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Error estimates for some composite corrected quadrature rules
Applied Mathematics Letters 22 (2009) 771–775
Contents lists available at ScienceDirect
Applied Mathematics Letters journal homepage: www.elsevier.com/locate/aml
Error estimates for some composite corrected quadrature rules Zheng Liu Institute of Applied Mathematics, School of Science, University of Science and Technology Liaoning, Anshan 114051, Liaoning, China
article
a b s t r a c t
info
Article history: Received 3 November 2006 Received in revised form 25 June 2008 Accepted 26 August 2008
The asymptotic behaviour of the error for a general quadrature rule is established and it is applied to some composite corrected quadrature rules. © 2008 Elsevier Ltd. All rights reserved.
Keywords: Corrected trapezoidal rule Corrected midpoint rule Corrected Simpson rule Composite corrected quadrature rule Error estimates
1. Introduction Let f : [a, b] → R be such that f 0 is absolutely continuous on [a, b]. It is well known that the trapezoidal rule T (f ) :=
b−a 2
[f (a) + f (b)]
(1)
and the midpoint rule M (f ) := (b − a)f
a+b
(2)
2
are the simplest quadrature formulae used to approximate the integral b
Z
f (t )dt a
which can serve as basic elements for constructing more sophisticated formulae by certain types of convex combinations, e.g., the classical Simpson rule is defined as S (f ) =
1 3
T (f ) +
2 3
M (F ) =
b−a 6
f (a) + 4f
a+b
2
+ f (b) .
(3)
Further, we may consider the corrected or perturbed trapezoidal rule CT (f ) :=
b−a 2
[f (a) + f (b)] −
( b − a) 2 12
[f 0 (b) − f 0 (a)]
E-mail address: [email protected]. 0893-9659/$ – see front matter © 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.aml.2008.08.016
(4)
772
Z. Liu / Applied Mathematics Letters 22 (2009) 771–775
and the corrected or perturbed midpoint rule
CM (f ) := (b − a)f
a+b
(b − a)2
+
2
24
[f 0 (b) − f 0 (a)].
(5)
It is well known that corrected quadrature formulae (4) and (5) have better estimates of error than corresponding original formulae (1) and (2). (See, e.g., [1–3].) Unfortunately, we could not obtain such a corrected version of the Simpson rule, since 1 3
CT (f ) +
2 3
CM (f ) =
b−a
f (a) + 4f
6
a+b
2
+ f (b)
does not involve the derivatives at the endpoints. Nevertheless, we did find in [4–6] the so-called corrected Simpson quadrature rule that improves on Simpson rule (3): CS (f ) =
=
7 15
8
CT (f ) +
15
CM (f )
b−a
7f (a) + 16f
30
a+b 2
(b − a)2 0 + 7f (b) − [f (b) − f 0 (a)].
(6)
60
It is not difficult to find that the corrected trapezoidal rule (4) and the corrected midpoint rule (5) are exact for polynomials of degree 3 or less, and the corrected Simpson rule (6) is exact for polynomials of degree 5 or less. a Now let a = t0 < t1 < · · · < tn = b be an equidistant subdivision of the interval [a, b] such that ti+1 − ti = h = b− , n i = 0, 1, . . . , n − 1. Then we have the composite corrected trapezoidal rule CTn (f ) =
b−a 2n
[f (t0 ) + 2f (t1 ) + · · · + 2f (tn−1 ) + f (tn )] −
(b − a)2 12n2
[f 0 (b) − f 0 (a)],
(7)
the composite corrected midpoint rule CMn (f ) =
b−a
f
n
t0 + t1
+f
2
t1 + t2
2
+ ··· + f
tn−1 + tn
+
( b − a) 2
−
( b − a) 2
2
24n2
[f 0 (b) − f 0 (a)]
(8)
[f 0 (b) − f 0 (a)]
(9)
and the composite corrected Simpson rule CSn (f ) =
b−a
30n
7[f (t0 ) + 2f (t1 ) + · · · + 2f (tn−1 ) + f (tn )]
+ 16 f
t0 + t1
2
+f
t1 + t2 2
+ ··· + f
tn−1 + tn
2
60n2
which correspond to the corrected quadrature rules (4)–(6), respectively. Motivated by [7,8], in this work, we will give a unified treatment for estimating the errors for the above mentioned composite corrected quadrature rules (7)–(9). 2. Main results We need the following result (see Theorem 10 in [8] or Theorem 3 in [9]): Lemma 1. Let f : [a, b] → R be such that its (r − 1)th derivative f (r −1) is of continuous bounded variation for some positive integer r. Then for any x ∈ [0, 1], we have b
Z
f (t )dt = h a
n−1 X
f (ti + xh) −
r X f (ν−1) (b) − f (ν−1) (a)
ν=1
i=0
ν!
