Dragomir, Sever S

Mabizela, Sizwe G

English

Name not available

Some Error Estimates in the Trapezoidal Quadrature Rule

Victoria University Institutional Repository (VUIR)

Victoria University Eprints Repository

SOME ERROR ESTIMATES IN THE TRAPEZOIDALQUADRATURE RULES. S. DRAGOMIR AND S. MABIZELAAbstract. We derive sharp error bounds in the Trapezoidal Rule for functionswhose first derivative satisfies a strong continuity condition or whose secondderivative is an Lp function.1. IntroductionThe classical error estimate in the Trapezoidal Rule asserts that if a function fhas a bounded second derivative on the interval [a, b] , then the Trapezoidal Ruleconverges to the true value of the definite integral∫ baf (x) dx with the rapidity ofat least 1n2 . If the second derivative of f is not bounded, then the Trapezoidal Rulemay or may not converge, and if it converges, it may do so very slowly.In some practical problems the integrand does not have a bounded second de-rivative, and therefore the classical error estimate of the Trapezoidal Rule is notavailable. In this paper we derive some error bounds in the Trapezoidal QuadratureRule for the following classes of functions:(1) Functions whose first derivative belongs to the Ho¨lder Space Cα [a, b] , wherefor α ∈ (0, 1] ,Cα [a, b] := {f : [a, b]→ R : |f (x)− f (y)| ≤ K |x− y|α , x, y ∈ [a, b]}for some finite K > 0,(2) Functions whose second derivative is an Lp−function.This paper is organized as follows: In this section we prove the key lemma(Lemma 1). This lemma is then applied to the classes of functions mentionedabove.In Section 2 we apply the results of Section 1 in deriving some inequalities ofTrapezoid type.In Section 3 we apply the results obtained in Section 2 to derive error boundsfor the Trapezoidal Quadrature Rule for the above classes of functions.Unless otherwise indicated, we shall assume throughout that [a, b] is a finiteinterval.The following lema will be useful in the sequel.Date: May, 1999.1991 Mathematics Subject Classification. Primary 26D15; Secondary 41A55.Key words and phrases. Trapezoid Inequality, Quadrature Formulae.12 S. S. DRAGOMIR AND S. MABIZELALemma 1. Let f : [a, b]→ R be an absolutely continuous function on [a, b] . Thenwe have the identityf (a) + f (b)2− 1b− a∫ baf (x) dx=12 (b− a)2∫ ba∫ ba[f ′ (x)− f ′ (y)] (x− y) dxdy.Proof. We have successively∫ ba∫ ba[f ′ (x)− f ′ (y)] (x− y) dxdy=∫ ba∫ ba[xf ′ (x) + yf ′ (y)− xf ′ (y)− yf ′ (x)] dxdy= 2∫ ba∫ ba[xf ′ (x)− xf ′ (y)] dxdy= 2∫ ba∫ baxf ′ (x) dxdy − 2∫ ba∫ baxf ′ (y) dxdy= 2 (b− a)[bf (b)− af (a)−∫ baf (x) dx]− (b2 − a2) [f (b)− f (a)]= (b− a)2 [f (a) + f (b)]− 2 (b− a)∫ baf (x) dx.Dividing both sides by 2 (b− a)2 yields the required result.2. Some Integral InequalitiesLet us start