Some Derivative-Free Quadrature Rules for Numerical Approximations of Cauchy Principal Value of Integrals
Rabindranath Das,^{1}Manoj Kumar Hota,^{2} and Manoranjan Bej^{3}
^{1}Department of Mathematics & Computer Science, G.M. Autonomous College, Sambalpur, Odisha, India
^{2}School of Applied Sciences, Department of Mathematics, KIIT University, Bhubaneswar, Odisha 751024, India
^{3}Department of Mathematics, GITA, Bhubaneswar, Odisha 752054, India
Academic Editor: H. J. Ruskin
Received11 Nov 2013
Accepted10 Dec 2013
Published17 Mar 2014
Abstract
Some derivative-free six-point quadrature rules for approximate evaluation of Cauchy principal value of integrals have been constructed in this paper. Rules are numerically verified by suitable integrals, their degrees of precision have been determined, and their respective errors have been asymptotically estimated.
1. Introduction
Recently Das and Hota [1] have constructed a derivative-free 8-point quadrature rule for numerical evaluation of complex Cauchy Principal Value of integrals of type
along the directed line segment , from the point to the point , and is assumed to be an analytic function in a domain containing .
The objective of this paper is to obtain some other quadrature rules having six-nodes not involving derivative of the function for numerical approximation of the complex CPV integrals given in (1) from the family of rules given by Das and Hota [1].
2. Formulation of Rules
Das and Hota [1] have given the following derivative-free 8-point parametric quadrature rule of degree of precision at most ten to approximate the integrals of the type given in (1):
where
the rule given in (2) is of precision eight for .
However, the rule may be reduced to a six point rule for suitable values of the parameter “.” without altering its algebraic degree of precision that is eight. These rules are as follows:(i);
for this value of ; the weight in (3) is zero and the rule given in (2) becomes a six point rule denoted by given as:
(ii);
in this case the weight and the corresponding rule (denoted by ) is:
(iii);
as in the case of two cases noted above, we found here that the weight for this value of ; and the rule denoted by becomes:
Each of these rules, that is, , and is a six-point rule. For the numerical integration of the integral (1) it is required to evaluate the function at six points instead of eight points as in the case, the rule proposed by Das and Hota [1]. Both the rules and have as nodes and hence they are closed type of rules.
It is pertinent to note here that the degree of precision of each of the rules , , and is eight which is the same as that of the rule except for the value of in which case it becomes a rule of precision ten; however, in this case, evaluation of function at 8 nodes is required in approximation of integrals.
3. Error Analysis
The error associated with the rule as given in (2) is
We assume here that the function is analytic in the disc
under this assumption, can be expanded in terms of the Taylor’s series about the point in the disc as
Where are the Taylor’s coefficients.
As the series given in (9) is uniformly convergent in , we obtain by integrating both sides of the series (9) term by term and obtain
Again by expanding each term of the rule given in (2) by Taylor’s series expansion about the point in the disc and then after simplification we obtain
Further, by substituting , , and in the expression given in (11) we have
respectively.
Denoting
and then making the appropriate substitutions for and in (15) we obtain after simplification the following expressions for , :which in turn imply
From the first term of the error expression given in (16), it is also evident that amongst the three rules (, , and ), shall integrate more accurately than the other two rules , . Further, in the approximation of the integral by rules and , the rule is better than the rule in the sense that the approximation obtained by is closer to the true value compared to that obtained by . This observation is also substantiated by the numerical results obtained in the approximation of some standard integrals by these rules given in Section 4.
3.1. Error Bounds
The error bounds of the two quadrature rules and constructed in this paper have been obtained here by following the technique due to Lether [3]. Since the derivation of error bound is similar to each of the two rules, we have derived the error bound of the rule only and it is given in Theorem 1, and the error bound of the rule is only stated in Theorem 2 following Theorem 1.
It is noted here that the error bound of the quadrature rule given in (4) cannot be determined in the same way as it is done for other two cases, that is, and , by following the technique due to Lether [3] for the reasons explained below.
Since denotes the truncation error in approximation of integral by the rule ,
and being a linear operator, we obtain from (9) the following:
by using the transformation , and from this we get
wherewhich is not of one sign for .
However, its asymptotic error estimates have been given in (17). Next we consider:
3.2. Error Bound of the Rule
Theorem 1. If is analytic in a closed disc
then
where
which for .
Proof. Here
Again since denotes the truncation error in approximation of integral by the rule , that is,
and being a linear operator, we obtain from (9) the following:
by using the transformation , , and from this we get
where
Now
which implies
Now by Cauchy-inequality [4],
where
which as . This completes the proof of the theorem.
It will not be out of place to mention here that the quantity is defined as error constant associated with a quadrature rule by Lether [5]. It is tacitly assumed that the function is an entire function so that the error-constant approaches to zero as approaches to infinite. In fact, we observe that is practically zero for as it is seen from the Table 18.
Hence, for all practical purposes we find that the function need to be analytic in a disc of finite radius .
Theorem 2. If is analytic in a closed disc
then
where
which for .
This corollary can also be established in the same vain as it is done in Theorem 1 since
where
The error constants , , have been numerically evaluated for values of and presented in Table 18 in the concluding part of this paper. A close look at the table reveals that the rule more accurately integrates a complex Cauchy principal value of integrals numerically than the rule . This assertion is evident from the numerical values of the error constants given in Table 18 and from their respective graphs drawn together in Figure 1 at the end of this paper. It is observed that
for any fixed but arbitrary value of .
Further, from the asymptotic error estimates of the rules , , and constructed in this paper for the numerical evaluation of complex CPV integrals, it also follows that
This observation is in fact very much noticeable from the numerical integration of the integral that we have taken and the results of numerical integration presented in tables given in Section 4.
4. Numerical Verification
4.1. Approximate Evaluation of Complex Cauchy Principal Value of Integrals
The integrals considered here are
Results of numerical integration of the above five CPV integrals are given in Tables 1, 2, 3, 4, and 5.
Rules
Integral
Approximate value of
Absolute error
Exact value
**
Rules
Integral
Approximate value of
Absolute error
Exact value
**
Rules
Integral
Approximate value of
Absolute error
Exact value
**
Rules
Integral
Approximate value of
Absolute error
0.0
0.0
Exact value
**
Rules
Integral
Approximate value of
Absolute error
Exact value
**
4.2. Approximations of Integrals of Analytic Functions over a Line Segment in the Complex Plane
In this subsection of numerical experimentation, quadrature rules , and constructed in this paper have been employed for approximate evaluation of integrals of analytic functions in complex variable over a line segment joining the points to in the complex plane . The integrals considered here are
Results of numerical integration of the above integrals are given in Tables 6, 7, and 8.
Rules
Integral
Approximate value of
Absolute error
Exact value
**
Rules
Integral
Approximate value of
Absolute error
Exact value
**
Rules
Integral
Approximate value of
Absolute error
Exact value
**
4.3. Evaluation of Some Real Cauchy Principal Value of Integrals
In this subsection of numerical verification, the quadrature rules , , and as formulated in this paper for numerical integration of complex Cauchy principal value integrals have been applied for the approximate evaluation of the following real Cauchy principal value of integrals. The results of approximations are depicted in Tables 9, 10, 11, and 12, respectively. The integrals considered here are
4.4. Approximate Evaluation of Real Definite Integrals
In this subsection of numerical experimentation, quadrature rules , , and constructed in this paper have been successfully employed for approximate evaluation of real definite integrals without having any kind of singularities. The rules have been practically verified with the help of the following integrals:
Results of numerical integration of the above five integrals are given in Tables 13, 14, 15, 16, and 17.