General Mathematics Vol. 11, No. 3–4 (2003), 35–44
On the Tricomi,s quadrature formula
Dumitru Acu
Dedicated to Professor Gheorghe Micula on his 60th birthday
Abstract
In this paper we obtain new results concerning the Tricomi, s
quadrature formula.
2000 Mathematical Subject Classification: 65D32
In [10] F. Tricomi introduced a interesting quadrature rule, analogous
to famous Richardson, s method [5].
The paper [8], [9], [1] complete with new results the Tricomi, s paper.
In this article we present some new results concerning the generalization
of the Tricomi, s quadrature formula, given in [1].
1. Let f be a function from AC n−1 [a, b] - the class of functions f whose
(n − 1)-th derivative f (n−1) is absolutely continuous in [a, b].
35
36
Dumitru Acu
We consider the elementary quadrature formula
Zb
(1)
f (x)dx =
n−1 X
m
X
(k)
Aki f(xi ) + R(f ),
k=0 i=1
a
with the algebraical degree of exactness n − 1 and with the remainder given
by
Zb
(2)
R(f ) =
(n)
f(ξ)
φ(x)dx, ξ ∈ (a, b),
a
where φ is the influence function ([7]).
We divide the interval [a, b] into a certain number p partial intervals
[uj−1 , uj ], j = 1, p, defined by the points
(3)
a = u0 < u1 < ... < up−1 < up = b
and we put dj = uj − uj−1 , j = 1, p.
We evaluate the integral
Zb
I=
f (x)dx,
a
using the generalized quadrature formula generated to the elementary formula (1) (see [7]). We obtain
(4)
with
(5)
I = Sp + Rp
¶k+1
µ
¶
p n−1 m µ
X
XX
dj
xj − a
(k)
Aki f
uj−1 + dj
Sp =
b−a
b−a
j=1 k=0 i=1
and the remainder
(6)
¶n+1 #
p µ
X
dj
(n)
f(ξ1 ) , ξ1 ∈ [a, b),
Rp = T
b−a
j=1
"
On the Tricomi, s quadrature formula
where
37
Zb
(7)
T =
φ(x)dx.
a
Now we divide the interval [a, b] into a certain number q, q > p, partial
intervals [vj−1 , vj ], j = 1, q, defined by the points
(8)
a = v0 < v1 < ... < vq−1 < vq = b,
with cj = vj − vj−1 , j = 1, q.
We evaluate the integral I using the generalized quadrature formula generated to the elementary formula (1) and the partition (8). We find:
(9)
I = Sq + Rq ,
with
(10)
¶k+1
µ
¶
q n−1 m µ
X
XX
cj
xi − a
(k)
Sq =
Aki f
vj−1 + cj
b−a
b−a
j=1 k=0 i=1
and the remainder
(11)
" q µ
¶n+1 #
X
cj
Rq = T
f (n) (ξ2 ), ξ2 ∈ (a, b).
b
−
a
j=1
If the function f is a polynomial of degree ≤ n, then f (n) (ξ1 ) = f (n) (ξ2 ).
In this situation, if we put

µ
¶n+1
p
P
dj


D=


b−a

j=1

(12)
and


´
p ³

P

cj n+1

 C=
,
b−a
j=1
then from (4), (6), (9), (11) it results
(13)
T f (n) (ξ1 ) = T f (n) (ξ2 ) =
Sq − Sp
.
D−C
38
Dumitru Acu
Using (9), (11) şi (13), we get
tn
(Sq − Sp ),
I = Sq +
1 − tn
(14)
where t =
p
n
C/D.
Now, for an arbitrary function f ∈ AC n−1 [a, b], we consider the quadarture formula
(15)
I = Sq +
tn
(Sq − Sp ) + Rp,q ,
1 − tn
where Sq , Sp and t are the above.
From (15), (4) and (9), we have
Rp,q =
1
(Rq − tn Rp ),
1 − tn
whence, using (6) and (11), we find the following forms for the remainder
of the formula (15):
(16)
Rp,q =
TC
(n)
(n)
n [f(ξ2 ) − f(ξ1 ) ],
1−t
(17)
Rp,q =
T Dtn (n)
(n)
[f − f(ξ1 ) ],
1 − tn (ξ2 )
(18)
Rp,q =
T CD (n)
(n)
[f(ξ2 ) − f(ξ1 ) ],
D−C
ξ1 , ξ2 ∈ (a, b)
In view of (16), (17) and (18), it results
|T |C (n)
Ω ,
1 − tn
(19)
|Rp,q | ≤
(20)
|T |Dtn (n)
Ω ,
|Rp,q | ≤
1 − tn
On the Tricomi, s quadrature formula
(21)
|Rp,q | ≤
39
|T |CD n
Ω ,
D−C
where Ω(n) is the oscillation of the function f (n) on [a, b].
If we take into account that for T usually have a relation by form
Zb
φ(x)dx| = |α||b − a|n+1 ,
|T | = |
a
where α is a constant, then from (9) - (21), we have
à p
!
X
|α|
Cjn+1 Ω(n)
(22)
j=1
|Rp,q | ≤
Ã
|α|
(23)
(24)
|Rp,q | ≤
p
X
à p
X
j=1
dn+1
Ω(n)
j
dn+1
j
1 − tn
!Ã q
X
j=1
p
X
!
j=1
|Rp,q | ≤
|α|
,
1 − tn
dn+1
−
j
,
!
Cjn+1
j=1
n
X
cn+1
j
j=1
· Ω(n) .
2. Particular cases.
2.1 If we consider