Bν (x)hν + R(nr ) (f ; x),
(10)
where R(nr ) (f ; x) =
hr r!
Z a
b
B˜ r
x−n
t −a b−a
df (r −1) (t )
and B˜ r (t ) := Br (t − [t ]) while Br (t ) is the rth Bernoulli polynomial.
(11)
Z. Liu / Applied Mathematics Letters 22 (2009) 771–775
773
Theorem 2. Suppose that for some real constants α 6= 0 and pj ∈ [0, 1], j = 0, 1, . . . , m − 1, with quadrature rule
j =0
pj = 1, the following
m−1
1
Z
Pm−1
X
f (t )dt = 0
pj f (xj ) + α[f 0 (1) − f 0 (0)]
(12)
j =0
is exact for any polynomial of degree ≤ r − 1 for some positive integer r ≥ 3. Let f : [a, b] → R be such that its (r − 1)th derivative f (r −1) is a continuous function of bounded variation on [a, b]. Then we have n −1 m −1 X X
b
Z
f (t )dt = h a
pj f (ti + xj h) + α h2 [f 0 (b) − f 0 (a)] + R(nr ) (f ),
(13)
i =0 j =0
where (r )
Rn (f ) = h
r
b
Z
Gr
n
a
t −a
b−a
df (r −1) (t ),
(14)
and Gr (t ) =
m−1 1 X
pj (B˜ r (xj − t ) − Br (xj )).
r ! j =0
(15)
Proof. We first note that for 1 ≤ ν ≤ r − 1, m−1
X
pj Bν (xj ) +
α[B0ν (1)
−
B0ν (0)]
1
Z
Bν (t )dt = 0
= 0
j =0
due to (12) being exact for any polynomial of degree ≤ r − 1 and a well known property of the Bernoulli polynomials. Moreover, since B0 (t ) = 1, B0n (t ) = nBn−1 (t ) and Bn (t + 1) − Bn (t ) = nt n−1 , n = 1, 2, . . . , we get m−1
X
pj Bν (xj ) =
j =0
0, −2α,
if 1 ≤ ν ≤ r − 1, ν 6= 2, if ν = 2.
(16)
Now setting x = xj in (10), multiplying both sides of (10) by pj , summing from j = 0 to m − 1, from (10), (16) and the following readily checked fact: f (r −1) (b) − f (r −1) (a) r!
Br (x)hr =
hr r!
b
Z
Br (x)df (r −1) (t ),
a
we obtain b
Z
f (t )dt = h
n −1 m −1 X X
a
r m −1 X X
pj f (ti + xj h) −
! pj Bν (xj )
i =0 j =0
ν=1
n−1 m −1 X X
r −1 m −1 X X
X f (ν−1) (b) − f (ν−1) (a) ν h + pj R(nr ) (f ; xj ), m−1
ν!
j =0
i.e., b
Z
f (t )dt = h a
pj f (ti + xj h) −
ν=1
i=0 j=0 m−1
−
X
pj Br (xj )
! pj Bν (xj )
ν!
j =0
f (r −1) (b) − f (r −1) (a)
j =0
f (ν−1) (b) − f (ν−1) (a) ν h
r!
m−1
hr +
X
pj R(nr ) (f ; xj ),
j =0
and it follows from (16) that b
Z
f (t )dt = h
n−1 m −1 X X
a
pj f (ti + xj h) + α h2 [f 0 (b) − f 0 (a)]
i=0 j=0 m−1
−
X j =0
pj
hr r!
