Volume 2008, Article ID 845934,15pages doi:10.1155/2008/845934
Research Article
Some Generalized Error Inequalities
and Applications
Fiza Zafar1 and Nazir Ahmad Mir2
1Centre for Advanced Studies in Pure and Applied Mathematics, Bahauddin Zakariya University,
Multan 60800, Pakistan
2Department of Mathematics, COMSATS Institute of Information Technology, Plot no. 30,
Sector H-8/1, Islamabad 44000, Pakistan
Correspondence should be addressed to Fiza Zafar,[email protected] Received 4 February 2008; Revised 30 May 2008; Accepted 29 July 2008
Recommended by Sever Dragomir
We present a family of four-point quadrature rule, a generalization of Gauss-two point, Simpson’s 3/8, and Lobatto four-point quadrature rule for twice-differentiable mapping. Moreover, it is shown that the corresponding optimal quadrature formula presents better estimate in the context of four-point quadrature formulae of closed type. A unified treatment of error inequalities for different classes of function is also given.
Copyrightq2008 F. Zafar and N. A. Mir. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
1. Introduction
We define
If b
a
fxdx. 1.1
The problem of approximating If is usually referred to as numerical integration or quadrature 1. Most numerical integration formulae are based on defining the approxi-mation by using polynomial or piecewise polynomial interpolation. Formulae using such interpolation with evenly spaced nodes are referred to as Newton-Cotes formulae. The Gaussian quadrature formulae, which are optimal and converge rapidly by selecting the node points carefully that need not be equally spaced, are investigated in2.
The deduction of the optimal quadrature formulae in the sense of minimal error bounds has not received the right attention as long as the work by Ujevi´csee6–8and9, pages 153–166, who used a new approach for obtaining optimal two-point and three-point quadrature formulae of open as well as closed type, has not appeared. Further, some error inequalities have also been presented by Ujevi´c to ensure the applications of these optimal quadrature formulae for different classes of functions.
In this paper, we present an approach similar to that of Ujevi´c’s6to present some improvements and generalizations in this context.
Let us first formulate the main problem. Consider
Kx, y, t ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
1 2t−α
2α
1, t∈a, x,
1 2t−β
2β
1, t∈x, y,
1 2t−γ
2γ
1, t∈y, b,
1.2
as defined in 6, where x, y ∈ a hb −a, b −hb− a, h ∈ 0,1/2, x < y, and
α, α1, β, β1, γ, γ1∈Rare parameters which are required to be determined.
We know that the exact value of the remainder term of the integralabKx, y, tftdt
may not be found, thus, we may proceed as
b
a
Kx, y, tftdt≤ max
t∈a,b|f t|b
a
|Kx, y, t|dt. 1.3
The main aim of this paper is to present a minimal estimation of the error bound 1.3 by appropriately choosing the variables and parameters involved. Moreover, it is worth mentioning that the family of quadrature formulae thus obtained hereafter is a generalization of that presented in6.
2. A generalized optimal quadrature formula
Consider the above stated error inequality problem fora −1,b 1,so that x, y ∈ −1 2h,1−2h.We will try to find out an optimal quadrature formula of the form
1
−1
ftdt−hf−1 1−hfx 1−hfy hf1
1
−1
Kx, y, tftdt, 2.1
whereKx, y, tis defined by1.2witha−1,b1, andx, y∈−12h,1−2hwithx < y,
h∈0,1/2.
The parameters α, a1, β, β1, γ, γ1 ∈ Rinvolved in Kx, y, t are required to be
Integrating by parts right-hand side of2.1, we have
1
−1
Kx, y, tftdt− 1
21α
2α
1
f−1 1 21−γ
2γ
1
f1
1
2{x−α
2−x−β2}α 1−β1
fx
1
2{y−β
2−y−γ2}β 1−γ1
fy−1αf−1
−1−γf1 α−βfx β−γfy
1
−1
ftdt.
