General Mathematics Vol. 13, No. 4 (2005), 73–80
An intermediate point property in the
quadrature formulas of type Gauss
Petrică Dicu
Abstract
In this paper we study a property of the intermediate point (see
[1], [2], [3], [6], [7]) from the quadrature formulas of Gauss type.
2000 Mathematics Subject Classification: 41A55, 65D32
Keywords and phrases: quadrature formulas of Gauss type
1. In [6] B. Jacobson studied a property of the intermediate point which
appear in the mean-value theorem for integrals. This property has been
studied for others mean-value formulas in the articles [1], [2], [3], and [7].
In this paper we will study this property for two particular cases of the
quadrature formulas of Gauss type.
The quadrature formulas of Gauss type have the form (see [5]).
Zb
(1)
f (x)dx = (b − a)[c1 f (x1 ) + c2 f (x2 ) + ... + cn f (xn )] + Rn (f )
a
73
74
Petrică Dicu
where f : [a, b] → R, f ∈ C 2n [a, b].
(2)
(n!)4 · 22n+1
Rn (f ) =
·
[(2n)!]3 (2n + 1)
b−1
2
2n+1
· f (2n) (ξb ), ξb ∈ (a, b),
the nodes xi , i = 1, n which appears in (1) are given by
a+b b−a
+
yi , i = 1, n
2
2
where yi , i = 1, n are the zeros of the Legendre polynomial
(3)
xi =
Ln (y) =
(4)
1
[(y 2 − 1)n ](n) , n ≥ 1
2 · n!
n
and the coefficients c1 , c2 , ..., cn from (1) are the solution of the system