b
Z a
Br (xj )df (r −1) (t ) +
m−1
X j =0
pj R(nr ) (f ; xj ).
j =0
774
Z. Liu / Applied Mathematics Letters 22 (2009) 771–775
Finally, from (11) we get b
Z
f (t )dt = h a
n −1 m −1 X X
pj f (ti + xj h) + α h2 [f 0 (b) − f 0 (a)] +
i=0 j=0
which completes the proof.
hr r!
b m−1
Z
X
a
pj (B˜ r
j =0
xj − n
t −a b−a
− Br (xj ))df (r −1) (t ),
Remark 3. The case α = 0 has already been studied in [8]. For α 6= 0, from the proof of Theorem 2, we can conclude that r −1 m −1 X X
ν=1
! pj Bν (xj )
f (ν−1) (b) − f (ν−1) (a) ν h = −α h2 [f 0 (b) − f 0 (a)]
ν!
j =0
only if r ≥ 3. Otherwise (for r = 1, 2) the expression on the left-hand side equals 0. Theorem 4. Let the assumptions of Theorem 2 hold. Then we have
|nr R(nr ) (f )| ≤ Cr (b − a)r
_b a
(f (r −1) ),
(17)
where
−1 m X ˜ pj (Br (xj − t ) − Br (xj )) . sup Cr := r ! 0
(18)
If further f (r −1) is absolutely continuous on [a, b], we then have lim nr R(nr ) (f ) = Kr (b − a)r
n→∞
Z
b
f (r ) (t )dt ,
(19)
a
where m−1 1 X
Kr := −
r ! j =0
pj Br (xj ).
(20)
The proof is similar to that for the Theorem 12 in [8] and so is omitted. Remark 5. If (12) has degree of precision r − 1 then Cν and Kν exist for all 3 ≤ ν ≤ r since in this case (12) is exact for any polynomial of degree ≤ ν − 1. We obtain from (16) and (20) that Kν = 0,
if 3 ≤ ν < r ,
(21)
which implies from (19) that lim nν R(ν) n (f ) = 0,
if 3 ≤ ν < r .
n→∞
3. Examples 1 Example 1. For the corrected trapezoid rule, we see that α = − 12 and it has degree of precision 3 (r = 4). Thus for any
function f such that f (ν−1) (ν = 3, 4) is continuous of bounded variation, we get Cν by direct calculations from (18) as
√
C3 =
3 216
,
C4 =
1 384
,
and, for any function f such that f (ν−1) (ν = 3, 4) is absolutely continuous, we obtain Kν from (21) and by a direct calculation from (20) as K3 = 0,
K4 =
1 720
and these imply that lim n3 R(f ; CTn ) = lim n3 R(n3) (f ) = 0
n→∞
n→∞
Z. Liu / Applied Mathematics Letters 22 (2009) 771–775
775
and lim n4 R(f ; CTn ) = lim n4 R(n4) (f ) =
n→∞
n→∞
1 720
(b − a)4 [f 000 (b) − f 000 (a)].
Example 2. For the corrected midpoint rule, we see that α =
1 24
and it has degree of precision 3 (r = 4). Thus for any
function f such that f (ν−1) (ν = 3, 4) is continuous of bounded variation, we get Cν by direct calculations from (18) as
√
C3 =
3 216
,
C4 =
1 384
,
and, for any function f such that f (ν−1) (ν = 3, 4) is absolutely continuous, we obtain Kν from (21) and by a direct calculation from (20) as K3 = 0,
K4 = −
7 5760
and these imply that lim n3 R(f ; CMn ) = lim n3 R(n3) (f ) = 0
n→∞
n→∞
and lim n4 R(f ; CMn ) = lim n4 R(n4) (f ) = −
n→∞
n→∞
7 5760
(b − a)4 [f 000 (b) − f 000 (a)].
