\n", + "## Parabolic Interpolation\n", "\n", "Near a minimum/maximum, the function resembles a parabola. With three data points, it is possible to fit a parabola to the function. So, given a lower and upper bound, we can choose the midpoint and fit a parabola to the 3 x,f(x) coordinates. \n", "\n", @@ -1202,7 +1184,7 @@ }, { "cell_type": "code", - "execution_count": 31, + "execution_count": 8, "metadata": { "collapsed": false }, @@ -1436,18 +1418,11 @@ }, { "cell_type": "code", - "execution_count": 32, + "execution_count": 9, "metadata": { "collapsed": false }, "outputs": [ - { - "name": "stdout", - "output_type": "stream", - "text": [ - "f3 = -0.038878\r\n" - ] - }, { "data": { "image/svg+xml": [ @@ -1653,18 +1628,11 @@ }, { "cell_type": "code", - "execution_count": 33, + "execution_count": 10, "metadata": { "collapsed": false }, "outputs": [ - { - "name": "stdout", - "output_type": "stream", - "text": [ - "f3 = -0.038878\r\n" - ] - }, { "data": { "image/svg+xml": [ @@ -1888,7 +1856,7 @@ }, { "cell_type": "code", - "execution_count": 37, + "execution_count": 24, "metadata": { "collapsed": false }, @@ -1897,24 +1865,962 @@ "name": "stdout", "output_type": "stream", "text": [ - "xmin = 3.2933\r\n" + "xmin = 0.32933\n", + "Emin = -2.7096e-04\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-0.0004\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t-0.0002\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t0\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t0.0002\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t0.0004\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t0.0006\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t0.2\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + 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"sigma = sigma*0.10; % nm/Angstrom\n", + "x=linspace(2.8,6,200)*0.10; % bond length in um\n", + "\n", + "Ex = lennard_jones(x,sigma,epsilon);\n", "\n", - "Etotal = @(dx,F) Au_x(xmin+dx)-F.*dx;\n" + "%[Emin,imin]=min(Ex);\n", + "\n", + "[xmin,Emin] = fminbnd(@(x) lennard_jones(x,sigma,epsilon),0.28,0.6)\n", + "\n", + "plot(x,Ex,xmin,Emin,'o')\n", + "ylabel('Lennard Jones Potential (aJ/atom)')\n", + "xlabel('bond length (nm)')\n", + "\n", + "Etotal = @(dx,F) lennard_jones(xmin+dx,sigma,epsilon)-F.*dx;\n" ] }, { "cell_type": "code", - "execution_count": null, + "execution_count": 114, "metadata": { - "collapsed": true + "collapsed": false }, - "outputs": [], - "source": [] + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "warning: Matlab-style short-circuit operation performed for operator |\n", + "warning: called from\n", + " goldmin at line 17 column 1\n", + "warning: Matlab-style short-circuit operation performed for operator |\n", + "warning: 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"output_type": "display_data" + } + ], + "source": [ + "N=50;\n", + "dx = zeros(1,N); % [in nm]\n", + "F_applied=linspace(0,0.0022,N); % [in nN]\n", + "for i=1:N\n", + " optmin=goldmin(@(dx) Etotal(dx,F_applied(i)),-0.001,0.035);\n", + " dx(i)=optmin;\n", + "end\n", + "\n", + "plot(dx,F_applied)\n", + "xlabel('dx (nm)')\n", + "ylabel('Force (nN)')" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "For this function, it is possible to take a derivative and compare the analytical result:" + ] + }, + { + "cell_type": "code", + "execution_count": 105, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "F =\n", + "\n", + "@(dx) 4 * epsilon * 6 * (sigma ^ 6 ./ (xmin + dx) .^ 7 - 2 * sigma ^ 12 ./ (xmin + dx) .^ 13)\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\t0\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t0.0005\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t0.001\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t0.0015\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t0.002\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t0.0025\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t0\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t0.01\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t0.02\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t0.03\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t0.04\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t0.05\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t0.06\n", + "\t\n", + "\n", + "\n", + "\n", + "\n", + "\t\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": [ + "dx_full=linspace(0,0.06,50);\n", + "F= @(dx) 4*epsilon*6*(sigma^6./(xmin+dx).^7-2*sigma^12./(xmin+dx).^13)\n", + "plot(dx_full,F(dx_full),dx,F_applied)" + ] + }, + { + "cell_type": "markdown", + "metadata": { + "collapsed": true + }, + "source": [ + "## Curve-fitting\n", + "Another example is minimizing error in your approximation of a function. If you have data (now we have Force-displacement data) we can fit this to a function, such as:\n", + "\n", + "$F(x) = K_{1}\\Delta x + \\frac{1}{2} K_{2}(\\Delta x)^{2}$\n" + ] + }, + { + "cell_type": "code", + "execution_count": 106, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "function SSE = sse_of_parabola(K,xdata,ydata)\n", + " % calculate the sum of squares error for a parabola given a function, func, and xdata and ydata\n", + " % output is SSE=sum of squares error\n", + " K1=K(1);\n", + " K2=K(2);\n", + " y_function = K1*xdata+1/2*K2*xdata.