General Mathematics Vol. 14, No. 2 (2006), 109–119
Optimal Quadrature Formulas in the Sense
of Nikolski 1
Ana Maria Acu
Abstract
In this paper we will perform a method to obtain a quadrature
formulas and we will study the optimality in sense of Nikolski for this
quadrature formulas.
2000 Mathematics Subject Classification: 26D15, 65D30
Key words and phrases: quadrature rule, numerical integration, error
bounds, optimal quadrature
1
Introduction
Let H be a linear space of real valued functions , defined and integrable
on a finite interval [a, b] ⊂ R and S : H → R be the integration operator
defined by
Z
b
S[f ] =
f (x)dx .
a
1
Received June 4, 2006
Accepted for publication (in revised form) July 7, 2006
109
110
Ana Maria Acu
Let
©
ª
Λ = λi | λi : H → R, i = 1, n
be a set of linear functionals. For f ∈ H, one considers the quadrature
formula
(1)
S[f ] = Qn [f ] + Rn [f ]
where
Qn [f ] =
n
X
Ai λi (f )
i=1
and Rn [f ] denotes the remainder term.
Remark 1.1. Usually , λi (f ), i = 1, n are the values of the function f or
of certain of its derivatives on the quadrature nodes from [a, b].
Definition 1.1. The quadrature formula (1) is called optimal in the sense
of Nikolski in the space H , if
Fn (H, A, X) = sup |Rn [f ]| ,
f ∈H
attains the minimum value with regard to A and X , where A = (A1 , . . . , An )
are the coefficients and X = (x1 , . . . , xn ) are the quadrature nodes.
2
Main Results
Let (∆m )m∈N be a division of [a, b] ,
∆m : a = x0 < x1 < x2 < · · · < xm−1 < xm = b
and (ξi )i=1,m a system of intermediate points ,
ξ1 < ξ2 < · · · < ξm , ξi ∈ [xi−1 , xi ] for i = 1, m .
Optimal Quadrature Formulas in the Sense of Nikolski
111
Theorem 2.1. If f ∈ C n−1 [a, b] , f (n−1) is absolutely continous , then
Z
b
(2)
f (t)dt =
a
n−1 X
m
X
Ak,i f (k) (xi ) + Rn [f ]
k=0 i=0
where
Z
b
n
(3)
Rn [f ] = (−1)
Kn (t)f (n) (t)dt
a

n

 (t − ξi ) , t ∈ [xi−1 , xi ) , i = 1, m − 1
n! n
Kn (t) =
(4)

