Numerical integration. Quentin Louveaux (ULiège - Institut Montefiore) Numerical analysis / 10
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1 Numericl integrtion Quentin Louveux (ULiège Institut Montefiore) Numericl nlysis / 10
2 Numericl integrtion We consider definite integrls Z b f (x)dx better to it use if known! A We do not ssume tht the indefinite integrl cn be computed Evlute the re under the grph f (x) f(x) b Quentin Louveux (ULiège Institut Montefiore) Numericl nlysis / 10
3 NewtonCotes formuls Ide : divide the intervl in n eqully spced intervls, compute the interpoltion polynomil of degree n nd integrte it Trpezoidl rule Divide the intervl in one intervl nd evlute the integrl from the liner interpolting function of the two boundry points f(x) of fcol Cb ) b Z b f (x)dx (b (f ()+f(b)) ) 2 height I verge the of bses Quentin Louveux (ULiège Institut Montefiore) Numericl nlysis / 10
4 NewtonCotes formuls Ide : divide the intervl in n eqully spced intervls, compute the interpoltion polynomil of degree n nd integrte it Simpson rule Divide the intervl in two intervls nd evlute the integrl from the liner interpolting function of the two boundry points + the middle puff? dnte f(x) +b 2 b Z b f (x)dx b f ()+4 3 f ( + b 2 )+1 3 f (b) Quentin Louveux (ULiège Institut Montefiore) Numericl nlysis / 10
5 Interpolte ( Its 2 qudrtic polynomil, f Cll Cthy, f Cthy ) ), from ( b fcb ) ), using geformffptu/eetettc#yii?ytfth ( t ) l b) PTI qudrtic polynomil fb Plt ) dt on equivlently Putt Pzlt ) ftpltldtfbpzltldtfpzltldt 1 coefficient of the formul
6 Intervl juxtposition Riemnn integrl : f#%tttf NewtonCotes formuls do not relly mke sense for lrge degree of the interpoltion polynomil! Apply NewtonCotes formuls to mny subintervls with low degree Composite trpezoidl rule Composite Simpson rule h ' " hint ":s 's h so Fortuntely here, the formuls will not be prone to numericl errors Quentin Louveux (ULiège Institut Montefiore) Numericl nlysis / 10
7 qgfbflxtdxbztk composite ) tf " ) ) fbfcxtdxbzfzfcitzfltttttsffbkpmn Composite Simpson : Trpezoid Sbfcxldx = rules If C smll { res pieces ) 194 hzffiitfktytfjhn?ifip!!j)tht2h+ 11 Iz =,hthh=b re lst piece) = Trpezoid hz ( fl ) gnffthltfctzhltefftzh ) tflb ) ) + = Simpson ftztcltffcthltyfftytffct#yy~fbflx1dx= ( Ifl ) 14g f Cth) t } Htch ) thzfctshlttzflb ) )
8 Next How question fst does : my composite rule converge to fbfcxldx? Answer : order of convergence for the formuls
9 Error nlysis Proposition Let f be twice continuously di erentible function on [, b] Let I := R b f (x)dx Let T h the pproximtion given by trpezoidl composite rule with step h For some 2 [, b], I T h = 1 12 (b )h2 f 00 ( ) for some 2 [, b] I 01h21 For n error with he, we obtin n error no ox tsmlln with yo Quentin Louveux (ULiège Institut Montefiore) Numericl nlysis / 10
10 t ) ) h Ide of the proof : Write the error term for one #n piece first fthfffxldx I formt) dx = fith Eltldt )=f"!e Wht is the error? interpoltion Ect ft It ) we will tht ssume Be is continuous int th th E =/ 2 1 terror of the 1st f " Cest ) fct It # piece It Hdt = f' =f ' (5) Is Ct Hdt f " b unknown How to evlute such n integrl since Seis unknown Lemm : f! f leg ) = flxlgcxldx fbgcxldx if g does not chnge sign on Cb ]
