General Mathematics Vol. 12, No. 1 (2004), 35–42
Optimal combined quadrature formulas in
Schmeisser,s sense
Dumitru Acu
Abstract
In this paper we study the optimal combined quadrature formulas
in Schmeisser, s sense ([8]).
2000 Mathematical Subject Classification: 65D32
1
Combined quadrature formulas
In [1] - [3] we introduced the combined quadrature formulas.
We consider the family of elementary quadrature formulas ([3])
(1)
Zb
a
f (x)dx =
mj
n−1 X
X
[j]
Ah,i f (h) (xi,j ) + Rj (f )
h=0 i=0
with
a = x0,j ≤ x1,j < x2,j < ... < xmj,j ≤ xmj +1,j = b
35
36
Dumitru Acu
and
Rj (xk ) = 0, k = 0, 1, 2, ..., n − 1
for j = 1, 2, ..., r. It results that the elementary quadrature formulas (1)
have the algebraical degree of exactness n − 1.
Now, we divide the interval [a, b] by the points
(2)
a = u0 < u1 < ... < ur−1 < ur = b
into the subintervals [dj−1 , uj ], j = 1, r, having the length dj = uj − uj−1 ,
j = 1, r. Having in view the identity
Zb
f (x)dx =
j=1 a
a
and computing the integral
a
r Zj
X
Ruj
f (x)dx
j−1
f (x)dx with the quadrature formula j by the
dj−1
family (1) of the quadrature formulas, j = 1, r, we obtain the quadrature
formula
Zb
(3)
f (x)dx =
a
¶h+1
µ
¶
mj µ
n−1 X
r X
X
dj
xi,j − a
[j] (h)
=
Ah,i f
uj−1 + dj
+ ϕ(t)
b−a
b−a
j=1 n=0 i=0
with
(4)
µ µ
¶¶
r
X
dj
x−a
Rj f uj−1 + dj
ρ(f ) =
b−a
b−a
j=1
The rule (3) with the remainder given (4) we call it the combined quadrature formula connected to the family of the elementary
formula (1).
Optimal combined quadrature formulas in Schmeisser, s sense
37
Remark 1. Every permutation of the elementary quadrature rules by the
family (1) determines a combined quadrature formula.
Remark 2. Evidently, when the all r the rule form the family (1) coincide
with the same elementary quadrature formula, then the combined quadrature
formulas reduces to the generalized composed quadrature formulas
which was studied in [5].
Remark 3. The combined quadrature formula (3) has the algebraical degree
of exactness n − 1.
Now, we suppose the function f to be from C n [a, b] - the set of all
functions f having on the interval [a, b] continuous derivatives up to the
order n.
Theorem 1. If every influence function φj (x), (see [5]), corresponding to
the quadrature formula j, j = 1, r, by the family (1) is semidefinite and
sign φ1 (x) = sign φ2 (α) = ... = sign φr (x), for any x from [a, b], then
for f ∈ C n [a, b] the remainder of combined quadrature formula (3) has the
form:
(5)


¶n+1 Zb
r µ
X
dj
ρ(f ) = 
φj (x)dx f (n) (ξ), ξ ∈ [a, b]
b
−
a
j=1
a
Proof. From (4) and the asumation of the theorem we have:
b
µ
¶n+1 Z
¶
r µ
X
x−a
dj
(n)
uj−1 + dj
φj (x)f
dx =
ρ(f ) =
b
−
a
b
−
a
j=1
a
b
¶n+1
Z
r µ
X
dj
(n)
=
f (ξj ) φj (x)dx =
b
−
a
j=1
a
38
Dumitru Acu
¯
¯
¯
¶n+1 ¯Zb
r µ
X
¯
¯
dj
¯


φj (x)dx¯¯ f (n) (ξj )
¯
b
¶n+1 Z
b−a
r µ
¯
¯
X
j=1
dj
a


¯
¯ b
=
=
φj (x)dx ·
¯
¶n+1 ¯Z
b−a
r µ
X
j=1
¯
¯
dj
a
¯ φj (x)dx¯
¯
¯
b
−
a
¯
¯
j=1
a


¶n+1 Zb
r µ
X
dj
=
φj (x) f (n) (ξ), ξ ∈ (a, b).
b
−
a
j=1
a
2
Optimal combined quadrature formulas in
Schmeisser,s sense
From Peano, s result we have that if the influence function (Peano, s ker-
nel) is semidefinite (it has constant sign), then the remainder of quadrature
formula with the algebraical degree of exactness n − 1 has the form
Rn (f ) = Cf (n) (ξ), ξ ∈ [a, b]
(6)
(see [4], [5], [6]).
In [8] G. Schmeisser formulated the problem of finding the quadrature
formula for which C has a minimum value.
We observe that in the conditions of the Theorem 1 the remainder of
combinated quadrature formulas (3) is the form (6).
For to find the optimal combined quadrature formula in Schmeisser, s
sense among the quadrature formulas (3) we must determine the parameters
d1 , d2 , ..., dr with
r
X
dj
= 1,
b
−
a
i=1
Optimal combined quadrature formulas in Schmeisser, s sense
39
such that the expression:
b
¶n+1 Z
r µ
X
dj
|C| =
|φj (x)| dx
b
−
a
i=1
a
has a minimum value. We have a problem of conditional extremum. Using
the method of lagrange multipliers we find:
Theorem 2. If are verified the conditions of Theorem 1, then for
b−a
(7)
di = v
, i = 1, r
u b
uZ
r
X
1
u
n
t
v
|φi (x)| dx ·
u
i=1 uZb
a
u
n
t
|φj (x)| dx
a
we obtain the optimal combined quadrature rule in Schmeisser sense with
¶n+1 Zb
r µ
X
dj
min
(8)
|φj (x)| dx =
d1 ,d2 ,...,dr
b−a
j=1
a
=


