Journal of Inequalities in Pure and
Applied Mathematics
SHARP ERROR BOUNDS FOR SOME
QUADRATURE FORMULAE AND APPLICATIONS
volume 7, issue 1, article 8,
2006.
NENAD UJEVIĆ
Department of Mathematics
University of Split
Teslina 12/III, 21000 Split
CROATIA.
Received 26 May, 2004;
accepted 10 November, 2005.
Communicated by: C.E.M. Pearce
EMail: ujevic@pmfst.hr
Abstract
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2000
Victoria University
ISSN (electronic): 1443-5756
103-04
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Abstract
In the article ”N. Ujević, A generalization of the pre-Grüss inequality and applications to some quadrature formulae, J. Inequal. Pure Appl. Math., 3(2), Art.
13, 2002” error bounds for some quadrature formulae are established. Here
we prove that all inequalities (error bounds) obtained in this article are sharp.
We also establish a new sharp averaged midpoint-trapezoid inequality and give
applications in numerical integration.
2000 Mathematics Subject Classification: Primary 26D10, Secondary 41A55.
Key words: Sharp error bounds, Quadrature formulae, Numerical integration.
Contents
1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2
Midpoint Inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
3
Trapezoid Inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
4
Averaged Midpoint-Trapezoid Inequality . . . . . . . . . . . . . . . . . 8
5
A Sharp Error Inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
6
Applications in Numerical Integration . . . . . . . . . . . . . . . . . . . 17
References
Sharp Error Bounds for Some
Quadrature Formulae and
Applications
Nenad Ujević
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1.
Introduction
In recent years a number of authors have considered error inequalities for some
known and some new quadrature rules. For example, this topic is considered in
[1] – [6] and [11] – [14].
In this paper we consider the midpoint, trapezoid and averaged midpointtrapezoid quadrature rules. These rules are also considered in [12], where some
new improved versions of the error inequalities for the mentioned rules are derived.
Here we first prove that all inequalities obtained in [12] are sharp. Second,
we specially consider the averaged midpoint-trapezoid quadrature rule. In [6]
it is shown that the last mentioned rule has a better estimation of error than the
well-known Simpson’s rule and in [13] it is shown that this rule is an optimal
quadrature rule. We give a new sharp error bound for this rule. Finally, we give
applications in numerical integration.
Sharp Error Bounds for Some
Quadrature Formulae and
Applications
Nenad Ujević
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2.
Midpoint Inequality
Let I ⊂ R be a closed interval and a, b ∈ Int I, a < b. Let f : I → R be an
absolutely continuous function whose derivative f 0 ∈ L2 (a, b). We define the
mapping
(
, t ∈ a, a+b
t − 2a+b
3
2
Φ(t) =
t − a+2b
, t ∈ a+b
,b
3
2
such that Φ0 (t) = Φ(t)/ kΦk2 , where
Z b
(b − a)3
2
2
.
kΦk2 =
(Φ(t)) dt =
36
a
Sharp Error Bounds for Some
Quadrature Formulae and
Applications
Nenad Ujević
We have
Z
b
Φ0 (t)f 0 (t)dt
a
Z b
2
a+b
3
=√
f (a) + f
+ f (b) −
f (t)dt .
2
b−a a
b−a
Q(f ; a, b) =
In [12] we can find the following midpoint inequality
Z b
(b − a)3/2
a
+
b
f
≤
√
(2.1)
(b
−
a)
−
f
(t)dt
C1 ,
2
2 3
a
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where
(
(2.2)
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C1 =
2
kf 0 k2
[f (b) − f (a)]2
− [Q(f ; a, b)]2
−
b−a
) 12
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Proposition 2.1. The inequality (2.1) is sharp in the sense that the constant
cannot be replaced by a smaller one.
1
√
2 3
Proof. We first define the mapping
( 1
t2 ,
t ∈ 0, 12
2
(2.3)
f (t) =
1 2
t − t + 12 , t ∈ 12 , 1
2
and note that f is a Lipschitzian function.