1 Example 3. For the corrected Simpson rule, we see that α = − 60 and it has degree of precision 5 (r = 6). Thus for any
function f such that f (ν−1) (ν = 3, 4, 5, 6) is continuous of bounded variation, we get Cν by direct calculations from (18) as
√
C3 =
7
+
19 19
,
C4 =
1
,
C5 =
1
,
C6 =
1
20 250 81 000 5760 58 320 230 400 and, for any function f such that f (ν−1) (ν = 3, 4, 5, 6) is absolutely continuous, we obtain Kν from (21) and by a direct calculation from (20) as K3 = K4 = K5 = 0,
K6 =
1 604 800
and these imply that lim nν R(f ; CSn ) = lim nν R(ν) n (f ) = 0,
n→∞
n→∞
for ν = 3, 4, 5,
and lim n6 R(f ; CSn ) = lim n6 R(n6) (f ) =
n→∞
n→∞
1 604 800
(b − a)4 [f (5) (b) − f (5) (a)].
Remark 6. It should be mentioned that the same estimates of error for the corrected Simpson rule were obtained in [6]. Furthermore, estimates of error for both classical and corrected trapezoidal rules were obtained in [10] and for the midpoint and corrected midpoint rules in [11]. Acknowledgment The author wishes to express his sincere thanks to the referee for valuable comments and suggestions. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11]
P. Cerone, On perturbed trapezoidal and midpoint rules, Korean J. Comput. Appl. Math. 2 (2002) 423–435. X.L. Cheng, J. Sun, A note on the perturbed trapezoid inequality, J. Inequal. Pure Appl. Math. 3 (2002). Article 29 http://jipam.vu.edu.au. Z. Liu, On sharp perturbed midpoint inequalities, Tamkang J. Math. 36 (2005) 131–136. N. Ujević, A.J. Roberts, A corrected quadrature formula and applications, ANZIAM J. 45 (E) (2004) E41–E56. N. Ujević, A generalization of the modified Simpson’s rule and error bounds, ANZIAM J. 47 (E) (2005) E1–E13. J. Pečarić, I. Franjić, Generalization of corrected Simpson’s formula, ANZIAM J. 47 (2006) 367–385. D. Cruz-Uribe, C.J. Neugebauer, Sharp error bounds for the trapezoidal rule and Simpson’s rule, J. Inequal. Pure Appl. Math. 3 (2002). Article 49 http://jipam.vu.edu.au. Q.B. Wu, S.J. Yang, A note to Ujević’s generalization of Ostrowski’s inequality, Appl. Math. Lett. 18 (2005) 657–665. I. Franjić, I. Perić, J. Pečarić, General Euler–Ostrowski formulae and applications to quadratures, Appl. Math. Comput. 177 (2006) 92–98. Lj. Dedić, M. Matić, J. Pečarić, On Euler trapezoid formulae, Appl. Math. Comput. 123 (2001) 37–62. Lj. Dedić, M. Matić, J. Pečarić, On Euler midpoint formulae, ANZIAM J. 46 (2005) 417–438.
Contents lists available at ScienceDirect
Applied Mathematics Letters journal homepage: www.elsevier.com/locate/aml
Error estimates for some composite corrected quadrature rules Zheng Liu Institute of Applied Mathematics, School of Science, University of Science and Technology Liaoning, Anshan 114051, Liaoning, China
article
a b s t r a c t
info
Article history: Received 3 November 2006 Received in revised form 25 June 2008 Accepted 26 August 2008
The asymptotic behaviour of the error for a general quadrature rule is established and it is applied to some composite corrected quadrature rules. © 2008 Elsevier Ltd. All rights reserved.
Keywords: Corrected trapezoidal rule Corrected midpoint rule Corrected Simpson rule Composite corrected quadrature rule Error estimates
1. Introduction Let f : [a, b] → R be such that f 0 is absolutely continuous on [a, b]. It is well known that the trapezoidal rule T (f ) :=
b−a 2
[f (a) + f (b)]
(1)
and the midpoint rule M (f ) := (b − a)f
a+b
(2)
2
are the simplest quadrature formulae used to approximate the integral b
Z
f (t )dt a
which can serve as basic elements for constructing more sophisticated formulae by certain types of convex combinations, e.g., the classical Simpson rule is defined as S (f ) =
1 3
T (f ) +
2 3
M (F ) =
b−a 6
f (a) + 4f
a+b
2
+ f (b) .