^2;\n", + " SSE = sum((ydata-y_function).^2);\n", + "end\n", + " " + ] + }, + { + "cell_type": "code", + "execution_count": 113, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "\n", + "The nonlinear spring constants are K1=0.16 nN/nm and K2=-5.98 nN/nm^2\n", + "The mininum sum of squares error = 7.35e-08\n" + ] + } + ], + "source": [ + "[K,SSE_min]=fminsearch(@(K) sse_of_parabola(K,dx,F_applied),[1,1]);\n", + "fprintf('\\nThe nonlinear spring constants are K1=%1.2f nN/nm and K2=%1.2f nN/nm^2\\n',K)\n", + "fprintf('The mininum sum of squares error = %1.2e',SSE_min)" + ] + }, + { + "cell_type": "code", + "execution_count": 108, + "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\t0\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t0.0005\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t0.001\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t0.0015\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t0.002\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t0.0025\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t0\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t0.005\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t0.01\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t0.015\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t0.02\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t0.025\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t0.03\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t0.035\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t0.04\n", + "\t\n", + "\n", + "\n", + "\n", + "\n", + "\t\n", + "\n", + "\n", + "\tgnuplot_plot_1a\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", + "\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", + "\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": [ + "plot(dx,F_applied,'o',dx,K(1)*dx+1/2*K(2)*dx.^2)" + ] } ], "metadata": { diff --git a/lecture_08/lecture_08.md b/lecture_08/lecture_08.md new file mode 100644 index 0000000..8a81a2a --- /dev/null +++ b/lecture_08/lecture_08.md @@ -0,0 +1,626 @@ + + +```octave +%plot --format svg +``` + +# Optimization + +Many problems involve finding a minimum or maximum based on given constraints. Engineers and scientists typically use energy balance equations to find the conditions of minimum energy, but this value may never go to 0 and the actual value of energy may not be of interest. + +The Lennard-Jones potential is commonly used to model interatomic bonding. + +$E_{LJ}(x)=4\epsilon \left(\left(\frac{\sigma}{x}\right)^{12}-\left(\frac{\sigma}{x}\right)^{6}\right)$ + +Considering a 1-D gold chain, we can calculate the bond length, $x_{b}$, with no force applied to the chain and even for tension, F. This will allow us to calculate the nonlinear spring constant of a 1-D gold chain. + +![TEM image of Gold chain](au_chain.jpg) + +Computational Tools to Study and Predict the Long-Term Stability of Nanowires. +By Martin E. Zoloff Michoff, Patricio Vélez, Sergio A. Dassie and Ezequiel P. M. Leiva + +![Model of Gold chain, from molecular dynamics simulation](Auchain_model.gif) + +[Single atom gold chain mechanics](http://www.uam.es/personal_pdi/ciencias/agrait/) + +### First, let's find the minimum energy $\min(E_{LJ}(x))$ + +## Brute force + + +```octave +setdefaults +epsilon = 0.039; % kcal/mol +sigma = 2.934; % Angstrom +x=linspace(2.8,6,200); % bond length in Angstrom + +Ex = lennard_jones(x,sigma,epsilon); + +[Emin,imin]=min(Ex); + +plot(x,Ex,x(imin),Emin,'o') +``` + + +![svg](lecture_08_files/lecture_08_3_0.svg) + + + +```octave +x(imin-1) +x(imin) +x(imin+1) +``` + + ans = 3.2824 + ans = 3.2985 + ans = 3.3146 + + +## Golden Search Algorithm + +We can't just look for a sign change for the problem (unless we can take a derivative) so we need a new approach to determine whether we have a maximum between the two bounds. + +Rather than using the midpoint of initial bounds, here the problem is more difficult. We need to compare the values of 4 function evaluations. The golden search uses the golden ratio to determine two interior points. + +![golden ratio](goldenratio.png) + +Start with bounds of 2.5 and 6 Angstrom. + + +```octave +% define Au atomic potential +epsilon = 0.039; % kcal/mol +sigma = 2.934; % Angstrom +Au_x= @(x) lennard_jones(x,sigma,epsilon); + +% calculate golden ratio +phi = 1/2+sqrt(5)/2; +% set initial limits +x_l=2.8; +x_u=6; + +% Iteration #1 +d=(phi-1)*(x_u-x_l); + +x1=x_l+d; % define point 1 +x2=x_u-d; % define point 2 + + +% evaluate Au_x(x1) and Au_x(x2) + +f1=Au_x(x1); +f2=Au_x(x2); +plot(x,Au_x(x),x_l,Au_x(x_l),'ro',x2,f2,'rs',x1,f1,'gs',x_u,Au_x(x_u),'go') +hold on; + +if f2x2 + plot(x4,f4,'*',[x1,x2],[f1,f2]) + x1=x2; + f1=f2; + else + plot(x4,f4,'*',[x3,x2],[f3,f2]) + x3=x2; + f3=f2; + end + x2=x4; f2=f4; +else + error('no minimum in bracket') +end +hold off +``` + + +![svg](lecture_08_files/lecture_08_11_0.svg) + + + +```octave +p = polyfit([x1,x2,x3],[f1,f2,f3],2); +x_fit = linspace(x1,x3,20); +y_fit = polyval(p,x_fit); + +plot(x,Au_x(x),x_fit,y_fit,[x1,x2,x3],[f1,f2,f3],'o') +hold on +if f2x2 + plot(x4,f4,'*',[x1,x2],[f1,f2]) + x1=x2; + f1=f2; + else + plot(x4,f4,'*',[x3,x2],[f3,f2]) + x3=x2; + f3=f2; + end + x2=x4; f2=f4; +else + error('no minimum in bracket') +end +hold off + +``` + + +![svg](lecture_08_files/lecture_08_12_0.svg) + + + +```octave +p = polyfit([x1,x2,x3],[f1,f2,f3],2); +x_fit = linspace(x1,x3,20); +y_fit = polyval(p,x_fit); + +plot(x,Au_x(x),x_fit,y_fit,[x1,x2,x3],[f1,f2,f3],'o') +hold on +if f2x2 + plot(x4,f4,'*',[x1,x2],[f1,f2]) + x1=x2; + f1=f2; + else + plot(x4,f4,'*',[x3,x2],[f3,f2]) + x3=x2; + f3=f2; + end + x2=x4; f2=f4; +else + error('no minimum in bracket') +end +hold off + +``` + + +![svg](lecture_08_files/lecture_08_13_0.svg) + + +Parabolic interpolation does not converge in many scenarios even though it it a bracketing method. Instead, functions like `fminbnd` in Matlab and Octave use a combination of the two (Golden Ratio and Parabolic) + +## Using the solutions to minimization for the nonlinear spring constant + +Now, we have two routines to find minimums of a univariate function (Golden Ratio and Parabolic). Let's use these to solve for the minimum energy given a range of applied forces to the single atom gold chain + +$E_{total}(\Delta x) = E_{LJ}(x_{min}+\Delta x) - F \cdot \Delta x$ + + +```octave +epsilon = 0.039; % kcal/mol +epsilon = epsilon*6.9477e-21; % J/atom +epsilon = epsilon*1e18; % aJ/J +% final units for epsilon are aJ + +sigma = 2.934; % Angstrom +sigma = sigma*0.10; % nm/Angstrom +x=linspace(2.8,6,200)*0.10; % bond length in um + +Ex = lennard_jones(x,sigma,epsilon); + +%[Emin,imin]=min(Ex); + +[xmin,Emin] = fminbnd(@(x) lennard_jones(x,sigma,epsilon),0.28,0.6) + +plot(x,Ex,xmin,Emin,'o') +ylabel('Lennard Jones Potential (aJ/atom)') +xlabel('bond length (nm)') + +Etotal = @(dx,F) lennard_jones(xmin+dx,sigma,epsilon)-F.*dx; + +``` + + xmin = 0.32933 + Emin = -2.7096e-04 + + + +![svg](lecture_08_files/lecture_08_16_1.svg) + + + +```octave +N=50; +dx = zeros(1,N); % [in nm] +F_applied=linspace(0,0.0022,N); % [in nN] +for i=1:N + optmin=goldmin(@(dx) Etotal(dx,F_applied(i)),-0.001,0.035); + dx(i)=optmin; +end + +plot(dx,F_applied) +xlabel('dx (nm)') +ylabel('Force (nN)') +``` + + warning: Matlab-style short-circuit operation performed for operator | + warning: called from + goldmin at line 17 column 1 + warning: Matlab-style short-circuit operation performed for operator | + warning: Matlab-style short-circuit operation performed for operator | + warning: Matlab-style short-circuit operation performed for operator | + warning: Matlab-style short-circuit operation performed for operator | + warning: Matlab-style short-circuit operation performed for operator | + warning: Matlab-style short-circuit operation performed for operator | + warning: Matlab-style short-circuit operation performed for operator | + warning: Matlab-style short-circuit operation performed for operator | + warning: Matlab-style short-circuit operation performed for operator | + warning: Matlab-style short-circuit operation performed for operator | + warning: 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Matlab-style short-circuit operation performed for operator | + warning: Matlab-style short-circuit operation performed for operator | + warning: Matlab-style short-circuit operation performed for operator | + warning: Matlab-style short-circuit operation performed for operator | + warning: Matlab-style short-circuit operation performed for operator | + warning: Matlab-style short-circuit operation performed for operator | + warning: Matlab-style short-circuit operation performed for operator | + warning: Matlab-style short-circuit operation performed for operator | + warning: Matlab-style short-circuit operation performed for operator | + warning: Matlab-style short-circuit operation performed for operator | + warning: Matlab-style short-circuit operation performed for operator | + warning: Matlab-style short-circuit operation performed for operator | + warning: Matlab-style short-circuit operation performed for operator | + warning: Matlab-style short-circuit operation performed for operator | + warning: Matlab-style short-circuit operation performed for operator | + warning: Matlab-style short-circuit operation performed for operator | + warning: Matlab-style short-circuit operation performed for operator | + warning: Matlab-style short-circuit operation performed for operator | + warning: Matlab-style short-circuit operation performed for operator | + warning: Matlab-style short-circuit operation performed for operator | + warning: Matlab-style short-circuit operation performed for