On the other hand, each Lipschitzian function is an absolutely continuous
function [10, p. 227].
Let us now assume that the inequality (2.1) holds with a constant C > 0, i.e.
Sharp Error Bounds for Some
Quadrature Formulae and
Applications
Z b
a
+
b
f
≤ C(b − a)3/2 C1 ,
(b
−
a)
−
f
(t)dt
2
Nenad Ujević
where C1 is defined by (2.2). Choosing a = 0, b = 1 and f defined by (2.3), we
get
Z 1
1
1
1
=
f (t)dt = , f
24
2
8
0
such that the left-hand side of (2.4) becomes
1
(2.5)
L.H.S.(2.4) = .
12
1
√
We also find that C1 = 2 3 such that the right-hand side of (2.4) becomes
C
(2.6)
R.H.S.(2.4) = √ .
2 3
From (2.4) – (2.6) we get C ≥ 2√1 3 , proving that C = 2√1 3 is the best possible
in (2.1).
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(2.4)
a
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3.
Trapezoid Inequality
Let I ⊂ R be a closed interval and a, b ∈ Int I, a < b. Let f : I → R be an
absolutely continuous function whose derivative f 0 ∈ L2 (a, b). We define the
mapping
(
, t ∈ a, a+b
t − 5a+b
6
2
χ(t) =
t − a+5b
, t ∈ a+b
,b
6
2
such that χ0 (t) = χ(t)/ kχk2 , where
Z b
(b − a)3
2
kχk2 =
(χ(t))2 dt =
.
36
a
Sharp Error Bounds for Some
Quadrature Formulae and
Applications
Nenad Ujević
We have
Z
b
χ0 (t)f 0 (t)dt
a
Z b
6
1
a+b
f (a) + 4f
+ f (b) −
f (t)dt .
=√
2
b−a a
b−a
P (f ; a, b) =
In [12] we can find the following trapezoid inequality:
Z b
f (a) + f (b)
(b − a)3/2
≤
√
(3.1)
(b
−
a)
−
f
(t)dt
C2 ,
2
2 3
a
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where
(
(3.2)
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C2 =
2
kf 0 k2
[f (b) − f (a)]2
−
− [P (f ; a, b)]2
b−a
) 12
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Proposition 3.1. The inequality (3.1) is sharp in the sense that the constant
cannot be replaced by a smaller one.
1
√
2 3
Proof. We define the mapping
1
1
f (t) = t2 − t.
2
2
It is obvious that f is an absolutely continuous function. Let us now assume
that the inequality (3.1) holds with a constant C > 0, i.e.
Z b
f (a) + f (b)
(3.4)
(b − a) −
f (t)dt ≤ C(b − a)3/2 C2 ,
2
a
(3.3)
where C2 is defined by (3.2).
Choosing a = 0, b = 1 and f defined by (3.3), we get
Z 1
1
f (t)dt =
and f (0) = f (1) = 0.
12
0
L.H.S.(3.4) =
1
.
12
1
√
,
2 3
proving that
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C
R.H.S.(3.4) = √ .
2 3
From (3.4) – (3.6) we get C ≥
(3.1).
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The right-hand side of (3.4) becomes
(3.6)
Nenad Ujević
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Thus, the left-hand side of (3.4) becomes
(3.5)
Sharp Error Bounds for Some
Quadrature Formulae and
Applications
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1
√
2 3
is the best possible in
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4.
Averaged Midpoint-Trapezoid Inequality
Let I ⊂ R be a closed interval and a, b ∈ Int I, a < b. Let f : I → R be
an absolutely continuous function whose derivative f 0 ∈ L2 (a, b). We now
consider a simple quadrature rule of the form
Z b
f (a) + 2f a+b
+ f (b)
2
(4.1)
(b − a) −
f (t)dt
4
a
Z b
a+b
1
f (a) + f (b)
=
f
+
(b − a) −
f (t)dt = R(f ).