a


d = d2 = ... = db = b −

p = ωp
 1
(25)
and



 c = c = ... = c = b − a = ω ,
1
2
q
q
q
then t = p/q.
40
Dumitru Acu
In this case, the formula (15) presents a generalization of the results
from [8] for the quadrature formulas which they contain the values of the
derivative the function f on the nodes.
For the remainder, from (23) we found
(26)
|Rp,q | ≤
|α|pωpn+1
tn
(n)
.
nΩ
1−t
For q = 2p we obtain a generalization of the Tricomi, s results.
2.2 If (25) are satisfied and the elementary quadrature formula (1)
contains only the values of the function f on the nodes, then (15) and (22)
- (24) give the results by [8]. For q = 2p we obtain the Tricomi, s results
([10]).
Examples.
In this section we present some examples of quadrature
formula by Tricomi, s type obtained by the particularizing of the elementary
rule (1).
3.1.
Assume that (1) is the rectangular formula ([7]). We find the
following Tricomi, s quadrature formula:
I = Sq +
with
Sp =
p
X
t2
(Sq − Sp ) + Rp,q
1 − t2
dj (a + d1 + d2 + ... + dj−1 +
j=1
Sq =
q
X
cj (a + c1 + c2 + ... + cj−1 +
j=1
vÃ
!
! Ã q
u q
X
u X
d3 ,
c3 /
t=t
j
j
j=1
j=1
dj
),
2
cj
),
2
On the Tricomi, s quadrature formula
1
|Rp,q | ≤
24
à p
X
41
!
d3j
t2
Ω(2) ,
1 − t2
·
j=1
where Ω(2) is the oscillation of the function f 00 on [a, b].
a
From here, for d1 = ... = dp = b − a = ωp , c1 = c2 = ... = cq = b −
q = ωq
b
we find the results by [8].
3.2 We consider (1) the Simpson, s formula.
It results the following Tricomi, s quadrature formula
I = Sq +
with
t4
(Sq − Sp ) + Rp,q ,
1 − t4
p−1
X dj + dj+1
dj
Sp = f (a) +
(a + d1 + ... + dj )+
2
6
j=1
µ
¶
p
X
2dj
dj
dp
+
f a + d1 + ... + dj−1 +
+ f (b),
3
3
6
j=1
q−1
X cj + cj+1
c1
f (a + c1 + ... + cj ) +
Sq = f (a) +
6
6
j=1
q
X
2cj ³
cj ´ Cq
+
f a + c1 + ... + cj+1 +
+
f (b),
3
3
6
j=1
vÃ
! Ã p
!
u q
X
u X
4
t=t
c5j /
d5j ,
j=1
|Rp,q | ≤
j=1
j=1
!
d5j
t4
(4)
4Ω ,
1−t
is the oscillation of the function f (4) on [a, b].
− a = 2ω and c = c = ... =
From here, for d1 = d2 = ... = dp = 2 b 2p
2p
1
2
2(b − a)
cq =
2q , we obtain the results by [8]. If q = 2p then we find the case
studied by Tricomi, s ([10]).
where Ω
(4)
1
2880
à p
X
42
Dumitru Acu
3.3. Finally, we assume that (1) is Newton, s quadrature formula ([2],
[3]).
We obtain the following Tricomi, s quadrature formula:
I = Sq +
with
t4
(Sq − Sp ) + Rp,q
1 − t4
p−1
X dj + dj+1
di
Sp = f (a) +
f (a + d1 + ... + dj )+
8
8
j=1
p
X 3dj
dp
+ f (b) +
f
8
8
j=1
µ
dj
a + d1 + ... + dj−1 +
3
¶
+
µ
¶
p
X
3dj
2dj
+
,
f a + d1 + ... + dj−1 +
8
3
j=1
q−1
X cj + cj+1
ci
Sq = f (a) +
f (a + c1 + ... + cj )+
8
8
j=1
X 3cj ³
cj ´
cq
+
f a + c1 + ... + cj−1 +
+ f (b) +
8
8
3
j=1
q
µ
¶
q
X
3cj
2cj
+
f a + c1 + ... + cj−1 +
,
8
3
j=1
vÃ
! Ã p
!
u q
X
u X
4
t=t
c5j /
d5j ,
j=1
1
|Rp,q | ≤
6480
Ã
j=1
p
X
j=1
!
d5j
t4
(4)
4Ω ,
1−t
where Ω(4) is the oscillation of the function f (4) on the [a, b].
On the Tricomi, s quadrature formula
43
References
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Buletinul Ştiinţific al Institutului de Învăţământ Superior din Sibiu,
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[3] Acu, D., O generalizare a formulei de quadratură a lui Newton, St.
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14, 2, 1969, 53 - 58 (in Romanian).
[6] Engels, H., Numerical Quadrature and Cubature, Academic Press, 1980
[7] Ghizzetti, A., Ossicioni, A., Quadrature formulae, Acad.-Verlag-Berlin,
1970
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[9] Micula, M., Micula, Gh., Sur la formule de quadrature de la Tricomi,
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(in Romanian).
44
Dumitru Acu
[10] Tricomi, F., Sul resto delle formule di quadrature numerica migliorate
con metode “extrapolazione”, Boll. dell Unione Math. Ital., 3, 14, 1959,
102 - 104 (in Romanian).
Department of Mathematics
“Lucian Blaga” University of Sibiu
Str. Dr. I. Raţiu, nr. 5-7
550012 Sibiu, Romania.
E-mail address: depmath@ulbsibiu.ro