X
 r


 j=1


3
1
n




1

v

u b

uZ

u
n
t |φj (x)| dx 
a
Particular cases
3.1
Generalized quadrature formulae
From Theorem 2 for the generalized quadrature formulae we find
b−a
d1 = d2 = ... = dr =
r
40
Dumitru Acu
and
min
d1 ,d2 ,...,dr
"
¶n+1 # Zb
r µ
X
dj
|φ(x)| dx =
b
−
a
j=1
a
=
1
rn
Zb
|φ(x)| dx,
a
where φ(x) are the influence functions corresponding to the quadrature
formula which generates the generalized quadrature formula.
3.2
Combined quadrature formula by type Simpson Newton
In [1] (see and [3]) we introduced a combined quadrature formula by
type Simpson - Newton.
Such, by applying the Simpson, s formula
·
µ
¶¸
Zb
b−a
a+b
f (x)dx =
f (a) + 4f
−
6
2
a
(b − a)5 (iv)
−
f (ξ1 ), ξ1 ∈ [a, b]
2880
to the interval [dj−1 , uj ], j = 1, k, 0 ≤ k ≤ r, and the Newton, s formula
Zb
a
·
µ
¶
µ
¶
¸
b−a
2(b − a)
b−a
f (a) + 3f a +
+ 3f a +
+ f (b) −
f (x)dx =
8
3
3
(b − a)5 (iv)
f (ξ2 ), ξ2 ∈ (a, b)
6480
to [uj−1 , uj ], j = k + 1, n, where f ∈ C 4 [a, b], we obtain the combined
−
quadrature formula
(9)
Zb
a
k−1
f (x)dx =
X dj + dj+1
d1
f (a) +
f (a + d1 + ... + dj )+
a
8
j=1
Optimal combined quadrature formulas in Schmeisser, s sense
k
X
2dj
+
j=1
+
µ
3
f (a + dj + ... + dj−1 +
dk dk+1
+
6
8
¶
41
dj
)+
2
f (a + d1 + ... + dk )+
r−1
X
dj + dj+1
f (a + d1 + ... + dj )+
+
8
j=k+1
µ
¶
r
X
3dj
dj
+
f a + d1 + ... + dj−1 +
+
8
3
j=k+1
¶
µ
r
X
dr
2dj
3dj
+ f (b) + ρsk ,Nr−k (f ),
f a + d1 + ... + dj−1 +
+
8
3
8
j=k+1
with
1
ρSk ,Nn−k (f ) = −
6!
Ã
k
r
1X 5 1 X 5
d +
d
4 j=1 j 9 j=k+1 j
!
f (iv) (ξ), ξ ∈ (a, b).
Using the Theorem 2 we find: among the all the combined quadrature
formulas (9) that which is optimal in Schmeisser, s sense is given by:
√
2(b − a)
√ , i = 1, k
di = √
k 2 + (r − k) 3
√
2(b − a)
√ , i = k + 1, k
di = √
k 2 + (r − k) 3
with
min
d1 ,...,dr

k
1X
1 
6! 4
j=1
d5j +
k+1
1X
9 j=1
5

dj  =
√
(b − a)5
√ , h = 0, r.
6![r 2 + (r − k) 3]4
42
Dumitru Acu
References
[1] Acu, D., On a combined quadrature formula, ST. CERC. MAT. Bucureşti, 24, 9, 1972, 1319 - 1328 (in Romanian)
[2] Acu, D., Extremal problems in the numerical integration of the functions, Doctor thesis (Cluj - Napoca), 1980 (in Romanian)
[3] Acu, D., Optimal combined quadrature formulae, Studia Univ. “Babeş
- Bolyai”, Math., XXVI, 3, 1981, 6 - 12.
[4] Engels, H., Numerical Quadrature and Cubature, Academic Press, 1980.
[5] Ghizzetti, A., and Ossicini, A., Quadrature formulae, Akademic - Verlag, Berlin 1970.
[6] Locher, F., Positivitat bei Quadraturformeln, Habilitationnchrift
in fachbereich Mathematik der Eberhard - Karls - Universität zu
Tübingen, 1972.
[7] Locher, F., Zur Struktur von Quadraturformeln, Numer. Math., 20, 317
- 326, 1973.
[8] Schmeisser, G., Optimal Quadratureformeln mit semidefiniten Karnen,
Numer. Math., 20, 1, 1972, 32 - 53.
Department of Mathematics
“Lucian Blaga” University of Sibiu
Str. Dr. I. Raţiu, nr. 5-7
550012 Sibiu, Romania.
E-mail address: depmath@ulbsibiu.ro