ON PERTURBED TRAPEZOIDAL AND MIDPOINT RULES
P. CERONE
Abstract. Explicit bounds are obtained for the perturbed, or corrected, trapezoidal and midpoint rules in terms of the Lebesque norms of the second derivative of the function. It is demonstrated that the bounds obtained are the same
for both rules although the perturbation or the correction term is different.
1. Introduction
Let f : [a, b] → R and define the functionals
Z b
(1.1)
I (f ) :=
f (t) dt
a
I (T ) (f ) :=
(1.2)
b−a
[f (a) + f (b)]
2
and
I (M ) (f ) := (b − a) f
(1.3)
a+b
2
.
Here I (T ) (f ) and I (M ) (f ) are the well known trapezoidal and midpoint rules used
to approximate the functional I (f ).
Atkinson [1] defined the corrected or perturbed trapezoidal and midpoint rules
by
(1.4)
P I (T ) (f ) := I (T ) (f ) −
c2 0
[f (b) − f 0 (a)]
3
P I (M ) (f ) := I (M ) (f ) +
c2 0
[f (b) − f 0 (a)]
6
and
(1.5)
respectively, where c = b−a
2 .
Atkinson [1] uses an asymptotic error estimate which does not readily produce
estimates of the bounds in using (1.4) and (1.5) to approximate (1.1) by (1.4)
or (1.5).
In a recent article Barnett and Dragomir [2] obtained explicit bounds
for I (f ) − P I (T ) (f ) in terms of the Lebesque norms of f 00 (t) − [f 0 ; a, b] where
0
0
(a)
[f 0 ; a, b] := f (b)−f
is the divided difference. If the Lebesque norms are defined
b−a
in the usual way such that by h ∈ Lp [a, b] we mean
! p1
Z
b
(1.6)
khkp :=
p
|h (t)| dt
, 1≤p<∞
a
Date: April 30, 2001.
1991 Mathematics Subject Classification. Primary 05C38, 15A15; Secondary 05A15, 15A18.
Key words and phrases. Keyword one, keyword two, keyword three.
1
2
P. CERONE
and
khk∞ := ess sup |h (t)| .
(1.7)
t∈[a,b]
Barnett and Dragomir [2] obtained the following theorem.
Theorem 1. Let f : [a, b] → R be differentiable and f 0 absolutely continuous on
[a, b] then
(1.8)

3

 (b − a) kf 00 − [f 0 ; a, b]k

if f 00 ∈ L∞ [a, b] ;

∞

24






 (b − a)2+ q1
(T )
00
− [f 0 ; a, b]kp , if f 00 ∈ Lp [a, b] ,
I (f ) − P I (f ) ≤
1 kf

q
8
(2q
+
1)




p > 1, p1 + 1q = 1;



2



 (b − a) kf 00 − [f 0 ; a, b]k1 ,
8
where [h; a, b] :=
h(b)−h(a)
b−a
is the divided difference.
Further, the Grüss integral inequality for a function h : [a, b] → R with −∞ <
m ≤ h (x) ≤ m < ∞ for almost every x ∈ [a, b] then (see for example [10, p. 296])
2
1
M −m
2
(1.9)
0≤
khk2 − M (h) ≤
,
b−a
2
Rb
1
where M (h) = b−a
h (t) dt, is the integral mean.
a
Barnett and Dragomir [2] also obtained the following results.
Let f : [a, b] → R, then
12
(b − a)3 1
2
(T )
00 2
0
√
(1.10) I (f ) − P I (f ) ≤
kf k2 − [f ; a, b]
, f 00 ∈ L2 [a, b]
b−a
8 5
and
(1.11)
(b − a)3
√ (Γ − γ) ,
I (f ) − P I (T ) (f ) ≤
16 5
γ ≤ f 00 (t) ≤ Γ a.e. t ∈ [a, b] .
Result (1.10) is obtained from the second inequality in (1.8) and (1.11) from (1.9)
and (1.10).
It is the intention of the current article to demonstrate that bounds for I (f ) − P I (T ) (f )
may be obtained involving the traditional Lebesque norms of kf 00 kp , p ≥ 1 where
k·kp are as defined by (1.6) and (1.7) rather than kf 00 − [f 0 ; a, b]kp as obtained in
(1.8). Further, bounds will be obtained for I (f ) − P I (M ) (f ) . These will be shown
to be the same as those obtained for the perturbed trapezoidal rule although the
correction term is different.
2. Identities and Inequalities for Trapezoidal Like Rules
Let the trapezoidal functional T (f ; a, b) be defined by
Z b
1
b−a
(T )
(2.1)
T (f ; a, b) := I (f ) − I (f ) =
f (t) dt −
[f (a) + f (b)]
b−a a
2
TRAPEZOIDAL AND MIDPOINT RULES
3
then it is well known that the identity
Z
1 b
(2.2)
T (f ; a, b) = −
(t − a) (b − t) f 00 (t) dt
2 a
holds. The following theorem was obtained in [9] using identity (2.2).
Theorem 2. Let f : [a, b] → R be a twice differentiable function on (a, b). Then
we have the estimate
(2.3)
|T (f ; a, b)|