(3)
Further, we may consider the corrected or perturbed trapezoidal rule CT (f ) :=
b−a 2
[f (a) + f (b)] −
( b − a) 2 12
[f 0 (b) − f 0 (a)]
E-mail address: [email protected]. 0893-9659/$ – see front matter © 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.aml.2008.08.016
(4)
772
Z. Liu / Applied Mathematics Letters 22 (2009) 771–775
and the corrected or perturbed midpoint rule
CM (f ) := (b − a)f
a+b
(b − a)2
+
2
24
[f 0 (b) − f 0 (a)].
(5)
It is well known that corrected quadrature formulae (4) and (5) have better estimates of error than corresponding original formulae (1) and (2). (See, e.g., [1–3].) Unfortunately, we could not obtain such a corrected version of the Simpson rule, since 1 3
CT (f ) +
2 3
CM (f ) =
b−a
f (a) + 4f
6
a+b
2
+ f (b)
does not involve the derivatives at the endpoints. Nevertheless, we did find in [4–6] the so-called corrected Simpson quadrature rule that improves on Simpson rule (3): CS (f ) =
=
7 15
8
CT (f ) +
15
CM (f )
b−a
7f (a) + 16f
30
a+b 2
(b − a)2 0 + 7f (b) − [f (b) − f 0 (a)].
(6)
60
It is not difficult to find that the corrected trapezoidal rule (4) and the corrected midpoint rule (5) are exact for polynomials of degree 3 or less, and the corrected Simpson rule (6) is exact for polynomials of degree 5 or less. a Now let a = t0 < t1 < · · · < tn = b be an equidistant subdivision of the interval [a, b] such that ti+1 − ti = h = b− , n i = 0, 1, . . . , n − 1. Then we have the composite corrected trapezoidal rule CTn (f ) =
b−a 2n
[f (t0 ) + 2f (t1 ) + · · · + 2f (tn−1 ) + f (tn )] −
(b − a)2 12n2
[f 0 (b) − f 0 (a)],
(7)
the composite corrected midpoint rule CMn (f ) =
b−a
f
n
t0 + t1
+f
2
t1 + t2
2
+ ··· + f
tn−1 + tn
+
( b − a) 2
−
( b − a) 2
2
24n2
[f 0 (b) − f 0 (a)]
(8)
[f 0 (b) − f 0 (a)]
(9)
and the composite corrected Simpson rule CSn (f ) =
b−a
30n
7[f (t0 ) + 2f (t1 ) + · · · + 2f (tn−1 ) + f (tn )]
+ 16 f
t0 + t1
2
+f
t1 + t2 2
+ ··· + f
tn−1 + tn
2
60n2
which correspond to the corrected quadrature rules (4)–(6), respectively. Motivated by [7,8], in this work, we will give a unified treatment for estimating the errors for the above mentioned composite corrected quadrature rules (7)–(9). 2. Main results We need the following result (see Theorem 10 in [8] or Theorem 3 in [9]): Lemma 1. Let f : [a, b] → R be such that its (r − 1)th derivative f (r −1) is of continuous bounded variation for some positive integer r. Then for any x ∈ [0, 1], we have b
Z
f (t )dt = h a
n−1 X
f (ti + xh) −
r X f (ν−1) (b) − f (ν−1) (a)
ν=1
i=0
ν!
Bν (x)hν + R(nr ) (f ; x),
(10)
where R(nr ) (f ; x) =
hr r!
Z a
b
B˜ r
x−n
t −a b−a
df (r −1) (t )
and B˜ r (t ) := Br (t − [t ]) while Br (t ) is the rth Bernoulli polynomial.