to the following result for convex mappings.Theorem 1. Let f : [a, b]→ R be a convex function on [a, b] . Thenf (a) + f (b)2− 1b− a∫ baf (x) dx ≥ max {|A| , |B| , |C|} ,(2.1)whereA : =1b− a∫ ba|xf ′ (x)| dx− 1(b− a)2∫ ba|f ′ (x)| dx ·∫ ba|x| dx,B : =1b− a∫ bax |f ′ (x)| dx− a+ b2· 1b− a∫ ba|f ′ (x)| dx, andC : =1b− a∫ ba|x| f ′ (x) dx− f (b)− f (a)(b− a)2∫ ba|x| dx.Moreover, the inequality (2.1) is sharp in the sense that it cannot be replaced by asmaller constant.Proof. Since f is a convex function on [a, b] , it follows that[f ′ (x)− f ′ (y)] (x− y) ≥ 0SOME ERROR ESTIMATES 3for almost every x, y ∈ (a, b) . Hence, for a.e. x, y ∈ (a, b) ,[f ′ (x)− f ′ (y)] (x− y)= |[f ′ (x)− f ′ (y)] (x− y)|≥ max {|[|f ′ (x)| − |f ′ (y)|] [|x| − |y|]| , |[|f ′ (x)| − |f ′ (y)|] [x− y]| ,|[f ′ (x)− f ′ (y)] [|x| − |y|]|} .Thus, ∫ ba∫ ba[f ′ (x)− f ′ (y)] [x− y] dxdy≥ max{∣∣∣∣∣∫ ba∫ ba[|f ′ (x)| − |f ′ (y)|] [|x| − |y|] dxdy∣∣∣∣∣ ,∣∣∣∣∣∫ ba∫ ba[|f ′ (x)| − |f ′ (y)|] [x− y] dxdy∣∣∣∣∣ ,∣∣∣∣∣∫ ba∫ ba[f ′ (x)− f ′ (y)] [|x| − |y|] dxdy∣∣∣∣∣}.A simple calculation shows thatA =2(b− a)2∫ ba∫ ba[|f ′ (x)| − |f ′ (y)|] [|x| − |y|] dxdy,B =2(b− a)2∫ ba∫ ba[|f ′ (x)| − |f ′ (y)|] [x− y] dxdy, andC =2(b− a)2∫ ba∫ ba[f ′ (x)− f ′ (y)] [|x| − |y|] dxdy.To prove the sharpness of the inequality (2.1) , let f (x) = 1x . Then f is differentiableand convex on (a, b) , where a > 0. Now,f (a) + f (b)2− 1b− a∫ baf (x) dx =a+ b2ab−(ln b− ln ab− a),and|B| =∣∣∣∣∣ 1b− a∫ ba1xdx− a+ b2· 1b− a∫ ba1x2dx∣∣∣∣∣=∣∣∣∣ ln b− ln ab− a − a+ b2ab∣∣∣∣ = a+ b2ab −(ln b− ln ab− a).We therefore get equality in the inequality (2.1) .The following inequality also holds.Theorem 2. Let f : [a, b] → R be an absolutely continuous function on [a, b] . Iff ′ ∈ Cα [a, b] , then we have the inequality∣∣∣∣∣f (a) + f (b)2 − 1b− a∫ baf (x) dx∣∣∣∣∣ ≤ K(α+ 2) (α+ 3) (b− a)α+1 ,for some finite constant K > 0.4 S. S. DRAGOMIR AND S. MABIZELAProof. Since f ′ ∈ Cα [a, b] , there is a finite constant K > 0 such that|f ′ (x)− f ′ (y)| ≤ K |x− y|α+1 for all x, y ∈ [a, b] .Using Lemma 1, we have∣∣∣∣∣f (a) + f (b)2 − 1b− a∫ baf (x) dx∣∣∣∣∣≤ 12 (b− a)2∫ ba∫ ba|f ′ (x)− f ′ (y)| |x− y| dxdy≤ K2 (b− a)2∫ ba∫ ba|x− y|α+1 dxdy=K(α+ 2) (α+ 3)(b− a)α+1 ,where the last equality follows from the fact that∫ ba∫ ba|x− y|α+1 dxdy =∫ ba(∫ by(x− y)α+1 dx+∫ ya(y − x)α+1 dx)dy=1α+ 2∫ ba[(b− y)α+2 + (y − a)α+2]dy=2 (b− a)α+3(α+ 2) (α+ 3).and the theorem is proved.The following corollary is natural.Corollary 1. Let f : [a, b] → R be an absolutely continuous function on [a, b] . Iff ′ is Lipschitz continuous on [a, b] , i.e.,|f ′ (x)− f ′ (y)| ≤ K |x− y| for all x, y ∈ [a, b] ,where K > 0, then we have the inequality∣∣∣∣∣f (a) + f (b)2 − 1b− a∫ baf (x) dx∣∣∣∣∣ ≤ K12 (b− a)2 .Moreover, the constant 112 is