kf 00 k∞
3


(b − a)
if f 00 ∈ L∞ [a, b] ;

12




1
1
2+ p
1
00
p
≤
, if f 00 ∈ Lp [a, b] ,
2 kf kq [B (q + 1, q + 1)] (b − a)

1
1



p + q = 1, p > 1


00
 kf k1
2
(b − a)
if f 00 ∈ L1 [a, b] ,
8
where B is the Beta function, that is,
Z 1
s−1
B (r, s) :=
tr−1 (1 − t)
dt, r, s > 0.
0
Let
(2.4)
P T (f ; a, b) := I (f ) − P I (T ) (f ) = T (f ; a, b) +
where c =
b−a
2 ,
c2 0
[f (b) − f 0 (a)] ,
3
then the following lemma holds.
Lemma 1. Let f : [a, b] → R be such that f 0 is absolutely continuous on [a, b] then
Z
1 b
κ (t) f 00 (t) dt
(2.5)
P T (f ; a, b) =
2 a
is valid with
(2.6)
κ (t) =
t−
a+b
2
2
−
c2
b−a
, c=
3
2
Proof. From (2.4) and (2.6) we have
Z
1 b
c2 0
P T (f ; a, b) = −
(t − a) (b − t) f 00 (t) dt +
[f (b) − f 0 (a)]
2 a
3
Z b 2
c
1
=
− (t − a) (b − t) f 00 (t) dt
3
2
a
Z b
1
c2 00
=
t2 − (a + b) t + ab + 2
f (t) dt
2 a
3
#
2
Z "
1 b
a+b
c2
=
t−
−
f 00 (t) dt
2 a
2
3
and so (2.5) holds with κ (t) as given by (2.6).
4
P. CERONE
Theorem 3. Let f : [a, b] → R be such that f 0 is absolutely continuous on [a, b]
then
(2.7)|P T (f ; a, b)|








≤







3
4c
√
9 3
kf 00 k∞ ,
c2
6
n
c2
6
kf 00 k1
√c B
3
if f 00 ∈ L∞ [a, b] ;
1
2, q
q o q1
R √3 2
+1 +2 1
u − 1 du kf 00 kp , if f 00 ∈ Lp [a, b] ,
1
1
p + q = 1, p > 1
if f 00 ∈ L1 [a, b] ,
where B is the beta function and c =
b−a
2 .
Proof. From identity (2.5) and using (2.6) we have
Z
1 b
(2.8)
|P T (f ; a, b)| ≤
|κ (t)| |f 00 (t)| dt
2 a
!
Z b
kf 00 kp
1
q
|κ (t)| dt q , p > 1.
≤
2
a
Now we need to examine the behaviour of κ (t) in order to proceed further. We
√c
notice from (2.6) that κ (a) = κ (b) = 23 c2 and κ (t) = 0 where t = a+b
2 ± 3.
Further,

 < 0, t < a+b
2 ;



a+b 
κ0 (t) = 2 t −
= 0, t = a+b
2 ;

