Hoover, Dorothy M.

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Journal of the Arkansas Academy of ScienceVolume 8 Article 231955Estimates of Error in Numerical IntegrationDorothy M. HooverUniversity of Arkansas, FayettevilleFollow this and additional works at: http://scholarworks.uark.edu/jaasPart of the Other Applied Mathematics CommonsThis article is available for use under the Creative Commons license: Attribution-NoDerivatives 4.0 International (CC BY-ND 4.0). Users are able toread, download, copy, print, distribute, search, link to the full texts of these articles, or use them for any other lawful purpose, without asking priorpermission from the publisher or the author.This Article is brought to you for free and open access by ScholarWorks@UARK. It has been accepted for inclusion in Journal of the Arkansas Academyof Science by an authorized editor of ScholarWorks@UARK. For more information, please contact scholar@uark.edu.Recommended CitationHoover, Dorothy M. (1955) "Estimates of Error in Numerical Integration," Journal of the Arkansas Academy of Science: Vol. 8 , Article23.Available at: http://scholarworks.uark.edu/jaas/vol8/iss1/23ON ESTIMATES OF ERROR IN NUMERICAL INTEGRATIONDOROTHY M. HOOVERUniversity of ArkansasINTRODUCTIONIn order to determine numerical approximations for definite integrals, theprincipal methods used are the trapezoidal rule and Simpson's rule. In this pa-per, upper bounds are established for the errors obtained by these two methods.The error resulting from use of the trapezoidal rule is expressed in terms of thesecond derivative of th": integrand, and that incurred in Simpson's rule is ex-pressed in terms of the fourth derivative. A difference formula, analogous toTaylor's formula with remainder, is established. This relation is then used toderive expressions for the errors in terms of second and fourth differences res-pectively.ERROR IN TERMS OF DERIVATIVESThe procedure is based on the Fundamental Theorem of the integral calculus.Let a subdivision of the interval from x-a to x- b into n parts be given bythe points a-xQ, x^, xn= b, with x^+^-x^= 28. The midpoint ofthe subinterval from x^ to x^ +jis designated by Xj. From the formula f(x)=f(xi) + (x-XjJf'Ui) +_(_x-XjJ 2fM (£) with f between x and xitit follows thatf(x)dx =[£(xi)] 28 + Ev with Ex < 83 Max If"( x)IJ x£ 3For one subdivision, the approximate area, A, as defined by the trapezoidalrule isA =i tfUi) + f(xi+1)] 28 (1.1)Hence A is given by this rule asA = [fU^] 28 + E2, where E2 < 83 Max lfM(x)I. (1.2)From ( 1. 1) and ( 1.2) itfollows that, in one subdivision, the actual error of thetrapezoidal rule satisfies the inequalityE<E! + E2 =iS3Max l£"(x)l (1.3)But 28-w, the width of one subdivision, so that 8-*;• HenceE<1 w3Max lf"(x)l.6 (1.4)Then, for n subdivisions, the total error obtained in the trapezoidal rule isF,t< n w3Max |fM(x)Ef j "( )|6(1.5)and since nw - b-aE f< (b-a) w^x |f"(x) \5^ 2Max|f ( ) (1.6)The error resulting from Simpson's Rule can be lumped into an expressioncontaining the fourth derivative. Developing the expansion of f(x) for two moreterms, the result isf(x) = f(^)+(x.Xi)fl(xi)+ (x-xi)2f>'(xi)+(x-xi)3f'"(xi)2! 3! 4!x and x^.204with x between x204Journal of the Arkansas Academy of Science, Vol. 8 [1955], Art. 23Published by Arkansas Academy of Science, 1955ESTIMATES OF ERROR 205For the two subdivisions from Xj_^ to Xj and from x.j to xj+j,we have/•xi+lf(x)dx = [f(x:)l 4S + te 3t"(*i)+ E3J xi-1Eo < 8 8 Max Iflv(x)I.3 ~ 15(2. 