Lecture HW 04 Roots

Name: insert name hereMECH 2220 Lecture Homework 4

Lecture HW 4 Root FindingAssigned: February 12, 2013Due: February 19, 2013 (11:00 a.m.)


Problem Statement #1:

Write a MATLAB script that will find the roots of a given equations using the BISECTION METHOD. Format your output to look similar to the examples given. You should write your output to a file. Set the maximum number of iterations to 100 and the tolerance to 0.000005.

Use a tolerance definition of are the interval endpoints for the nth iteration

Use the script to solve the given equations. Use the interval endpoints as your initial a and b. Eq1 f(x) = ex - (x2 + 4) = 0 on [2,3]

Eq2 f(x) = x - 2-x = 0 on [1/3,1]

Eq3 f(x) = x3 - 1.8999 x2 + 1.5796 x - 2.1195 = 0 on [1,2].

Code for problem #1:

function [est_x,est_y] = bisectionmethod(a,b,f)maxIterations = 100;minTolerance = 0.000005;fileID=fopen('bisect.txt','wt+');fprintf(fileID,'n\t\ta\t\t\tb\t\t\tp\t\t\tf(p)\n'); if f(a)*f(b) > 0 error('Bad bounds!')endif f(a)*f(b) == 0 error('One is already 0. Bad bounds!')endn=1;


p=(a+b)/2;while n<=maxIterations && abs(f(p( max([n-1,1]) )))>=minTolerance % if f(a(n))*f(b(n)) < 0 p(n)=(a(n)+b(n))/2; if f(a(n))*f(p(n))<0 %Good bounds at a and p. a(n+1)=a(n); b(n+1)=p(n); elseif f(b(n))*f(p(n))<0 %Now b and p are good bounds. a(n+1)=p(n); b(n+1)=b(n); end fprintf(fileID,'%3.0f\t\t%3.6f\t\t%3.6f\t\t%3.6f\t\t%3.6f\n',n,a(n),b(n),p(n),f(p(n))); n=n+1;endest_x = p(n-1);est_y = f(p(n-1));fprintf(fileID,'The root to the equation on [%2.3f,%2.3f] was found to be %3.6f where f(%3.6f)=%3.6f.\n',a(1),b(1),p(n-1),p(n-1),f(p(n-1))); fclose(fileID);bisectionmethod(2,3,@eq1)bisectionmethod((1/3),1,@eq2)bisectionmethod(1,2,@eq3)

function y = eq1(x)y=exp(x)-(x^2+4);


end

function y = eq2(x)y = x-2^(-x);end

function y = eq3(x)y=x^3-1.8999*x^2+1.5796*x-2.1195;end

Solution for problem #1:n a b p f(p) 1 2.000000 3.000000 2.500000 1.932494 2 2.000000 2.500000 2.250000 0.425236 3 2.000000 2.250000 2.125000 -0.142728 4 2.125000 2.250000 2.187500 0.127747 5 2.125000 2.187500 2.156250 -0.010732 6 2.156250 2.187500 2.171875 0.057680 7 2.156250 2.171875 2.164063 0.023269 8 2.156250 2.164063 2.160156 0.006218 9 2.156250 2.160156 2.158203 -0.002270 10 2.158203 2.160156 2.159180 0.001971 11 2.158203 2.159180 2.158691 -0.000151 12 2.158691 2.159180 2.158936 0.000910 13 2.158691 2.158936 2.158813 0.000380 14 2.158691 2.158813 2.158752 0.000115 15 2.158691 2.158752 2.158722 -0.000018 16 2.158722 2.158752 2.158737 0.000048 17 2.158722 2.158737 2.158730 0.000015 18 2.158722 2.158730 2.158726 -0.000001The root to the equation on [ 2, 3] was found to be 2.158726 where f(2.158726)=-0.000001.

