Some Derivative-Free Quadrature Rules for Numerical Approximations of Cauchy Principal Value of Integrals
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.
Recently Das and Hota  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 .
2. Formulation of Rules
Das and Hota  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);
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 . 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 . 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  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 , 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 . 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
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.
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.
The rules constructed in this paper successfully integrate to at least six decimal places of the complex Cauchy principal value integrals we have numerically evaluated in this paper. Each of the rules does not require evaluation of the function at ; the point of singularity. Also, it is not required to evaluate derivative of the integrand at any of its nodes, which is a positive advantage over the existing rules found in the literature. [6–8].
Again, these rules can also be applied for numerical integration of:(i)Complex definite integrals on the line segment having end points and in the complex plane .(ii)These rules one can apply for the numerical integration of real definite integrals, and(iii)It is also shown that the rules constructed numerically integrate real Cauchy principal value integrals to at least six decimal places accurately.
The numerical values of the error constants and associated with the quadrature rules and meant for numerical evaluation of the complex Cauchy principal value of integrals as given (5) and (6), respectively, in this paper are depicted in Table 18.
From this table it is quite apparent that the error constant corresponding to the rule moves faster towards zero than that of the rules although both the rules are six-point rule with the same degree of precision. This observation is substantiated by the results obtained in the approximation of the integrals given in Section 4.
It is worth mentioning here that three mixed quadrature rules can be constructed following the technique suggested by Das and Pradhan . The degree of precision of each of these mixed quadrature rules will be 10. The authors are investigating two parameter rules of the same problem, that is, approximation of integral given in (1).
Conflict of Interests
The authors declare that they have done this research work on their own interest. No organization/institution has given financial assistance to carry this research work. The authors declare that they have no conflict of interests. Also, this paper has not been published in any other journal.
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