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A mixed quadrature rule for numerical integration of analytic functions.

1. Introduction

There are several rules for the approximate evaluation of real definite integral

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)

However there are only few quadrature rules for evaluating an integral of type

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2)

Where L is a directed line segment from the point ([z.sub.0] - h)to([z.sub.0] + h)in the complex plane C and f(z) is analytic in certain domain [OMEGA] containing L. Using the transformation z = [z.sub.0] + ht, t [member of] [-1,1] (due to lether(1976)), we transformed the integral(2) to the form

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3)

and made the approximation of the integral by applying standard quadrature rule meant for approximate evaluation of real definite integral(1). The rules so formed are termed as TRANSFORMED RULES for numerical integration of (2).

R.N. Das and G. Pradhan (1996)[1] have constructed a quadrature rule for approximate evaluation of (1) from two quadrature rules of different type but of equal precision. Such rules are termed as 'MIXED QUADRATURE RULES'.

In this light S.K. Mohanty and R.B. Dash [5] have constructed two Mixed quadrature rules, i,e [SM.sub.1](f) and [SM.sub.2](f) each is of precision seven. The [SM.sub.1](f) rule is formed by using Bool's quadrature (BL(f)) and Gauss- Legendre quadrature (GL(f)) rule's and the [SM.sub.2](f) rule is formed by using Bool's quadrature (BL(f)) and Birkhoff-Young quadrature(BY(f)) rules, where

[SM.sub.1](f) = 1/49 [24 BL(f) + 25GL(f)] ...(4)

[SM.sub.2](f) = 1/7 [8 BL(f) - BY (f)] ...(5)

and BL(f) = h/45 [7{ + h) + f([z.sub.0] - h)} + f([z.sub.0] - h) + 12([z.sub.0]) + 32{f([z.sub.0] - [h/2])} + f([z.sub.0] - [h/2])} ...(6)

GL(f) = h/9[5 f([z.sub.0] + h [square root of 3/5]) + 8 f([z.sub.0]) + 5f([z.sub.0] - h [square root of 3/5]) ... (7)

BY(f) = h/15 [4{f([z.sub.0] + h) + f([z.sub.0] - h)} + 24f([z.sub.0]) - {f([z.sub.0] + ih) + f([z.sub.0] - ih)}] ... (8)

Each of the above three quadrature rules (6), (7) & (8) is of precision five.

In this paper we desire to construct a mixed quadrature rule of precision nine in the same vein of [5] for the approximation of the integral (2).

2. Formulation of the Rule

For the construction of the desired rule we choose (4) and (5). Each of the rules under consideration is of precision seven.

Denoting the truncation error by [E.sub.SM1] and [E.sub.SM2] in approximating the integral (2) by rules(4) and (5) respectively we have

I(f) = [SM.sub.1](f) + [E.sub.SM1] ...(9)

and I(f) = [SM.sub.2] + [E.sub.SM2] ...(10)

f is infinitely differentiable, since it is assumed to be analytic in certain domain [OMEGA] containing the line segment L. So by using Taylor's expansion the truncation errors associated with the quadrature rules under reference can be expressed as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] ... (11)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] ...(12)

Now multiplying (9) and (10) by 39/105(9!) and 7/105(9!) respectively and then adding we obtained

I(f) = 1/12 [13.[SM.sub.1](f) - [SM.sub.2](f)] + [E.sub.SM3] = [SM.sub.3](f) + [ESM.sub.3]

Where [SM.sub.3](f) = 1/12 [13[SM.sub.1](f) - [SM.sub.2](f)] = 1/588 [325GL(f) + 256BL(f) + 7 BY (f)] ...(13)

is desired quadrature rule of precision nine for the approximate evaluation of and truncation error committed in this approximation is given by

[E.sub.SM3] = 1/812[13[E.sub.SM1] - [E.sub.SM2]] ... (14)

The Rule (13) may be called MIXED TYPE QUADRATURE RULE as it is constructed from two different types of rules of same precision.

3. Error Analysis

Let f(z) is analytic in the disc [[OMEGA].sub.R] = {z: [absolute value of z - [z.sub.0]] [less than or equal to] R > h}

So that the points [z.sub.0], [z.sub.0] [+ or -] h, [z.sub.0] [+ or -] h[square root of 3/5] are all interior in the disc [[OMEGA].sub.R]. Now using Taylor's series expansion.

f(z) = [[infinity].summation over (n=0)] [a.sub.n] [(z - [z.sub.0]).sup.n]: [a.sub.n] = [1/n!] [f.sup.n]([z.sub.0])

in (13) we obtained after simplification

[E.sub.SM3] = 2h[[-103949.[h.sup.10]/182700(11!)] [f.sup.10]([z.sub.0]) + [7488866.[h.sup.12]/3528000] [f.sup.12]([z.sub.0]) + ...] ...(15)

From (15) we have the following theorem.

Theorem

If f is assumed to be analytic in the domain [OMEGA] [contains] L then

[E.sub.SM3](f) = o([h.sup.11])

Error Comparison

It is shown by Lether(1976) that

[absolute value of [E.sub.GL]] [less than or equal to] [absolute value of [E.sub.BL]] ...(16) Again from (11), (12)

[absolute value of [E.sub.SM1]] [less than or equal to] [absolute value of [E.sub.GL]] and [absolute value of [E.sub.SM2]] [less than or equal to] [absolute value of [E.sub.GL]] From (15) [absolute value of [E.sub.SM3]] [less than or equal to] [absolute value of [E.sub.GL]] and [absolute value of [E.sub.SM3]] [less than or equal to] [absolute value of [E.sub.SM2]]

4. Numerical Verification

Let us calculate approximate value of the integral [I.sub.1] and [I.sub.2] by using BL(f), GL(f), [SM.sub.1](f), [SM.sub.2](f) and [SM.sub.3](f).

For [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
    Quadrature      Approximation         Approximation value
    rules           value of([I.sub.1])   of([I.sub.2])

a   BL(f)           i(1.6828781)          i(1.0421911)
b   GL(f)           i(1.6830035)          i(1.0421901)
c   [SM.sub.1](f)   i(1.6829421)          i(1.0421906)
d   [SM.sub.2](f)   i(1.6829440)          i(1.04219061)
e   [SM.sub.3](f)   i(1.6829419238)       i(1.0421905917)
Exact value         i(1.6829419239)       i(1.0421905917)


References

[1] Das, R.N and Pradhan, G. 1997. A mixed quadrature rule for numerical integration of analytic functions. Bull Cal. Math. Soc. 89:37-42.

[2] Acharya, B.P and Das, R.N. 1983. Compound Birkhoff-Young rule for numerical integration of analytic functions. Int. J. Math. Educ. Sci. Technol 14, 1, 101.

[3] Birkhoff, G. and Young, D. 1950. Numerical quadrature of analytic and harmonic functions. J. Maths, Phys, 29, 217.

[4] Das, R.N and Pradhan, G. 1996. A Mixed quadrature rule for approximate evalution of real definite integrals. Int, J. Math. Educ. Sci. Techonol.

[5] Mohanty, Sanjit Ku. and Dash, R.B. 2008. Bulletin of Pure and Applied Sciences, Vol.27E(No.2)2008:P,373-376.

Sanjit Ku. Mohanty and Rajani B. Dash

Department of Mathematics, Ravenshaw University, Orissa, India
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Author:Mohanty, Sanjit Ku.; Dash, Rajani B.
Publication:International Journal of Computational and Applied Mathematics
Article Type:Report
Geographic Code:9INDI
Date:Aug 1, 2009
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