1)whereIn these two subdivisions, A, as defined by Simpson's rule isA = 28 [ft^.p + 4f(Xi) + f(xi+1)]3= j?£ [f(x<-28) + 4f(x,) + f(x;+2S)] , and since3 1 li[f(xi-2S)+4f(xi)+f(xi+28)] = 6f(xi)+i^f"( i^)+l^ [f'v(xi+012S)+f |V(xi-5228)]O<0j<1, 0<52<l.the value for A isA = 4-8 [f(x£)] +i_83 f'Cxj) + E4 (2.2)E^l S 5 Max If« v(x)l9withFrom (2. 1) and (2.2) it follows that an upper bound for the actual error inSimpson's rule for the two subdivisions is given byE< Eo+ E, =iiS 5Max Ifiv(x)I.45(2.3)Hence, when expressed in terms of w,E < _w^Max lflv(x)l.45(2.4)For the entire interval, the error in Simpson's rule is then bounded as folJows:E<.< JLw5Max If'v( x) I " , and, as nw = b-aE < (b-alwW lf'v (x*" 45(2.5)Thus E t and Es give upper bounds in terms of derivatives for the trapezoidalrule and Simpson's rule respectively.DIFFERENCE INTERPOLATION FORMULA WITH REMAINDERWhen it is difficult to obtain the derivatives of the integrand, Taylor'sFormula with remainder may be replaced by a difference interpolation formulawith remainder. An expression for the remainder after one term (Mean Value Theo-rem for differences) for the difference interpolation formula will be extablish-ed. Likewise, an expression for the remainder after two terms will be determined.THEOREM. If the function f( x) is continuous in the interval a<x<a+pS, then witheach value of x in that interval there is associated a value i) satisfying\-r\\ < |p| such thatf(a+p8) = f(a)+p A f(a)+ P(p-D A2f(a)+ . . . +p(p-D- • -(p-n+1) Anf(a+778).2! nlPROOF. The following equations are valid:pf(a+pS) - f(a) =2 {f(a+nS)-f [a+(n-l)S]}n=l=I{Af[a+(n-l)8]}.n=l205Journal of the Arkansas Academy of Science, Vol. 8 [1955], Art. 23http://scholarworks.uark.edu/jaas/vol8/iss1/23206 ARKANSAS ACADEMY OF SCIENCEThe continuity of f(x) implies the continuity of Af(x). Therefore, the last sum-mation over p terms may be replaced by the expression pAf(x), with a_<x<a+p8.This relationship can also be expressed asf(a+pS) = f(a) + p [f(x+S)- £(x)]which is designated as the Mean Value Theorem for differences.COROLLARY. If f(a+p8) = f(a) = 0, then f(x+S) = f(x), where x is some interme-diary x.Consider now the quantity K defined by the equationf(a+pS) = f(a) + pAf(a) + £iEl_LL K.2«Tf a + pS- b , the equation takes the formf(b) - f(a) - b^a Af(a) (-^) 8 2," K=°Oefine F(x) as follows:F(x) = f(b) - f(x)-(k2L)Af(x) -(^(-2-rrr^K •8 e 2:Then F^a) and F(b) are both zero. Butvalue x between A and B for which A Fby the above corollary there exists a(x) vanishes.When F(x) is evaluated at (x+S) and thesimplified, the result isdifference F(x+8) - F(x) determined andAF(x) = A2f(x) [1 - £z*] - [1 - hz*.] K = 0.o oSolving for K we obtainK = A2f(x), with a <x<b = a+pS.The general result for a remainder after n terms can be established byanalogous reasoning.ERRORS IN TERMS OF DIFFERENCESThe difference interpolation formula with remainder in the second differ-ence isf(xi+£S) = f(xi)+^A fft.juHS-D A2f(xi+77S) , 0<ti<^2!Let x = (x^S). Then(3.1)/Xj+8 if(x)dx = ftGi+^S)Sd^ = 2Sf(xi) + E{ , with Ki'<£ MaxlA2f| (3.2)Xl-8 J-lFor the trapezoidal ruleA =4- [f(Xi ) + f(xi+1)] 28 takes the formA =i- [f(x-) -A-.