J. Comp. & Math. Sci. Vol.3 (2), 207-215 (2012)
A Mixed Quadrature Rule by Blending Clenshaw-Curtis and Lobatto Quadrature Rules for Approximation of Real Definite Integrals in Adaptive Environment RAJANI B. DASH and DEBASISH DAS Department of Mathematics, Ravenshaw University, Cuttack-753003, Odisha, India (Received on : March 26, 2012) ABSTRACT A mixed quadrature rule blending Clenshaw- Curtis five point rule and Lobatto four point rule is formed. This mixed rule of precision 7 has been tested in adaptive environment and it is found to be more effective than its constituent Clenshaw-Curtis five point rule. This rule is also more effective than Gauss-Legendre four point rule which is of precision 7. Keywords: Clenshaw-Curtis 5- point rule ( RCC5 ( f ) ) , Lobatto 4point rule
( RLob4 ( f ) ) ,
mixed quadrature rule ( RCC5 Lob4 ( f ) ) ,
Gauss-Legendre 4-point rule ( RGl4 ( f ) ) , Adaptive quadrature.
1. INTRODUCTION Real definite integrals of the type b
I ( f ) = âˆŤ f ( t ) dt
(1.1)
a
have been successfully approximated by several Authors9,10,11 by applying the mixed quadrature rule. The method involves construction of a symmetric quadrature rule of higher precision as a linear combination of two other rules of equal lower precision.
If we consider a Lobatto rule and a Clenshaw-Curtis rule having same precision, Clenshaw-Curtis rule is better than Lobatto rule. An n-point Lobatto rule is of precision 2n-3, while the precision of an n-point Clenshaw-Curtis rule is n. In general, Lobatto type rule is of higher precision than that of Clenshaw-Curtis type rule when same number of abscissae are used. In this paper, taking the advantage of the fact that Lobatto 4-point rule and Clenshaw-Curtis 5- point rule are of same precision (i.e. precision 5), we formed a
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mixed quadrature rule of higher precision (i.e. precision 7) taking linear combination of these rules. The mixed rule so formed has been tested on different integrals giving better results than its constituent ClenshawCurtis quadrature rule in adaptive environment. This rule is also more effective than Gauss-Legendre four point rule which is of precision 7. 2. THE CLENSHAW-CURTIS QUADRATURE RULE The Clenshaw-Curtis method4 essentially approximates a function f(t) over any interval [α − h, α + h ] using the Chebyshev polynomials Tr ( x ) of degree n n
f ( t ) = F ( x ) = ∑ ' ar Tr ( x ) r =0
( −1 ≤ x ≤ 1)
(2.1)
where, ar are the expansion coefficients and ∑ ' denotes a finite sum whose first term is to be halved before beginning to sum. That is, F ( x) =
1 a0T0 ( x ) + a1T1 ( x ) + a2T2 ( x ) + ... + anTn ( x ) 2
(
Tr ( xi ) = cos r cos −1 ( xi )
)
r≥0
(2.4)
riπ = cos n
Then n n n ∑ '' f ( α + hxi ) Tr ( xi ) = ∑ '' ∑ ' ak Tk ( xi ) Tr ( xi )
i =0
i = 0 k =0
n n kiπ riπ = ∑ ' ak ∑ '' cos ⋅ cos n n k =0 i =0
(2.5)
The notation ∑ '' means that the first and last terms are to be halved before summation begins. The orthogonality of the cosine function5 with respect to the points iπ xi = cos n
is expressed by
n n π π ki ri ∑ '' cos ⋅ cos = n n 2 i =0 0 n
r = k = 0 or n 0<r =k < n
(2.6)
r≠k
Substituting Eq(2.6) into Eq(2.5), gives
(2.2) Collocating with points, iπ xi = cos n
,
f ( α + hx )
at the n + 1
( i = 0,1,..., n )
(2.3)
one can evaluate the expansion coefficients ar . The Chebyshev polynomials expressed as
