An. Şt. Univ. Ovidius Constanţa
Vol. 17(3), 2009, 1–13
SOME NEW FOUR - POINT QUADRATURE
FORMULAS
Ana Maria Acu, Arif Rafiq and Florin Sofonea
Abstract
In this paper we present a new family of four-point quadrature formulas of close type. These quadrature formulas can be considered as
generalizations of Gauss, Newton, Simpson and Lobatto quadrature formulas for different classes of functions. The optimal quadrature formulas in the sense of minimal errors are obtained. An analysis of error
inequalities for different classes of functions is also given.
1
Introduction
In our paper, we intend to introduce a family of four-point quadrature formulas. For this purpose, we consider the following two classes of functions
(
)
Z b¯
¯
¯ (n) ¯p
n,p
n−1
(n−1)
H [a, b] = f ∈ C
[a, b], f
abs. cont.,
¯f (x)¯ dx < ∞ ,
a
1 ≤ p < ∞, (
)
¯
¯
¯
¯
(n)
H n,∞ [a, b] = f ∈ C n−1 [a, b], f (n−1) abs. cont., sup ¯f (x)¯ < ∞ ,
x∈[a,b]
The norms on these spaces of functions are the usual ones:
(Z
)1/p
¯p
b¯
° (n) °
¯
¯
(n)
°f ° :=
, 1 ≤ p < ∞, f ∈ H n,p [a, b],
¯f (x)¯ dx
p
a
¯
¯
° (n) °
°f ° := sup ¯¯f (n) (x)¯¯ , f ∈ H n,∞ [a, b].
∞
x∈[a,b]
Key Words: Quadrature rule, Error inequalities, Numerical integration.
Mathematics Subject Classification: 65D30 , 65D32.
Received: April 2009
Accepted: October 2009
1
2
ANA MARIA ACU, ARIF RAFIQ AND FLORIN SOFONEA
In order to compare our results to others obtained by some authors, we recall
them. In [3], N. Ujevic obtained the following optimal quadrature formula in
the sense of minimal error:
Theorem 1.[3] Let I ⊂ R be an open interval such that [−1, 1] ⊂ I and let
f : I → R be a twice differentiable function such that f ′′ ∈ L2 (1, 1). Then we
have
Z 1
√
√
(1)
f (t)dt = f ( 6 − 3) + f (3 − 6) + R[f ],
−1
and
|R[f ]| ≤
r
°
°
°
√ °
98
°
°
°
°
− 8 6 °f (2) ° ≃ 0.0639 °f (2) ° .
5
2
2
(2)
In [4], F. Zafar, N.A. Mir presented a family of four-point quadrature
formulas, a generalization of Gauss two-point, Newton Simpson and Lobatto
four-point quadrature formula for twice-differentiable mapping.
Theorem 2.[4] Let I ⊂ R be an open interval such that [−1, 1] ⊂ I and let
f : I → R be a twice differentiable function such that f ′′ (t) is bounded and
integrable. Then,
Z 1
h
p
f (t)dt = hf (−1) + (1 − h)f (−4 + 4h + 2 3 − 6h + 4h2 )
(3)
−1
i
p
+ (1 − h)f (4 − 4h − 2 3 − 6h + 4h2 ) + hf (1) + R[f ],
·
¸
°
°
1
where |R[f ]| ≤ 2∆(h) °f (2) °∞ , h ∈ 0,
, and ∆(h) is defined as
2
p
52 3
83
83
∆(h) =
h − 44h2 + h −
+ 8(1 − h)2 4h2 − 6h + 3
3
2
6
i 23
p
2h 2
8h − 14h + 7 − 4(1 − h) 4h2 − 6h + 3 .
+
3
2
Main results
Our goal is to give a new family of four-point quadrature rule. The estimation
of the remainder term can be obtained and it is better than in other formulas.
¸
1
3,2
Theorem 3. If f ∈ H [−1, 1] and h ∈ (−∞,
, then we have
3
!
Ãs
!
à s
Z 1
3h−1
3h−1
+(1−h)f
+hf (1)+R[f ],
f (t)dt = hf (−1)+(1−h)f −
3(h−1)
3(h−1)
−1
(4)
3
SOME NEW FOUR - POINT QUADRATURE FORMULAS
|R[f ]| ≤
where
p
°
°
°
°
∆3 (h) °f (3) °
(5)
2
£
2
−1134h4 + 2646h3 − 2097h2 + 666h − 74
2
8505(h − 1)
#
r
√
¢
3h − 1 ¡
4
3
2
378h − 1008h + 924h − 336h + 42 .
+ 3
h−1
∆3 (h) = −
Proof. We seek a quadrature formula of the type
Z
Z 1
f (t)dt + hf (−1) + (1 − h)f (x) + (1 − h)f (y) + hf (1) =
−
−1
1
p3 (t)f (3) (t)dt,
−1
¸
1
where x, y ∈ [−1, 1], x < y, h ∈ (−∞,
. We define
3