2.2
For the representation2.1, we require from2.2
1 2x−α
2α
1−1
2x−β
2−β 10,
1 2y−β
2β
1−1
2y−γ
2−γ 1 0,
1 21α
2α
10,
1 21−γ
2γ
10,
β−γ−1−h, α−β−1−h,
1αh,
1−γh.
2.3
This gives through simple calculations:
α−1−h, γ 1−h, β0,
γ1−
1 2h
2,
α1−
1 2h
2,
β1 1
2−h 1−hx
1
2−h−1−hy.
Henceforth,
y−x. 2.5
So, we have
Kx, t ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
1
2t 1−h
2−1
2h
2, t∈−1, x,
1 2t
2 1−hx−h1
2, t∈x, y, 1
2t−1−h
2−1
2h
2, t∈y,1.
2.6
We further see that
1
−1
Kx, tftdt≤ f∞ 1
−1
|Kx, t|dt. 2.7
We are now required to find anxthat minimizes1−1|Kx, t|dt.
We next define
Gx 1
−1
|Kx, t|dt
1
2
x
−1
|t 1−h2−h2|dt y
x
12t2 1−hx−h 1
2
dt1
2
1
y
|t−1−h2−h2|dt,
2.8
and consider the problem
minimizeGx, x∈−12h,1−2h, h∈ 0,1
2
. 2.9
Hence, we would like to find a global minimizer ofG. Recall that a global minimizer is a point
x∗that satisfies
Gx∗≤Gx ∀x∈−12h,1−2h, h∈ 0,1
2
We now consider the following cases.
iLetx∈−1−2h,h−1/2/1−h.Then by symmetry, we may consider
G1x −
1 2
−12h
−1
t 1−h2−h2dt
1
2
x
−12h
t 1−h2−h2dt
1
2
−√2h−1−21−hx
x
t221−hx−2h1dt
−1
2
0
−√2h−1−21−hx
t221−hx−2h1dt
1
6− 1
21−hx
2−4
31−h
2h−1−21−hxx
4
3
h−1
2
2h−1−21−hx4
3h
3−h
2.
2.11
We may note that
Gx 2G1x. 2.12
Combining2.11and2.12with2.1and2.7, we get
1
−1
ftdt−hf−1 1−hfx 1−hf−x hf1
≤ 1
3−1−hx
2−8
31−h
2h−1−21−hxx
8
3
h−1
2
2h−1−21−hx8
3h
3−h
f
∞.
2.13
Moreover, simple calculations show thatG1x 0 for
x1,2−44h±2
3−6h4h2. 2.14
It is not difficult to find that
Thus,x1is the local minimizer ofGxforx∈−1−2h,h−1/2/1−h.We have
G1x1
52 3 h
3−44h283
2 h− 83
6 81−h
24h2−6h3
2
38h
2−14h7
8h2−14h7−41−h4h2−6h3
− 8
31−h
8h2−14h7−41−h4h2−6h34h2−6h3,
2.16
such that
Gx1 2G1x1. 2.17
iiNext, we check the pointx3 h−1/2/1−h.We find that minh∈0,1/2G1x1<
minh∈0,1/2G1x3.
Thus, from the above considerations, we find thatx∗−44h2
3−6h4h2is the
global minima ofG. Therefore, we get the following conclusion.
Theorem 2.1. LetI ⊂ R be an open interval such that−1,1 ⊂ I and letf : I→Rbe a twice differentiable function such thatfis bounded and integrable. Then,
1
−1
ftdt
hf−11−hf−44h23−6h4h21−hf4−4h−23−6h4h2hf1Rf,
2.18
where
|Rf| ≤2Δhf∞, 2.19
h∈0,1/2, andΔhis defined as
Δh 52
3 h
3−44h2 83
2 h− 83
6 81−h
24h2−6h3
2
38h
2−14h7
8h2−14h7−41−h4h2−6h3
−8
31−h
8h2−14h7−41−h4h2−6h34h2−6h3.
