Engineering Analysis ENG 3420 Fall 2009.ppt

:HEC439BOfficehours:Tu-Th11:00-12:001**Lecture26Lecture26ScheduleThelasthomeworkHW5andthelastprojectaredueonTuesdayNovember24!!StudentswhohaveoptedforresearchprojectsinsteadofthefinalshouldpresenttheirprojectsonTuesdayDecember1stWe’,10AMto12::NumericalintegrationToday:urateestimationofintegrals(chapter18)RombergintegrationGaussquadratureAdaptivequadratureNextTimeNumericaldiufferentiation(chapter19).2RichardsonextrapolationRichardextrapolationputeathird,(h2)estimatesI(h1)andI(h2)arecalculatedforanintegralusingstepsizesofh1andh2,respectively,animprovedO(h4)estimatemaybeformedusing:Whentheintervalishalved(h2=h1/2),es:3Richardsonextrapolation(cont’d)WhentherearetwoO(h4)estimatesandtheintervalishalved(hm=hl/2),animprovedO(h6)estimatemaybeformedusing:WhentherearetwoO(h6)estimatesandtheintervalishalved(hm=hl/2),animprovedO(h8)estimatemaybeformedusing:uracyincreases,,whereij+1,k-1andij,k-urateintegrals,respectively,andij,:67GaussquadratureGaussquadrature-LegendreformulasTheGauss-LegendreformulasOptimizeestimatestointegralsforfunctionsoverintervalsfrom--:wheretheciandxiarecalculatedtoensurethatthemethodexactlyintegratesupto(2n-1)thorderpolynomialsovertheintervalfrom-’s1/,automaticallyadjustthestepsizesothatsmallstepsaretakeninregionsofsharpvariationsandlargerstepsaretak