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ME3255S2017/lecture_07/lecture_07.ipynb

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{
"cells": [
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"cell_type": "code",
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"%plot --format svg"
]
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"cell_type": "code",
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"source": [
"setdefaults"
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"# Roots: Open methods\n",
"## Newton-Raphson"
]
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{
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"metadata": {},
"source": [
"First-order approximation for the location of the root (i.e. assume the slope at the given point is constant, what is the solution when f(x)=0)\n",
"\n",
"$f'(x_{i})=\\frac{f(x_{i})-0}{x_{i}-x_{i+1}}$\n",
"\n",
"$x_{i+1}=x_{i}-\\frac{f(x_{i})}{f'(x_{i})}$\n",
"\n",
"Use Newton-Raphson to find solution when $e^{-x}=x$"
]
},
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"execution_count": 2,
"metadata": {
"collapsed": false
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"outputs": [
{
"name": "stdout",
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"text": [
"x_r = 0.50000\n",
"error_approx = 1\n"
]
}
],
"source": [
"f= @(x) exp(-x)-x;\n",
"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_i=x_r;\n"
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"text": [
"x_r = 0.56631\n",
"error_approx = 0.11709\n"
]
}
],
"source": [
"x_r = x_i-f(x_i)/df(x_i)\n",
"error_approx = abs((x_r-x_i)/x_r)\n",
"x_i=x_r;"
]
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"cell_type": "code",
"execution_count": 4,
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"text": [
"x_r = 0.56714\n",
"error_approx = 0.0014673\n"
]
}
],
"source": [
"x_r = x_i-f(x_i)/df(x_i)\n",
"error_approx = abs((x_r-x_i)/x_r)\n",
"x_i=x_r;"
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"execution_count": 5,
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"name": "stdout",
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"text": [
"x_r = 0.56714\n",
"error_approx = 2.2106e-07\n"
]
}
],
"source": [
"x_r = x_i-f(x_i)/df(x_i)\n",
"error_approx = abs((x_r-x_i)/x_r)\n",
"x_i=x_r;"
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{
"cell_type": "markdown",
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"source": [
"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. \n",
"\n",
"$f(m)=\\sqrt{\\frac{gm}{c_{d}}}\\tanh(\\sqrt{\\frac{gc_{d}}{m}}t)-v(t)$.\n",
"\n",
"to use the Newton-Raphson method, we need the derivative $\\frac{df}{dm}$\n",
"\n",
"$\\frac{df}{dm}=\\frac{1}{2}\\sqrt{\\frac{g}{mc_{d}}}\\tanh(\\sqrt{\\frac{gc_{d}}{m}}t)-\n",
"\\frac{g}{2m}\\mathrm{sech}^{2}(\\sqrt{\\frac{gc_{d}}{m}}t)$"
]
},
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"cell_type": "code",
"execution_count": 6,
"metadata": {
"collapsed": true
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"setdefaults\n",
"g=9.81; % acceleration due to gravity\n",
"m=linspace(50, 200,100); % possible values for mass 50 to 200 kg\n",
"c_d=0.25; % drag coefficient\n",
"t=4; % at time = 4 seconds\n",
"v=36; % speed must be 36 m/s\n",
"f_m = @(m) sqrt(g*m/c_d).*tanh(sqrt(g*c_d./m)*t)-v; % anonymous function f_m\n",
"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)*t).^2;"
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"outputs": [
{
"name": "stdout",
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"text": [
"root = 142.74\n",
"ea = 8.0930e-06\n",
"iter = 48\n"
]
}
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"[root,ea,iter]=newtraph(f_m,df_m,140,0.00001)"
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"## Secant Methods"
]
},
{
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"metadata": {},
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"Not always able to evaluate the derivative. Approximation of derivative:\n",
"\n",
"$f'(x_{i})=\\frac{f(x_{i-1})-f(x_{i})}{x_{i-1}-x_{i}}$\n",
"\n",
"$x_{i+1}=x_{i}-\\frac{f(x_{i})}{f'(x_{i})}$\n",
"\n",
"$x_{i+1}=x_{i}-\\frac{f(x_{i})}{\\frac{f(x_{i-1})-f(x_{i})}{x_{i-1}-x_{i}}}=\n",
" x_{i}-\\frac{f(x_{i})(x_{i-1}-x_{i})}{f(x_{i-1})-f(x_{i})}$\n",
" \n",
"What values should $x_{i}$ and $x_{i-1}$ take?\n",
"\n",
"To reduce arbitrary selection of variables, use the\n",
"\n",
"## Modified Secant method\n",
"\n",
"Change the x evaluations to a perturbation $\\delta$. \n",
"\n",