11 f vlue E ( one piece ) = HI " I Si ) Totl error = HI = b f " C So ) men theorem ( mhnnihhuvls ),Et been Know tht The = I fb t fly )d + czh ' t Cuh " t Cll only even terms We cn use Richrdson extrpoltion
12 Romberg lgorithm We know the order of convergence! we cn pply the Richrdson extrpoltion The error terms only contin even power Th = It Czh? t Cy 44 The = It p4t h I 0,0 I & 3 h 2 I 1,0! I 1,1 compute h= b, & Tlk & b h 4 I 2,0! I 2,1! I 2,2 & & & b h 8 I 3,0! I 3,1! I 3,2! I 3,3 Tb & & & & h 16 I 4,0! I 4,1! I 4,2! I 4,3! I 4,4 O(h 2 ) O(h 4 ) O(h 6 ) O(h 8 ) O(h 10 ) Trpezoidl Simpson Boole c h2 t h " Quentin Louveux (ULiège Institut Montefiore) Numericl nlysis / 10
13 Selecting the points for better ccurcy If we fix the number clls to of f, t wht should we evlute points?f Question : If we cn select given number of points to perform the integrtion, wht is the best choice? We wnt to minimize the integrl of the error! We work on bsis of formul of the type Z b we f (x)dx 0 f (x 0 )+ + n f (x n ), interpolte f with one polynomil I I i = R b l i(x)dx li (x) ispolynomilusedinechtermofthelgrngeinterpoltion I The formul is exct if f is polynomil of degree n Tht is the degree of the formul lrgest degree of polynomil tht the is integrted exctly by formul Quentin Louveux (ULiège Institut Montefiore) Numericl nlysis / 10
14 Theorem of GussLegendre GussLegendre qudrtures Let q be nonzero polynomil of degree n +1suchtht,forll0pple k pple n, Z b x k q(x)dx =0 Let x 0, x 1,,x n be the roots of q(x) Then the formul t Z b rel nx re lwys f (x)dx i f (x i ), nd E C, b) i=0 q 5 Legendre poleyn tftwgmffiey where i = R b l i(t)dt is exct for ny polynomil of degree less thn or equl to 2n + 1 Quentin Louveux (ULiège Institut Montefiore) Numericl nlysis / 10
15 r Proof Given : g f is Legendre polynomil : polynomil of degree E 2h 11 1 f : f= p g t r qtuotient tremin der deg Ct # = htt 2h 11 deg (g) C by ssumption) deg Cp ) In fils fbflxldx fb = plxlgcxldx deg C r ) E n plxilglxiltrlxi ) If Xi is the t fb rlxldx root of g fcxi ) on = o by ssumption g Sbrlxldx = tzoircxi ) = Ezo ifcxi ) = fbfcxldx ( ti ) D
16 x' xtn Xtc Eucliden division of polynomils X " in quotient X ' x' x 's th ex T reminder X4tn=(x4x ) ( x quotient ' ) + ( deg = deg t deg L ) reminder deg I reminder ) deglxtxti )
17 sit Tr± Exmple 9141 = x C 5 23 ) 955 9, =3 t Wht re the three best points to integrte over [ 1, 1]? q(x) q =5x q 3 3x is Legendre polynomil over [ 1, 1] with roots 3 5, 0, 3 5 C C n ] 1 = 5 9, 2 = 8 9, 3 = 5 9 It is more ccurte thn using 1, 0 nd 1 i, t / Quentin Louveux (ULiège Institut Montefiore) Numericl nlysis / 10
18 o wht is the Legendre polynomil of degree 3 over El, D? guk got fig Cxldx = o f! If 90=920 9 X, +92kt 93K then xqlxldy fix ' qcxldx = 0 fin/dx=f!gcxlx2dx=o utomticlly 9141=9, X = 219ft to I,lqx2t9zx4dx= ( 9 x3+ x5 )!,
19 if l For ny Ig f ( function ) t Ig f to ) t Ig f ( tf ) pproximtes better f! f Cx) dx thn } f C ) t I, flo ) tf fu )
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