operator | + warning: Matlab-style short-circuit operation performed for operator | + warning: Matlab-style short-circuit operation performed for operator | + warning: Matlab-style short-circuit operation performed for operator | + warning: Matlab-style short-circuit operation performed for operator | + warning: Matlab-style short-circuit operation performed for operator | + warning: Matlab-style short-circuit operation performed for operator | + warning: Matlab-style short-circuit operation performed for operator | + warning: Matlab-style short-circuit operation performed for operator | + warning: Matlab-style short-circuit operation performed for operator | + warning: Matlab-style short-circuit operation performed for operator | + warning: Matlab-style short-circuit operation performed for operator | + warning: Matlab-style short-circuit operation performed for operator | + warning: Matlab-style short-circuit operation performed for operator | + warning: Matlab-style short-circuit operation performed for operator | + warning: Matlab-style short-circuit operation performed for operator | + warning: Matlab-style short-circuit operation performed for operator | + warning: Matlab-style short-circuit operation performed for operator | + warning: Matlab-style short-circuit operation performed for operator | + warning: Matlab-style short-circuit operation performed for operator | + warning: Matlab-style short-circuit operation performed for operator | + warning: Matlab-style short-circuit operation performed for operator | + warning: Matlab-style short-circuit operation performed for operator | + warning: Matlab-style short-circuit operation performed for operator | + warning: Matlab-style short-circuit operation performed for operator | + warning: Matlab-style short-circuit operation performed for operator | + warning: Matlab-style short-circuit operation performed for operator | + warning: Matlab-style short-circuit operation performed for operator | + warning: Matlab-style short-circuit operation performed for operator | + warning: Matlab-style short-circuit operation performed for operator | + warning: Matlab-style short-circuit operation performed for operator | + warning: Matlab-style short-circuit operation performed for operator | + warning: Matlab-style short-circuit operation performed for operator | + warning: Matlab-style short-circuit operation performed for operator | + warning: Matlab-style short-circuit operation performed for operator | + warning: Matlab-style short-circuit operation performed for operator | + + + +![svg](lecture_08_files/lecture_08_17_1.svg) + + +For this function, it is possible to take a derivative and compare the analytical result: + + +```octave +dx_full=linspace(0,0.06,50); +F= @(dx) 4*epsilon*6*(sigma^6./(xmin+dx).^7-2*sigma^12./(xmin+dx).^13) +plot(dx_full,F(dx_full),dx,F_applied) +``` + + F = + + @(dx) 4 * epsilon * 6 * (sigma ^ 6 ./ (xmin + dx) .^ 7 - 2 * sigma ^ 12 ./ (xmin + dx) .^ 13) + + + + +![svg](lecture_08_files/lecture_08_19_1.svg) + + +## Curve-fitting +Another example is minimizing error in your approximation of a function. If you have data (now we have Force-displacement data) we can fit this to a function, such as: + +$F(x) = K_{1}\Delta x + \frac{1}{2} K_{2}(\Delta x)^{2}$ + + + +```octave +function SSE = sse_of_parabola(K,xdata,ydata) + % calculate the sum of squares error for a parabola given a function, func, and xdata and ydata + % output is SSE=sum of squares error + K1=K(1); + K2=K(2); + y_function = K1*xdata+1/2*K2*xdata.^2; + SSE = sum((ydata-y_function).^2); +end + +``` + + +```octave +[K,SSE_min]=fminsearch(@(K) sse_of_parabola(K,dx,F_applied),[1,1]); +fprintf('\nThe nonlinear spring constants are K1=%1.2f nN/nm and K2=%1.2f nN/nm^2\n',K) +fprintf('The mininum sum of squares error = %1.2e',SSE_min) +``` + + + The nonlinear spring constants are K1=0.16 nN/nm and K2=-5.98 nN/nm^2 + The mininum sum of squares error = 7.35e-08 + + + +```octave +plot(dx,F_applied,'o',dx,K(1)*dx+1/2*K(2)*dx.^2) +``` + + +![svg](lecture_08_files/lecture_08_23_0.svg) + diff --git 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-x=xopt;fx=f(xopt,varargin{:}); \ No newline at end of file +x=xopt;fx=f(xopt,varargin{:}); diff --git a/lecture_08/lecture_08.ipynb b/lecture_08/lecture_08.ipynb index d99107a..32b973c 100644 --- a/lecture_08/lecture_08.ipynb +++ b/lecture_08/lecture_08.ipynb @@ -2,7 +2,41 @@ "cells": [ { "cell_type": "code", - "execution_count": 1, + "execution_count": 6, + "metadata": { + "collapsed": true + }, + "outputs": [], + "source": [ + "function h = parabola(x)\n", + " y=1;\n", + " h=sum(x.^2+y.^2-1);\n", + "end\n", + " " + ] + }, + { + "cell_type": "code", + "execution_count": 9, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "ans = 0\r\n" + ] + } + ], + "source": [ + "fzero(@(x) parabola(x),1)" + ] + }, + { + "cell_type": "code", + "execution_count": 10, "metadata": { "collapsed": false }, @@ -35,7 +69,7 @@ "Computational Tools to Study and Predict the Long-Term Stability of Nanowires.