n a b p f(p) 1 0.333333 1.000000 0.666667 0.036706 2 0.333333 0.666667 0.500000 -0.207107 3 0.500000 0.666667 0.583333 -0.084087 4 0.583333 0.666667 0.625000 -0.023420 5 0.625000 0.666667 0.645833 0.006710 6 0.625000 0.645833 0.635417 -0.008338 7 0.635417 0.645833 0.640625 -0.000810


8 0.640625 0.645833 0.643229 0.002951 9 0.640625 0.643229 0.641927 0.001071 10 0.640625 0.641927 0.641276 0.000130 11 0.640625 0.641276 0.640951 -0.000340 12 0.640951 0.641276 0.641113 -0.000105 13 0.641113 0.641276 0.641195 0.000013 14 0.641113 0.641195 0.641154 -0.000046 15 0.641154 0.641195 0.641174 -0.000017 16 0.641174 0.641195 0.641184 -0.000002The root to the equation on [ 0.333, 1] was found to be 0.641184 where f(0.641184)=-0.000002.

n a b p f(p) 1 1.000000 2.000000 1.500000 -0.649875 2 1.500000 2.000000 1.750000 0.185731 3 1.500000 1.750000 1.625000 -0.278558 4 1.625000 1.750000 1.687500 -0.058767 5 1.687500 1.750000 1.718750 0.060302 6 1.687500 1.718750 1.703125 -0.000016 7 1.703125 1.718750 1.710938 0.029946 8 1.703125 1.710938 1.707031 0.014916 9 1.703125 1.707031 1.705078 0.007437 10 1.703125 1.705078 1.704102 0.003708 11 1.703125 1.704102 1.703613 0.001845 12 1.703125 1.703613 1.703369 0.000914 13 1.703125 1.703369 1.703247 0.000449 14 1.703125 1.703247 1.703186 0.000216 15 1.703125 1.703186 1.703156 0.000100 16 1.703125 1.703156 1.703140 0.000042 17 1.703125 1.703140 1.703133 0.000013 18 1.703125 1.703133 1.703129 -0.000002The root to the equation on [1.000,2.000] was found to be 1.703129 where f(1.703129)=-0.000002.



Do problem 5.9 (5.8 in the third edition) in the text


%Call functions for 5.8global osfosf = 8;T_8=bisectionmethod(0,35,@problem8)osf=10;T_10=bisectionmethod(0,35,@problem8)osf=14;T_14=bisectionmethod(0,35,@problem8)


function y=problem8(x)global osfy=-log(osf)-139.34411+1.575701e5/(x+273.15)-6.642308e7/(x+273.15)^2+1.2438e10/(x+273.15)^3-8.621949e11/(x+273.15)^4;end

Solution for problem #2:n a b p f(p) 1 0.000000 35.000000 17.500000 0.178671 2 17.500000 35.000000 26.250000 0.009568 3 26.250000 35.000000 30.625000 -0.067487 4 26.250000 30.625000 28.437500 -0.029485 5 26.250000 28.437500 27.343750 -0.010097 6 26.250000 27.343750 26.796875 -0.000300 7 26.250000 26.796875 26.523438 0.004625 8 26.523438 26.796875 26.660156 0.002160 9 26.660156 26.796875 26.728516 0.000929 10 26.728516 26.796875 26.762695 0.000314 11 26.762695 26.796875 26.779785 0.000007 12 26.779785 26.796875 26.788330 -0.000147 13 26.779785 26.788330 26.784058 -0.000070 14 26.779785 26.784058 26.781921 -0.000032 15 26.779785 26.781921 26.780853 -0.000012 16 26.779785 26.780853 26.780319 -0.000003The root to the equation on [0.000,35.000] was found to be 26.780319 where f(26.780319)=-0.000003.