1
1


(t − a1 )3 + (t − a2 )2 + a3 , t ∈ [−1, x],

 6
2


1
1
p3 (t) =
(t − b1 )3 + (t − b2 )2 + b3 , t ∈ [x, y],
 6
2




 1 (t − c1 )3 + 1 (t − c2 )2 + c3 , t ∈ [y, 1],
6
2
(6)
where ai , bi , ci ∈ R, i ∈ {1, 2, 3} are parameters which have to be determined.
Integrating by parts, we obtain
¸
Z x·
Z 1
1
1
(t − a1 )3 + (t − a2 )2 + a3 f (3) (t)dt
p3 (t)f (3) (t)dt =
2
−1 6
−1 ·
¸
¸
Z 1·
Z y
(t−c1 )3 (t−c2 )2
(t−b1 )3 (t−b2 )2
+
+b3 f (3) (t)dt+
+
+c3 f (3) (t)dt
+
6
2
6
2
y
·x
¸
1
1
1
1
3
2
3
2
=
(x − a1 ) + (x − a2 ) + a3 − (x − b1 ) − (x − b2 ) − b3 f ′′ (x)
2
6
2
¸
·6
1
1
1
1
3
2
3
2
+ (y − b1 ) + (y − b2 ) + b3 − (y − c1 ) − (y − c2 ) − c3 f ′′ (y)
2
·6
¸6
· 2
¸
(−1 − a1 )3
(1 − c1 )3
(−1 − a2 )2
(1 − c2 )2
−
+
+ a3 f ′′ (−1)+
+
+ c3 f ′′ (1)
6
2
6
2
¸
·
¸
·
(y−b1 )2
(x−b1 )2
(y−c1 )2
(x−a1 )2
′
+a2 +
−b2 f (x)+ −
+b2 +
−c2 f ′ (y)
+ −
2
2¸
2
2
·
·
¸
1
1
+ (−1 − a1 )2 − 1 − a2 f ′ (−1) − (1 − c1 )2 + 1 − c2 f ′ (1)
2
2
4
ANA MARIA ACU, ARIF RAFIQ AND FLORIN SOFONEA
+(−a1 + b1 )f (x) + (−b1 + c1 )f (y) + a1 f (−1) + (2 − c1 )f (1) −
Z
1
f (t)dt.
−1
We require that
1
1
1
1
(x − a1 )3 + (x − a2 )2 + a3 − (x − b1 )3 − (x − b2 )2 − b3 = 0,
6
2
6
2
1
1
1
1
(y − b1 )3 + (y − b2 )2 + b3 − (y − c1 )3 − (y − c2 )2 − c3 = 0,
6
2
6
2
1
1
1
1
(−1 − a1 )3 + (−1 − a2 )2 + a3 = 0, (1 − c1 )3 + (1 − c2 )2 + c3 = 0,
6
2
6
2
1
1
1
1
− (x−a1 )2 +a2 + (x−b1 )2 −b2 = 0, − (y−b1 )2 +b2 + (y−c1 )2 −c2 = 0,
2
2
2
2
1
1
(−1 − a1 )2 − 1 − a2 = 0, (1 − c1 )2 + 1 − c2 = 0,
2
2
−a1 + b1 = 1 − h, −b1 + c1 = 1 − h, a1 = h, 2 − c1 = h.
From the above relations we find
1
1
1
1
1
1
a1 = h, a2 = − + h + h2 , a3 =
− h3 − h2 − h4 ,
2
2
24 3
4
8
s
b1 = 1, b2 = h+(1−h)
c1 = 2 − h, c2 =
s
"
#
3h − 1
3h − 1
1
−3h + 2 + 3(h − 1)
h,
, b3 =
3(h − 1)
3
3(h − 1)
1
1
1
1
1
3
− h + h2 , c3 =
− h2 + h3 − h4 ,
2
2
24 4
3
8
s
x=−
3h − 1
,y=
3(h − 1)
s
3h − 1
.
3(h − 1)
Therefore we have
!
Ãs
!
3h−1
3h−1
+(1−h)f
+hf (1)+R[f ],
f (t)dt = hf (−1)+(1−h)f −
3(h−1)
3(h−1)
−1
Z
1
à s
where
R[f ] = −
Z
1
p3 (t)f (3) (t)dt,
−1
(7)
SOME NEW FOUR - POINT QUADRATURE FORMULAS
5
s
#
"