\n", "By Martin E. Zoloff Michoff, Patricio Vélez, Sergio A. Dassie and Ezequiel P. M. Leiva \n", "\n", - "![Model of Gold chain, from molecular dynamics simulation](Auchain_model.gif)\n", + "![Model of Gold chain, from molecular dynamics simulation](Auchain_model.png)\n", "\n", "[Single atom gold chain mechanics](http://www.uam.es/personal_pdi/ciencias/agrait/)\n", "\n", @@ -46,7 +80,7 @@ }, { "cell_type": "code", - "execution_count": 2, + "execution_count": 11, "metadata": { "collapsed": false }, @@ -225,7 +259,7 @@ }, { "cell_type": "code", - "execution_count": 3, + "execution_count": 12, "metadata": { "collapsed": false }, @@ -263,7 +297,7 @@ }, { "cell_type": "code", - "execution_count": 4, + "execution_count": 13, "metadata": { "collapsed": false }, @@ -497,7 +531,7 @@ }, { "cell_type": "code", - "execution_count": 5, + "execution_count": 14, "metadata": { "collapsed": false }, @@ -720,7 +754,7 @@ }, { "cell_type": "code", - "execution_count": 6, + "execution_count": 15, "metadata": { "collapsed": false }, @@ -943,7 +977,7 @@ }, { "cell_type": "code", - "execution_count": 7, + "execution_count": 16, "metadata": { "collapsed": false }, @@ -1174,17 +1208,14 @@ "\n", "Near a minimum/maximum, the function resembles a parabola. With three data points, it is possible to fit a parabola to the function. So, given a lower and upper bound, we can choose the midpoint and fit a parabola to the 3 x,f(x) coordinates. \n", "\n", - "$ x_{4} =x_{2} - \\frac{1}{2} \\frac{ \\left(x_{2} -x_{1} \\right)^{2} \\left[f\n", - " \\left(x_{2} \\right)-f \\left(x_{3} \\right) \\right]- \\left(x_{2} -x_{3} \\right)^{2}\n", - " \\left[f \\left(x_{2} \\right)-f \\left(x_{1} \\right) \\right]}{ \\left(x_{2} -x_{1} \\right) \\left[f \\left(x_{2} \\right)-f \\left(x_{3} \\right)\n", - " \\right]- \\left(x_{2} -x_{3} \\right) \\left[f \\left(x_{2} \\right)-f \\left(x_{1} \\right) \\right]}$\n", + "$$ x_{4} =x_{2} - \\frac{1}{2}\\frac{\\left(x_{2} -x_{1}\\right)^{2} \\left(f\\left(x_{2} \\right)-f \\left(x_{3} \\right) \\right)- \\left(x_{2} -x_{3} \\right)^{2}\\left(f \\left(x_{2} \\right)-f \\left(x_{1} \\right) \\right)}{ \\left(x_{2} -x_{1} \\right) \\left(f \\left(x_{2} \\right)-f \\left(x_{3} \\right)\\right)- \\left(x_{2} -x_{3} \\right) \\left(f \\left(x_{2} \\right)-f \\left(x_{1} \\right) \\right)}$$\n", "\n", "Where $x_{4}$ location of the maximum or minimum of the parabola. " ] }, { "cell_type": "code", - "execution_count": 8, + "execution_count": 23, "metadata": { "collapsed": false }, @@ -1418,7 +1449,7 @@ }, { "cell_type": "code", - "execution_count": 9, + "execution_count": 24, "metadata": { "collapsed": false }, @@ -1628,7 +1659,7 @@ }, { "cell_type": "code", - "execution_count": 10, + "execution_count": 25, "metadata": { "collapsed": false }, @@ -1851,12 +1882,14 @@ "\n", "Now, we have two routines to find minimums of a univariate function (Golden Ratio and Parabolic). Let's use these to solve for the minimum energy given a range of applied forces to the single atom gold chain\n", "\n", - "$E_{total}(\\Delta x) = E_{LJ}(x_{min}+\\Delta x) - F \\cdot \\Delta x$" + "$E_{total}(\\Delta x) = E_{LJ}(x_{min}+\\Delta x) - F \\cdot \\Delta x$\n", + "\n", + "$1 aJ = 10^{-18} J = 1~nN* 1~nm = 10^{-9}N * 10^{-9} m$" ] }, { "cell_type": "code", - "execution_count": 24, + "execution_count": 26, "metadata": { "collapsed": false }, @@ -1865,6 +1898,11 @@ "name": "stdout", "output_type": "stream", "text": [ + "warning: Matlab-style short-circuit operation performed for operator |\n", + "warning: called from\n", + " goldmin at line 17 column 1\n", + "warning: Matlab-style short-circuit operation performed for operator |\n", + "warning: Matlab-style short-circuit operation performed for operator |\n", "xmin = 0.32933\n", "Emin = -2.7096e-04\n" ] @@ -2037,7 +2075,7 @@ "\n", "%[Emin,imin]=min(Ex);\n", "\n", - "[xmin,Emin] = fminbnd(@(x) lennard_jones(x,sigma,epsilon),0.28,0.6)\n", + "[xmin,Emin] = goldmin(@(x) lennard_jones(x,sigma,epsilon),0.28,0.6)\n", "\n", "plot(x,Ex,xmin,Emin,'o')\n", "ylabel('Lennard Jones Potential (aJ/atom)')\n", @@ -2048,168 +2086,11 @@ }, { "cell_type": "code", - "execution_count": 114, + "execution_count": 31, "metadata": { "collapsed": false }, "outputs": [ - { - "name": "stdout", - "output_type": "stream", - "text": [ - "warning: Matlab-style short-circuit operation performed for operator |\n", - "warning: called from\n", - " goldmin at line 17 column 1\n", - "warning: Matlab-style short-circuit operation performed for operator |\n", - "warning: Matlab-style short-circuit operation performed for operator |\n", - "warning: Matlab-style short-circuit operation performed for operator |\n", - "warning: Matlab-style short-circuit operation performed for operator |\n", - "warning: Matlab-style short-circuit operation performed for operator |\n", - "warning: Matlab-style short-circuit operation performed for operator |\n", - "warning: Matlab-style short-circuit operation