2.20
Proof. From the above discussion, we find that2.18holds with
Rf 1
−1
andKx, tis given by2.6withy−x. We further have
|Rf| ≤ f∞ 1
−1
|K−44h23−6h4h2, t|dt
G−44h23−6h4h2f
∞.
2.22
SinceG−44h2
3−6h4h2 2G
1−44h2
3−6h4h2, thus2.19holds.
We would like now to mention here some special cases of2.13.
Remark 2.2. As it has been mentioned in 6, we recapture the Gauss two-point quadrature formula forh0 andx−√3/3.
Remark 2.3. It may be noted that forh 1/6 and x −√5/5,we get Lobbato four-point quadrature rule as follows:
1
−1
ftdt 1
6 f−1 5f
−
√
5 5
5f √
5 5
f1
R1f, 2.23
where
|R1f| ≤C1f∞, 2.24
andC11/81 4/27
−63√5√5−2≈0.0418.
Remark 2.4. Forh1/4 andx−1/3,we get 3/8 Simpson’s rule as follows:
1
−1
ftdt 1
4 f−1 3f
−1
3
3f
1 3
f1
R2f, 2.25
where
|R2f| ≤C2f∞, 2.26
andC21/24≈0.0417.
Remark 2.5. Keeping in view the above special cases, 2.13 may be considered as a generalization of Gauss two-point, Simpson’s 3/8 and Lobatto four-point quadrature rule for twice differentiable mappings.
Corollary 2.7. Let the assumptions ofTheorem 2.1hold. Then, one has the following optimal quadra-ture rule:
1
−1
ftdt 1
5 f−1 4f
−2
5
4f
2 5
f1
R3f, 2.27
|R3f| ≤C3f∞, 2.28
whereC314/375≈0.0373.
Remark 2.8. The comparison of2.23,2.25, and2.27shows that the latter presents a much better estimate in the context of four-point quadrature rules of closed type.
By considering the problem on the intervala, b,the following theorem is obvious.
Theorem 2.9. LetI ⊂ Rbe an open interval such thata, b ⊂ I and letf : I→Rbe a twice-differentiable function such thatfis bounded and integrable. Then,
b
a
ftdt 1
2b−ahfa 1−hfx1 1−hfx2 hfb Rf, 2.29
where
x1 b−a
2 x
∗ ab
2 , x2−
b−a
2 x
∗ab
2 , 2.30
with
x∗−44h2
3−6h4h2,
|Rf| ≤ 1
4Δhb−a
3f
∞,
2.31
h∈0,1/2, andΔhis as defined above.
3. Generalized error inequalities
From the basic properties of theLpa, bspaces forp 1,2,∞,we know thatL2a, bis a
Hilbert space with the inner product defined as
f, g2
b
a
ftgtdt. 3.1
We now defineX L2a, b,·,·2.In the spaceX, the norm·2 is defined in the usual
manner as
f2
b
a f2tdt
1/2
Let us also considerY L2a, b,·,·,where the inner product·,·is defined by
f, g 1
b−a b
a
ftgtdt 3.3
with the corresponding norm · defined by
ff, f. 3.4
We know that the Chebyshev functional is defined as
Tf, g f, g − f, eg, e, 3.5
wheref, g ∈L2a, bande 1 which satisfies the pre-Gr ¨uss inequality4, page 296or5,
page 209:
T2f, g≤Tf, fTg, g. 3.6
Let us denote
σf σf;a, b
b−aTf, f 3.7
as defined in6. Moreover, the spaceL1a, bis a Banach space with the norm
f1
b
a
|ft|dt, 3.8
and the spaceL∞a, bis also a Banach space with the norm
f∞ess sup
t∈a,b|ft|. 3.9
So, iff∈L1a, bandg ∈L∞a, b,then we have
|f, g2| ≤ f1g∞. 3.10
Finally, we define
Jf Jf;a, b;h
b
a
ftdt−1
2b−ahfa 1−hfx1 1−hfx2 hfb,
3.11
We would also like to mention the following lemma10.
