Forward Divided Difference - Numerical Methods - Lecture Slides, Slides of Mathematical Methods for Numerical Analysis and Optimization

Download Forward Divided Difference - Numerical Methods - Lecture Slides and more Slides Mathematical Methods for Numerical Analysis and Optimization in PDF only on Docsity! Forward Divided Difference Docsity.com Definition ix ix . ( ) ( ) ( ) x xfxxf x xf Δ Δ 0Δ lim −+ → =′ f(x) Slope at x y Docsity.com 2Δ =t ttt ii Δ1 +=+ 18216 =+= ( ) ( ) ( ) 2 161816 νν −=a ( ) ( ) ( )188.91821001014 1014ln200018 4 4 −      −× × =ν sm /02.453= ( ) ( ) ( )168.91621001014 1014ln200016 4 4 −      −× × =ν sm /07.392= Example (contd.) Docsity.com Hence The exact value of ( )16a can be calculated by differentiating ( ) t t t 8.9 21001014 1014ln2000 4 4 −      −× × =ν as Example (contd.) ( ) 2/475.3007.39202.453 2 )16()18(16 sma =−=−= νν [ ] 2/674.29)16( 3200 4.294040)()( sma t tt dt dta = +− −− == ν Docsity.com The absolute relative true error is Example (contd.) 100×−= TrueValue eValueApproximatTrueValue tε %6993.2 100 674.29 475.30674.29 = × − = Docsity.com Effect of Step Size on Approximate Error -5 -4 -3 -2 -1 0 0 2 4 6 8 10 12 Number of times step size halved, n E a Initial step size=0.05 Docsity.com Effect of Step Size on Absolute Relative Approximate Error 0 1 2 3 4 5 6 0 1 2 3 4 5 6 7 8 9 10 11 12 Number of times step size halved, n |E a| % Initial step size=0.05 Docsity.com Effect of Step Size on Least Number of Significant Digits Correct 0 1 2 3 4 1 2 3 4 5 6 7 8 9 10 11 Number of times step size halved, n Le as t nu m be r o f s ig ni fic an t d ig its co rr ec t Initial step size=0.05 Docsity.com