"$x_{i+1}=x_{i}-\\frac{f(x_{i})(\\delta x_{i})}{f(x_{i}+\\delta x_{i})-f(x_{i})}$"
]
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"name": "stdout",
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"text": [
"root = 142.74\n",
"ea = 3.0615e-07\n",
"iter = 7\n"
]
}
],
"source": [
"[root,ea,iter]=mod_secant(f_m,1,50,0.00001)"
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"ans = 1.1185e+04\r\n"
]
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"</g>\n",
"<g color=\"black\" fill=\"none\" stroke=\"currentColor\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\" stroke-width=\"0.50\">\n",
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"\t\t<text><tspan font-family=\"{}\">100000</tspan></text>\n",
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"\t\t<text><tspan font-family=\"{}\">30</tspan></text>\n",
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"\t<path d=\"M88.5,16.7 L88.5,362.4 L535.0,362.4 L535.0,16.7 L88.5,16.7 Z \" stroke=\"black\"/></g>\n",
"<g color=\"black\" fill=\"none\" stroke=\"currentColor\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\" stroke-width=\"0.50\">\n",
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"\t\t<text><tspan font-family=\"{}\">principle amount left ($)</tspan></text>\n",
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"\t<g id=\"gnuplot_plot_1a\"><title>gnuplot_plot_1a</title>\n",
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"text/plain": [
"<IPython.core.display.SVG object>"
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},
"metadata": {},
"output_type": "display_data"
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],
"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_numerical*12*30"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"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. "
]
},
{
"cell_type": "code",
"execution_count": 19,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"Amt = 566.14\r\n"
]
}
],
"source": [
"% Amortization calculation\n",
"A = @(P,r,n) P*(r*(1+r)^n)./((1+r)^n-1);\n",
"Amt=A(30000,0.05/12,5*12)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Matlab's function\n",
"\n",
"Matlab and Octave combine bracketing and open methods in the `fzero` function. "
]
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"'fzero' is a function from the file /usr/share/octave/4.0.0/m/optimization/fzero.m\n",
"\n",
" -- Function File: fzero (FUN, X0)\n",
" -- Function File: fzero (FUN, X0, OPTIONS)\n",
" -- Function File: [X, FVAL, INFO, OUTPUT] = fzero (...)\n",
" Find a zero of a univariate function.\n",
"\n",
" FUN is a function handle, inline function, or string containing the\n",
" name of the function to evaluate.\n",
"\n",
" X0 should be a two-element vector specifying two points which\n",
" bracket a zero. In other words, there must be a change in sign of\n",
" the function between X0(1) and X0(2). More mathematically, the\n",
" following must hold\n",
"\n",
" sign (FUN(X0(1))) * sign (FUN(X0(2))) <= 0\n",
"\n",
" If X0 is a single scalar then several nearby and distant values are\n",
" probed in an attempt to obtain a valid bracketing. If this is not\n",
" successful, the function fails.\n",
"\n",
" OPTIONS is a structure specifying additional options. Currently,\n",
" 'fzero' recognizes these options: \"FunValCheck\", \"OutputFcn\",\n",
" \"TolX\", \"MaxIter\", \"MaxFunEvals\". For a description of these\n",
" options, see *note optimset: XREFoptimset.\n",
"\n",
" On exit, the function returns X, the approximate zero point and\n",
" FVAL, the function value thereof.\n",
"\n",
" INFO is an exit flag that can have these values:\n",
"\n",
" * 1 The algorithm converged to a solution.\n",
"\n",
" * 0 Maximum number of iterations or function evaluations has\n",
" been reached.\n",
"\n",
" * -1 The algorithm has been terminated from user output\n",
" function.\n",
"\n",
" * -5 The algorithm may have converged to a singular point.\n",
"\n",
" OUTPUT is a structure containing runtime information about the\n",
" 'fzero' algorithm. Fields in the structure are:\n",
"\n",
" * iterations Number of iterations through loop.\n",
"\n",
" * nfev Number of function evaluations.\n",
"\n",
" * bracketx A two-element vector with the final bracketing of the\n",
" zero along the x-axis.\n",
"\n",
" * brackety A two-element vector with the final bracketing of the\n",
" zero along the y-axis.\n",
"\n",
" See also: optimset, fsolve.\n",
"\n",
"Additional help for built-in functions and operators is\n",
