Numerical Integration - (4.3)
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1 Numericl Itegrtio - (.). Te Degree of Accurcy of Qudrture Formul: Te degree of ccurcy of qudrture formul Qf is te lrgest positive iteger suc tt x k dx Qx k, k,,,...,. Exmple fxdx 9 f f,,. Fid te degree of ccurcy of tis qudrture formul Qf 9 f f. Cosider fx, fx x, fx x,... Itiskow tt x k dx k k k. k fx k k k 9 f f 9 x 9 x 9 x So te degree of ccurcy is Exmple Suppose tt te qudrture formul fxdx c f c f c f is exct for ll polyomils of degree less t d equl to. Fid c, c d c Qf c f c f c f : c, c. fx, fx x, fx x, c c c c c c c Sustitute it ito : c Sustitute ot c d c ito : c fxdx f f f. Numericl Qudrture: Te sic metod ivolved i pproximtig fxdx is clled umericl qudrture. Cosider Wt re,,...,? fxdx i fx i. i
2 . Usig Lgrge Iterpoltig Polyomils: If te fx fx il,ix f c x x i! i i fxdx fx i L,ixdx i L,ixdx, i,,..., i Te correspodig pproximtio error is R it f c! i x x idx. Trpezoidl Rule: Cosider te st degree Lgrge iterpoltig polyomil P x wit dt pirs, f d, f. Let. Pxdx P x f x f x f x f x f x dx f f x x f x f f f f Weigted Me Vlue Teorem for Itegrls: (Pge 9, Teorem. Suppose tt F is cotiuous o,, gxdx exists, d gx does ot cge sig o,. Te tere exists umer c i, wit Fxgxdx Fc gxdx. Te correspodig pproximtio error: Cosider Fx f c, d gx x x. gx does ot cge sig o,. f R trp c x x dx! f c x x dx f c x x dx f c x x dx f c f c If f x M for ll x i,, te x x f c f c R trp M. Usig Tylor Polyomils d Differece Formuls: Toug we c derive umericl qudrture usig te formul: i fx i L,ixdx, te remider formul f R it c x x idx! i my ot e esy to use to derive te pproximtio error. To derive te pproximtio error for te
3 Trpezoidl Rule, we use te Weigted Me Vlue Teorem wic s coditio: gx does ot cge sig o,. Tis coditio is ot lwys stisfied for f c! x x i i. For exmple, to derive te pproximtio error for Simpso s Rule ( pirs of dt, d degree Lgrge polyomil) f R it c x x! x x x x dx Here gx! x x x x x x wic does cge sig over te itervl x, x. By Tylor Teorem, we kow fx P x R x, d fxdx Pxdx Rxdx were P x fx i f x i! R x f c! x x i Te pproximtio error for is x x i f x i! x x i f x i! fxdx Pxdx x x i... f x i! f R it c! x x i dx. Oserve tt te sig of f c! x x i does ot cge o te itervl,. So y te Weigted Me-Vlue Teorem we kow tere exists umer c i, suc tt f c! x x i dx f c x xi dx! f c! x i x i For te pproximtio error for Simpso s Rule: coose x i x, d use x, x, x : x x, x x, Note tt te itervl, x, x is symmetric out x x.so, x x x m x x dx x x m x x dx x x m dx x x x i Cosider fxdx Pxdx ifm is odd x x x x m dx m x x m m m if m is eve x x fx f x! x x f x! x x f x! x x dx fx f x Approximte f x y te differece formul:
4 f x we ve te umericl itegrl formul: Qf fx fx fx fx d te pproximtio error: x f R it c x! fx fx fx f c fx fx fx x x dx fx fx fx fx f c f c! 6 f c f c f c were c is i x, x. Numericl itegrtio formuls d pproximtio errors: Nme Approximtio formul Approximtio error Trpezoidl Rule f f, f c Simpso s Rule fx fx fx, 9 f c / Exmple Approximte six dx y te Trpezoidl Rule, d Simpso s Rule. Estimte correspodig pproximtio errors. Let fx six Trpezoidl Rule: y T si 6 x si y S x.97x y six, x, ds y T, dsdots y S, T f f si 6.76, S True error: f f f si 6 si.6 6
5 R T / six dx R S / six dx Approximtio errors: f x six x cos x f x cos x x six x f x 6six x cos x x six x...6. x y f x y f x f x f, f x f 6 si cos si Err T f. 7 Err S 9 f Newto-Cotes Formuls: Tese formuls re derived usig Lgrge iterpoltig polyomils. Severl umericl itegrtio formuls re listed o Pge 9-7: formuls (.)-(.).. poit closed Newto - Cotes formuls: x, x,, x i i. poit ope Newto - Cotes formuls: x, x,, x i x i. Te Midpoit formuls re ope Newto - Cotes formuls. / Exmple Use te Midpoit Rule (.) d.9 to pproximte six dx d estimte te pproximtio errors.. (.):
6 , x, x 6 T fx fx si si 6.. (.9): 6. x i i 6, i,, T fx fx fx si si 6 6 si. True errors: 6 R / six dx R / six dx.. 67 Approximtio errors:. R f R f
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