 (t − ξm ) , t ∈ [xm−1 , b]
n!
and for k = 0, n − 1
(xi − ξi )k+1 − (xi − ξi+1 )k+1
, i = 1, m − 1
(k + 1)!
(a − ξ1 )k+1
Ak,0 = (−1)k+1
(k + 1)!
(b
−
ξm )k+1
Ak,m = (−1)k
(k + 1)!
Ak,i = (−1)k
Proof. We prove (2) by induction. For n = 1 we have
"m−1 Z
#
Z b
Z b
X xi
K1 (t)f 0 (t)dt = −
(t − ξi )f 0 (t)dt +
(t − ξm )f 0 (t)dt =
−
a
xi−1
i=1
=−
"m−1
X
i=1
¯xi
¯
(t − ξi )f (t)¯
xi−1
xm−1
¯b
¯
+ (t − ξm )f (t)¯
xm−1
Z
b
−
#
f (t)dt =
a
"m−1
X
=−
(xi − ξi )f (xi )−
i=1
−
# Z
b
(xi−1 − ξi )f (xi−1 ) + (b − ξm )f (b) − (xm−1 − ξm )f (xm−1 ) +
f (t)dt =
m−1
X
i=1
=−
a
"m−1
X
i=1
#
((xi − ξi ) − (xi − ξi+1 )) f (xi ) − (a − ξ1 )f (a) + (b − ξm )f (b)
112
Ana Maria Acu
Z
b
+
f (t)dt = −
a
Therefore
Z
b
a
a
i=1
i=1
m
X
xi
xi−1
b
A0,i f (xi ) +
f (t)dt .
a
Z
b
A0,i f (xi ) −
K1 (t)f 0 (t)dt
a
i=0
For n = 2 we have
Z b
m−1
XZ
00
K2 (t)f (t)dt =
=
Z
i=0
f (t)dt =
m−1
X
m
X
(t − ξi )2 00
f (t)dt +
2!
Z
b
xm−1
(t − ξm )2 00
f (t)dt
2!
Z b
(t − ξm )2 0 ¯¯b
(t − ξi )2 0 ¯¯xi
+
−
K1 (t)f 0 (t)dt
f (t)¯
f (t)¯
2!
2!
xi−1
xm−1
a
=
m−1
X
i=1
m−1
X (xi−1 − ξi )2
(xi − ξi )2 0
f (xi ) −
f 0 (xi−1 )+
2!
2!
i=1
(b − ξm )2 0
(xm−1 − ξm )2 0
f (b) −
f (xm−1 )−
2!
2!
Z b
m−1
X (xi − ξi )2 − (xi − ξi+1 )2
0
f 0 (xi )−
−
K1 (t)f (t)dt =
2!
a
i=1
+
(a − ξ1 )2 0
(b − ξm )2 0
f (a) +
f (b)−
2!
2!
Z b
Z b
m
m
X
X
0
0
f (t)dt .
K1 (t)f (t)dt = −
A1,i f (xi ) −
A0,i f (xi ) +
−
−
a
i=0
a
i=0
Therefore
Z
b
f (t)dt =
a
1 X
m
X
Z
Ak,i f
(k)
b
(xi ) +
K2 (t)f 00 (t)dt .
a
k=0 i=0
Now suppose that (2) holds for an arbitrary n. We have to prove that (3)
holds for n → n + 1. We have
"m−1 Z
Z b
X
Kn+1 (t)f (n+1) (t)dt = (−1)n+1
(−1)n+1
a
i=1
xi
xi−1
(t − ξi )n+1 (n+1)
f
(t)dt +
(n + 1)!
Optimal Quadrature Formulas in the Sense of Nikolski
113
¸
(t − ξm )n+1 (n+1)
+
f
(t)dt =
(n + 1)!
xm−1
"m−1
¯xi
X (t − ξi )n+1
¯
= (−1)n+1
f (n) (t)¯ +
(n
+
1)!
xi−1
i=1
Z
b
¸
Z b
(t − ξm )n+1 (n) ¯¯b
(n)
f (t)¯
+
−
Kn (t)f (t)dt =
(n + 1)!
xm−1
a
"m−1
m−1
X (xi − ξi )n+1
X (xi−1 − ξi )n+1
n+1
(n)
= (−1)
f (xi ) −
f (n) (xi−1 ) +
(n + 1)!
(n + 1)!
i=1
i=1
(b − ξm )n+1 (n)
f (b) −
(n + 1)!
¸
Z b
(xm−1 − ξm )n+1 (n)
n
f (xm−1 ) + (−1)
Kn (t)f (n) (t)dt =
−
(n + 1)!
a
+
n+1
= (−1)
m−1
X
i=1
(xi − ξi )n+1 − (xi − ξi+1 )n+1 (n)