sharp.Proof. Take α = 1 in Theorem 2.To prove the sharpness of the constant 112 , let f (x) =x22 . Then|f ′ (x)− f ′ (y)| = |x− y| ≤ K |x− y|for all x ∈ [a, b] , where K = 1. Consequently,f (a) + f (b)2− 1b− a∫ baf (x) dx =112(b− a)2 .and the proof is completed.The following theorem also holds.SOME ERROR ESTIMATES 5Theorem 3. Let f : [a, b] → R be such that f ′ is absolutely continuous on [a, b] .If f ′′ ∈ Lp (a, b) , where 1 ≤ p <∞, then we have the estimation∣∣∣∣∣f (a) + f (b)2 − 1b− a∫ baf (x) dx∣∣∣∣∣≤ 12 (b− a)2∫ ba∫ baKp (x, y) |x− y|2−1p dxdy≤ p2 (b− a)2− 1p(3p− 1) (4p− 1) ‖f′′‖p ,whereKp (x, y) =∣∣∣∣∫ yx|f ′′ (t)|p dt∣∣∣∣ 1p , x, y ∈ [a, b] .The first inequality is sharp.Proof. By Lemma 1,f (a) + f (b)2− 1b− a∫ baf (x) dx=12 (b− a)2∫ ba∫ ba[f ′ (x)− f ′ (y)] (x− y) dxdy=12 (b− a)2∫ ba∫ ba(∫ yxf ′′ (η) dη)(x− y) dxdy.Also, using Ho¨lder’s Inequality for p > 1, 1/p + 1/q = 1, we have that, for allx, y ∈ [a, b] ,∣∣∣∣∫ yx|f ′′ (η)| dη∣∣∣∣ ≤ ∣∣∣∣∫ yx|f ′′ (η)|p dη∣∣∣∣ 1p ∣∣∣∣∫ yxdη∣∣∣∣ 1q = |x− y| 1q Kp (x, y) .6 S. S. DRAGOMIR AND S. MABIZELANow, ∣∣∣∣∣f (a) + f (b)2 − 1b− a∫ baf (x) dx∣∣∣∣∣≤ 12 (b− a)2∫ ba∫ ba∣∣∣∣∫ yxf ′′ (η) dη∣∣∣∣ |y − x| dxdy≤ 12 (b− a)2∫ ba∫ ba|x− y|1− 1q∣∣∣∣∫ yx|f ′′ (η)|p dη∣∣∣∣ 1p dxdy=12 (b− a)2∫ ba∫ ba|x− y|2− 1p∣∣∣∣∫ yx|f ′′ (t)|p dt∣∣∣∣ 1p dxdy≤ 12 (b− a)2∫ ba∫ ba|x− y|2− 1p(∫ ba|f ′′ (t)|p dt) 1pdxdy=‖f ′′‖p2 (b− a)2 ·2p2 (b− a)4− 1p(3p− 1) (4p− 1)=p2 (b− a)2− 1p(3p− 1) (4p− 1) ‖f′′‖p .To prove the sharpness of the first inequality, let f (x) = x22 , for x ∈ [a, b] . Then∣∣∣∣∣f (a) + f (b)2 − 1b− a∫ baf (x) dx∣∣∣∣∣ = (b− a)212 ,and12 (b− a)2∫ ba∫ baKp (x, y) |x− y| dxdy=12 (b− a)2∫ ba∫ ba|x− y| 1p |x− y|2− 1p dxdy=12 (b− a)2∫ ba∫ ba|x− y|2 dxdy=(b− a)212.The case p = 1 goes likewise and we omit the details.3. Error Bounds in the Trapezoidal Quadrature RuleIn this section we derive error bounds in the Trapezoidal Quadrature Rule forthe classes of functions mentioned in Section 1.Theorem 4. Let f : [a, b] → R be an absolutely continuous function such thatf ′ ∈ Cα [a, b] . If P : a = x0 < x1 < x2 < ... < xn = b is a partition of the interval[a, b] , hi = xi − xi−1 for i = 1, 2, ..., n, andTn (f, P ) :=n∑i=1f (xi−1) + f (xi)2· hiSOME ERROR ESTIMATES 7then|Rn (f, P )| =∣∣∣∣∣∫ baf (x) dx− Tn (f, P )∣∣∣∣∣ ≤ K(α+ 2) (α+ 3)n∑i=1hα+2ifor some finite constant K > 0.Proof. We just have to apply Theorem 2 to f on each subinterval [xi−1, xi] , fori = 1, 2, 3, ..., n. This gives|Rn (f, P )| =∣∣∣∣∣∫ baf (x) dx− Tn (f, P )∣∣∣∣∣=∣∣∣∣∣n∑i=1∫ xixi−1f (x) dx−n∑i=1f (xi−1) + f (xi)2hi∣∣∣∣∣=∣∣∣∣∣n∑i=1hi(f (xi−1) + f (xi)2− 1hi∫ xixi−1f (x) dx)∣∣∣∣∣≤n∑i=1hi∣∣∣∣∣f (xi−1) + f (xi)2 − 1hi∫ xixi−1f (x) dx∣∣∣∣∣ ≤≤n∑i=1hiK(α+ 2) (α+ 3)(xi − xi−1)α+1=K(α+ 2) (α+ 3)n∑i=1hα+2i .and the theorem is completely proved.The