2
2
2
a
It is not difficult to see that (4.1) is a convex combination of the midpoint
quadrature rule and the trapezoid quadrature rule. In [6] it is shown that (4.1)
has a better estimation of error than the well-known Simpson’s quadrature rule
(when we estimate the error in terms of the first derivative f 0 of integrand f ). In
[12] the following inequality is proved
Z b
f (a) + 2f a+b + f (b)
(b − a)3/2
2
√
(4.2)
(b − a) −
f (t)dt ≤
C3 ,
4
4 3
a
where
Sharp Error Bounds for Some
Quadrature Formulae and
Applications
Nenad Ujević
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"
2
(4.3) C3 = kf 0 k2 −
[f (b) − f (a)]2
b−a
2 # 12
a+b
1
f (a) − 2f
+ f (b)
−
.
b−a
2
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Proposition 4.1. The inequality (4.2) is sharp in the sense that the constant
cannot be replaced by a smaller one.
1
√
4 3
Proof. We first define the mapping
( 1 2 1
t − 4 t,
t ∈ 0, 12
2
(4.4)
f (t) =
1 2
3
1
1
t
−
t
+
,
t
∈
,
1
2
4
4
2
and note that f is a Lipschitzian function.
Let us now assume that the inequality (4.2) holds with a constant C > 0, i.e.
Z b
f (a) + 2f a+b + f (b)
2
(4.5)
(b − a) −
f (t)dt ≤ C(b − a)3/2 C3 ,
4
a
where C3 is defined by (4.3). Choosing a = 0, b = 1 and f defined by (4.4), we
get
Z 1
1
1
f (t)dt = − , f (0) = f (1) = f
=0
48
2
0
such that the left-hand side of (4.5) becomes
1
(4.6)
L.H.S.(4.5) = .
48
1
We also find that C3 = 4√3 such that the right-hand side of (4.5) becomes
(4.7)
C
R.H.S.(4.5) = √ .
4 3
From (4.5) – (4.7) we get C ≥
in (4.2).
4
1
√
, proving that C =
3
Sharp Error Bounds for Some
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Nenad Ujević
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1
√
4 3
is the best possible
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5.
A Sharp Error Inequality
In [12] we can find the following inequality
S(f, g)2 ≤ S(f, f )S(g, g),
(5.1)
where
Z
(5.2)
S(f, g) =
a
b
Z b
Z b
1
f (t)g(t)dt −
f (t)dt
g(t)dt
b−a a
a
Z b
Z b
1
f (t)Ψ(t)dt
−
g(t)Ψ(t)dt
kΨk2 a
a
Sharp Error Bounds for Some
Quadrature Formulae and
Applications
Nenad Ujević
and Ψ satisfies
b
Z
Ψ(t)dt = 0,
(5.3)
Contents
a
while
2
Z
kΨk =
b
2
Ψ (t)dt.
a
In [14] we can find a variant of the following lemma.
Lemma 5.1. Let f ∈ C 1 [a, c], g ∈ C 1 [c, b] be such that f (c) = g(c). Then
(
f (t), t ∈ [a, c]
h(t) =
g(t), t ∈ [c, b]
is an absolutely continuous function.
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Theorem 5.2. Let f : [0, 1] → R be an absolutely continuous function whose
derivative f 0 ∈ L2 (0, 1). Then
(5.4)
Z
0
1
1
1
f (0) + 2f
+ f (1) f (t)dt −
4
2
s
2
2
1
1
1
2
.
kf 0 k − 2 f
− f (0) − 2 f (1) − f
≤ √
2
2
4 3
The inequality (5.4) is sharp in the sense that the constant
placed by a smaller one.