(a − ξ1 )n+1 (n)
f (xi )+(−1)n
f (a)+
(n + 1)!
(n + 1)!
− ξm )n+1 (n)
f (b) + (−1)n
(n + 1)!
n+1 (b
+(−1)
=−
m
X
An,i f
(n)
i=0
(xi ) −
m
n−1 X
X
Z
b
Kn (t)f (n) (t)dt =
a
Z
Ak,i f
(k)
b
(xi ) +
f (t)dt .
a
k=0 i=0
Therefore
Z b
Z b
n X
m
X
(k)
n+1
f (t)dt =
Ak,i f (xi ) + (−1)
Kn+1 (t)f (n+1) (t)dt .
a
k=0 i=0
a
Remark 2.1. The quadrature formulas of type (2) with equidistant knots
had obtain from this method in [1], [2], [3], [5], [6], [7], [9], [10].
Remark 2.2. If ξ1 = a and ξm = b then quadrature formula (2) is open
type.
114
Ana Maria Acu
Next, we will study the optimality in sense of Nikolski for this quadrature
¯
n
o
¯
n,p
n−1
(n)
p
formulas. Let H [a, b] = f : [a, b] → R¯f ∈ C [a, b] , f ∈ L [a, b] .
If f ∈ H n,p [a, b] for rest term we have the evaluation
£
¤1
|Rn [f ]| ≤ Mn[p] [f ] p
(5)
where
Z
Mn[p] [f ]
b
=
a
·Z
b
|Kn (t)|q dt
¸ 1q
a
¯ (n) ¯p
¯f (t)¯ dt , 1 + 1 = 1
p q
with remark that in cases p = 1 and p = ∞ this evaluation is
|Rn [f ]| ≤ Mn[1] [f ] sup |Kn (t)|
(6)
t∈[a,b]
Z
(7)
|Rn [f ]| ≤
where
Mn∞ [f ]
b
|Kn (t)| dt
a
R b ¯ (n) ¯
¯f (t)¯ dt
a
¯
¯
Mn∞ [f ] = sup ¯f (n) (t)¯ .
[1]
Mn [f ] =
t∈[a,b]
The quadrature formula (2) is optimal in the sense of Nikolski in H n,p [a, b],
if
Z
b
|Kn (t)|q dt ,
a
1 1
+ =1
p q
attains the minimum value.
Theorem 2.2. If f ∈ H n,p [a, b] , p > 1 ,then quadrature formula of the
form (2), optimal with regard to the error, is
Z
b
f (x)dx =
a
n−1 X
m
X
k=0 i=0
A∗k,i f (k) (x∗i ) + R∗n [f ]
Optimal Quadrature Formulas in the Sense of Nikolski
where , for k = 0, n − 1
(b − a)k+1
2k+1 mk+1 (k + 1)!
(b − a)k+1
A∗k,i = [1 + (−1)k ] k+1 k+1
, i = 1, m − 1
2 m (k + 1)!
(b − a)k+1
A∗k,m = (−1)k k+1 k+1
2 m (k + 1)!
b−a
∗
xi = a +
i , i = 1, m − 1
m
A∗k,0 =
with
|R∗n [f ]|
(b − a)n
≤
·
(2m)n n!
µ
b−a
qn + 1
¶ 1q
£
¤1
· Mn[p] [f ] p .
Proof. We will determine the parameters A and X for which
Z b
F (A, X) =
|Kn (t)|q dt
a
attains the minimum value . We have
¯
Z b ¯
m−1
X Z xi ¯¯ (t − ξi )n ¯¯q
¯ (t − ξm )n ¯q
¯
¯
¯ dt
¯
F (A, X) =
¯ n! ¯ dt +
¯
¯
n!
x
x
m−1
i−1
i=1
" m
#
m
X
X
1
=
(xi − ξi )qn+1 +
(ξi − xi−1 )qn+1 .
(qn + 1)(n!)q i=1
i=1
The optimal nodes constitute the solution of the system