2




> 0, t > a+b
2 .
a+b
a+b
Also, κ (t) is a symmetric function about 2 since κ 2 + x = κ a+b
2 − x so
that from (2.8)
c
Z a+b
Z b
√
2 + 3
q
q
(2.9)
kκkq =
[−κ (t)] dt +
κq (t) dt
a+b
2
:
a+b
c
√
2 + 3
= 2 [I1 (q) + I2 (q)] .
Further, from (2.6)
a+b
c
√
2 + 3
Z
I1 (q) =
Let
√c u
3
(2.10)
R1
since 0
Also,
=t−
a+b
2 ,
"
a+b
2
2 # q
c2
a+b
− t−
dt
3
2
then
Z 1 2 q
c
c
c2q+1 1
1
2 q
I1 (q) = √
1−u
du = q+ 1 · B
,q + 1
3
2
3 0
3 2 2
q
1 − u2 du = 12 B 12 , q + 1 .
Z
I2 (q)
b
a+b
c
√
2 + 3
"
a+b
t−
2
2
c2
−
3
#q
dt
TRAPEZOIDAL AND MIDPOINT RULES
and substituting
√c v
3
=t−
a+b
2
gives
c
I2 (q) = √
3
(2.11)
5
c2
3
q Z
1
2
q
v − 1 dv.
0
Combining (2.10) and (2.11) into (2.9) gives from (2.8) the second inequality in
(2.7).
The first inequality is obtained by taking q = 1 in the second inequality of (2.7)
as may be noticed from (2.8).
Thus
#
" Z √3
Z
1 b
c3
1
2
√ B
,2 + 2
u − 1 du
|κ (t)| dt =
2
2 a
6 3
1
c3 4 4
4c3
√
=
+
= 5 .
6 3 3 3
32
Now, for the final inequality, from (2.8) we obtain
|P T (f ; a, b)| ≤
1
sup |κ (t)| kf 00 k1
2 t∈[a,b]
and so from the behaviour of κ (t) discussed earlier
2 2 1 2
2
sup |κ (t)| = max
c , c = c2 .
3
3
3
t∈[a,b]
The following corollary involving the Euclidean norm is of particular importance.
Corollary 1. Let f ; [a, b] → R be such that f 00 ∈ L2 [a, b]. Then we have the
inequality
√ 5
5
2c 2
(b − a) 2
00
√
(2.12)
|P T (f ; a, b)| ≤ √ kf k2 =
kf 00 k2 ,
3 5
6 5
where c =
b−a
2 .
Proof. Taking p = q = 2 in (2.7) gives
(
#) 12
" Z √3
c2
c
1
2
√ B
|P T (f ; a, b)| ≤
u2 − 1 du
kf 00 k2
,3 + 2
6
2
3
1
which, upon using the facts that
Γ 21 Γ (3)
16
1
=
B
,3 =
7
15
2
Γ 2
and
√
Z
1
3
2
u − 1 du = 4
2
!
√
3 3−2
15
gives the stated result (2.12) after some simplification.
6
P. CERONE
3. Identities and Inequalities for Midpoint Like Rules
Let, from (1.1) and (1.3) the midpoint functional, M (f ; a, b) be defined by
Z b
1
a+b
(M )
(3.1)
M (f ; a, b) := I (f ) − I
(f ) =
f (t) dt − (b − a) f
b−a a
2
then the identity
Z
(3.2)
M (f ; a, b) =
b
φ (t) f 00 (t) dt
a
is well known, where
(3.3)
φ (t) =

2
(t − a)


,



2

2


(b − t)


,
2
a+b
t ∈ a,
,
2
t∈
a+b
,b .
2
The following theorem concerning the classical midpoint functional (3.1) with
bounds involving the Lp [a, b] norms of the second derivative is known (see [8] and
[6]).
Theorem 4. Let f : [a, b] → R be such that f 0 is absolutely continuous on [a, b].
Then

3
 (b−a) kf 00 k∞
if f 00 ∈ L∞ [a, b] ;