performed for operator |\n", - "warning: Matlab-style short-circuit operation performed for operator |\n", - "warning: Matlab-style short-circuit operation performed for operator |\n", - 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"warning: Matlab-style short-circuit operation performed for operator |\n", - "warning: Matlab-style short-circuit operation performed for operator |\n", - "warning: Matlab-style short-circuit operation performed for operator |\n", - "warning: Matlab-style short-circuit operation performed for operator |\n", - "warning: Matlab-style short-circuit operation performed for operator |\n", - "warning: Matlab-style short-circuit operation performed for operator |\n", - "warning: Matlab-style short-circuit operation performed for operator |\n", - "warning: Matlab-style short-circuit operation performed for operator |\n", - "warning: Matlab-style short-circuit operation performed for operator |\n", - "warning: Matlab-style short-circuit operation performed for operator |\n", - "warning: Matlab-style short-circuit operation performed for operator |\n", - "warning: Matlab-style short-circuit operation performed for operator |\n", - "warning: Matlab-style short-circuit operation performed for operator |\n", - "warning: Matlab-style short-circuit operation performed for operator |\n", - "warning: Matlab-style short-circuit operation performed for operator |\n", - "warning: Matlab-style short-circuit operation performed for operator |\n", - "warning: Matlab-style short-circuit operation performed for operator |\n", - "warning: Matlab-style short-circuit operation performed for operator |\n", - "warning: Matlab-style short-circuit operation performed for operator |\n", - "warning: Matlab-style short-circuit operation performed for operator |\n", - "warning: Matlab-style short-circuit operation performed for operator |\n", - "warning: Matlab-style short-circuit operation performed for operator |\n", - "warning: Matlab-style short-circuit operation performed for operator |\n", - "warning: Matlab-style short-circuit operation performed for operator |\n", - "warning: Matlab-style short-circuit operation performed for operator |\n", - "warning: Matlab-style short-circuit operation performed for operator |\n", - "warning: Matlab-style short-circuit operation performed for operator |\n", - "warning: Matlab-style short-circuit operation performed for operator |\n", - "warning: Matlab-style short-circuit operation performed for operator |\n", - "warning: Matlab-style short-circuit operation performed for operator |\n", - "warning: Matlab-style short-circuit operation performed for operator |\n", - "warning: Matlab-style short-circuit operation performed for operator |\n", - "warning: Matlab-style short-circuit operation performed for operator |\n", - "warning: Matlab-style short-circuit operation performed for operator |\n", - "warning: Matlab-style short-circuit operation performed for operator |\n", - "warning: Matlab-style short-circuit operation performed for operator |\n", - "warning: Matlab-style short-circuit operation performed for operator |\n", - "warning: Matlab-style short-circuit operation performed for operator |\n", - "warning: Matlab-style short-circuit operation performed for operator |\n", - "warning: Matlab-style short-circuit operation performed for operator |\n", - "warning: Matlab-style short-circuit operation performed for operator |\n", - "warning: Matlab-style short-circuit operation performed for operator |\n", - "warning: Matlab-style short-circuit operation performed for operator |\n", - "warning: Matlab-style short-circuit operation performed for operator |\n", - "warning: Matlab-style short-circuit operation performed for operator |\n", - "warning: Matlab-style short-circuit operation performed for operator |\n", - "warning: Matlab-style short-circuit operation performed for operator |\n", - "warning: Matlab-style short-circuit operation performed for operator |\n", - "warning: Matlab-style short-circuit operation performed for operator |\n", - "warning: Matlab-style short-circuit operation performed for operator |\n", - "warning: Matlab-style short-circuit operation performed for operator |\n", - "warning: Matlab-style short-circuit operation performed for operator |\n", - "warning: Matlab-style short-circuit operation performed for operator |\n", - "warning: Matlab-style short-circuit operation performed for operator |\n", - "warning: Matlab-style short-circuit operation performed for operator |\n", - "warning: Matlab-style short-circuit operation performed for operator |\n", - "warning: Matlab-style short-circuit operation performed for operator |\n", - "warning: Matlab-style short-circuit operation performed for operator |\n", - "warning: Matlab-style short-circuit operation performed for operator |\n", - "warning: Matlab-style short-circuit operation performed for operator |\n", - "warning: Matlab-style short-circuit operation performed for operator |\n" - ] - }, { "data": { "image/svg+xml": [ @@ -2289,30 +2170,45 @@ "\n", "\n", "\t\t\n", - "\t\t-0.01\n", + "\t\t0\n", "\t\n", "\n", "\n", - "\t\t\n", - "\t\t0\n", + "\t\t\n", + "\t\t0.005\n", "\t\n", "\n", "\n", - "\t\t\n", + "\t\t\n", "\t\t0.01\n", "\t\n", "\n", "\n", - "\t\t\n", + "\t\t\n", + "\t\t0.015\n", + "\t\n", + "\n", + "\n", + "\t\t\n", "\t\t0.02\n", "\t\n", "\n", "\n", - "\t\t\n", + "\t\t\n", + "\t\t0.025\n", + "\t\n", + "\n", + "\n", + "\t\t\n", "\t\t0.03\n", "\t\n", "\n", "\n", + "\t\t\n", + "\t\t0.035\n", + "\t\n", + "\n", + "\n", "\t\t\n", "\t\t0.04\n", "\t\n", @@ -2337,7 +2233,7 @@ "\n", "\n", "\n", - "\t\n", + "\t\n", "\t\n", "\n", "\n", @@ -2360,6 +2256,7 @@ ], "source": [ "N=50;\n", + "warning('off')\n", "dx = zeros(1,N); % [in nm]\n", "F_applied=linspace(0,0.0022,N); % [in nN]\n", "for i=1:N\n", @@ -2376,12 +2273,16 @@ "cell_type": "markdown", "metadata": {}, "source": [ - "For this function, it is possible to take a derivative and compare the analytical result:" + "For this function, it is possible to take a derivative and compare the analytical result:\n", + "\n", + "dE/dx = 0 = d(E_LJ)/dx - F\n", + "\n", + "F= d(E_LJ)/dx" ] }, { "cell_type": "code", - "execution_count": 105, + "execution_count": 32, "metadata": { "collapsed": false }, @@ -2438,68 +2339,73 @@ "\n", "\n", "\t\n", - "\t\t\n", + "\t\t\n", "\t\n", "\n", "\n", "\n", "\n", - "\t\t\n", + "\t\t\n", + "\t\t-0.0005\n", + "\t\n", + "\n", + "\n", + "\t\t\n", "\t\t0\n", "\t\n", "\n", "\n", - "\t\t\n", + "\t\t\n", "\t\t0.0005\n", "\t\n", "\n", "\n", - "\t\t\n", + "\t\t\n", "\t\t0.001\n", "\t\n", "\n", "\n", - "\t\t\n", + "\t\t\n", "\t\t0.0015\n", "\t\n", "\n", "\n", - "\t\t\n", + "\t\t\n", "\t\t0.002\n", "\t\n", "\n", "\n", - "\t\t\n", + "\t\t\n", "\t\t0.0025\n", "\t\n", "\n", "\n", - "\t\t\n", + "\t\t\n", "\t\t0\n", "\t\n", "\n", "\n", - "\t\t\n", + "\t\t\n", "\t\t0.01\n", "\t\n", "\n", "\n", - "\t\t\n", + "\t\t\n", "\t\t0.02\n", "\t\n", "\n", "\n", - "\t\t\n", + "\t\t\n", "\t\t0.03\n", "\t\n", "\n", "\n", - "\t\t\n", + "\t\t\n", "\t\t0.04\n", "\t\n", "\n", "\n", - "\t\t\n", + "\t\t\n", "\t\t0.05\n", "\t\n", "\n", @@ -2511,18 +2417,18 @@ "\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", "\n", @@ -2563,7 +2469,7 @@ }, { "cell_type": "code", - "execution_count": 106, + "execution_count": 33, "metadata": { "collapsed": false }, @@ -2582,7 +2488,7 @@ }, { "cell_type": "code", - "execution_count": 113, + "execution_count": 34, "metadata": { "collapsed": false }, @@ -2605,7 +2511,7 @@ }, { "cell_type": "code", - "execution_count": 108, + "execution_count": 35, "metadata": { "collapsed": false }, @@ -2755,7 +2661,7 @@ "\t\n", "\t\n", "\t\n", - "\t\n", + "\t\n", "\t\n", "\t\n", "\t\n", @@ -2797,7 +2703,7 @@ "\t\n", "\tgnuplot_plot_2a\n", "\n", - "\t\n", + "\t\n", "\t\n", "\n", "\n", @@ -2821,6 +2727,15 @@ "source": [ "plot(dx,F_applied,'o',dx,K(1)*dx+1/2*K(2)*dx.^2)" ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "collapsed": true + }, + "outputs": [], + "source": [] } ], "metadata": { diff --git a/lecture_08/lecture_08.md b/lecture_08/lecture_08.md index 8a81a2a..79edc0a 100644 --- a/lecture_08/lecture_08.md +++ b/lecture_08/lecture_08.md @@ -19,7 +19,7 @@ Considering a 1-D gold chain, we can calculate the bond length, $x_{b}$, with no Computational Tools to Study and Predict the Long-Term Stability of Nanowires. By Martin E. Zoloff Michoff, Patricio Vélez, Sergio A. Dassie and Ezequiel P. M. Leiva -![Model of Gold chain, from molecular dynamics simulation](Auchain_model.gif) +![Model of Gold chain, from molecular dynamics simulation](Auchain_model.png) [Single atom gold chain mechanics](http://www.uam.es/personal_pdi/ciencias/agrait/) diff --git a/lecture_08/lecture_08.pdf b/lecture_08/lecture_08.pdf new file mode 100644 index 0000000000000000000000000000000000000000..c251694ccac6915a741b7756bb7c7e1364e65c7d GIT binary patch literal 447457 zcmd42byQqm(=LcxaCd0jq0!*dxVvkhaR~%>2*Dw^yGxLuA-KD{6M_@meLBDQd*8Wh z{%b$0DtyXtwWXjLVpS%GXkD6~80C-*2oE=mqcrw_I$LPG4C zULV2ix+))R!RD^)8g3t4|K7_vm|20@b!;qLttoll1KDN4HdfZIl-%5$>>ww5Cl`&6 zX69gaNwB+(IatQU%!^&Z#`Tjb*ahU|@X^T;?C6T}o?Y6;-WBY^E^Tk-3YG+$J6V9) z6~T^HkRE}YKwc3MluxcMU^534&y|yz?`Xkdgn?@Ld6eRxJ?eoWXu^euDEhTuA|GS&QDT9%y zAYZcodfwlM{;T0{>%XP=E75;i|0UvIIcfg=XQY2i`H#f^ZvAg}S@1DD)Gn(8y?`k^ z#Um?+OHvC{7{)J$=&zjrABy#_$N!g#{@1+eKV$swk^RqjV*f`4|4W+z{bzR!Ocqvqy(~YyyxfOHDZ@Bb8s*-;AG=O zfgF^O!-!qU%+TP 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syllabus --- README.md | 4 ++-- 1 file changed, 2 insertions(+), 2 deletions(-) diff --git a/README.md b/README.md index 8059993..73ca14a 100644 --- a/README.md +++ b/README.md @@ -74,8 +74,8 @@ general, I will not post homework solutions. |3|1/31||Consistent Coding habits| | |2/2|5|Root Finding| |4|2/7|6|Root Finding con’d| -| |2/9|7|Optimization| -|5|2/14||Intro to Linear Algebra| +| |2/9|7| **Snow Day**| +|5|2/14|| Optimization | | |2/16|8|Linear Algebra| |6|2/21|9|Linear systems: Gauss elimination| | |2/23|10|Linear Systems: LU factorization|