(
p(t) =
t − 14 , t ∈ 0, 12
t − 34 , t ∈ 12 , 1
and
(
(5.6)
Ψ(t) =
t ∈ 0, 12
.
t − 1, t ∈ 12 , 1
t,
It is not difficult to verify that
Z 1
Z
(5.7)
p(t)dt =
0
4 3
cannot be re-
Sharp Error Bounds for Some
Quadrature Formulae and
Applications
Nenad Ujević
Proof. We define the functions
(5.5)
1
√
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Ψ(t)dt = 0.
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We also have
Z
2
1
kpk =
(5.8)
p2 (t)dt =
1
,
48
Ψ2 (t)dt =
1
,
12
0
Z
2
kΨk =
(5.9)
1
0
Z
1
p(t)Ψ(t)dt =
(5.10)
0
1
.
48
Sharp Error Bounds for Some
Quadrature Formulae and
Applications
From (5.1), (5.2) and (5.3) we get
Z 1
2
Z 1
Z 1
1
0
0
(5.11)
p(t)f (t)dt −
p(t)Ψ(t)dt
f (t)Ψ(t)dt
kΨk2 0
0
0
"
Z 1
2 #
1
p(t)Ψ(t)dt
≤ kpk2 −
kΨk2
0
"
Z 1
2
Z 1
2 #
1
2
× kf 0 k −
f 0 (t)dt −
f 0 (t)Ψ(t)dt
.
2
kΨk
0
0
Integrating by parts, we obtain
Z 1
Z
0
(5.12)
p(t)f (t)dt =
0
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1
2
1
3
0
f (t)dt +
f 0 (t)dt
t−
1
4
0
2
Z 1
1
1
=
f (0) + 2f
+ f (1) −
f (t)dt
4
2
0
1
t−
4
Z
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and
Z
1
Z
0
1
2
0
1
(t − 1)f 0 (t)dt
tf (t)dt +
f (t)Ψ(t)dt =
(5.13)
Z
0
1
2
0
Z 1
1
=f
−
f (t)dt.
2
0
We introduce the notations
Z
(5.14)
i=
1
f (t)dt,
0
(5.15)
1
1
f (0) + 2f
+ f (1) .
q=
4
2
From (5.11) – (5.15) and (5.8) – (5.10) it follows that
2
1
1
(5.16)
(q − i) −
f
−i
4
2
"
2 #
1
1
2
≤
kf 0 k − [f (1) − f (0)]2 − 12 f
−i
64
2
or
Sharp Error Bounds for Some
Quadrature Formulae and
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Nenad Ujević
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4
1
(5.17) i2 − 2qi + q 2 +
[f (1) − f (0)]2
3
48
2
1
1
0 2
− kf k + 16 f
− 32f
q ≤ 0.
2
2
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If we now introduce the notations
β = −2q,
(5.18)
2
1
4 2
1
1
2
0 2
(5.19) γ = q +
[f (1) − f (0)] − kf k + 16 f
− 32f
q
3
48
2
2
then we have
Sharp Error Bounds for Some
Quadrature Formulae and
Applications
i2 + βi + γ ≤ 0.
(5.20)
Thus, i ∈ [i1 , i2 ], where
Nenad Ujević
p
−β − β 2 − 4γ
i1 =
,
2
i2 =
−β +
p
β 2 − 4γ
.
2
In other words,
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−
β
−
2
p
β 2 − 4γ
β
≤i≤− +
2
2
p
β 2 − 4γ
2
or
p 2
β
i + ≤ β − 4γ .
2
2
(5.21)
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We have
"
2
2 #
1
1
1
2
0 2
(5.22) β − 4γ =
kf k − 2 f
− f (0) − 2 f (1) − f
.
12
2
2
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From (5.21) and (5.22) we easily find that (5.4) holds.
We have to prove that (5.4) is sharp. For that purpose, we define the function
( 1 2 1
1
t − 4 t + 32
, t ∈ 0, 12
2
(5.23)
f (t) =
1 .