24






2+ 1
(b−a) q
00
if f 00 ∈ Lp [a, b] ,
(3.4)
|M (f ; a, b)| ≤
1 kf kp ,

8(2q+1) q


1
1


p + q = 1, p > 1


2
 (b−a)
kf 00 k1
if f 00 ∈ L1 [a, b] ,
8
The first inequality in (3.4) is the one that is traditionally most well known.
Further, from (1.5) and (3.1) define the perturbed or corrected midpoint functional
as
c2 0
[f (b) − f 0 (a)] ,
(3.5)
P M (f ; a, b) := I (f ) − P I (M ) (f ) = M (f ; a, b) −
6
where c = b−a
2 .
The following lemma concerning P M (f ; a, b) holds.
Lemma 2. Let f : [a, b] → R be such that f 0 is absolutely continuous on [a, b].
Then
Z
1 b
(3.6)
P M (f ; a, b) =
χ (t) f 00 (t) dt,
2 a
where

a+b

2
1 2

(t
−
a)
−
c
,
t
∈
a,
,

3


2
(3.7)
χ (t) =


a+b

2

,b .
 (b − t) − 13 c2 , t ∈
2
with c =
b−a
2 .
TRAPEZOIDAL AND MIDPOINT RULES
7
Proof. From (3.5) and (3.7) we have, on utilising (3.2),
P M (f ; a, b)
c2 0
= M (f ; a, b) −
[f (b) − f 0 (a)]
6
Z b
c2 00
=
φ (t) −
f (t) dt
6
a
Z c2 00
1 b
=
2φ (t) −
f (t) dt.
2 a
3
Now, from the definition of φ (t) from (3.3) gives

c2
a+b

2

(t
−
a)
−
,
t
∈
a,
,



3
2
c2
2φ (t) −
=

3

c2
a+b

2

,b .
 (b − t) − , t ∈
3
2
and the result as stated in (3.6) readily follows and the lemma is thus proved.
Theorem 5. The Lebesgue norms for the perturbed midpoint functional P M (f ; a, b)
as given by (3.6) are the same as those for the perturbed trapezoid function P T (f ; a, b)
given by (2.7).
Proof. To prove the theorem it suffices to demonstrate that
kχkp = kκkp , p ≥ 1.
(3.8)
The properties of κ were investigated in the proof of Theorem 3. Now for χ (t) .
2
χ (a) = χ (b) = − c3 and χ (t) = 0 when t = a + √c3 , b − √c3 for t ∈ [a, b]. Further,
2 2
a+b
χ (t) is continuous at t = a+b
= 3 c . Also
2 and χ
2

a+b


,
2
(t
−
a)
>
0,
t
∈
a,



2
χ0 (t) =


a+b


,b .
 −2 (b − t) < 0, t ∈
2
a+b a+b
As a matter of fact, for t ∈ a, 2 , χ (t) = κ a + a+b
2 − t and for t ∈
2 ,b ,
χ (t) = κ b + a+b
t . That is, χ (t) and κ (t) are symmetric about 3a+b
2 −
4 , the
a+b
a+b
midpoint of a, 2 and a+3b
the
midpoint
of
,
b
.
Thus,
(3.8)
holds
and the
4
2
theorem is valid as stated.
Remark 1. The bound given in (2.12) also holds for P M (f ; a, b) given the results
of Theorem 4.
4. Perturbed Rules from the Chebychev Functional
For g, h : [a, b] → R the following T (g, h) is well known as the Chebychev
functional. Namely,
T (g, h) = M (gh) − M (g) M (h) ,
(4.1)
where M (g) =
1
b−a
Rb
a
g (t) dt is the integral mean.
8
P. CERONE
The Chebychev functional (4.1) is known to satisfy a number of identities including
1
T (g, h) =
b−a
(4.2)
b
Z
h (t) [g (t) − M (g)] dt.
a
Further, a number of sharp bounds for |T (g, h)| exist, under various assumptions
about g and h, including (see [5] for example)
(4.3)

