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ME3255S2017/lecture_07/.ipynb_checkpoints/lecture_07-checkpoint.ipynb

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{
"cells": [
{
"cell_type": "code",
"execution_count": 1,
"metadata": {
"collapsed": true
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"outputs": [],
"source": [
"%plot --format svg"
]
},
{
"cell_type": "code",
"execution_count": 2,
"metadata": {
"collapsed": true
},
"outputs": [],
"source": [
"setdefaults"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# Roots: Open methods\n",
"## Newton-Raphson"
]
},
{
"cell_type": "markdown",
"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 $e^{x}=x$"
]
},
{
"cell_type": "code",
"execution_count": 20,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"error_approx = 1\r\n"
]
}
],
"source": [
"f= @(x) exp(-x)-x;\n",
"df= @(x) -exp(-x)-1;\n",
"\n",
"x_i= 0;\n",
"error_approx = abs((x_r-x_i)/x_r)\n",
"x_r=x_i;\n"
]
},
{
"cell_type": "code",
"execution_count": 21,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"x_r = 0.50000\n",
"error_approx = 1\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;"
]
},
{
"cell_type": "code",
"execution_count": 22,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"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;"
]
},
{
"cell_type": "code",
"execution_count": 23,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"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;"
]
},
{
"cell_type": "markdown",
"metadata": {},
"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",
"$\\frac{df}{dm}=\\frac{1}{2}\\sqrt{\\frac{g}{mc_{d}}}\\tanh(\\sqrt{\\frac{gc_{d}}{m}}t)-\n",
"\\frac{g}{2m}\\mathrm{sech}^{2}(\\sqrt{\\frac{gc_{d}}{m}}t)$"
]
},
{
"cell_type": "code",
"execution_count": 7,
"metadata": {
"collapsed": true
},
"outputs": [],
"source": [
"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;"
]
},
{
"cell_type": "code",
"execution_count": 8,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"ans = 142.74\r\n"
]
}
],
"source": [
"newtraph(f_m,df_m,140,0.00001)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Secant Methods"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"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",
"$x_{i+1}=x_{i}-\\frac{f(x_{i})(\\delta x_{i})}{f(x_{i}+\\delta x_{i})-f(x_{i})}$"
]
},
{
"cell_type": "code",
"execution_count": 9,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"ans = 142.74\r\n"
]
}
],
"source": [
"mod_secant(f_m,1e-6,50,0.00001)"
]
},
{
"cell_type": "code",
"execution_count": 10,
"metadata": {
"collapsed": false
},
"outputs": [
{
"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"
]
}
],
"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)"
]
},
{
"cell_type": "code",
"execution_count": 11,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"error: 'Amt' undefined near line 1 column 1\r\n"
]
}
],
"source": [
"Amt*12*5"
]
},
{
"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": 12,
"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. "
]
},
{
"cell_type": "code",
"execution_count": 13,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"'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"
]
},
{
"cell_type": "code",
"execution_count": 14,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"ans = 563.79\r\n"
]
}
],
"source": [
"fzero(@(A) car_payments(A,30000,0.05,5,0),500)"
]
},
{
"cell_type": "markdown",
"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"
]
},
{
"cell_type": "code",
"execution_count": 15,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
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"warning: axis: omitting non-positive data in log plot\n",
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" 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",
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]
},
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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",
"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_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",
"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')"
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},
{
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"metadata": {
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{
"name": "stdout",
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"text": [
"ea_ms =\n",
"\n",
" Columns 1 through 6:\n",
"\n",
" 2.3382e+03 1.9171e-14 1.9171e-14 1.9171e-14 1.9171e-14 1.9171e-14\n",
"\n",
" Columns 7 through 12:\n",
"\n",
" 1.9171e-14 1.9171e-14 1.9171e-14 1.9171e-14 1.9171e-14 1.9171e-14\n",
"\n",
" Columns 13 through 18:\n",
"\n",
" 1.9171e-14 1.9171e-14 1.9171e-14 1.9171e-14 1.9171e-14 1.9171e-14\n",
"\n",
" Columns 19 and 20:\n",
"\n",
" 1.9171e-14 1.9171e-14\n",
"\n"
]
}
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},
{
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"text": [
"warning: axis: omitting non-positive data in log plot\n",
"warning: called from\n",
" __line__ at line 120 column 16\n",
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" __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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"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": 18,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"ea_bs =\n",
"\n",
" Columns 1 through 6:\n",
"\n",
" 9.5357e+03 -4.7554e-01 -2.1114e-01 6.0163e-02 -2.4387e-03 6.1052e-04\n",
"\n",
" Columns 7 through 12:\n",
"\n",
" 2.2891e-04 -9.5367e-06 2.3842e-06 8.9407e-07 -2.2352e-07 9.3132e-09\n",
"\n",
" Columns 13 through 18:\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",
"\n",
"ans = 16.208\n"
]
}
],
"source": [
"ea_bs\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": 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
}