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])
b
Z
a
f(x)dx=
n−1
X
h=0 mj
X
i=0
A[j]h,if(h)(xi,j) +Rj(f) (1)
with
a=x0,j ≤x1,j < x2,j < ... < xmj,j ≤xmj+1,j =b 35
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 a=u0 < u1 < ... < ur−1 < ur =b (2)
into the subintervals [dj−1, uj], j = 1, r, having the length dj = uj −uj−1, j = 1, r. Having in view the identity
b
Z
a
f(x)dx=
r
X
j=1 aj
Z
aj−1
f(x)dx
and computing the integral
uj
R
dj−1
f(x)dxwith the quadrature formulajby the family (1) of the quadrature formulas, j = 1, r, we obtain the quadrature formula
b
Z
a
f(x)dx= (3)
=
r
X
j=1 n−1
X
n=0 mj
X
i=0
µ dj
b−a
¶h+1
A[j]h,if(h) µ
uj−1+dj
xi,j −a b−a
¶
+ϕ(t) with
ρ(f) =
r
X
j=1
dj
b−aRj
µ f
µ
uj−1+dj
x−a b−a
¶¶
(4)
The rule (3) with the remainder given (4) we call it the com- bined quadrature formula connected to the family of the elementary formula (1).
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 Cn[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 ∈ Cn[a, b] the remainder of combined quadrature formula (3) has the form:
ρ(f) =
r
X
j=1
µ dj
b−a
¶n+1 b
Z
a
φj(x)dx
f(n)(ξ), ξ∈[a, b]
(5)
Proof. From (4) and the asumation of the theorem we have:
ρ(f) =
r
X
j=1
µ dj
b−a
¶n+1 b
Z
a
φj(x)f(n) µ
uj−1+dj
x−a b−a
¶ dx=
=
r
X
j=1
µ dj
b−a
¶n+1
f(n)(ξj)
b
Z
a
φj(x)dx=
=
r
X
j=1
µ dj
b−a
¶n+1 b
Z
a
φj(x)dx
·
r
X
j=1
µ dj
b−a
¶n+1¯
¯
¯
¯
¯
¯
b
Z
a
φj(x)dx
¯
¯
¯
¯
¯
¯
f(n)(ξj)
r
X
j=1
µ dj
b−a
¶n+1¯
¯
¯
¯
¯
¯
b
Z
a
φj(x)dx
¯
¯
¯
¯
¯
¯
=
=
r
X
j=1
µ dj
b−a
¶n+1 b
Z
a
φj(x)
f(n)(ξ), ξ ∈(a, b).
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 exactnessn−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
i=1
dj
b−a = 1,
such that the expression:
|C|=
r
X
i=1
µ dj
b−a
¶n+1 b
Z
a
|φj(x)|dx
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
di = b−a
n
v u u u t
b
Z
a
|φi(x)|dx·
r
X
i=1
1
n
v u u u t
b
Z
a
|φj(x)|dx
, i= 1, r (7)
we obtain the optimal combined quadrature rule in Schmeisser sense with
d1,dmin2,...,dr
r
X
j=1
µ dj
b−a
¶n+1 b
Z
a
|φj(x)|dx= (8)
= 1
r
X
j=1
1
n
v u u u t
b
Z
a
|φj(x)|dx
n
3 Particular cases
3.1 Generalized quadrature formulae
From Theorem 2 for the generalized quadrature formulae we find d1 =d2 =...=dr = b−a
r
and
d1,dmin2,...,dr
" r X
j=1
µ dj
b−a
¶n+1# b Z
a
|φ(x)|dx=
= 1 rn
b
Z
a
|φ(x)|dx,
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
b
Z
a
f(x)dx= b−a 6
·
f(a) + 4f
µa+b 2
¶¸
−
−(b−a)5
2880 f(iv)(ξ1), ξ1 ∈[a, b]
to the interval [dj−1, uj], j = 1, k, 0≤k ≤r, and the Newton,s formula
b
Z
a
f(x)dx= b−a 8
·
f(a) + 3f µ
a+b−a 3
¶ + 3f
µ
a+ 2(b−a) 3
¶
+f(b)
¸
−
−(b−a)5
6480 f(iv)(ξ2), ξ2 ∈(a, b)
to [uj−1, uj], j = k+ 1, n, where f ∈ C4[a, b], we obtain the combined quadrature formula
b
Z
a
f(x)dx= d1
a f(a) +
k−1
X
j=1
dj+dj+1
8 f(a+d1+...+dj)+
(9)
+
k
X
j=1
2dj
3 f(a+dj +...+dj−1+dj
2)+
+ µdk
6 +dk+1 8
¶
f(a+d1+...+dk)+
+
r−1
X
j=k+1
dj +dj+1
8 f(a+d1 +...+dj)+
+
r
X
j=k+1
3dj
8 f µ
a+d1 +...+dj−1+ dj
3
¶ +
+
r
X
j=k+1
3dj
8 f µ
a+d1+...+dj−1+2dj
3
¶ +dr
8f(b) +ρsk,Nr
−k(f), with
ρSk,Nn−k(f) =−1 6!
Ã1 4
k
X
j=1
d5j + 1 9
r
X
j=k+1
d5j
!
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:
di =
√2(b−a) k√
2 + (r−k)√
3, i= 1, k di =
√2(b−a) k√
2 + (r−k)√
3, i=k+ 1, k with
d1min,...,dr
1 6!
1 4
k
X
j=1
d5j +1 9
k+1
X
j=1 5
dj
= (b−a)5 6![r√
2 + (r−k)√
3]4, h= 0, r.
References
[1] Acu, D., On a combined quadrature formula, ST. CERC. MAT. Bu- cure¸sti, 24, 9, 1972, 1319 - 1328 (in Romanian)
[2] Acu, D., Extremal problems in the numerical integration of the func- tions, Doctor thesis (Cluj - Napoca), 1980 (in Romanian)
[3] Acu, D., Optimal combined quadrature formulae, Studia Univ. “Babe¸s - 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 - Ver- lag, Berlin 1970.
[6] Locher, F., Positivitat bei Quadraturformeln, Habilitationnchrift in fachbereich Mathematik der Eberhard - Karls - Universit¨at zu T¨ubingen, 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. Rat¸iu, nr. 5-7
550012 Sibiu, Romania.
E-mail address: [email protected]