T_8 =

26.7803

n a b p f(p) 1 0.000000 35.000000 17.500000 -0.044472 2 0.000000 17.500000 8.750000 0.150964 3 8.750000 17.500000 13.125000 0.049486 4 13.125000 17.500000 15.312500 0.001624 5 15.312500 17.500000 16.406250 -0.021637 6 15.312500 16.406250 15.859375 -0.010061 7 15.312500 15.859375 15.585938 -0.004232


8 15.312500 15.585938 15.449219 -0.001307 9 15.312500 15.449219 15.380859 0.000158 10 15.380859 15.449219 15.415039 -0.000575 11 15.380859 15.415039 15.397949 -0.000209 12 15.380859 15.397949 15.389404 -0.000026 13 15.380859 15.389404 15.385132 0.000066 14 15.385132 15.389404 15.387268 0.000020 15 15.387268 15.389404 15.388336 -0.000003The root to the equation on [0.000,35.000] was found to be 15.388336 where f(15.388336)=-0.000003.

T_10 =

15.3883

n a b p f(p) 1 0.000000 35.000000 17.500000 -0.380945 2 0.000000 17.500000 8.750000 -0.185509 3 0.000000 8.750000 4.375000 -0.075641 4 0.000000 4.375000 2.187500 -0.017313 5 0.000000 2.187500 1.093750 0.012737 6 1.093750 2.187500 1.640625 -0.002363 7 1.093750 1.640625 1.367188 0.005168 8 1.367188 1.640625 1.503906 0.001398 9 1.503906 1.640625 1.572266 -0.000484 10 1.503906 1.572266 1.538086 0.000456 11 1.538086 1.572266 1.555176 -0.000014 12 1.538086 1.555176 1.546631 0.000221 13 1.546631 1.555176 1.550903 0.000104 14 1.550903 1.555176 1.553040 0.000045 15 1.553040 1.555176 1.554108 0.000016 16 1.554108 1.555176 1.554642 0.000001The root to the equation on [0.000,35.000] was found to be 1.554642 where f(1.554642)=0.000001.



Do problem 5.13 (same number for second and third edition) in the text.


function dydx = problem13(x)w0=2.5;L=600;I=30e3;E=50e3;


dydx=w0/(120*E*I*L) * (-5*x^4+6*L^2*x^2-L^4);

bisectionmethod(0,550,@problem13)

Solution for problem #3:n a b p f(p) 1 0.000000 550.000000 275.000000 0.000119 2 0.000000 275.000000 137.500000 -0.002096 3 137.500000 275.000000 206.250000 -0.001082 4 206.250000 275.000000 240.625000 -0.000493 5 240.625000 275.000000 257.812500 -0.000188 6 257.812500 275.000000 266.406250 -0.000034 7 266.406250 275.000000 270.703125 0.000042 8 266.406250 270.703125 268.554688 0.000004The root to the equation on [0.000,550.000] was found to be 268.554688 where f(268.554688)=0.000004.


Sample Output for Equation 3:

Joe Ragan Bisection Routine

The root to the equation f= x^3-1.8999*x^2+1.5796*x-2.1195 =0 on [1,2] was found to be 1.703133 where f(1.703133)=0.000013.

The 17 iterations for the bisection routine were recorded as:

n a b p f(p) 1 1.000000 2.000000 1.500000 -0.649875 2 1.500000 2.000000 1.750000 0.185731 3 1.500000 1.750000 1.625000 -0.278558 4 1.625000 1.750000 1.687500 -0.058767 5 1.687500 1.750000 1.718750 0.060302 6 1.687500 1.718750 1.703125 -0.000016 7 1.703125 1.718750 1.710938 0.029946 8 1.703125 1.710938 1.707031 0.014916 9 1.703125 1.707031 1.705078 0.007437 10 1.703125 1.705078 1.704102 0.003708 11 1.703125 1.704102 1.703613 0.001845 12 1.703125 1.703613 1.703369 0.000914 13 1.703125 1.703369 1.703247 0.000449 14 1.703125 1.703247 1.703186 0.000216 15 1.703125 1.703186 1.703156 0.000100 16 1.703125 1.703156 1.703140 0.000042 17 1.703125 1.703140 1.703133 0.000013

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Lecture HW 04 Roots