1
3h
−
1
1
1
1
1
1

3
2
2

,
t − t h + t + t − th + − h, t ∈ −1, −


6
2
2
2
6 2
3(h − 1)



s
Ã
!
!
à s
r


√
3h−1 2
3h−1
3h−1
1
p3 (t) =
−6h+2 3(h − 1)
,
+t +3 t, t ∈ −
,

6
h−1
3(h−1)
3(h−1)


s
"
#




1
1 1
1 3 1 2 1 2
3h − 1

 t − t + t h + t − th − + h, t ∈
,1 .

6
2
2
2
6 2
3(h − 1)
The remainder term (7) has the following evaluation
°
°
°
°
|R[f ]| ≤ kp3 k2 °f (3) ° ,
2
but
2
kp3 k2
=
Z
+
Z
−
−1
q
−
+
q
Z
q
3h−1
3(h−1)
3h−1
3(h−1)
3h−1
3(h−1)
1
q
3h−1
3(h−1)
¤2
1 £3
t − 3t2 h + 3t2 + 3t − 6th + 1 − 3h dt
36
"
#2
r
r
√
√
3h − 1
3h
−
1
1
−6h + 2 3
h + t2 − 2 3
+ 3 t2 dt
36
h−1
h−1
¤2
1 £3
t + 3t2 h − 3t2 + 3t − 6th − 1 + 3h dt
36
£
2
−1134h4 + 2646h3 − 2097h2 + 666h − 74
8505(h − 1)2
#
r
√
¢
3h − 1 ¡
4
3
2
378h − 1008h + 924h − 336h + 42 .
+ 3
h−1
=−
From the above relations we obtain the estimation (5).
Remark 1.For h = 0 we obtain the Gauss two-point quadrature formula
à √ !
Ã√ !
Z 1
3
3
f (t)dt = f −
+f
+ R[f ],
3
3
−1
where
|R[f ]| ≤
r
°
°
°
4 √ °
148
°
°
°
°
−
3 °f (3) ° ≃ 0.0172 °f (3) ° .
8505 405
2
2
1
we get Lobatto four-point quadrature formula as follows
6
"
à √ !
Ã√ !
#
Z 1
5
5
1
f (t)dt =
f (−1) + 5f −
+ 5f
+ f (1) + R[f ],
6
5
5
−1
Remark 2.For h =
6
ANA MARIA ACU, ARIF RAFIQ AND FLORIN SOFONEA
where
|R[f ]| ≤
r
−
°
°
°
79 °
1 √
° (3) °
°
°
5+
°f ° ≃ 0.0056 °f (3) ° .
675
23625
2
2
1
we get Newton Simpson ′ s rule as follows:
4
·
µ
¶
µ ¶
¸
Z 1
1
1
1
f (−1) + 3f −
+ 3f
+ f (1) + R[f ],
f (t)dt =
4
3
3
−1
Remark 3.For h =
where
|R[f ]| ≤
r
°
°
°
11 °
°
°
° (3) °
°f ° ≃ 0.0085 °f (3) ° .
153090
2
2
Remark 4.From the above special cases, (4) may be considered as a generalization of Gauss two-point, Newton Simpson and Lobatto four-point quadrature
