Crescenzi Robert Math 112-010 Matlab Assignment 1
Questions:
Professor Horntrop March 03, 2011
b.)WhileN=10andN=100,noagreementbetweentheexactanswerandeach approximationwaspresentinadditiontoLeftendpointapproximationandRightapproximation ofN=1000.Ontheotherhand,whileN=1000fortrapezoidalrule,therewasafourdigit agreementbetweenthetrapezoidalapproximationandthetruevalue.Thetruevalueequatedto 2701.0114andthetrapezoidalapproximationequatedto2701.043792whicharenearlyidentical toeachother. c.)Inthiscase,theleftendpointapproximationandrightendpointapproximationarenot asaccurateasthetrapezoidalapproximationbyacertainfactorinrespecttothevalueofN. Numerically,whileN=10,theleftendpointapproximationisapprox.53%andtheright endpointapproximationisapprox.53%off.WhenN=100,theleftendpointapproximationand therightendpointapproximationisonlyoffbyabout6%.Finally,whenN=1000,theleft endpointapproximationandtherightendpointapproximationisonlyoffbyabout0.6%.Thus, thetrapezoidalrulesaccuracyisdependentonwhatkindofintervalsthatthesubjectacquires. Theaccuracywillincreasebythefactorof6foreveryintervalthatisused.Therightendpoint approximationandleftendpointapproximationwilleitherbebiggerthanorlessthanthe approximationoftrapezoidalruleaccordingtotheslopeofthefunctiongiven.Thetrapezoidal rulelimitsthatestimationduetousingslopeofthefunctioninbetweenthedeltaxinsteadof usingstraightlineonthefunctiontoapproximate
Codes: a=1; N=10; b=3; h=(ba)/N; x=[a:h:b]; f=exp(3*x); IL=0; IR=0; IT=0; fork=1:N; IL=IL+f(k);
IR=IR+f(k+1); IT=IT+(f(k)+f(k+1))/2; end; IL=IL*h; IR=IR*h; IT=IT*h; fprintf('WhenN=%i,wefind:\n',N); fprintf('Leftendpointapproximation=%f.\n',IL); fprintf('Rightendpointapproximation=%f.\n',IR); fprintf('Trapezoidalapproximation=%f.\n',IT); a=1; b=3; N=100; h=(ba)/N; x=[a:h:b]; f=exp(3*x); IL=0; IR=0; IT=0; fork=1:N; IL=IL+f(k); IR=IR+f(k+1); IT=IT+(f(k)+f(k+1))/2; end; IL=IL*h; IR=IR*h; IT=IT*h; fprintf('WhenN=%i,wefind:\n',N); fprintf('Leftendpointapproximation=%f.\n',IL); fprintf('Rightendpointapproximation=%f.\n',IR); fprintf('Trapezoidalapproximation=%f.\n',IT); a=1; b=3; N=1000; h=(ba)/N; x=[a:h:b]; f=exp(3*x); IL=0; IR=0; IT=0;
fork=1:N; IL=IL+f(k); IR=IR+f(k+1); IT=IT+(f(k)+f(k+1))/2; end; IL=IL*h; IR=IR*h; IT=IT*h; fprintf('WhenN=%i,wefind:\n',N); fprintf('Leftendpointapproximation=%f.\n',IL); fprintf('Rightendpointapproximation=%f.\n',IR); fprintf('Trapezoidalapproximation=%f.\n',IT); Output:WhenN=10,wefind: Leftendpointapproximation=1397.004446. Rightendpointapproximation=4638.218102. Trapezoidalapproximation=3017.611274. WhenN=100,wefind: Leftendpointapproximation=2542.191133. Rightendpointapproximation=2866.312499. Trapezoidalapproximation=2704.251816. WhenN=1000,wefind: Leftendpointapproximation=2684.837724. Rightendpointapproximation=2717.249861. Trapezoidalapproximation=2701.043792.