following three corollaries are natural consequences.Corollary 2. Assume that the hypotheses of Theorem 4 hold and that P is a regularpartition of [a, b] , i.e., xi = a+ i(b−an)for i = 0, 1, 2, ..., n. Then|Rn (f, P )| ≤ K (b− a)α+2(α+ 2) (α+ 3)nα+1for some finite constant K > 0.Moreover, if ε > 0 is given, thennε :=( K (b− a)α+2(α+ 2) (α+ 3) ε) 1α+1+ 1where, for r ∈ R, [r] is the integer part of r, is the smallest natural number forwhich we have|Rn (f)| ≤ ε for n ≥ nε.The second part of Corollary 2 says that if we use a regular (or uniform) partitionof [a, b] and we want the magnitude of the error to be less than some preassignedtolerance ε, then we need to use at leastnε :=( K (b− a)α+2(α+ 2) (α+ 3) ε) 1α+1+ 18 S. S. DRAGOMIR AND S. MABIZELApartition points.Corollary 3. Let f : [a, b] → R be an absolutely continuous function such that f ′is Lipschitz continuous on [a, b] . If P : a = x0 < x1 < x2 < ... < xn = b is apartition of the interval [a, b] , hi = xi − xi−1 for i = 1, 2, ..., n, andTn (f, P ) :=n∑i=1f (xi−1) + f (xi)2· hiThen|Rn (f, P )| ≤ K12n∑i=1h3i .Moreover, the constant 112 is sharp.Corollary 4. Assume that the hypotheses of Corollary 3 hold and that P is aregular partition of [a, b] , i.e., xi = a+ i(b−an)for i = 0, 1, 2, ..., n. Then|Rn (f, P )| =∣∣∣∣∣∫ baf (x) dx− Tn (f, P )∣∣∣∣∣ ≤ L (b− a)312n2 .Moreover, if ε > 0 is given, thennε :=√L (b− a)312ε+ 1is the smallest natural number for which we have|Rn (f)| ≤ ε for n ≥ nε.In the case of functions which are twice differentiable and the second derivativeis integrable, we have the following error estimate.Theorem 5. Let f : [a, b]→ R be a twice differentiable function on (a, b) such thatf ′′ ∈ L1 (a, b) and P : a = x0 < x1 < x2 < ... < xn = b is a partition of the interval[a, b] . Then we have the following estimation|Rn (f, P )|(3.1)≤ 12n∑i=11hi∫ xixi−1∫ xixi−1K1 (x, y) |x− y| dxdy≤ ‖P‖26‖f ′′‖1 ,where K1 (x, y) =∣∣∫ yx|f ′′ (t)| dt∣∣ with x, y ∈ [a, b] , and ‖P‖ = max1≤i≤n {xi − xi−1} .The first inequality in (3.1) is sharp.SOME ERROR ESTIMATES 9Proof. Applying Theorem 3 on each subinterval [xi−1, xi] for i = 1, 2, ..., n, weobtain|Rn (f, P )|=∣∣∣∣∣n∑i=1∫ xixi−1f (x) dx−n∑i=1f (xi−1) + f (xi)2hi∣∣∣∣∣=∣∣∣∣∣n∑i=1hi(1hi∫ xixi−1f (x) dx− f (xi−1) + f (xi)2)∣∣∣∣∣≤n∑i=1hi∣∣∣∣∣(1hi∫ xixi−1f (x) dx− f (xi−1) + f (xi)2)∣∣∣∣∣≤n∑i=1hi12h2i∫ xixi−1∫ xixi−1K1 (x, y) |x− y| dxdy≤n∑i=1h2i6∫ xixi−1|f ′ (t)| dt≤ ‖P‖26n∑i=1∫ xixi−1|f ′′ (t)| dt≤ ‖P‖26‖f ′′‖1 ,which completes the proof.Assume that the hypotheses of Theorem 5 hold and that P is a regular partitionof [a, b] . Then|Rn (f, P )| ≤ (b− a)26n2‖f ′′‖1 .If ε > 0 is given, thennε :=[(b− a)√‖f ′′‖16ε]+ 1is the smallest natural number for which we have|Rn (f)| ≤ ε for n ≥ nε.Finally, we haveTheorem 6. Let f : [a, b] → R be so that f ′ is an absolutely continuous functionon (a, b) and such that f ′′ ∈ Lp (a, b) , where 1 < p < ∞, and P : a = x0 < x1 <x2 < ... < xn = b is a partition of the interval [a, b] . Then we have the followingestimation|Rn (f, P )|≤ 12n∑i=11hi∫ xixi−1∫ xixi−1Kp (x, y) |x− y|2−1p dxdy≤ p2(3p− 1) (4p− 1)(n∑i=1h2p−1p−1i)1− 1p‖f ′′‖p ,10 S. S. DRAGOMIR AND S. MABIZELAwhereKp (x, y) =∣∣∣∣∫ yx|f ′′ (t)|p dt∣∣∣∣ 1p , x, y ∈ [a, b] .The first inequality is sharp.A naturally corollary of this theorem isCorollary 5. Assume that the hypothesis of Theorem 6 hold and that P is a regularpartition of [a, b] . Then|Rn (f, P )| ≤ p2 (b− a)2− 1pn2 (3p− 1) (4p− 1) ‖f′′‖p .If ε > 0 is given, thennε :=p√√√√ (b− a)2− 1p ‖f ′′‖p(3p− 1) (4p− 1) ε+ 1is the smallest natural number for which we have|Rn (f)| ≤ ε for n ≥ nε.References[1] S. S. Dragomir, On the trapezoid inequality for absolutely continuous mappings, submitted.[2] S. S. Dragomir, On the trapezoid formula for Lipschitzian mappings and applications, TamkangJournal of Math. (in press)[3] S. S. Dragomir, On the trapezoid quadrature formula for mappings of bounded variation andapplications, Extracta Mathematicae, (in press).[4] S. S. Dragomir, P. Cerone and A. Sofo, Some remarks on the trapezoid rule in numericalintegration, submitted.[5] S. S. Dragomir and T. C. Peachey, New estimation of the remainder in the trapezoidal formulawith applications, submitted.[6] S. S. Dragomir, P. Cerone and C. E. M Pearce, Generalization of the trapezoid inequality formappings of bounded variation and applications, submitted.[7] N. S. Barnett, S. S. Dragomir, and C. E. M. Pearce, A quasi-trapezoid inequality for doubleintegrals submitted.[8] S. S. Dragomir and A. McAndrew, On trapezoid inequality via a Gru¨ss type result and appli-cations, submitted.(S. S. Dragomir), School of Communications and Informatics, Victoria University ofTechnology, PO Box 14428, Melbourne City, MC 8001 AustraliaE-mail address: sever@matilda.vu.edu.au(S. Mabizela), Department of Mathematics and Applied Mathematics, University ofCapetown, Rondebosch, 7700, SOUTH AFRICAE-mail address: sizwe@maths.uct.ac.za

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Generalization of the trapezoid inequality for mappings of bounded variation and applications,

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On the trapezoid formula for Lipschitzian mappings and applications,

On the trapezoid inequality for absolutely continuous mappings,

On the trapezoid quadrature formula for mappings of bounded variation and applications, Extracta Mathematicae,

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Some Error Estimates in the Trapezoidal Quadrature Rule

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