formula for the class of functions H 3,2 [−1, 1].
From the class of quadrature formulas (4) we will determine the optimal
quadrature formula in the sense of minimal error.
Corollary 1. If f ∈ H 3,2 [−1, 1], then one has the following optimal quadrature
rule
√
√
√
Z 1
´ 2− √2
2− 2
2+1 ³ √ ´ 2+1 ³√
f (t)dt =
f (−1)+
f 1− 2 +
f
f (1)+R[f ],
2−1 +
3
3
3
3
−1
(8)
s
√
°
°
°
°
4 5 2 − 7 ° (3) °
°
°
√
(9)
|R[f ]| ≤
°f ° ≃ 0.0046 °f (3) ° .
945 (1 + 2)3
2
2
Proof. We seek h such that ∆3 (h) → min. For that purpose, we calculate
h√ ¡
¢
4
1
q
3 162h4 − 459h3 + 441h2 − 165h + 21
·
1215 (h − 1)3 3h−1
h−1
#
r
¢
3h − 1 ¡
4
3
2
+
−162h + 513h − 567h + 252h − 37 .
h−1
∆′3 (h) = −
√
2− 2
From the equation
= 0 we find the solution h =
. We have
3
Ã
√
√ !
4 5 2−7
2− 2
√ .
=
∆3
3
945 (1 + 2)3
∆′3 (h)
7
SOME NEW FOUR - POINT QUADRATURE FORMULAS
√
2− 2
We conclude that h =
is the point of global minimum and we obtain
3
the quadrature formula (8) and the estimation of the remainder term (9).
In the following Theorem we obtain the estimation of the remainder term
of our quadrature formula for the functions from H 2,2 [−1, 1]. Moreover, we
will compare this estimation with the result obtained by N. Ujevic in Theorem
1.
¸
1
Theorem 4. If f ∈ H 2,2 [−1, 1] and h ∈ (−∞,
, then we have
3
!
Ãs
!
à s
Z 1
3h−1
3h−1
+(1−h)f
+hf (1)+R[f ],
f (t)dt = hf (−1)+(1−h)f −
3(h−1)
3(h−1)
−1
(10)
°
°
p
°
°
|R[f ]| ≤ ∆2 (h) °f (2) °
(11)
2
where
"
#
r
¢
9h−3 ¡
2
3
2
3
2
∆2 (h) =
−90h +180h −120h+17+
30h −70h +50h−10 .
135(h − 1)
h−1
Proof. We define
s
#
"


1
3h
−
1
1

2

t − th + t + − h, t ∈ −1, −


2
2
3(h − 1)



s
Ã
!
!
à s
r


√
3h−1
3h−1
3h−1
1
2
p2 (t) =
−6h+2 3(h−1)
+3t +3 , t ∈ −
,

6
h−1
3(h−1)
3(h−1)