1 2
3
9
t
−
t
+
,
t
∈
,1
2
4
32
2
From Lemma 5.1 we see that the above function is absolutely continuous. If we
substitute the above function in the left-hand side of (5.4) then we get
1
.
48
If we substitute the above function in the right-hand side of (5.4) then we get
(5.24)
L.H.S.(5.4) =
1
(5.25)
R.H.S.(5.4) = .
48
From (5.24) and (5.25) we conclude that (5.4) is sharp.
Sharp Error Bounds for Some
Quadrature Formulae and
Applications
Nenad Ujević
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Theorem 5.3. Let f : [a, b] → R be an absolutely continuous function whose
derivative f 0 ∈ L2 (a, b). Then
Z b
a
+
b
b
−
a
(5.26) f (t)dt −
f (a) + 2f
+ f (b) 4
2
a
2
(b − a)3/2
2
a+b
0 2
√
≤
kf k −
f
− f (a)
b−a
2
4 3
2 ! 12
2
a+b
−
f (b) − f
.
b−a
2
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√
The above inequality is sharp in the sense that the constant 1/(4 3) cannot be
replaced by a smaller one.
Remark 1. We have better estimates than (5.26). For example, we have the
inequality
Z b
b − a
a
+
b
(5.27) f (a) + 2f
+ f (b) −
f (t)dt
4
2
a
1
≤ kf 0 k∞ (b − a)2 .
8
However, note that the estimate (5.27) can be applied only if f 0 is bounded. On
the other hand, the estimate (5.26) can be applied for absolutely continuous
functions if f 0 ∈ L2 (a, b).
There are many examples where we cannot apply the estimate (5.27) but we
can apply (5.26).
R1√
3
Example 5.1. Let us consider the integral 0 sin t2 dt. We have
f (t) =
√
3
sin t2
and
2t cos t2
f 0 (t) = √
3
3 sin2 t2
such that f 0 (t) → ∞, t → 0 and we cannot apply the estimate (5.27). On the
other hand, we have
Z 1
Z
4
t2 cos t2 1 dt
16
2
0
√
[f (t)] dt ≤ max
≤
,
3
9 t∈[0,1] sin t2 0
9
sin t2
0
i.e. kf 0 k2 ≤
4
3
Sharp Error Bounds for Some
Quadrature Formulae and
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Nenad Ujević
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and we can apply the estimate (5.26).
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6.
Applications in Numerical Integration
Let π = {x0 = a < x1 < · · · < xn = b} be a given subdivision of the interval
[a, b] such that hi = xi+1 − xi = h = (b − a)/n. We define
(6.1) σn (f ) =
n−1 X
b−a
i=0
n
2
kf 0 k2 − (f (xi+1 ) − f (xi ))2
−
f (xi ) − 2f
xi + xi+1
2
2 # 12
,
+ f (xi+1 )
Sharp Error Bounds for Some
Quadrature Formulae and
Applications
Nenad Ujević
(6.2) ηn (f ) =
"
n−1
X
b−a
i=0
n
2
kf 0 k2
2
xi + xi+1
−2 f
− f (xi )
2
− 2 f (xi+1 ) − f
xi + xi+1
2
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2 # 12
and
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12
1
0 2
2
ωn (f ) = (b − a) kf k2 − (f (b) − f (a))
.
n
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Theorem 6.1. Let π be a given subdivision of the interval [a, b] and let the
Page 17 of 22
(6.3)
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assumptions of Theorem 5.2 hold. Then
Z
n−1 b
xi + xi+1
hX
(6.4) f (t)dt −
f (xi ) + 2f
+ f (xi+1 ) a
4 i=0
2
b−a
b−a
≤ √ σn (f ) ≤ √ ωn (f ),
4 3n
4 3n
where σn (f ) and ωn (f ) are defined by (6.1) and (6.3), respectively.