Lemma 3.1. Let
ft ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩
f1t, t∈a, x1,
f2t, t∈x1, x2,
f3t, t∈x2, b,
3.12
where a < x1 < x2 < b, f1 ∈ C1a, x1, f2 ∈ C1x1, x2, f3 ∈ C1x2, bf1x1 f2x1,and
f2x2 f3x2. If
sup
t∈a,x1
|f
1t|<∞, sup
t∈x1,x2
|f
2t|<∞, sup
t∈x2,b
|f
3t|<∞, 3.13
then the functionfis an absolutely continuous function.
Theorem 3.2. Letf :−1,1→Rbe a function such thatf∈L1−1,1.If there exists a real number
γ1,such thatγ1≤ft, t∈−1,1,then
|Jf;−1,1;h| ≤2Δ0hS−γ1, 3.14
and if there exists a real numberΓ1,such thatft≤Γ1,t∈−1,1,then
|Jf;−1,1;h| ≤2Δ0hΓ1−S, 3.15
whereJf;−1,1;his defined by3.11,S f1−f−1/2, andh∈0,1/2.If there exist real numbersγ1, Γ1,such thatγ1≤ft≤Γ1,t∈−1,1,then
|Jf;−1,1;h| ≤ 1
2Δ1hΓ1−γ1, 3.16
Δ0handΔ1hare defined as
Δ0h 2
4h2−6h3−31−h,
Δ1h 58h2−98h49−281−h
4h2−6h3.
3.17
Proof. In order to prove3.16, let us define
p1t
⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩
t1−h, t∈−1, x, t, t∈x, y, t−1−h, t∈y,1,
wherex−44h2
3−6h4h2andy−x.Note that sincep
1, e20, thus
p1, f2−Jf;−1,1;h,
f−Γ1γ1
2 , p1
2
f, p
12.
3.19
From3.10,
f−Γ1γ1
2 , p1
2
≤f−Γ1γ1
2
∞p11
≤ 1
2Δ1hΓ1−γ1,
3.20
as
f−Γ1γ1
2
∞≤
Γ1−γ1
2 ,
p1158h2−98h49−281−h
√
4h2−6h3.
3.21
From3.19and3.20, it may be observed that3.16holds. Further, it can be seen that
|f−γ
1, p12| ≤ p1∞f−γ11
2Δ0hS−γ1,
3.22
since
p1∞2
4h2−6h3−31−h,
f−γ
11
1
−1
ft−γ1dt
f1−f−1−2γ1
2S−γ1.
3.23
Hence,3.14holds. In the similar manner, we can prove3.15.
Remark 3.3. It may be noted thatΔ0hhas its minimum value 0.396 ath0.259.In a similar
way, it may be observed that 1/2Δ1hattains its minimum value 0.1698 ath0.296.
Theorem 3.4. Letf:a, b→Rbe a function, such thatf∈L1a, b.If there exists a real number
γ1,such thatγ1≤ft,t∈a, b,then
|Jf;a, b;h| ≤ 1
2Δ0hS−γ1b−a
and if there exists a real numberΓ1,such thatft≤Γ1, t∈a, b,then
|Jf;a, b;h| ≤1
2Δ0hΓ1−Sb−a
2, 3.25
whereJf;a, b;his defined by3.11andS fa−fb/b−aandh∈0,1/2.If there exist real numbersγ1,Γ1, such thatγ1≤ft≤Γ1,t∈a, b,then
|Jf;a, b;h| ≤ 1
8Δ1hΓ1−γ1b−a
2, 3.26
Δ0handΔ1hare as defined in3.17.
Theorem 3.5. Letf : −1,1→Rbe an absolutely continuous function, such thatf ∈L2−1,1.
Then,
|Jf;−1,1;h| ≤
Δ2hσf;−1,1, 3.27
whereσf;−1,1is defined by3.7and
Δ2h −56h3154h2−146h
146
3 −281−h
24h2−6h3 3.28
forh∈0,1/2.