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  1. Create a new github repository called ‘roots_and_optimization’.

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    1. Add rcc02007 and pez16103 as collaborators.

      2. -

    2. Clone the repository to your computer.

      3. -

    3. Copy your projectile.m function into the ‘roots_and_optimization’ folder. Disable the plotting routine for the solvers

      4. -

    4. Use the four solvers falsepos.m, incsearch.m, newtraph.m and mod_secant.m to solve for the angle needed to reach h=1.72, with an initial speed of 1.5 m/s.

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    5. The newtraph.m function needs a derivative, calculate the derivative of your function with respect to theta, dprojectile_dtheta.m. This function should output d(h)/d(theta).

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    6. In your README.md file, you will document the following under the heading # Homework #3:

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      1. Compare the number of iterations that each function needed to reach an accuracy of 0.00001%. Include a table in your README.md with:
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      | solver | initial guess(es) | ea | number of iterations|
-| --- | --- | --- | --- |
-|falsepos   |  |  |  |
-|incsearch  |  |  |  |
-|newtraph   |  |  |  |
-|mod_secant |  |  |  |
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      1. Compare the convergence of the 4 methods. Plot the approximate error vs the number of iterations that the solver has calculated. Save the plot as convergence.png and display the plot in your README.md with:
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      ![Plot of convergence for four numerical solvers.](convergence.png)

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      1. In the README.md provide a description of the files used to create the table and the convergence plot.
        2. -
      7. -
    2. -

  2. The Newton-Raphson method and the modified secant method do not always converge to a solution. One simple example is the function f(x) = x*exp(-x^2). The root is at 0, but using the numerical solvers, newtraph.m and mod_secant.m, there are certain initial guesses that do not converge.

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    1. Calculate the first 5 iterations for the Newton-Raphson method with an initial guess of x_i=2.

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    2. Add the results to a table in the README.md with:

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    ### divergence of Newton-Raphson method
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-| iteration | x_i | approx error |
-| --- | --- | --- |
-| 0 | 2 | n/a |
-| 1 |   |     |
-| 2 |   |     |
-| 3 |   |     |
-| 4 |   |     |
-| 5 |   |     |
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    1. Repeat steps a-b for an initial guess of 0.2. (But change the heading from ‘divergence’ to ‘convergence’)
      2. -
    3. -

  3. Commit your changes to your repository. Sync your local repository with github. Then copy and paste the “clone URL” into the following Google Form Homework #3

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