Tr ( xi )
can be
n ar n ∑ '' f ( α + hxi )Tr ( xi ) = ar i =0 2 0 n
r=k =n 0≤r =k < n r≠k
Hence 2 n '' ∑ f ( α + hxi ) Tr ( xi ) n i =0 ar = n 1 '' f α + hx T x ( i) r ( i) n ∑ i =0
( r = 0,1,...., n − 1) (r = n)
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(2.7)
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Rajani B. Dash, et al., J. Comp. & Math. Sci. Vol.3 (2), 207-215 (2012)
Denoting the integral of f ( t ) over the interval [α − h, α + h ] by I, and replacing t by α + hx , we get
−1
r =0
−1
r =0
1 2 n '' ∑ f ( α + hxi ) Tr ( xi ) ∫ Tr ( x ) dx n i =0 −1
Since 1
−2
−1
r2 −1
( r = even ) ,
n
Assuming I ≈ I n , we write 1
''
∫ Tr ( x ) dx =
1
I = h ∫ f ( α + hx ) dx
n
n
In = h ∑
we get I n = h ∑ '' wi f ( α + hxi ) i =0
I n = h ∑ ' ar ∫ Tr ( x ) dx
(2.8)
(2.9(a))
where
Substituting the values of ar (as given in Eq 2.8), we get
wi = −
4 n '' 1 Tr ( xi ) ∑ n r =0 r 2 − 1
( i = 0,1,. . ., n) (2.9(b))
r =even
With n = 4, I4 =
h h h f (α + h) + 8 f α + + 12 f ( α ) + 8 f α − + f ( α − h) 15 2 2
(2.10)
3. CONSTRUCTION OF THE MIXED QUADRATURE RULE OF PRECISION SEVEN We choose the Clenshaw-Curtis five point rule 1
I ( f ) = ∫ f ( x ) dx ≈ RCC5 ( f ) = −1
1 ( ) −1 1 f −1 + 8 f + 12 f ( 0 ) + 8 f + f (1) 15 2 2
(3.1)
and the Lobatto four point rule 1 1 −1 1 I ( f ) = ∫ f ( x ) dx ≈ RLob4 ( f ) = f ( −1) + 5 f +5f + f (1) 6 5 5 −1
(3.2)
Each of the rules (3.1) and (3.2) is of precision five. Let ECC5 ( f ) and ELob4 ( f )
I ( f ) = RCC5 ( f ) + ECC5 ( f )
(3.3)
denote the errors in approximating the integral I ( f ) by the rules (3.1) and (3.2), respectively. Using Maclaurin’s expansion of functions in Eqs (3.1) and (3.2) we get
and I ( f ) = RLob4 ( f ) + ELob4 ( f )
(3.4)
where ECC5 ( f ) =
1 1 ( ) ( ) f 6 (0) + f 8 (0) + ⋯ 315 × 5! 360 × 7!
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Rajani B. Dash, et al., J. Comp. & Math. Sci. Vol.3 (2), 207-215 (2012)
E Lob4 ( f ) = −
32 128 ( ) ( ) f 6 (0) − f 8 (0) − ⋯ 525 × 6! 1125 × 8!
2 ( ) f 8 (0) 1575 × 7!
ECC5 Lob4 ( f ) ≈
Now multiplying the equations (3.3) and (3.4) by 1 and 1 respectively, and then
Proof
16
From Eq (3.7)
adding the resulting equations we obtain.
ECC5 Lob4 ( f ) =
5
I( f )=
1 1 16 RCC5 ( f ) + 5 RLob4 ( f ) + 16 ECC5 ( f ) + 5 E Lob4 ( f ) 21 21
or, I ( f ) = RCC5 Lob4 ( f ) + ECC5 Lob4 ( f )
(3.5)
ECC5 Lob4 ( f )
So,
−2 ( ) f 8 (0) + ⋯ 1575 × 7! 2 ( ) ≈ f 8 (0) 1575 × 7!
Theorem 4.2
where
The bound for the truncation error
1 RCC5 Lob4 ( f ) = 16 RCC5 ( f ) + 5 RLob4 ( f ) 21
(3.6)
ECC5 Lob4 ( f ) = I ( f ) − RCC5 Lob4 ( f )
is given by ECC5 Lob4 ( f ) ≤
2M η2 − η1 99225
This is the desired quadrature rule of precision seven for the approximate evaluation of I ( f ) . The truncation error generated in this approximation is given by
where, M = max f ( 7 ) ( x )
1 2 ( ) 16 ECC5 ( f ) + 5 E Lob4 ( f ) = − ECC5 Lob4 ( f ) = f 8 (0) + ⋯ 21 1575 × 7!
We have
(3.7) The rule (3.6) may be called a mixed type rule as it is constructed from two different types of rules of the same precision (i.e, precision 5)
η1, η2 ∈ [ −1,1]
−1≤ x ≤1
Proof ECC5 ( f ) ≈
1 ( ) f 6 ( η2 ) 315 × 5!
η2 ∈ [ −1,1]
E Lob4 ( f ) ≈
−32 ( ) f 6 ( η1 ) 525 × 6!
η1 ∈ [ −1,1]
Hence, 1 16 ECC5 ( f ) + 5 E Lob4 ( f ) 21
4. ERROR ANALYSIS
ECC5 Lob4 ( f ) =
An asymptotic error estimate and an error bound of the rule (3.6) are given in theorems 4.1 and 4.2 respectively.