(11)
Z. Liu / Applied Mathematics Letters 22 (2009) 771–775
773
Theorem 2. Suppose that for some real constants α 6= 0 and pj ∈ [0, 1], j = 0, 1, . . . , m − 1, with quadrature rule
j =0
pj = 1, the following
m−1
1
Z
Pm−1
X
f (t )dt = 0
pj f (xj ) + α[f 0 (1) − f 0 (0)]
(12)
j =0
is exact for any polynomial of degree ≤ r − 1 for some positive integer r ≥ 3. Let f : [a, b] → R be such that its (r − 1)th derivative f (r −1) is a continuous function of bounded variation on [a, b]. Then we have n −1 m −1 X X
b
Z
f (t )dt = h a
pj f (ti + xj h) + α h2 [f 0 (b) − f 0 (a)] + R(nr ) (f ),
(13)
i =0 j =0
where (r )
Rn (f ) = h
r
b
Z
Gr
n
a
t −a
b−a
df (r −1) (t ),
(14)
and Gr (t ) =
m−1 1 X
pj (B˜ r (xj − t ) − Br (xj )).
r ! j =0
(15)
Proof. We first note that for 1 ≤ ν ≤ r − 1, m−1
X
pj Bν (xj ) +
α[B0ν (1)
−
B0ν (0)]
1
Z
Bν (t )dt = 0
= 0
j =0
due to (12) being exact for any polynomial of degree ≤ r − 1 and a well known property of the Bernoulli polynomials. Moreover, since B0 (t ) = 1, B0n (t ) = nBn−1 (t ) and Bn (t + 1) − Bn (t ) = nt n−1 , n = 1, 2, . . . , we get m−1
X
pj Bν (xj ) =
j =0
0, −2α,
if 1 ≤ ν ≤ r − 1, ν 6= 2, if ν = 2.
(16)
Now setting x = xj in (10), multiplying both sides of (10) by pj , summing from j = 0 to m − 1, from (10), (16) and the following readily checked fact: f (r −1) (b) − f (r −1) (a) r!
Br (x)hr =
hr r!
b
Z
Br (x)df (r −1) (t ),
a
we obtain b
Z
f (t )dt = h
n −1 m −1 X X
a
r m −1 X X
pj f (ti + xj h) −
! pj Bν (xj )
i =0 j =0
ν=1
n−1 m −1 X X
r −1 m −1 X X
X f (ν−1) (b) − f (ν−1) (a) ν h + pj R(nr ) (f ; xj ), m−1
ν!
j =0
i.e., b
Z
f (t )dt = h a
pj f (ti + xj h) −
ν=1
i=0 j=0 m−1
−
X
pj Br (xj )
! pj Bν (xj )
ν!
j =0
f (r −1) (b) − f (r −1) (a)
j =0
f (ν−1) (b) − f (ν−1) (a) ν h
r!
m−1
hr +
X
pj R(nr ) (f ; xj ),
j =0
and it follows from (16) that b
Z
f (t )dt = h
n−1 m −1 X X
a
pj f (ti + xj h) + α h2 [f 0 (b) − f 0 (a)]
i=0 j=0 m−1
−
X j =0
pj
hr r!
b
Z a
Br (xj )df (r −1) (t ) +
m−1
X j =0
pj R(nr ) (f ; xj ).
j =0
774
Z. Liu / Applied Mathematics Letters 22 (2009) 771–775
Finally, from (11) we get b
Z
f (t )dt = h a
n −1 m −1 X X
pj f (ti + xj h) + α h2 [f 0 (b) − f 0 (a)] +
i=0 j=0
which completes the proof.
hr r!
b m−1
Z
X
a
pj (B˜ r
j =0
xj − n
t −a b−a
− Br (xj ))df (r −1) (t ),
Remark 3. The case α = 0 has already been studied in [8]. For α 6= 0, from the proof of Theorem 2, we can conclude that r −1 m −1 X X
ν=1
! pj Bν (xj )
f (ν−1) (b) − f (ν−1) (a) ν h = −α h2 [f 0 (b) − f 0 (a)]
ν!
j =0
only if r ≥ 3. Otherwise (for r = 1, 2) the expression on the left-hand side equals 0. Theorem 4. Let the assumptions of Theorem 2 hold. Then we have
|nr R(nr ) (f )| ≤ Cr (b − a)r
_b a
(f (r −1) ),
(17)
where
−1 m X ˜ pj (Br (xj − t ) − Br (xj )) . sup Cr := r ! 0
(18)
If further f (r −1) is absolutely continuous on [a, b], we then have lim nr R(nr ) (f ) = Kr (b − a)r
n→∞
Z
b
f (r ) (t )dt ,
(19)
a
where m−1 1 X
Kr := −
r ! j =0
pj Br (xj ).
