= ( ) Applying the trapezoidal rule to each subinterval, we get

Transcription

1 Trpezoidl Rule For sigle pplictio of the trpezoidl rule, estimte of the error i the pproximtio for the itegrl I b dx is give by Et f ( b ) ξ, where ξ lies somewhere i the itervl [ b, ]. Note tht, ulike the expressios for error we developed for iterpoltig polyomils, there is o vlue of ξ [ b, ] tht will mke this error estimte exct. It is oly estimte, but it is still quite useful. Now, suppose we estimte the itegrl by brekig the itervl b, ito subitervls d pplyig the trpezoidl rule to ech subitervl. I prticulr, let x < x < L < x b, d rewrite the itegrl s I x i+ xi dx. Applyig the trpezoidl rule to ech subitervl, we get I xi+ xi dx x x i+ i f( xi)+ f( xi+ ). [ ] up This is the formul for the multiple-pplictio trpezoidl rule with rbitrrily spced itermedite poits. If we mke the dditiol simplifictio tht the itermedite poits { x, x, K, x } re eqully

2 spced, so tht ech subitervl hs size h b becomes ( i+ i) I x x b b i f( xi)+ f( xi+ ) + i+ i + +, the formul This is the multiple-pplictio trpezoidl rule for eqully spced poits. Notice tht oce gi, this formul c be regrded s the product of the legth of the itervl times estimte of the verge vlue of the fuctio o the itervl. Applyig the error estimte give bove to ech of the subitervls, we get the followig estimte of the error i the itegrl pproximtio E b b b f b h f, f ( ξ ) i f ( ξi).

3 where ξ i represets poit i the subitervl [ xi, xi], d f f ( ξi) represets estimte of the verge vlue of the secod derivtive of the fuctio f( x) o the itervl [ b, ]. Notice tht this implies tht the rte of decrese i the error is secod-order. Tht is, the error i the estimte of the itegrl usig the trpezoidl rule is proportiol to h or. Simpso's Rules The geerl method of pproximtig fuctio with iterpoltig polyomil fit to eqully-spced poits d the itegrtig leds to whole clss of pproximte itegrtio formuls tht re clled Simpso's rules. The trpezoidl rule is, i fct, Simpso's rule, but it is seldom referred to i tht wy. Oe of the most importt d most commoly used itegrtio formuls is clled Simpso's / rule, d it is bsed o usig secod-order fuctio pproximtio. To derive the sigle-pplictio versio of Simpso's / rule, we use the Lgrge iterpoltig polyomil of degree two. Hece, we eed to hve vlues of the fuctio t three poits x < x < x b, which re eqully spced i the itervl [ b, ]. Lettig h ( b ), we get

4 I b b dx f ( x) dx x x x x x x x x x) x) x x x x f( ( ) ( ) x)+ f( x) ( x x )( x x ) ( x x )( x x ) dx x) x) + f( x) ( x x) ( x x) x h) x h) x )x h) f( x ) h h x) x h) + h x) hx ( x)+ h f ( x) h h x) hx) + f ( x) h ( f( x) f( x)+ f( x) ) 4 x x ( ) h h + f( x) 8h h h [ f ( x )+ 4 f ( x )+ f ( x ) ] 4 ( b ) 6 dx x) hx) ( + ) ( + ) + ( ) h 4 h dx x) dx h With some dditiol effort, it c be show tht sigle pplictio of Simpso's / rule hs tructio error give by 4

5 5 ( b ) Et f ( 4) ( ξ) 88 b h 4 f ( 4) ( ξ ), 8 where ξ [ b, ]. It is importt to ote tht this error is proportiol to h 4 isted of h, s oe might guess without doig the lgebr. Proceedig s we did for the trpezoidl rule, it is strightforwrd to show tht the multiple-pplictio form of Simpso's / rule becomes I b 4 i i 5,, 46,, Similrly, the estimte of the tructio error for the multiple-pplictio form of Simpso's / rule becomes b E h f 8 4 4, where f ( 4) represets the verge vlue of the fourth derivtive of f( x) o the itervl [ b, ]. Becuse the rte of decrese i the tructio error for Simpso's / rule is fourth-order rther th just third-order, it is method tht is prticulrly ttrctive. Ufortutely, to pply this rule, ot oly do you eed eqully-spced poits, but you lso eed odd umber of dt poits. For situtios where there re eve umber of poits, oe geerlly resorts to sigle pplictio of Simpso's /8 rule, which requires four dt poits d is bsed o usig third-degree iterpoltig. 5

6 polyomil. The formul for sigle pplictio of Simpso's /8 rule is give by I b ( ) 8 The error estimte for sigle pplictio of this rule is give by where h ( b ). E b h 4 f ( 4 ) ξ, 8 t. 6