=
2 ( 6) ( ) f ( η2 ) − f 6 ( η1 ) 99225
=
2 η2 ( 7 ) ∫ f ( x ) dx 99225 η
Theorem 4.1 f ( x) Let be a sufficiently differentiable function in the closed inter [ −1,1] . Then the error ECC5 Lob4 ( f ) associated
with the rule RCC5 Lob4 ( f ) is given by
(assuming η1 < η2 )
1
So, ECC5 Lob4 ( f ) =
2 η2 ( 7 ) 2 η2 ( 7 ) f ( x ) dx ≤ ∫ ∫ f ( x ) dx 99225 η 99225 η 1
or, ECC5 Lob4 ( f )
2M ≤ η2 − η1 99225
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1
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5. NUMERICAL VERIFICATION Table 5.1 : Approximation of some real definite Integrals using Clenshaw-Curtis 5-point quadrature rule ( RCC5 ( f ) ) in adaptive environment
Integrals
Exact value
( RCC
5
( f ))
by
adaptive quadrature method π2
I1 =
∫ 0
dx 1 + cos x
Steps required
Approximate value Clenshaw-Curtis 5-Point rule
Error = exact value
(
− RCC5 ( f )
)
1 dx 2 1 25 + x 0
0.9999998240
2
0.000000176
0.000001
0.2746801533
0.2746801889
6
0.0000000356
0.000001
0.7237976340
0.7237975921
5
0.0000000419
0.000001
0.1534264097
0.1534259448
2
0.0000004649
0.000001
0.375
0.3749998904
4
0.0000001095
0.000001
1.2091995761
1.2091995748
3
0.0000000013
0.000001
0.2031547017
0.2031547362
2
0.0000000344
0.1
0.0999998888
2
0.0000001111
0.000001
0.379885493041
0.3798854926
2
0.0000000004
0.000000001
0.1569287647
0.156928511427
3
0.0000002533
2
x dx 3 0 1+ x
I3 = ∫
π4
I4 =
∫
tan 3 x dx
ε1 =
1
1
I2 = ∫
Maximum admissible absolute error ( ε )
ε2 = ε3 = ε4 =
0 π2
I5 =
sin x
∫ (1 + cos x )
3
dx
ε5 =
0
1
1 dx 2 0 1− x + x
I6 = ∫ 2
1 dx 4 1 1+ x
I7 = ∫ 1
I8 = ∫ 0
1
( 3 x + 2 )2
dx
1
1 dx x 0 1+ e
I9 = ∫ 2
I10 = ∫ 1
dx x(6(ln x) + 7(ln x) + 2) 2
Journal of Computer and Mathematical Sciences Vol. 3, Issue 2, 30 April, 2012 Pages (131-247)
ε6 = ε7 = 0.000001
ε8 = ε9 = ε10 = 0.000001
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Table 5.2 : Approximation of some real definite Integrals using Gauss-Legendre 4-point rule ( RGL4 ( f )) in adaptive environment
Integrals
Exact value
( RGL
( f )) 4
(of 7 precision) by adaptive quadrature method π2
I1 =
dx
∫ 1 + cos x
Steps required
Approximate value Gauss-Legendre 4-point rule
Error = exact value
(
- RGL
4
( f ))
Maximum admissible absolute error ( ε )
ε1 =
1
0.9999999369
1
0.0000000631
0.2746801533
0.2746801576
3
0.0000000043
0.000001
0.7237976340
0.7237977153
3
0.0000000813
0.000001
0.1534264097
0.1534260928
1
0.0000003169
0.000001
0.375
0.3749999636
2
0.0000000363
0.000001
1.2091995761
1.2091997489
1
0.0000001728
0.000001
0.2031547017
0.2031548460
0
0.0000001442
0.1
0.0999998683
1
0.0000001316
0.000001
0.379885493041
0.379885493045
1
0.00000000000383
0.00000000 1
0.1569287647
0.156928728003
2
0.0000000367
0.000001
0
1
1 dx 2 0 1 + 25 x
I2 = ∫ 2
x dx 3 0 1+ x
I3 = ∫
π4
I4 =
∫ tan
3
x dx
ε2 = ε3 = ε4 =
0 π2
I5 =
sin x
∫ (1 + cos x )
3
dx
ε5 =
0
1
1 dx 2 1 − x + x 0
I6 = ∫ 2
1 dx 4 1 1+ x
I7 = ∫ 1
I8 = ∫ 0
1
( 3x + 2 )
2
dx
1
1 dx x 0 1+ e
I9 = ∫ 2
I10 = ∫ 1
dx x(6(ln x) + 7(ln x) + 2) 2
Journal of Computer and Mathematical Sciences Vol. 3, Issue 2, 30 April, 2012 Pages (131-247)
ε6 = ε7 = 0.000001
ε8 = ε9 = ε10 = 0.000001
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Rajani B. Dash, et al., J. Comp. & Math. Sci. Vol.3 (2), 207-215 (2012)
Table 5.3 : Approximation of some real definite Integrals using Mixed quadrature rule ( RCC5 Lob4 ( f )) in adaptive environment.