s
"
#




1
3h − 1
1

 t2 + th − t + − h, t ∈
,1

2
2
3(h − 1)
Since
1
1
"
à s
3h−1
f (t)dt − hf (−1)+(1−h)f −
p2 (t)f (t)dt =
3(h−1)
−1
−1
!
#
Ãs
3h−1
+ hf (1) ,
+ (1−h)f
3(h−1)
Z
′′
Z
!
we can write
à s
!
Ãs
!
Z 1
3h−1
3h−1
f (t)dt = hf (−1)+(1−h)f −
+(1−h)f
+hf (1)+R[f ],
3(h−1)
3(h−1)
−1
8
ANA MARIA ACU, ARIF RAFIQ AND FLORIN SOFONEA
where
R[f ] =
Z
1
p2 (t)f ′′ (t)dt.
(12)
−1
The remainder term (12) has the following evaluation
|R[f ]| ≤ kp2 k2 kf ′′ k2 ,
but
2
kp2 k2
q
3h−1
3(h−1)
¶2
1 2
1
t − th + t + − h dt
2
2
−1
q
Ã
!2
r
3h−1
Z 3(h−1)
√
3h − 1
1
−6h + 2 3(h − 1)
+ 3t2 + 3 dt
+
q
3h−1 36
h−1
− 3(h−1)
¶2
µ
Z 1
1
1 2
+ q
t + th − t + − h dt
3h−1
2
2
3(h−1)
"
#
r
¢
2
9h−3 ¡
3
2
3
2
−90h +180h −120h+17+
30h −70h +50h−10 .
=
135(h−1)
h−1
=
Z
−
µ
From the above relations we obtain the estimation (11).
Remark 5.For h = 0 we obtain the Gauss two-point quadrature formula
à √ !
Ã√ !
Z 1
3
3
f (t)dt = f −
+f
+ R[f ],
3
3
−1
where
|R[f ]| ≤
r
−
°
°
°
34
4√ °
°
°
°
°
3 °f (2) ° ≃ 0.0689 °f (2) ° .
+
135 27
2
2
1
we get Lobatto four-point quadrature formula as follows
6
"
à √ !
Ã√ !
#
Z 1
5
5
1
f (t)dt =
f (−1) + 5f −
+ 5f
+ f (1) + R[f ],
6
5
5
−1
Remark 6.For h =
where
|R[f ]| ≤
r
−
°
°
°
11
1√ °
°
°
°
°
5 °f (2) ° ≃ 0.0365 °f (2) ° .
+
135 27
2
2
(13)
9
SOME NEW FOUR - POINT QUADRATURE FORMULAS
1
we get Newton Simpson ′ s rule as follows:
4
·
µ
¶
µ ¶
¸
Z 1
1
1
1
f (t)dt =
f (−1) + 3f −
+ 3f
+ f (1) + R[f ],
4
3
3
−1
Remark 7.For h =
where
|R[f ]| ≤
r
°
°
°
1 °
°
°
° (2) °
°f ° ≃ 0.0351 °f (2) ° .
810
2
2
(14)
Remark 8.From the above special cases, (10) may be considered as a generalization of Gauss two-point, Newton Simpson and Lobatto four-point quadrature
formula for the class of functions H 2,2 [−1, 1].
Remark 9.We see that the estimation (13) and (14) are better then the estimation obtained by N. Ujevic in Theorem 1.
Corollary 2.If f ∈ H 2,2 [−1, 1], then one has the following optimal quadrature
rule
Z
 s

√
√ ¶
√
√ ¶
√
√
µ
µ
3 39 33
1 39 33 
−5+ 3 9+3 3 3
√
√
f (t)dt =
f (−1)+ +
f −
−
−
+
4 12
4
4 12
4
3+ 3 9+3 3 3
−1