+ f(Xi)+A2] 28, where A, = Af(Xi ) = A f(Xj-8) (3.3)and A2= Af(xi)Simplifying, we obtainA = 28 f(Xi) + E2 with E'< 8 Max lA2fl. (3.4)Thus the error in one subdivision in terms of differences for the trapezoidalrule can be expressed by the inequalityE'<E' + E 11 2<|S Max |A2FI =1w Max |A2Fl . (3.5)206Journal of the Arkansas Academy of Science, Vol. 8 [1955], Art. 23Published by Arkansas Academy of Science, 1955ESTIMATES OF ERROR 207IFor n subdivisions, the resulting total error in the trapezoidal rule is thenEtilM MaxlA2FI , so thatE* < l(b-a) MaxlA2Fl. (3.6)*- "" 3For Simpson's Rule the error can be expressed in terms of a fourth differ-ence. When the difference interpolation formula with remainder is developedthrough two more terms, the result isf(xi+£8) = f(x,)+^Af(x.)+ 1} A2 f(xi)+illliUlll)A3 f(xi)+2 ¦ '+-fU-l)(g-2)(£-3) A4 f(x;+778) where 0 <v<£ • ( 4. 1}4.1Therefore/,x1+28 -2f(x)dx = [ftxi^SJSdf = 4Sf(xi)+-|SA 2f(xi)-l8A3f(xi) + E3 (4- 2)J-2where E3'<Mis Max IA4flJ ~ 45For Simpson's RuleA = |S [f(Xi-28) + 4f(x£) + f(xi+2S)] becomesA =|S {[f(Xi) -Ax -A2] + 4f(Xi) + [f(Xi) + A3 + A4]},(4.3)where \ = f[xi+(n-2)S] - f£xi+(n-3)S] , n = 1, 2, 3, 4.SinceS2A4 = A3 +A2 = A 3- A2 + .A4A:= A3-2A| + 3A3 - 3aJ- A*A = 48 f(xj)+-|S A 2-^.s A+ E4 (4.4)where Ea < Max IA4fl.4 ~ 3Then, for the two subdivisions, the upper bound for the error in Simpson'ssatisfies the following relations:RuleE1 < E3 +E^ orE 1 < 2M 8 Max |A4f| = ii2 w Max lA4fl. (4.5)45 45We obtain finally, for the entire range a< x < bE' < 142 (JL) w Max lA4f | orb ~ 45 2E' < 11 (b-a) Max lA4fls ~~ 45 (4.6)These two results for E't (3.6) and Es (4.6) establish, in terms of differ-ences, upper bounds for the errors in the trapezoidal rule and Simpson's rulerespecti vely.207Journal of the Arkansas Academy of Science, Vol. 8 [1955], Art. 23http://scholarworks.uark.edu/jaas/vol8/iss1/23

Estimates of Error in Numerical Integration

UARK (University of Arkansas )

Journal of the Arkansas Academy of Science Volume 8 Article 23 1955 Estimates of Error in Numerical Integration Dorothy M. Hoover University of Arkansas, Fayetteville Follow this and additional works at: https://scholarworks.uark.edu/jaas  Part of the Other Applied Mathematics Commons Recommended Citation Hoover, Dorothy M. (1955) "Estimates of Error in Numerical Integration," Journal of the Arkansas Academy of Science: Vol. 8 , Article 23. Available at: https://scholarworks.uark.edu/jaas/vol8/iss1/23 This article is available for use under the Creative Commons license: Attribution-NoDerivatives 4.0 International (CC BY-ND 4.0). Users are able to read, download, copy, print, distribute, search, link to the full texts of these articles, or use them for any other lawful purpose, without asking prior permission from the publisher or the author. This Article is brought to you for free and open access by ScholarWorks@UARK. It has been accepted for inclusion in Journal of the Arkansas Academy of Science by an authorized editor of ScholarWorks@UARK. For more information, please contact scholar@uark.edu. ON ESTIMATES OF ERROR IN NUMERICAL INTEGRATIONDOROTHY M. HOOVERUniversity of ArkansasINTRODUCTIONIn order to determine numerical approximations for definite integrals, theprincipal methods used are the trapezoidal rule and Simpson's rule. In this pa-per, upper bounds are established for the errors obtained by these two methods.The error resulting from use of the trapezoidal rule is expressed in terms of thesecond derivative of th": integrand, and that incurred in Simpson's rule is ex-pressed in terms of the fourth derivative. A difference formula, analogous toTaylor's formula with remainder, is