A Derivative Free Quadrature Rule for Numerical Approximations of

Applied Mathematical Sciences, Vol. 6, 2012, no. 111, 5533 – 5540

A Derivative Free Quadrature Rule for Numerical Approximations of Complex Cauchy Principal Value Integrals R. N. Das School of Mathematics & Computing Gandhi Institute for Technological Advancement Badaraghunathpur, Jaanla, Pin: 752054 Bhubaneswar, India rabi_das08 @rediffmail.com

M. K. Hota School of Mathematics & Computing Gandhi Institute for Technological Advancement Badaraghunathpur, Jaanla, Pin: 752054 Bhubaneswar, India [email protected]

Abstract Some interpolatory types of rules have been constructed for approximate evaluation of complex Cauchy principal value integrals and the error bound have been obtained. The rules have been numerically verified in case of certain complex Cauchy principal value integrals.

Keywords: Cauchy principal value, degree of precision, error bound.

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R. N. Das and M. K. Hota

1. Introduction Complex Cauchy principal value integrals of the type: z0 + h f ( z) I(f ) = ∫ dz z − z0 z0 − h

(1.2)

along the directed line segment L , from the point z0 − h to z0 + h and f ( z ) is

assumed to be an analytic function in a domain Ω containing L , occur very often in contour integration, which in turn, is an essential tool in applied mathematics. As far as it is known, the numerical evaluation of the complex CPV integrals is still getting sufficient attention of many researchers. Acharya and Das [2], Milovanovic, Acharya and Pattanaik [13] etc, has given significant contributions for the same. However, the rules developed by them require evaluation of the derivative of the function f ( z ) at z = z0 which is, as a result not suitable for evaluation over computer. Thus, the objective of this paper is to obtain interpolatory types of rules not involving derivative of the function for the numerical approximation of the complex CPV integrals given in equation (1.2), which numerically integrates the complex CPV integrals more accurately than by the existing rules found in the literature.

2. Formulation of rules For the construction of the rule the nodes that we have chosen here are:

z0 , z k = z 0 + ( i ) h k

for k = 1, 2,3, 4

along with the following nodes:

z p = z0 + ( i ) α h p

for p = 5,6,7,8.

and the rule based on these points is denoted by R ( f ;α ) and is given by:

R ( f ; α ) = w0 f ( z0 ) + w1 ⎡⎣ f ( z0 + h ) − f ( z0 − h ) ⎤⎦ + w2 ⎡⎣ f ( z0 + ih ) − f ( z0 − ih ) ⎤⎦ + w3 ⎡⎣ f ( z0 + α h ) − f ( z0 − α h ) ⎤⎦ + w4 ⎡⎣ f ( z0 + iα h ) − f ( z0 − iα h ) ⎤⎦ ; i = −1. (2.1)

The weights w0 , w1 , w2 , w3 , w4 and α are to be determined so that:

I

(( z − z ) ) = R (( z − z ) ;α ); for k =,1,3,5,7 k

k

0

0

It may be noted that,

I

(( z − z ) ) = R (( z − z ) ); 2k

0

2k

0

for k = 1,2,........

(2.2)

Derivative free quadrature rule

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since the rule proposed in equation (2.1) is a fully symmetric quadrature rule. Thus, to determine the weights in equation (2.1) in terms of α we make use of the identities given in equation (2.2) and we have the following set of five linear equations in weights w0 , w1 , w2 , w3 and w4 . 1⎫ w0 = 0, w1 + iw2 + α w3 + iα w4 = 1, w1 − iw2 + α 3 w3 − iα 3 w4 = ⎪ 3 ⎪ . (2.3) ⎬ 1 1⎪ 5 5 7 7 w1 + iw2 + α w3 + iα w4 = and w1 − iw2 + α w3 − iα w4 = 5 7 ⎪⎭

On solving the set of equations given in (2.3) on assumption that α ∉ Δ = {0,1} we have ⎡ 18 − 70α 4 w0 = 0, w1 = ⎢ ⎢⎣105 1 − α 4

(

⎤ ⎡ 35α 4 − 3 ⎥ , w2 = i ⎢ ⎥⎦ ⎢⎣105 1 − α 4

⎤ ⎡ 10 + 42α 2 ⎥ , w3 = ⎢ ⎥⎦ ⎢⎣105α 3 1 − α 4

⎤ ⎡ 10 − 42α 2 ⎥ and w4 = i ⎢ ⎥⎦ ⎢⎣105α 3 1 − α 4

⎤ ⎥ ⎥⎦

(2.4)

) ( ) ( ) ( ) Thus, for α ∉ Δ , the rule R ( f ;α ) given in equation (2.1) with weights w0 ,

w1 , w2 , w3 and w4 as given in equation (2.4) represents a one parameter (α )

family of nine point rules integrating all polynomials of degree at most eight. Further, if E ( f ;α ) denotes the error in approximating the integral

I ( f ) given in equation (1.2) by the rule R ( f ;α ) as stated in equation (2.1)

with weights prescribed by equation (2.4), then

I ( f ) = R ( f ;α ) + E ( f ;α ) .