√
√ ¶ s
√
√ ¶
√
√ µ
µ
1 3 9 3 3  −5+ 3 9+3 3 3 3 3 9 3 3
√
√
+
+
+
−
−
f
f (1)+R[f ],(15)
4 12
4
4 12
4
3+ 3 9+3 3 3
1
where
v Ã
!
u
√
√
°
°
°
°
3
3
u
3
9
3
° (2) °
°
°
−
−
|R[f ]| ≤ t∆2
°f ° ≃ 0.0313 °f (2) ° ,
4 12
4
2
2
(16)
Ã
√
√ !
h √
√
3 39 33
1
1
3
3
√
√
∆2
=
− 3873 − 129 − 2532
−
−
·
3
3
4 12
4
180 (3 + 9 + 3 3)2
q
q
³ √
´¸
√
√
√
√
√
3
3
3
3
3
3
+ −5 + 9 + 3 3 3 + 9 + 3 3 · 35 3 + 105 + 25 9 .
Proof. We seek h such that ∆2 (h) → min. For that purpose, we calculate
h √
√
√
√
2
1
q
∆′2 (h) =
36 3h3 − 78 3h2 + 52 3h − 10 3
27 3h−1 (h − 1)2
h−1
#
r
3h − 1
3
2
+
(−36h + 90h − 72h + 17) .
h−1
10
ANA MARIA ACU, ARIF RAFIQ AND FLORIN SOFONEA
From
the√equation ∆′2 (h) = 0 we find the point of global minimum
√
3
9 33
3
and we obtain the quadrature formula (15) and the estimation
h= − −
4 12 4
of the remainder term (16).
p
∆3 (h),
Remark 10.In Figure 1. we can see the graphics of functions
f (h) = )
(
√
p
p
1 1 2− 2
and
g(h) =
∆2 (h) and the points (h, ∆3 (h)), h ∈ 0, , ,
6 4
3
)
(
√
√
p
1 1 3 39 33
(h, ∆2 (h)), h ∈ 0, , , −
.
−
6 4 4 12
4
0.14
sqrt(Delta 3(h)),−−−−−−sqrt(Delta 2(h))
sqrt(Delta 3(0)),sqrt(Delta 2(0))
sqrt(Delta 3(1/6)),sqrt(Delta 2(1/6))
sqrt(Delta 3(1/4)),sqrt(Delta 2(1/4))
sqrt(Delta 3((2−sqrt(2))/3),sqrt(Delta 2(3/4−91/3/12−31/3/4)
0.12
0.1
0.08
0.06
0.04
0.02
0
−0.5
−0.4
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
0.4
Figure 1.
In the following Theorem we obtain the estimation of the remainder term
of our quadrature formula for the functions from H 2,∞ [−1, 1].
¸
1
2,∞
, then we have
Theorem 5.If f ∈ H
[−1, 1] and h ∈ (−∞,
3
à s
!
Ãs
!
Z 1
3h−1
3h−1
f (t)dt = hf (−1)+(1−h)f −
+(1−h)f
+hf (1)+R[f ],
3(h−1)
3(h−1)
−1
(17)
°
°
° (2) °
|R[f ]| ≤ ∆(h) °f °
(18)
∞
SOME NEW FOUR - POINT QUADRATURE FORMULAS
11
where
∆(h) =