established. This relation is then used toderive expressions for the errors in terms of second and fourth differences res-pectively.ERROR IN TERMS OF DERIVATIVESThe procedure is based on the Fundamental Theorem of the integral calculus.Let a subdivision of the interval from x-a to x-b into n parts be given bythe points a-xQ, x^ , xn = b, with x^+^-x^= 28. The midpoint ofthe subinterval from x^ to x^ +jis designated by Xj. From the formula f(x)=f(xi)+ (x-XjJf'Ui) +_(_x-XjJ 2fM (£) with f between x and xitit follows thatf(x)dx =[£(xi)] 28+ Ev with Ex < 83 Max If"( x)IJ x£ 3For one subdivision, the approximate area, A, as defined by the trapezoidalrule isA =i tfUi) + f(xi+1)] 28 (1.1)Hence A is given by this rule asA = [fU^] 28 + E2, where E2 < 83 Max lfM(x)I. (1.2)From ( 1. 1) and ( 1.2) itfollows that, in one subdivision, the actual error of thetrapezoidal rule satisfies the inequalityE<E! + E2=iS3Max l£"(x)l (1.3)But 28-w, the width of one subdivision, so that 8- *;• HenceE<1 w3Max lf"(x)l.6(1.4)Then, for n subdivisions, the total error obtained in the trapezoidal rule isF,t< n w3Max |fM(x)Efj "( )|6(1.5)and since nw - b-aE f< (b-a) w^x |f"(x) \5^ 2Max |f ( ) (1.6)The error resulting from Simpson's Rule can be lumped into an expressioncontaining the fourth derivative. Developing the expansion of f(x) for two moreterms, the result isf(x) = f(^)+(x.Xi)fl(xi)+(x-xi)2f>'(xi)+(x-xi)3f'"(xi)2! 3! 4!x and x^.204with x between x204Journal of the Arkansas Academy of Science, Vol. 8 [1955], Art. 23Published by Arkansas Academy of Science, 1955ESTIMATES OF ERROR 205For the two subdivisions from Xj_^ to Xj and from x.j to xj+j,we have/•xi+lf(x)dx = [f(x:)l 4S + te 3t"(*i)+ E3J xi-1Eo < 8 8 Max Iflv(x)I.3 ~15(2. 1)whereIn these two subdivisions, A, as defined by Simpson's rule isA = 28 [ft^.p + 4f(Xi)+ f(xi+1)]3= j?£ [f(x<-28) + 4f(x,) + f(x;+2S)] , and since3 1 li[f(xi-2S)+4f(xi)+f(xi+28)]= 6f(xi)+i^f"( î)+l^ [f'v(xi+012S)+f|V(xi-5228)]O<0j<1, 0<52<l.the value for A isA= 4-8 [f(x£)] +i_83 f'Cxj) + E4 (2.2)E^l S 5 Max If« v(x)l9withFrom (2. 1) and (2.2) it follows that an upper bound for the actual error inSimpson's rule for the two subdivisions is given byE< Eo+ E, =iiS 5Max Ifiv(x)I.45(2.3)Hence, when expressed in terms of w,E < _w^Max lflv(x)l.45(2.4)For the entire interval, the error in Simpson's rule is then bounded as folJows:E<.< JLw5Max If'v( x) I " , and, as nw = b-aE < (b-alwW lf'v (x*"45(2.5)Thus E t and Es give upper bounds in terms of derivatives for the trapezoidalrule and Simpson's rule respectively.DIFFERENCE INTERPOLATION FORMULA WITH REMAINDERWhen it is difficult to obtain the derivatives of the integrand, Taylor'sFormula with remainder may be replaced by a difference interpolation formulawith remainder. An expression for the remainder after one term (Mean Value Theo-rem for differences) for the difference interpolation formula will be extablish-ed. Likewise, an expression for the remainder after two terms will be determined.THEOREM. If the function f( x) is continuous in the interval a<x<a+pS, then witheach value of x in that interval there is associated a value i) satisfying\-r\\ < |p| such thatf(a+p8) = f(a)+p A f(a)+ P(p-D A2f(a)+ . . . +p(p-D- • -(p-n+1) Anf(a+778).2! nlPROOF. The following equations are valid:pf(a+pS)-f(a) =2 {f(a+nS)-f [a+(n-l)S]}n=l=I{Af[a+(n-l)8]}.n=l205Journal of the Arkansas Academy of Science, Vol. 8 [1955], Art. 