1
1
g, h ∈ L2 [a, b]
[T (g, g)] 2 [T (h, h)] 2 ,







1
 Au − Al
[T (h, h)] 2 ,
Al ≤ g (t) ≤ Au , t ∈ [a, b]
|T (g, h)| ≤
2






Bu − Bl

 Au − Al
, Bl ≤ h (t) ≤ Bu , t ∈ [a, b] (Grüss).
2
2
It will be demonstrated how (4.1) may be used to obtain perturbed results which
(4.2) will provide an identity with which to obtain bounds. The following theorem
holds.
Theorem 6. Let f : [a, b] → R be such that f 0 is absolutely continuous, then
3
(4.4) |P T (f ; a, b)|
12
≤
(b − a)
√
12 5
≤
(b − a)
√ (Bu − Bl ) , Bl ≤ f 00 (t) ≤ Bu , t ∈ [a, b] ,
24 5
1
2
2
kf 00 k2 − [f 0 ; a, b]
b−a
f 00 ∈ L2 [a, b]
,
3
where P T (f ; a, b) is the perturbed trapezoidal rule defined by (2.4).
Proof. Let g (t) = − 12 (t − a) (b − t), the trapezoidal kernel and h (t) = f 00 (t) then
from (4.1)
(4.5)
00
(b − a) T (g (t) , f (t))
Z
b
Z
00
g (t) f (t) dt − M (g)
=
a
b
f 00 (t) dt
a
= T (f ; a, b) +
c2 0
[f (b) − f 0 (a)] ,
3
2
where M (g) = − c3 .
Now, from (4.2)
(4.6)
(b − a) T (g (t) , f 00 (t))
b
Z
c2
f 00 (t) g (t) +
dt
3
=
a
=
1
2
Z
b
κ (t) f 00 (t) dt
a
and so (4.4) and (4.5) produce identities (2.5) – (2.6).
TRAPEZOIDAL AND MIDPOINT RULES
9
Thus, from (4.3) and (4.4) we get
(4.7)

1
1

[(b − a) T (g, g)] 2 [(b − a) T (f 00 , f 00 )] 2 , g, h ∈ L2 [a, b]







1
 Au − Al
[(b − a) T (f 00 , f 00 )] 2 ,
Al ≤ g (t) ≤ Au ,
|P T (f ; a, b)| ≤
2







Bu − Bl
Au − Al


(b − a) ,
Bl ≤ f 00 (t) ≤ Bu .
2
2
Here, from (4.1)
[(b − a) T (h, h)]
1
2
"Z
# 21
b
2
2
h (t) dt − (b − a) M (h)
=
a
= (b − a)
1
2
12
1
2
2
khk2 − M (h) .
b−a
Specifically,
(4.8) [(b − a) T (f 00 , f 00 )]
1
2
2 # 12
0
0
1
f
(b)
−
f
(a)
2
= (b − a)
kf 00 k2 −
b−a
b−a
12
1
1
2
2
= (b − a) 2
kf 00 k2 − [f ; a, b]
b−a
1
2
"
and
(4.9)
1
[(b − a) T (g (t) , g (t))] 2

!2  12
Rb
Z b

g
(t)
dt
a
=
g 2 (t) dt −
 a

b−a
=

1
Z b
(b − a) 2  1
2
2
(t − a) (b − t) dt −
2
b−a a
Rb
a
(t − a) (b − t) dt
b−a
!2  12

5
=
=
1
(b − a) 2 B (3, 3) − B 2 (2, 2) 2
2
"
5
5
2 # 12
2
(b − a) 2 (2!)
1
(b − a) 2
√ .
−
=
2
5!
3!
12 5
Utilising (4.8) and (4.9) into the first result in (4.7) gives the first result (4.4). For
the second result in (4.4) we utilise (1.9) giving the stated coarser bound.
2
Remark 2. Even though Al = − c2 ≤ g (t) ≤ 0 = Au , it is not worthwhile using
this in the second and third inequality of (4.5) as this would produce a coarser bound
than those stated in Theorem 6.
Remark 3. The results of Theorem 6 as represented by (4.4) are tighter than (1.10)
and (1.11). For a different proof of the sharpness of (4.4) see [3].
The following bounds for the perturbed midpoint rule holds.
10
P. CERONE
Corollary 2. Let f : [a, b] → R be such that f 0 is absolutely continuous, then
12
3 (b − a)
1
2
2
√
|P M (f ; a, b)| ≤
kf 00 k2 − [f 0 ; a, b]
,
f 00 ∈ L2 [a, b]
12 5 b − a
3
≤
(b − a)
√ (Bu − Bl ) , Bl ≤ f 00 (t) ≤ Bu , t ∈ [a, b] .
24 5
Proof. The proof follows readily from Theorem 5 since
1
T (φ, φ) = kχk2 ,
2
where φ (t) and χ (t) are as given by (3.3) and (3.7) respectively.
Acknowledgement 1. The work for this paper was done while the author was on
sabbatical at La Trobe University, Bendigo.
References
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