"
#s
r
r


9h−3
9h−3
4

−
3−6h+2
(h−1)
(1−h), h ≤ 0,
−9+18h+6


27
h−1
h−1



s

"
#
r
r
µ ¶
1
2
9h−3
9h−3
= 8 3 4
,
h −
(1−h)· (h−1)
−2h+1 , h ∈ 0,
−9+18h+6



3
9
h−1
3
h−1
4


¸
·


8
1 1


 h3 , h ∈
.
,
3
4 3
Proof. From the proof of Theorem 4 we have the quadrature formula (17)
and the remainder term
Z 1
p2 (t)f (2) (t)dt
R[f ] =
−1
has the following evaluation
°
°
°
°
|R[f ]| ≤ kp2 k1 · °f (2) °
,
∞
with
"
#s
r
r
4
9h−3
9h−3
kp2 k1 = −
−9+18h+6
3−6h+2
(h−1)
(1−h), h ≤ 0,
27
h−1
h−1
s
"
#
r
r
µ ¶
9h−3
9h−3
1
2
8 3 4
(1−h)· (h−1)
−2h+1 , h ∈ 0, ,
kp2k1 = h −
−9+18h+6
3
9
h−1
3
h−1
4
¸
·
8 3
1 1
kp2 k1 = h , h ∈
.
,
3
4 3
Remark 11.For h = 0 we obtain the Gauss two-point quadrature formula
à √ !
Ã√ !
Z 1
3
3
f (t)dt = f −
+f
+ R[f ],
3
3
−1
where
|R[f ]| ≤ −
4
9
q
µ
¶°
°
°
°
√
2√
°
°
°
°
−9 + 6 3 −
3 + 1 °f (2) ° ≃ 0.0811 °f (2) ° .
3
∞
∞
1
Remark 12.For h = we get Lobatto four-point quadrature formula as fol6
lows
"
à √ !
Ã√ !
#
Z 1
5
5
1
f (t)dt =
f (−1) + 5f −
+ 5f
+ f (1) + R[f ],
6
5
5
−1
12
ANA MARIA ACU, ARIF RAFIQ AND FLORIN SOFONEA
where
|R[f ]| ≤
µ
¶
q
°
°
°
√ °
1
4 √
°
°
°
°
+ ( 5 − 2) −6 + 3 5 °f (2) ° ≃ 0.0418 °f (2) ° .
81 27
∞
∞
1
we get Newton Simpson ′ s rule as follows:
4
·
µ
¶
µ ¶
¸
Z 1
1
1
1
f (t)dt =
f (−1) + 3f −
+ 3f
+ f (1) + R[f ],
4
3
3
−1
Remark 13.For h =
where
°
°
°
1 °
° (2) °
°
°
°f ° ≃ 0.0417 °f (2) ° .
24
∞
∞
r
1
3h − 1
,h≤ ,
Remark 14.From the graphics of functions f (h) = ±
3(h
−
1)
3
¸
·
√
1
g(h) = ±(4 − 4h − 2 3 − 6h + 4h2 ), h ∈ 0, , we see that the set of values
2
of nodes from our quadrature formula include the set of values of nodes from
quadrature formula (3) obtained by F. Zafar and N.A. Mir.
|R[f ]| ≤
1
0.8
0.6
0.4
0.2
0
−0.2
−0.4
4−4h−2sqrt(3−6h+4h 2)
−4+4h+2sqrt(3−6h+4*h 2)
sqrt((3h−1)/(3(h−1)))
−sqrt((3h−1)/(3(h−1)))
−0.6
−0.8
−1
−2
−1.5
−1
Figure 2.
−0.5
0
0.5
SOME NEW FOUR - POINT QUADRATURE FORMULAS
13
References
[1] A. M. Acu, A. Baboş, An error analysis for a quadrature formula, The 14the International Conference The Knowledge-based organization Technical
Sciences Computer Science, Modelling & Simulation and E-learning Technologies. Physics, Mathematics and Chemistry. Conference Proceedings 8,
Sibiu, 27-29 November, 2008, ISSN: 1843-6722, 290-298
[2] A. Branga, Natural splines of Birkhoff type and optimal approximation, J.
Comput. Analysis Appl., 7(2005), no. 1, 81-88.
[3] N. Ujevic, Error inequalities for a quadrature formula of open type, Rev.
Colombiana de Mat., 37(2003), 93-105.
[4] F. Zafar, N.A.Mir, Some generalized error inequalities and applications,
Journal of Inequalities and Applications, Volume 2008, Article ID 845934,
15 pages.
Ana Maria Acu and Florin Sofonea
Lucian Blaga” University, Department of Mathematics
E-mail: acuana77@yahoo.com, sofoneaflorin@yahoo.com
Arif Rafiq
COMSATS Institute of Information Technology,
Department of Mathematics,
Defense Road, Off Raiwind Road, Lohore-Pakistan e-mail:
arafiq@comsats.edu.pk