23https://scholarworks.uark.edu/jaas/vol8/iss1/23206 ARKANSAS ACADEMY OF SCIENCEThe continuity of f(x) implies the continuity of Af(x). Therefore, the last sum-mation over p terms may be replaced by the expression pAf(x), with a_<x<a+p8.This relationship can also be expressed asf(a+pS)=f(a) + p [f(x+S)-£(x)]which is designated as the Mean Value Theorem for differences.COROLLARY. If f(a+p8) = f(a) = 0, then f(x+S) = f(x), where x is some interme-diary x.Consider now the quantity K defined by the equationf(a+pS) = f(a) + pAf(a) + £iEl_LL K.2«Tf a + pS-b , the equation takes the formf(b)-f(a)-b^a Af(a) (-^) 8 2," K=°Oefine F(x) as follows:F(x) = f(b) - f(x) -(k2L)Af(x) -(^(-2-rrr^K •8 e 2:Then F^a) and F(b) are both zero. Butvalue x between A and B for which A Fby the above corollary there exists a(x) vanishes.When F(x) is evaluated at (x+S) and thesimplified, the result isdifference F(x+8)- F(x) determined andAF(x) = A2f(x) [1 - £z*] - [1 - hz*.] K = 0.o oSolving for K we obtainK = A2f(x), with a <x<b = a+pS.The general result for a remainder after n terms can be established byanalogous reasoning.ERRORS IN TERMS OF DIFFERENCESThe difference interpolation formula with remainder in the second differ-ence isf(xi+£S)= f(xi)+^A fft.juHS-D A2f(xi+77S) , 0<ti<^2!Let x= (x^S). Then(3.1)/Xj+8 if(x)dx = ftGi+^S)Sd^ = 2Sf(xi) + E{ , with Ki'<£ MaxlA2f| (3.2)Xl-8 J-lFor the trapezoidal ruleA =4- [f(Xi ) + f(xi+1)] 28 takes the formA =i- [f(x-) -A-.+ f(Xi)+A2] 28, where A,=Af(Xi )=A f(Xj-8)(3.3)and A2= Af(xi)Simplifying, we obtainA = 28 f(Xi) + E2 with E'< 8 Max lA2fl. (3.4)Thus the error in one subdivision in terms of differences for the trapezoidalrule can be expressed by the inequalityE'<E' + E 11 2<|S Max |A2FI =1w Max |A2Fl . (3.5)206Journal of the Arkansas Academy of Science, Vol. 8 [1955], Art. 23Published by Arkansas Academy of Science, 1955ESTIMATES OF ERROR 207IFor n subdivisions, the resulting total error in the trapezoidal rule is thenEtilM MaxlA2FI , so thatE* < l(b-a) MaxlA2Fl. (3.6)*- "" 3For Simpson's Rule the error can be expressed in terms of a fourth differ-ence. When the difference interpolation formula with remainder is developedthrough two more terms, the result isf(xi+£8)= f(x,)+^Af(x.)+ 1} A2 f(xi)+illliUlll)A3 f(xi)+2 ¦'+-fU-l)(g-2)(£-3) A4 f(x;+778) where 0 <v<£•( 4. 1}4.1Therefore/,x1+28 -2f(x)dx = [ftxi^SJSdf = 4Sf(xi)+-|SA 2f(xi)-l8A3f(xi) + E3 (4- 2)J -2where E3'<Mis Max IA4flJ ~ 45For Simpson's RuleA = |S [f(Xi-28) + 4f(x£) + f(xi+2S)] becomesA =|S {[f(Xi) -Ax -A2] + 4f(Xi) + [f(Xi) + A3 + A4]},(4.3)where \= f[xi+(n-2)S]- f£xi+(n-3)S] , n = 1, 2, 3, 4.SinceS2A4= A3+A2 = A 3- A2 + .A4A:=A3-2A| + 3A3-3aJ- A*A = 48 f(xj)+-|S A 2 -^.s A + E4 (4.4)where Ea < Max IA4fl.4 ~ 3Then, for the two subdivisions, the upper bound for the error in Simpson'ssatisfies the following relations:RuleE1 < E3 +E^ orE 1 < 2M 8 Max |A4f| = ii2 w Max lA4fl. (4.5)45 45We obtain finally, for the entire range a< x < bE' < 142 (JL) w Max lA4f | orb ~ 45 2E' < 11 (b-a) Max lA4fls ~~ 45(4.6)These two results for E't (3.6) and Es (4.6) establish, in terms of differ-ences, upper bounds for the errors in the trapezoidal rule and Simpson's rulerespecti vely.207Journal of the Arkansas Academy of Science, Vol. 8 [1955], Art. 23https://scholarworks.uark.edu/jaas/vol8/iss1/23

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Estimates of Error in Numerical Integration

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