(

E ( z − z0 )

and it is found that:

9

4 1 ⎡⎣1− 4α ⎤⎦ ≠0 ;α = − 9 5

)

Hence, the rule R ( f ;α ) integrates all polynomials of degree eight or less. However, E

(( z − z ) ;α ) = 0; 9

0

if α 2 =

1 . 3

So the rule R ( f ;α ) given in equation (2.1) boils down to a rule of degree of precision ten if α =

w0 = 0 , w1 =

by:

1 . Thus, in this case the weights are: 3

23 i 27 3 9 3 , w2 = , w3 = and w4 = − i 210 105 35 70

and the rule is given

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⎛ 1 ⎞ 23 i ⎡⎣ f ( z0 + h ) − f ( z0 − h ) ⎤⎦ + ⎡ f ( z0 + ih ) − f ( z0 − ih ) ⎤⎦ R⎜ f ; ⎟= 105 ⎣ 3 ⎠ 210 ⎝ +

27 3 35

⎡ ⎢f ⎣⎢

⎛ ⎛ h ⎞ h ⎞⎤ 9 3 ⎜ z0 + ⎟ − f ⎜ z0 − ⎟⎥ − i 70 3⎠ 3 ⎠ ⎥⎦ ⎝ ⎝

corresponding to α =

⎡ ⎢f ⎣⎢

⎛ ⎛ h ⎞ h ⎞⎤ ⎜ z0 + i ⎟ − f ⎜ z0 − i ⎟ ⎥ ( 2. 5) 3⎠ 3 ⎠ ⎥⎦ ⎝ ⎝

1 . 3

3. Error analysis The error

E ( f ;α )

associated with the rule

E ( f ;α ) = I ( f ) − R ( f ;α ) .

R ( f ;α )

is given by

(3.1).

We assume here that the function

f ( z ) is analytic in the disc:

Ω = { z ∈ : z − z 0 ≤ ρ = r h ; r > 1} .

f ( z ) can be expanded in terms of the Taylor’s series about the point z = z0 in the disc Ω as: Under the above assumption,

f ( z) =

∞

∑a (z − z )

n=0

n

0

n

f ( n ) ( z0 ) ; where an = ( n )!

(3.2)

As the series given in (3.2) is uniformly convergent in Ω , we obtain by integrating both sides of the series (3.2) term by term and obtain: I ( f ) = 2hf ′ ( z0 ) +

2h3 2h5 ( 5 ) 2h 7 ( 7 ) ′′′ f ( z0 ) + f ( z0 ) + f ( z0 ) + ..... (3.3) 3 ( 3) ! 5 ( 5)! 7 ( 7 )!

Again by expanding each term of the rule R ( f ;α ) given in equation (2.5) by Taylor’s expansion about the point simplification we obtain:

z = z0 in the disc Ω and then after

⎛ 1 ⎞ 2h3 2h5 ( 5 ) 2h 7 ( 7 ) ′ ′′′ R⎜ f ; = 2 hf z + f z + f z + f ( z0 ) + ......(3.4) ( ) ( ) ( ) ⎟ 0 0 0 3 ( 3) ! 5 ( 5)! 7 ( 7 )! 3⎠ ⎝

Therefore, equations (3.1), (3.3) and (3.4) jointly imply: ⎛ 1 ⎞ E⎜ f ; ⎟ =− 3⎠ ⎝

⎡⎛ 128 ⎞ h11 (11) ⎤ ⎡ 256 ⎞ h13 (13) ⎤ f f ( z0 )⎥ − ⎢⎛⎜ ( z0 ) ⎥ − ....... ⎢⎜ ⎟ ⎟ ⎣⎢⎝ 2079 ⎠ (11)! ⎦⎥ ⎣⎢⎝ 5265 ⎠ (13)! ⎦⎥

Hence, this leads to following theorem.

(3.5)


⎛

Theorem: 3.1 If E ⎜ f ; ⎝

1 ⎞ ⎟ 3⎠

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⎛

is the error associated with the rule R ⎜ f ;

for the approximation of the complex CPV integral I ( f ⎛

in the disc: Ω = { z ∈ : z − z0 ≤ ρ = r h ; r > 1} , then E ⎜ f ; ⎝

⎛

meant

and f ( z ) is analytic

1 ⎞ 11 ⎟ ∈ O (h ) 3⎠

Error bound: To find the error bound of the rule R ⎜ f ; ⎝

)

⎝

1 ⎞ ⎟ 3⎠

.