Integrals
Exact value
Mixed quadrature rule
( RCC Lob 5
4
( f ))
by adaptive quadrature method π2
I1 =
dx
∫ 1 + cos x
Steps required
Approximate value Maximum admissible absolute error ( ε )
Error = exact value -
( RCC Lob 5
4
( f ))
ε1 =
1
1.0000000557
1
0.0000000557
0.2746801533
0.2746801499
3
0.0000000034
0.000001
0.7237976340
0.7237975642
3
0.0000000698
0.000001
0.1534264097
0.1534266904
1
0.0000002807
0.000001
0.375
0.3750000320
2
0.0000000320
0.000001
1.2091995761
1.2091994278
1
0.0000001483
0.000001
0.2031547017
0.2031548435
0
0.0000001417
0.1
0.1000001175
1
0.0000001175
0.000001
0.379885493041
0.379885493038
1
0.000000000003
0.000000001
0.1569287647
0.1569287973
2
0.0000000325
0.000001
0
ε2 =
1
1 dx 2 0 1 + 25 x
I2 = ∫
ε3 =
2
x dx 3 1 + x 0
I3 = ∫
ε4 =
π4
I4 =
∫ tan
3
x dx
0 π2
I5 =
sin x
∫ (1 + cos x )
3
dx
ε5 =
0
ε6 =
1
1 dx 2 1 − x + x 0
I6 = ∫
ε7 =
2
1 dx 4 1 1+ x
I7 = ∫ 1
1
I8 = ∫
( 3x + 2 )
0
2
dx
0.000001
ε8 = ε9 =
1
1 dx x 0 1+ e
I9 = ∫ 2
I10 = ∫ 1
dx x(6(ln x) + 7(ln x) + 2) 2
ε10 = 0.000001
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Rajani B. Dash, et al., J. Comp. & Math. Sci. Vol.3 (2), 207-215 (2012)
which gives only a theoretical error bound, as η1, η2 are unknown points in [ −1,1] . From the equation it is evident that the error in the approximation will be less if the points η1, η2 are close to each other.
Corollary The error bound for the truncation error ECC5 Lob4 ( f ) is given by ECC5 Lob4 ( f ) ≤
4M 99225
Proof We know from the theorem 4.2
ECC5 Lob4 ( f ) ≤
where ,
2M η2 − η1 99225
η1 , η2 ∈ [ −1,1]
M = max f
(7 )
−1≤ x ≤1
η1 − η2 ≤ 2
Choosing
( x)
, we have
ECC5 Lob4 ( f ) ≤
4M 99225
CONCLUSION The mixed quadrature rule (precision 7) obtained in this paper has been tested on different real definite integrals in adaptive environment. Approximation of some real integrals using Cleanshaw-Curtis 5-point rule, Gauss-Legendre 4-point rule and the mixed quadrature rule in adaptive environment are presented in Tables 5.1, 5.2 and 5.3 respectively.
We observe that the mixed quadrature rule reduces the number of steps in comparison to the Clenshaw-Curtis 5point rule in adaptive mode. Though this mixed rule and Gauss-Legendre 4-point rule are of same precision (Precision 7), the mixed rule is giving better result than the Gauss-Legendre 4-point rule.
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Clenshaw-Curtis and Gauss-Legendre Quadrature rules for approximation of real definite integrals in adaptive environment’, Proceedings of the international Multi-conference of Engineers and computer Scientist,Vol-1, 202-205, Hongkong. (2011). 10. Rajani Ballav Dash and Debasish Das, “Identification of some Clenshaw-Curtis quadrature rules as Mixed quadrature of
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Fejer and Newton-Cote type of rules, ‘Int. J of Mathematical Sciences and Applications, Vol-1 No.3 PP.1493-1496, mind Reader Publications. 11. R. N. Das and G. Pradhan, ‘A mixed quadrature rule for approximate evaluation of real definite integrals’. Int. J. Math. Educ. Sci. Technol, Vol.27, No.2, 279-283, (1996).
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