Revisited Optimal Error Bounds for Interpolatory Integration Rules

Abstract

We present a unified way to obtain optimal error bounds for general interpolatory integration rules. The method is based on the Peano form of the error term when we use Taylor’s expansion. These bounds depend on the regularity of the integrand. The method of integration by parts “backwards” to obtain bounds is also discussed. The analysis includes quadrature rules with nodes outside the interval of integration. Best error bounds for composite integration rules are also obtained. Some consequences of symmetry are discussed.

1. Introduction

Establishing numerical integration rules and their error bounds is an old subject; for example, see the following classic textbooks [1–4]. However, in recent years, several papers have presented error bounds for the midpoint, trapezoidal, and Simpson’s rules; see [5–10]. In these papers, questions were raised about the generality of the results in terms of the optimality of the bounds and regularity of the function. They also suggest that consequences of the symmetry on integration rules should be more fully investigated. Based on Peano form of the error term when we use Taylor’s expansion, we present a unified way to obtain optimal error bounds for general interpolatory integration rules. These bounds depend on the regularity of the integrand. Other similar approaches based on kernels have been recently presented in [11] for Newton-Cotes quadrature rules and in [12] for Gaussian weighted quadrature rules.

Let us start by a transformation of the definite integral as follows:where , , and . Then we consider the method of undetermined coefficients to approximate the following expression:The method of undetermined coefficients consists in finding a -dimensional weight vector associated with a given -dimensional vector of distinct coordinates (or nodes) such that is approximated by its discrete version given byThe truncation error of the process is defined byand the method is based on the requirement thatA general analysis of the method of undetermined coefficients was recently published in [13].

To study the truncation error, two approaches both based on Taylor’s expansions for absolutely continuous functions and Peano’s kernels are presented. In the first approach, the “direct method,” we use (4) for and a Taylor expansion of the integrand. While in the second approach, the method of integration by parts “backwards” [4, 14–16], we use Taylor’s expansion not only for integrand but also for . It is shown that both methods lead to the same best error bounds.

Coming back to the definite integral given in (1), we getwhere , for . However, to compute this definite integral we can also use a composite rule, for which we also present optimal error bounds.

The plan of the paper is the following. In Section 2, we obtain best error bounds using Taylor’s expansions and Peano’s kernels. Section 2.3 is devoted to the “direct method,” and Section 2.4 presents the “method of integration by parts backwards.” Examples are presented in Section 3. Composite integration rules is the object of Section 4. Finally we consider symmetric rules in Section 5.

We use for the th derivative of for , where . Let ; if is defined on a set , is its -norm on , and if is a vector in , its -norm is .

2. Truncation Error

2.1. Introduction

Let and set . Let us observe that , or , means that at least one is strictly greater that . Hence it allows the possibility of having numerical integration formulae with nodes outside the interval of integration ; see [17], for example. In that situation, the function has to be defined on an interval which contains .

For the method of undetermined coefficients it is required that at least for polynomial of degree less than or equal to , but might hold for some polynomials of degree higher than ; see [13]. It happens for Simpson’s rule () which is also exact for polynomials of degree , or also for points Gaussian rule which is exact for polynomials of degree . Let us define the degree of accuracy (or precision) of the approximation process (3) as the largest integer such that holds for any polynomial of degree . These rules are also called interpolatory quadrature formulae.

2.2. Taylor’s Expansions

Let and . For , we simply use , , and . Let and be two extended real numbers such that and . Let be the set of continuously differentiable functions up to order on and let . Let be the set of absolutely continuous function on defined by if and only if and (a),(b).Taylor’s expansion of around of order iswhere is its associated kernelfor any , in ; see [18, 19]. This kernel is a piecewise polynomial function of degree . In this expression, if is a subset of , then and for any nonnegative integer If we set , , and , then the kernel becomesfor any and in .

2.3. Direct Method

For the truncation error analysis, let , for . Using Taylor’s expansion (9) of of order and the fact that the process is exact for polynomials of degree , we obtain Here is the Peano kernel associated with the process , given by Let be the conjugate of such that . From Holder’s inequality, we obtainbecause , for any .

Let us observe that if and , then . It follows thatwhere the constantdoes not depend on . So we have established the following results.