1 ⎞ ⎟ we are following the 3⎠

technique due to Lether [11]. From equation (3.2), we obtained:

1 ⎞ ⎛ E⎜ f ; ⎟= 3⎠ ⎝

∞

∑a μ μ =5

2 +1

h2 μ +1E ( t 2 μ +1 )

(3.6)

by substituting z = z0 + ht; t ∈ [ −1,1] and from this we get: 1 ⎞ ⎛ E⎜ f ; ⎟= 3⎠ ⎝ where

∞

∑ 2a μ μ =5

h2 μ +1 B ( μ )

2 +1

⎡ 1 ⎡ 23 ( −1) μ ⎧⎪ ( −1) μ ⎫⎪⎤ ⎤ 1 ⎢ −⎢ − + B (μ ) = ⎨3 + ⎬⎥ ⎥ . 2 ⎭⎪⎥ ⎥ ⎢ 2 μ + 1 ⎢⎣ 210 105 35 ( 3μ − 2 ) ⎩⎪ ⎦⎦ ⎣

Now it can be claimed that B ( μ ) < 0, μ ≥ 5 . Thus, by Cauchy-inequality [5]; ∞ 1 ⎞ 1 ⎛ E⎜ f ; 2 M E ( t 2 μ +1 ) = 2Mer ≤ ∑ ⎟ 2 μ +1 3⎠ μ =5 r ⎝

⎛ ⎛ t ⎞ −1 ⎞ ⎛ r +1⎞ where M = Max f ( z ) and er = E ⎜ ⎜1 − ⎟ ⎟ = r ln ⎜ ⎟− ⎜⎝ r ⎠ ⎟ z =ρ ⎝ r −1 ⎠ ⎝ ⎠

⎡ r ( 21r 2 + 25 ) 27r ( 9r 2 + 1) ⎤ ⎢ ⎥ + 35 ( 9r 4 − 1) ⎥⎦ ⎢⎣ 105 ( r 4 −1)

which → 0 as r → ∞ is known as the error constant.

4. Numerical verifications Section: A(Approximation of complex Cauchy principal value integrals). To test the accuracy of the formula, we have taken the following integrals and their approximation values are given in Table- 4.1.

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Table-4.1 Integrals i

I1 =

∫

(1 + z ) e z dz

3.5751081 * i

(1 + z cos z ) dz

2.3504024 *i

z

−i

I2 =

i ∫ −i

Approximation by

I3 =

∫

(1+ i ) / 4 3(1+ i ) / 2

I4 =

∫

(1+ i ) / 2

Exact value 3.5751081 * i

tan −1 z dz z

−0.5066134 + 0.4927643*i

−0.5066134 + 0.4927643*i

sin z dz z − (1 + i )

1.8175597 − 0.2057226 *i

1.8175587 − 0.2057225 * i

i

I5 =

)

2.3504024*i

z

( −1+ i ) / 4

(

R f ;1 / 3

ez dz −i z

1.8921661* i

∫

1.8921661* i

Section-B (Approximations of integrals of analytic functions over a line segment in the complex plane).

Table-4.2

Integrals

Approximate value Exact value by the rule R f ;1/ 3

(

i

J1 = ∫ e z dz = −i

i

ze z dz −i z

∫

i/2

J2 =

∫

i/2

cos z dz =

−i / 2

(1+ i ) /

J3 =

∫

z cos z dz z −i / 2

∫

(1+ i ) /

2

− (1+ i ) / 2

ze z dz =

∫

− (1+ i ) /

2

z 2e z dz z 2

)

1.6829419i

1.6829419i

1.0421906

1.0421906

0.5168307 + 0.4226121 * i

0.5168305 + 0.4226120 * i


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Section-C (Evaluation of some real Cauchy principal value integrals). Table-4.3 Integrals

Approximate value by Exact value the rule R ( f ;1 / 3 )

1

Q1 =

ex dx x −1

∫

x +1 dx 2 +1

1/ 4

Q2 =

∫ x(x

−1/ 4

1/ 2

Q3 =

)

0.4899573122

0.4899573126

− 0.2554128196

( x + 1) e x dx

1

∫

2.11450175

dx

∫ x ( x + 2)

−1/ 2

Q4 =

2.1145017523

4.4649041567

x

−1

Conclusion: It is claimed that the rule

1 ⎞ ⎛ R⎜ f ; ⎟ 3⎠ ⎝

− 0.25541281956

4.46490415664

given in equation (2.5) integrates all

the test integrals (usually considered by other researchers in this line) accurately 1 ⎞ ⎛ R⎜ f ; ⎟ up to seven decimal places. The approximate evaluation by the rule 3 ⎝

⎠

requires value of z = z0 the function at eight symmetrical points and does not need to be evaluated at f ( z ) since its coefficient is zero; which is an advantage over the rules already constructed by our predecessors [2], [3] and [13]. In addition to this it is pertinent to note here that the existing rules requires evaluation of the derivative at singular point where as the rule constructed in this paper does not require the evaluation of derivatives at any of the nodes, which may also be considered as an additional advantage over other existing rules. Further, as it is shown in Table-4.3, the formula constructed for numerical integration of complex CPV integrals may be fruitfully used for numerical evaluation of real CPV integrals.

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[2]

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5540


[3]

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[4]

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[5]

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[10]

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Received: May, 2012