Proof. We have
Z xi+1
xi + xi+1
h
(6.5)
f (xi ) + 2f
+ f (xi+1 ) −
f (t)dt
4
2
xi
Z xi+1
=
Ki (t)f 0 (t)dt,
xi
where
Ki (t) =

 t−
3xi +xi+1
,
4
t ∈ xi , xi +x2 i+1
 t−
xi +3xi+1
,
4
t∈
xi +xi+1
,
x
i+1
2
From Proposition 4.1 we obtain
Z xi+1
h
f (xi ) + 2f xi + xi+1 + f (xi+1 ) −
f
(t)dt
4
2
xi
h3/2 h 0 2 1
≤ √ kf k2 − (f (xi+1 ) − f (xi ))2
h
4 3
Sharp Error Bounds for Some
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1
−
h
f (xi ) − 2f
xi + xi+1
2
2 # 12
+ f (xi+1 )
.
If we sum (6.5) over i from 0 to n − 1 and apply the above inequality then
we get
Z
n−1 b
X
x
+
x
h
i
i+1
f (t)dt −
f (xi ) + 2f
+ f (xi+1 ) a
4 i=0
2
" n−1
h3/2 X 0 2 1
kf k2 − (f (xi+1 ) − f (xi ))2
≤ √
h
4 3 i=0
2 # 12
xi + xi+1
1
.
−
f (xi ) − 2f
+ f (xi+1 )
h
2
From the above relation and the fact h = (b−a)/n we see that the first inequality
in (6.4) holds.
Using the Cauchy inequality we have
21
n−1 X
1
2
0 2
(6.6)
kf k2 − (f (xi+1 ) − f (xi ))
h
i=0
# 12
"
n−1
X
1
2
(f (xi+1 ) − f (xi ))2
≤ n kf 0 k2 −
b − a i=0
12
1 1
2
0 2
≤ n kf k2 −
(f (b) − f (a))
.
b−an
Sharp Error Bounds for Some
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Since
2
xi + xi+1
f (xi ) − 2f
+ f (xi+1 )
2
1
2
≤ kf 0 k2 − (f (xi+1 ) − f (xi ))2 ,
h
we easily conclude that the second inequality in (6.4) holds, too.
2
kf 0 k2
1
1
− (f (xi+1 ) − f (xi ))2 −
h
h
Remark 2. The second inequality in (6.4) is coarser than the first inequality. It
may be used to predict the number of steps needed in the compound rule for a
given accuracy of the approximation. Of course, we shall use the first inequality
in (6.4) to obtain the error bound. Note also that in this last case
R b we use the
same values f (xi ) to calculate the approximation of the integral a f (t)dt and
to obtain the error bound and recall that function evaluations are generally
considered the computationally most expensive part of quadrature algorithms.
Theorem 6.2. Under the assumptions of Theorem 6.1 we have
Z
n−1 b
hX
xi + xi+1
f (t)dt −
f (xi ) + 2f
+ f (xi+1 ) a
4
2
i=0
b−a
b−a
≤ √ ηn (f ) ≤ √ ωn (f ),
4 3n
4 3n
where ηn (f ) is defined by (6.2).
Proof. The proof of this theorem is similar to the proof of Theorem 6.1. Here
we use Theorem 5.3.
Sharp Error Bounds for Some
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References
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[3] P. CERONE AND S.S. DRAGOMIR, Midpoint-type rules from an inequalities point of view, Handbook of Analytic-Computational Methods in Applied Mathematics, Ed.: G. Anastassiou, CRC Press, New York, (2000),
135–200.
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Sharp Error Bounds for Some
Quadrature Formulae and
Applications
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[6] S.S. DRAGOMIR, P. CERONE AND J. ROUMELIOTIS, A new generalization of Ostrowski’s integral inequality for mappings whose derivatives are bounded and applications in numerical integration and for special
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Sharp Error Bounds for Some
Quadrature Formulae and
Applications
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[14] N. UJEVIĆ, Two sharp Ostrowski-like inequalities and applications, Meth.
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