Theorem 1. Let the real number be fixed and . If , for any polynomial of degree , then (16), and equivalently (18), holds for any , where .

Theorem 2. In (17), the kernel is given by and, in (19), the kernel is given by

Remark 3. It can be proved that the bound given by (16) and (17), or equivalently by (18) and (19), is the best possible one; see [13].

Remark 4. For the function used in (1), we have where . Obviously, is supposed to be defined for any .

2.4. Integration by Parts “Backwards”

The method of integration by parts “backwards,” which is reported to have initially appeared in [4, 14–16], was used in [6–10] to find truncation error estimates for the midpoint, trapezoidal, and Simpson’s rules. These rules possess a property of symmetry, which help in finding optimal bounds in these cases. However, the method of integration by parts “backwards” can be applied to any rule that can be obtained by the method of undetermined coefficients. We present an analysis of the truncation error based on this method.

The process is based on Taylor’s expansions of and we suppose that for . We haveso . For , we have Also, for ,Using Taylor’s expansions of order 2 for which is in and of order for which is in , we obtainwhereWe remark that

Taylor’s expansion of order for or integration by parts “backwards” leads to

As mentioned in Remark 3, the construction used in [13] to show the optimality of the bounds leads also to the following result.

Theorem 5. Let and be given; the kernels and are such that almost everywhere.

As a consequence both methods lead to the same best error bounds.

3. Examples

In this section we present several examples. In some cases, constants are computed for (and ) and are compared to constants already existing in the references.

Example 1. Midpoint rule (or one point Gauss rule): , , and hence . Also, , and . The quadrature formula is For , we have so we obtain For example, and , which correspond to values obtained in [7, 9].

Example 2. Trapezoidal rule: , , and . Also, and . The quadrature formula is For , we have We obtain For example, and , which correspond to values obtained in [5, 6, 9].

Example 3. First Simpson’s rule: , , and . Also, and . The quadrature formula is For , we have We obtain For and , we get , , , and , which correspond to constants obtained in [5, 7–10].

Example 4. Second Simpson’s rule: , , and . Also, and . The quadrature formula is For , we have

Example 5. A -point Gauss rule: , , and . Also, and . The quadrature formula is For , we have

Example 6. A -point Gauss rule: , , and . Also, and . The quadrature formula is For , we have

Example 7. First nonstandard rule: , , and . Also, and . The quadrature formula is For , we have

Example 8. Second nonstandard rule: , , and . Also, and . The quadrature formula is For , we have

Example 9. Third nonstandard rule: , , and . Also, and . The quadrature formula is For , we have

Example 10. Gauss-Radau rule: , , and . Also, and . The quadrature formula is For , we have

4. Composite Rules

For an integral , where , a composite rule uses a partition of in subintervals and applies a formula on each subinterval. To simplify, we consider subintervals of equal length . To allow the possibility that , which cause an overlap between subintervals, we suppose also that , where and , for small enough or equivalently large enough. Let us set for , and , for . Then The composite rule is then defined byThe truncation error is But To measure the overlap for , let us define by then and we obtain

5. Symmetric Interpolatory Rules

In case of symmetry with respect to , more precisely when we have As a consequence, we have , for . Moreover, for any monomials of odd degree; then is odd and is an even function. Moreover, for , since , for two nonnegative integers and , it follows that

For some polynomials, we can evaluate exactly the truncation error, as mentioned in [10] for Simpson’s rule. Indeed, let where is a polynomial of degree . Then we have It follows, for the composite rule, that So we have Moreover we obtain becauseThen we have a general explanation of the result mentioned in [10] for Simpson’s rule applied to quartic and quintic polynomials.

Example 11. Midpoint rule: , , and ; and for we have

Example 12. Trapezoidal rule: , , and ; and for we have

Example 13 (see [10]). First Simpson rule: , , and ; and for we have

Example 14. Second Simpson rule: , , and ; and for we have

Example 15. First nonstandard rule: , , and ; and for we have

Similar results for monomial of degree require the evaluation of for , which is not simple.

Competing Interests

The author declares that he has no competing interests.

Acknowledgments

This work has been financially supported by an individual discovery grant from NSERC (Natural Sciences and Engineering Research Council of Canada).