∂F (A, X)
1


=
[(xk − ξk )qn − (ξk+1 − xk )qn ] = 0 , k = 1, m − 1

q
∂x
(n!)
k
(8)
1
∂F (A, X)


=
[−(xk − ξk )qn + (ξk − xk−1 )qn ] = 0 , k = 1, m

∂ξk
(n!)q
From (8) we obtain
(9)
(10)
ξk =
xk−1 + xk
, k = 1, m
2
xk+1 − 2xk + xk−1 = 0 , k = 1, m − 1 .
115
116
Ana Maria Acu
From recurrent relation (10) we obtain
(11)
xk = a +
b−a
k , k = 1, m − 1
m
From (9) and(11) follows that
(b − a)k+1
2k+1 mk+1 (k + 1)!
(b − a)k+1
Ak,i = [1 + (−1)k ] k+1 k+1
, i = 1, m − 1
2 m (k + 1)!
(b − a)k+1
Ak,m = (−1)k k+1 k+1
.
2 m (k + 1)!
Ak,0 =
Because the quadratic form
φ=
m−1
m
XX
i=1 j=1
+
m−1
X m−1
X
i=1 j=1
m m−1
X X ∂ 2 F (A, X)
∂ 2 F (A, X)
ai bj +
bj ai
∂xi ∂ξj
∂ξj ∂xi
j=1 i=1
m
m
X X ∂ 2 F (A, X)
∂ 2 F (A, X)
ai aj +
bi bj
∂xi xj
∂ξi ∂ξj
i=1 j=1
in cross point (A, X) is positive, namely
)
µ
¶qn−1 (m−1
X£
¤
b−a
qn
φ=
·
·
(ai − bi )2 + (ai − bi+1 )2 + b21 + b2m
(n!)q
2m
i=1
then F (A, X) attains the minimum value for the knots X ∗ = (x∗i )i=1,m−1
m
and coefficients A∗ = (A∗k,i )n−1
k=0 i=0 , where
b−a
i , i = 1, m − 1
m
(b − a)k+1
A∗k,0 = k+1 k+1
2 m (k + 1)!
(b − a)k+1
A∗k,i = [1 + (−1)k ] k+1 k+1
, i = 1, m − 1
2 m (k + 1)!
(b − a)k+1
A∗k,m = (−1)k k+1 k+1
2 m (k + 1)!
x∗i = a +
Optimal Quadrature Formulas in the Sense of Nikolski
117
Finally, we have
F (A∗ , X ∗ ) = min F (A, X) =
A,X
(b − a)qn+1
,
(qn + 1)(n!)q (2m)qn
and
¶ 1q
µ
n
£ [p] ¤ p1
(b
−
a)
b
−
a
|R∗n [f ]| ≤
·
·
Mn [f ] .
(2m)n n!
qn + 1
In this way we prove follow result:
Theorem 2.3. The quadrature formula of the form (2) is optimal in the
sense of Nikolski for p = ∞ if it has the coefficients and knots
(b − a)k+1
= k+1 k+1
2 m (k + 1)!
(b − a)k+1
A∗k,i = [1 + (−1)k ] k+1 k+1
, i = 1, m − 1
2 m (k + 1)!
(b − a)k+1
A∗k,m = (−1)k k+1 k+1
2 m (k + 1)!
b
−
a
x∗i = a +
i , i = 1, m − 1
m
A∗k,0
and there is for rest term evaluation
|R∗n [f ]| ≤
(b − a)n+1
· Mn∞ [f ] .
(n + 1)!(2m)n
The optimal quadrature formulas had obtain by S.M. Nikolski (see [8]).In
[4] T. Cătinaş and G. Coman obtain the optimal quadrature formulas using
ϕ- function method.
For example if f ∈ H 1,2 [0, 1] then quadrature formula of the form (2),
optimal with regard to the error, is
"
#
Z 1
m−1
X µi¶
1
· f (0) + 2
f
f (x)dx =
+ f (1) + R∗1 [f ] ,
2m
m
0
i=1
where
|R∗1 [f ]| ≤
1
√ kf 0 k2 .
2m 3
118
Ana Maria Acu
For f ∈ H 2,2 [0, 1] we have
"
#
Z 1
m−1
X µi¶
1
1
1
f (x)dx =
· f (0) + 2
f
+ f (1) + 2 f 0 (0)− 2 f 0 (1)+R∗2 [f ] ,
2m
m
8m
8m
0
i=1
where
|R∗2 [f ]| ≤
1
√ kf 00 k2 .
8m2 5
References
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[2] A. M. Acu, Some News Quadrature Rules of Close Type , AAMA (to
appear).
[3] G. A. Anastassiou,
Ostrowski Type Inequalities,
Proc. Amer.
Math.Soc.,Vol 123,No 12,(1995),3775-3781.
[4] T. Cătinaş, G. Coman, Optimal Quadrature Formulas Based on the ϕfunction Method, Studia Univ. ”Babes-Bolyai”, Mathematica, Volume
LII, Number 6,January 2005.
[5] P. Cerone, S. S. Dragomir, Midpoint-type Rules from an Inequalities
Point of View,Handbook of Analytic-Computational Methods in Applied Mathematics, Editor: G. Anastassiou, CRC Press, New York,
(2000),135-200.
[6] P. Cerone, S. S. Dragomir, Trapezoidal-type Rules from an Inequalities Point of View,Handbook of Analytic-Computational Methods in
Applied Mathematics, Editor: G. Anastassiou, CRC Press, New York,
(2000),65-134.
Optimal Quadrature Formulas in the Sense of Nikolski
119
[7] Lj. Dedić, M. Matić, J. Pečarić, On Euler trapezoid formulae,Appl.
Math. Comput.,123(2001),37-62.
[8] S. M. Nikolski, Formule de cuadratura,Editura tehnica, Bucuresti , 1964.
[9] C. E. M. Pearce, J. Pečarić, N. Ujević, S. Varošanec, Generalizations of
some inequalities of Ostrowski-Grüss type, Math. Inequal. Appl., 3(1),
(2000), 25-34.
[10] N. Ujević, Error Inequalities for a Generalized Quadrature Rule , General Mathematics, Vol. 13, No. 4(2005), 51-64.
University ”Lucian Blaga” of Sibiu
Department of Mathematics
Str. Dr. I. Rat.iu, No. 5-7
550012 - Sibiu, Romania
E-mail address: acuana77@yahoo.com