(20)
The proof is similar to that for the Theorem 12 in [8] and so is omitted. Remark 5. If (12) has degree of precision r − 1 then Cν and Kν exist for all 3 ≤ ν ≤ r since in this case (12) is exact for any polynomial of degree ≤ ν − 1. We obtain from (16) and (20) that Kν = 0,
if 3 ≤ ν < r ,
(21)
which implies from (19) that lim nν R(ν) n (f ) = 0,
if 3 ≤ ν < r .
n→∞
3. Examples 1 Example 1. For the corrected trapezoid rule, we see that α = − 12 and it has degree of precision 3 (r = 4). Thus for any
function f such that f (ν−1) (ν = 3, 4) is continuous of bounded variation, we get Cν by direct calculations from (18) as
√
C3 =
3 216
,
C4 =
1 384
,
and, for any function f such that f (ν−1) (ν = 3, 4) is absolutely continuous, we obtain Kν from (21) and by a direct calculation from (20) as K3 = 0,
K4 =
1 720
and these imply that lim n3 R(f ; CTn ) = lim n3 R(n3) (f ) = 0
n→∞
n→∞
Z. Liu / Applied Mathematics Letters 22 (2009) 771–775
775
and lim n4 R(f ; CTn ) = lim n4 R(n4) (f ) =
n→∞
n→∞
1 720
(b − a)4 [f 000 (b) − f 000 (a)].
Example 2. For the corrected midpoint rule, we see that α =
1 24
and it has degree of precision 3 (r = 4). Thus for any
function f such that f (ν−1) (ν = 3, 4) is continuous of bounded variation, we get Cν by direct calculations from (18) as
√
C3 =
3 216
,
C4 =
1 384
,
and, for any function f such that f (ν−1) (ν = 3, 4) is absolutely continuous, we obtain Kν from (21) and by a direct calculation from (20) as K3 = 0,
K4 = −
7 5760
and these imply that lim n3 R(f ; CMn ) = lim n3 R(n3) (f ) = 0
n→∞
n→∞
and lim n4 R(f ; CMn ) = lim n4 R(n4) (f ) = −
n→∞
n→∞
7 5760
(b − a)4 [f 000 (b) − f 000 (a)].
1 Example 3. For the corrected Simpson rule, we see that α = − 60 and it has degree of precision 5 (r = 6). Thus for any
function f such that f (ν−1) (ν = 3, 4, 5, 6) is continuous of bounded variation, we get Cν by direct calculations from (18) as
√
C3 =
7
+
19 19
,
C4 =
1
,
C5 =
1
,
C6 =
1
20 250 81 000 5760 58 320 230 400 and, for any function f such that f (ν−1) (ν = 3, 4, 5, 6) is absolutely continuous, we obtain Kν from (21) and by a direct calculation from (20) as K3 = K4 = K5 = 0,
K6 =
1 604 800
and these imply that lim nν R(f ; CSn ) = lim nν R(ν) n (f ) = 0,
n→∞
n→∞
for ν = 3, 4, 5,
and lim n6 R(f ; CSn ) = lim n6 R(n6) (f ) =
n→∞
n→∞
1 604 800
(b − a)4 [f (5) (b) − f (5) (a)].
Remark 6. It should be mentioned that the same estimates of error for the corrected Simpson rule were obtained in [6]. Furthermore, estimates of error for both classical and corrected trapezoidal rules were obtained in [10] and for the midpoint and corrected midpoint rules in [11]. Acknowledgment The author wishes to express his sincere thanks to the referee for valuable comments and suggestions. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11]
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