Proof. Letp1be the same as defined above. We have
p1, f2−Jf;−1,1;h, 3.29
sincep1, e20, ifa, b −1,1.Moreover,f, g 1/2f, g2and
p1, fTf, p1. 3.30
From3.6, it follows that
Tf, p1≤
Tp1, p1
Tf, f
1
2p12σf;−1,1
1
2
Δ2hσf;−1,1,
3.31
as
p122−56h3154h2−146h
146
3 −281−h
24h2−6h3. 3.32
Remark 3.6.
Δ2hattains its minimum value 0.2799 ath0.2957.
Theorem 3.7. Letf:a, b→Rbe an absolutely continuous function, such thatf∈L2a, b.Then,
|Jf;a, b;h| ≤ 1
2√2
Δ2hσf;a, bb−a3/2, 3.33
whereσf;a, bis defined by3.7andΔ2his as defined above.
4. Applications in numerical integration
Let π {x0 a < x1 < · · · < xn b} be a subdivision of the intervala, b,such that hixi1−xih b−a/n.From3.11, we have
Jf Jf;xi, xi1;δ
xi1
xi
ftdt−h
2δfxi 1−δfx1i 1−δfx2i δfxi1,
4.1
where
x1i h
2x∗
xixi1
2 , x2−
h
2x∗
xixi1
2 ,
x∗−44δ2√3−6δ4δ2, δ∈ 0,1
2
.
4.2
Summing-up the above relation from 1 ton−1,we get
n−1
i0
Jf;xi, xi1;δ
b
a
ftdt−h
2
n−1
i0
δfxi 1−δfx1i 1−δfx2i δfxi1.
4.3
Let us denote
Sf;a, b;δ n−1
i0
Jf;xi, xi1;δ. 4.4
Theorem 4.1. Let the assumptions ofTheorem 2.9hold, then
|Sf;a, b;δ| ≤ 1
4n2Δδf
∞b−a3, 4.5
Theorem 4.2. Let the assumptions ofTheorem 3.4hold, then it follows that
|Sf;a, b;δ| ≤1
8Δ1δ
Γ1−γ1
n b−a
2,
|Sf;a, b;δ| ≤ 1
2nΔ0δS−γ1b−a
2,
4.6
and if there exists a real numberΓ1,such thatft≤Γ1, t∈a, b,then
|Sf;a, b;δ| ≤ 1
2nΔ0δΓ1−Sb−a
2, 4.7
whereSf;a, b;δis defined by4.4,Δ0δ,Δ1δare defined by3.17andS fa−fb/b−
a.However,πis the uniform subdivision ofa, b.
Theorem 4.3. Let the assumptions ofTheorem 3.7hold, then it follows that
|Sf;a, b;δ| ≤ b−a
3/2
2√2n
Δ2δσf, 4.8
whereSf;a, b;δis defined by4.4,σfis defined by3.7, andΔ2δis as defined by3.28.
However,πis the uniform subdivision ofa, b.
Proof. ApplyingTheorem 3.7on the intervalxi, xi1,
xi1
xi
ftdt−h
2δfxi 1−δfx1i 1−δfx2i δfxi1
≤ 1
2√2
Δ2δh3/2
xi1
xi
ft2dt− 1
hfxi1−fxi
2
1/2
.
4.9
Summing overifrom 0 ton−1,
|Sf;a, b;δ| ≤ 1
2√2
Δ2δh3/2
n−1
i0
xi1
xi
ft2dt−1
hfxi1−fxi
21/2. 4.10
Using Cauchy-Schwartz inequality and the relationh b−a/n,we obtain the required inequality:
|Sf;a, b;δ| ≤ 1
2√2
Δ2δb−a 3/2
n3/2 n
1/2 f2 2−
n b−a
n−1
i0
fxi1−fxi2 1/2
≤ 1
2√2
Δ2δ
b−a3/2 n f
2 2−
fb−fa2 b−a
1/2
.
Acknowledgment
The authors are thankful to the referees for giving valuable comments and suggestions for the preparation of final version of this paper.
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