"available in the online version of the manual. Use the command\n",
"'doc <topic>' to search the manual index.\n",
"\n",
"Help and information about Octave is also available on the WWW\n",
"at http://www.octave.org and via the help@octave.org\n",
"mailing list.\n"
]
}
],
"source": [
"help fzero"
]
},
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"text": [
"ans = 563.79\r\n"
]
}
],
"source": [
"fzero(@(A) car_payments(A,30000,0.05,5,0),500)"
]
},
{
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"metadata": {},
"source": [
"## Comparison of Solvers\n",
"\n",
"It's helpful to compare to the convergence of different routines to see how quickly you find a solution. \n",
"\n",
"Comparing the freefall example\n"
]
},
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"warning: axis: omitting non-positive data in log plot\n",
"warning: called from\n",
" __line__ at line 120 column 16\n",
" line at line 56 column 8\n",
" __plt__>__plt2vv__ at line 500 column 10\n",
" __plt__>__plt2__ at line 246 column 14\n",
" __plt__ at line 133 column 15\n",
" semilogy at line 60 column 10\n",
"warning: axis: omitting non-positive data in log plot\n",
"warning: axis: omitting non-positive data in log plot\n",
"warning: axis: omitting non-positive data in log plot\n"
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"N=20;\n",
"iterations = linspace(1,400,N);\n",
"ea_nr=zeros(1,N); % appr error Newton-Raphson\n",
"ea_ms=zeros(1,N); % appr error Modified Secant\n",
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"ea_bs=zeros(1,N); % appr error bisect method\n",
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" [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",
"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": 22,
"metadata": {
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},
"outputs": [
{
"name": "stdout",
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"text": [
"ea_nr =\n",
"\n",
" Columns 1 through 8:\n",
"\n",
" 6.36591 0.06436 0.00052 0.00000 0.00000 0.00000 0.00000 0.00000\n",
"\n",
" Columns 9 through 16:\n",
"\n",
" 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000\n",
"\n",
" Columns 17 through 20:\n",
"\n",
" 0.00000 0.00000 0.00000 0.00000\n",
"\n"
]
}
],
"source": [
"ea_nr"
]
},
{
"cell_type": "code",
"execution_count": 26,
"metadata": {
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{
"name": "stdout",
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"text": [
"warning: axis: omitting non-positive data in log plot\n",
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" __plt__ at line 133 column 15\n",
" semilogy at line 60 column 10\n",
"warning: axis: omitting non-positive data in log plot\n",
"warning: axis: omitting non-positive data in log plot\n",
"warning: axis: omitting non-positive data in log plot\n"
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"text/plain": [
"<IPython.core.display.SVG object>"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"N=20;\n",
"f= @(x) x^10-1;\n",
"df=@(x) 10*x^9;\n",
"iterations = linspace(1,50,N);\n",
"ea_nr=zeros(1,N); % appr error Newton-Raphson\n",
"ea_ms=zeros(1,N); % appr error Modified Secant\n",
"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,df,0.5,0,iterations(i));\n",
" [root_ms,ea_ms(i),iter_ms]=mod_secant(f,1e-6,0.5,0,iterations(i));\n",
" [root_fp,ea_fp(i),iter_fp]=falsepos(f,0,5,0,iterations(i));\n",
" [root_bs,ea_bs(i),iter_bs]=bisect(f,0,5,0,iterations(i));\n",
"end\n",
" \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": 27,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"ea_nr =\n",
"\n",
" Columns 1 through 7:\n",
"\n",
" 99.03195 11.11111 11.11111 11.11111 11.11111 11.11111 11.11111\n",
"\n",
" Columns 8 through 14:\n",
"\n",
" 11.11111 11.11111 11.11111 11.11109 11.11052 11.10624 10.99684\n",
"\n",
" Columns 15 through 20:\n",
"\n",
" 8.76956 2.12993 0.00000 0.00000 0.00000 0.00000\n",
"\n",
"ans = 16.208\n"
]
}
],
"source": [
"ea_nr\n",
"newtraph(f,df,0.5,0,12)"
]
},
{
"cell_type": "code",
"execution_count": 19,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"ans = 1.9683e+23\r\n"
]
}
],
"source": [
"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,
"metadata": {
"collapsed": true
},
"outputs": [],
"source": []
}
],
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"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"
}
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"mimetype": "text/x-octave",
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