c1 + c2 + ... + cn = 1






c1 y1 + c2 y2 + ... + cn yn = 0



 c y 2 + c y 2 + ... + c y 2 = 1
1 1
2 2
n n
3
3
3
3

c1 y1 + c2 y2 + ... + cn yn = 0





 c1 y14 + c2 y24 + ... + cn yn4 = 19



 ....................................
(5)
Remark 1. The quadrature formula (1) has the algebraic degree of exactness 2n − 1.
2. Now, let us consider the quadrature formula (1) for the particular case
n = 2. Taking into account of the formulas (2), (3), (4) and (5) we obtain:
a+b b−a
a+b b−a
1
+
y1 , x 2 =
+
y2 ,
c1 = c2 = , x 1 =
2
2
2
2
2
where y1 , y2 are the zeros of the polynom L2 (y) = 21 (3y 2 − 1), and R2 (f ) =
1 b − a 5 · f (IV ) (ξ), ξ ∈ (a, b).
135
2
An intermediate point property...
75
In this case we have that if f : [a, b] → R, f ∈ C 4 [a, b] then for any
x ∈ (a, b] there is cx ∈ (a, x) such that
Zx
(6)
f (t)dt =
a
1
= (x − a) f
2
1
a+x x−a
a+x x−a
−
y1 + f
+
y1 +
2
2
2
2
2
5
1
x−a
f (IV ) (cx )
+
135
2
with y12 = 31 .
We now prove the following theorem
Theorem 1. If f ∈ C 6 [a, b] and f (5) (a) 6= 0 then for the intermediate point
cx which appears in formula (6) we have
cx − a
1
=
x→a x − a
2
lim
Proof. Let us consider F, G : [a, b] → R defined by
Zx
F (x) =
1
f (t)dt−(x−a) f
2
a+x x−a
a+x x−a
1
−
y1 + f
+
y1 −
2
2
2
2
2
a
1
−
135
x−a
2
5
f (IV ) (a),
G(x) = (x − a)6 .
We have that F and G are six times derivable on [a, b], G(i) (x) 6= 0,
i = 1, 5 for any x ∈ (a, b] and F (k) (a) = 0, G(k) (a) = 0, k = 1, 5.
By using successive l, Hospital rule, we obtain
F (V I) (x)
F (x)
= lim (V I) .
lim
x→a G
x→a G(x)
(x)
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Petrică Dicu
Now, from
F
(V I)
(x) = f
(V )
(x) − 3f
(V )
a+x x−a
−
y1
2
2
1 y1
−
2
2
5
−
5
1 y1
a+x x−a
−
−3f
+
y1
+
2
2
2
2
"
6 #
(x − a) (V I) a + x x − a
1 y1
−
f
−
y1
−
−
2
2
2
2
2
"
6 #
(x − a) (V I) a + x x − a
1 y1
−
,
f
+
y1
+
2
2
2
2
2
(V )
G(V I) (x) = 6!
we obtain
F (x)
1
= F (V I) (a)
x→a G(x)
6!
lim
(7)
It is easily to see that
"
F (V I) (a) = f (V ) (a) 1 − 3
1 y1
−
2
2
5
−3
1 y1
+
2
2
5 #
=
=f
(V )
3
1
2
4
(a) 1 −
1 + 10y1 + 5y1 = f (V ) (a).
16
12
Therefore
(8)
lim
x→a
F (x)
1
=
· f (V ) (a).
G(x)
6!12
On the other hand, we have
F (x)
=
G(x)
=
1
135
x−a
2
5
f (IV ) (cx ) − f (IV ) (a)
(x − a)6
1
f (IV ) (cx ) − f (IV ) (a) cx − a
·
·
c2 − a
x−a
135 · 25
=
An intermediate point property...
77
whence
(9)
F (x)
1
cx − a
=
· f (V ) (a) · lim
.
5
x→a G(x)
x→a x − a
135 · 2
From the relation (8) and (9) we obtain
lim
cx − a
1
=
x→a x − a
2
lim
which is exactly the assertion Theorem 1.
3. The second particular case of the quadrature formula (1) is that one in
which n = 3.
The nodes and coefficients of the corresponding quadrature formula can
be obtained, from (3), (4) and (5). We find L3 (y) = 21 (5y 3 − 3y)
5
8
5
c1 = , c2 = , c3 =
18
18
18
and
x1 =
a+b b−a
a+b
a+b b−a
−
y1 , x2 =
, x3 =
+
y1
2
2
2
2
2
with y12 = 53 .
From relation (2) we obtain
1
R3 (f ) =
15750
b−a
2
7
f (6) (ξ).
In this case we have that if f : [a, b] → R, f ∈ C 6 [a, b] then for any
x ∈ (a, b] there is cx ∈ (a, x) such that
Zx
(10)
f (t)dt =
a
a+x
a+x x−a
a+x x−a
−
y1 + 8f
+
y1 +
+ 5f
2
2
2
2
2
7
1
x−a
+
f (6) (cx ).
15750
2
Our main result is contained in the following theorem.
(x − a)
=
5f
18
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Petrică Dicu
Theorem 2. If f ∈ C 8 [a, b] and f (7) (a) 6= 0 then for the intermediate point
cx which appears in formula (10) we have:
cx − a
1
= .
x→a x − a
2
lim
Proof. Let us consider F, G : [a, b] → R defined by
Zx
F (x) =
f (t)dt−
a
(x − a)
−
5f
18
a+x x−a
a+x x−a
a+x
+ 5f
−
y1 + 8f
+
y1 −
2
2
2
2
2
7
x−a
1
−
f (6) (a),
15750
2
G(x) = (x − a)8 .
Since F and G are eight times derivable on [a, b], G(i) 6= 0, i = 1, 7 for
any x ∈ (a, b] and F (k) (a) = 0, G(k) (a) = 0, k = 1, 7.
By using successive l, Hospital rule, we obtain
F (x)
F (8) (x)
= lim (8)
x→a G(x)
x→a G (x)
lim
Now, from
"
7
4
a+x x−a
1 y1
(8)
(7)
(7)
F (x) = f (x) −
5f
−
y1
−
+
9
2
2
2
2
7 #
a
+
x
1 (7) a + x
x
−
a
1
y
1
+ f
+ 5f (7)
+
y1
+
−
16
2
2
2
2
2
"
8
1 y1
1 (8) a + x
(x − a)
a+x x−a
(8)
−
y1
−
+ f
−
5f
+
18
2
2
2
2
32
2
8 #
x
−
a
1
y
a
+
x
1
+
y1
+
,
+ +5f (8)
2
2
2
2
An intermediate point property...
79
G(8) (x) = 8!
we obtain
F (x)
1
= F (8) (a),
x→a G(x)
8!
lim
(11)
where
(
" #)
7
7
1
4
y
1
y
1
1
1
+5
+
F (8) (a) = f (7) (a) 1 −
5
−
+
=
9
2
2
2
2
16
4 5
5
1
(7)
7
7
= f (a) 1 −
(1 − y1 ) + 7 (1 + y1 ) +
=
9 27
16
2
4 5
1
(7)
2
4
6
= f (a) 1 −
(1 + 21y1 + 35y1 + 7y1 ) +
=
9 26
16
4 891
1 (7)
(7)
= f (a) 1 − ·
=
f (a).
9 400
100
Hence
F (x)
1
lim
(12)
=
· f (7) (a).
x→a G(x)
100 · 8!
On the other hand
7
1
x − a (6)
f (cx ) − f (6) (a)
F (x)
15750
2
=
=
G(x)
(x − a)8
1
f (6) (cx ) − f (6) (a) cx − a
=
·
·
,
cx − a
x−a
15750 · 27
whence
F (x)
1
cx − a
=
· f (7) (a) · lim
7
x→a G(x)
x→a x − a
15750 · 2
From the relation (12) and (13) we obtain
cx − a
1
lim
= .
x→a x − a
2
4. Open problem. For intermediate point which appear in quadrature
(13)
lim
formulas of Gauss type (1) - (2) we have
ξb − a
1
lim
= ,
b→a b − a
2
any natural number n.
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Petrică Dicu
References
[1] D. Acu, About intermediate point in the mean-value theorems, The annual session of scientifical comunications of the Faculty of Sciences in
Sibiu, 17-18 May, 2001 (in romanian).
[2] D. Acu, P. Dicu, About some properties of the intermediate point in certain mean-value formulas, General mathematics, vol. 10, no. 1-2, 2002,
51-64.
[3] P. Dicu, An intermediate point property in some of classical generalized
formulas of quadrature, General Mathematics, vol. 12, no. 1, 2004, 61-70.
[4] A. Ghizzetti, A. Ossicini, Quadrature Formulae, Birkhäuser Verlag Basel
und Stuttgart, 1970.
[5] D. V. Ionescu, Numerical quadratures, Bucharest, Technical Editure,
1957 (in romanian).
[6] B. Jacobson, On the mean-value theorem for integrals, The American
Mathematical Monthly, vol. 89, 1982, 300-301.
[7] E. C. Popa, An intermediate point property in some the mean-value
theorems, Astra Matematică, vol. 1, nr. 4, 1990, 3-7 (in romanian).
”Lucian Blaga” University of Sibiu
Department of Mathematics
Str. Dr. I. Rat.iu, No. 5-7
550012 - Sibiu, Romania
E-mail address: petrica.dicu@ulbsibiu.ro