Abstract
We present an analysis of corrected quadrature rules based on the method of undetermined coefficients and its associated degree of accuracy. The correcting terms use weighted values of the first derivative of the function at the endpoint of the subinterval in such a way that the composite rules contain only two new values. Using Taylor’s expansions and Peano’s kernels we obtain best truncation error bounds which depend on the regularity of the function and the weight parameter. We can minimize the bounds with respect to the parameter, and we can find the best parameter value to increase the order of the error bounds or, equivalently, the degree of accuracy of the rule.
1. Introduction
The problem of correcting known quadrature formulas by using derivative values is an old one (see [1] and the references therein) and continues regularly to be analyzed [2, 3]. Here we propose an analysis of corrected quadrature rules using the method of undetermined coefficients and its associated degree of accuracy (or precision). Using Taylor’s expansions for absolutely continuous functions, we obtain Peano’s kernels and, as a consequence, best error bounds. The bounds, expressed in terms of -norms, depend on the regularity of the function and also on a parameter . For a given regularity of the function and a -norm, we can minimize the bound with respect to . Moreover, under certain conditions, it is possible to find a value of the parameter to increase the degree of accuracy (and the order of error bounds) of the method.
Let us start by considering the integral we want to approximate: This form of the integral is motivated by the composite quadrature formula considered in Section 7 which consists in adding terms of the form . The expression also represents any definite integral subject to a linear transformation of the argument.
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 the finite sum In this expression we use to fix the position of the nodes relatively to the interval for any .
For weighted endpoint corrected quadrature rules, the approximation of (1) is given by The endpoint correction term is and is the weight or the parameter. In (3), the weights ’s depend on . The truncation error of the process is and the method is based on the requirement that for a regular enough function.
The plan of the paper is the following. In the next section we present preliminaries about polynomials and Taylor’s expansions. The method of undetermined coefficients adapted to include correction terms is the object of Section 3. In Section 4 we obtain optimal error bounds using Taylor’s expansions and Peano’s kernels. In Section 5 we present the way we can choose the parameter to increase the degree of accuracy and the order of the error bound of the method. Examples are given in Section 6. Composite rules are presented in Section 7 with numerical examples.
Throughout this paper we will use for the th derivative of for . Let ; if is defined on a set , will be its -norm on , and if is a vector in , its -norm will be .
2. Preliminaries
2.1. Small and Big Notations
Let be a function such that . We say that is small of around and write , if for any there exists a such that which holds for . We say that is big of around and write , if there exist a constant and a such that which holds for .
2.2. Polynomials and Small Notation
The next lemma is a direct consequence of the small notation for polynomials and will be useful to obtain necessary conditions in Section 4.
Lemma 1. Let be a positive real number and . Let be a polynomial of degree such that Then, where is a polynomial of degree .
2.3. Taylor’s Expansion
Let ; for we will simply use , , and ; for we will 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 . Let be the set of absolutely continuous functions on be defined by if and only ifand(a),(b), for all , .Taylor’s expansion of around of order is where is the kernel for any , in (see [4, 5]). This kernel is a piecewise polynomial function of degree . In this expression, if is a set, then and, for any nonnegative integer , If we set and , then the kernel becomes for any , in .
3. The Method of Undetermined Coefficients
3.1. Direct Approach
We observe that given by (5) is linear with respect to , and if is a polynomial of degree with respect to , then is a polynomial of degree with respect to . From Lemma 1, condition (6) implies that for any polynomial of degree . Then using the standard basis , we have to solve the linear system where is the Vandermonde matrix associated with , and the entries of are The solution of this system is where is the -column vector, the transpose of , and if , if , for . Let us remark that does not depend on for and (cases which include the midpoint and trapezoidal rules).
It might happen that for some 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 .
3.2. A Decomposition
Using (19), let us set Then is the solution of where the entries of are and is the solution of where the entries of are Let us write Then the following result follows directly.
Theorem 2. The following two conditions are equivalent:(A)for any , there exists a unique such that for any polynomial of degree ,(B)there exists a unique such that for any polynomial of degree , and there exists a unique such that for any polynomial of degree .
Moreover, one has and
4. Truncation Error
Since the method requires the values of and for , we assume that is at least in . Let such that and . Since the process is exact for polynomials of degree , using a Taylor expansion (12) of order , we obtain where is the Peano kernel given by Following (26), we have where since . We observe that where So where does not depend on . Since we have this result says that
Since an or an is an , (37) means that . In summary we have proved the following theorem which presents not only a sufficient condition but also a necessary condition to obtain the desired error order.
Theorem 3. A necessary and sufficient condition to have for any is that for any polynomial of degree .
As an extension of this result, the next one shows the dependance of the error in terms of the regularity of the function and the -norm of .
Theorem 4. If for any polynomial of degree , then (34), (35), and (37) hold for any and .
From (35), for fixed and , is a continuous function with respect to . Moreover then we have the following result.
Theorem 5. For any given and (then is also given), the constant reaches its minimum value with respect to .
Let us set then for all .
Remark 6. It can be shown that bounds given by (34) and (35) are best bounds. Indeed, using a standard construction [6], we can find a function , which depends on and (then is given), such that
5. Increasing the Degree of Accuracy and the Order of Error Bounds
To increase the degree of accuracy of the process and consequently increase the order of error bounds, we consider the decomposition (26). Then we have the following result.
Theorem 7. Let (i) be the degree of accuracy of ;(ii) be the degree of accuracy of ;(iii) be the degree of accuracy of .One can increase the degree of accuracy of if and only if , for
Proof. Using (26), we observe that . So the parameter can be used to increase the degree of accuracy of the formula as follows: (i)if : let (a)if , since , then , , and for any , so we cannot increase the degree of accuracy;(b)if , then , , so we can increase of the degree of accuracy;(ii)if , then for any , so we cannot increase the degree of accuracy.So the result follows.
6. Examples
In this section we analyze four corrected quadrature rules: the midpoint and trapezoidal rules and two Simpson’s rules: with and with . We identify such that for all when it exists. Moreover for the midpoint and trapezoidal rules we compute the best for given by (39).
6.1. Midpoint Rule
Let and ; then the quadrature formula is where with and with . For , the kernel is given by Then , where
For this method, and , so . We obtain , and for any we have . We have computed some constants with (or ), for () using and , and for () using as reported in Table 1. In summary(i)for , we have ;(ii)for , we have .
6.2. Trapezoidal Rule
Let and ; then the quadrature formula is where with and with . For the kernel is given by Then , where
For this method and , so . We obtain , and for any we have . We have computed some constants with (or ), for () using and , and for () using as reported in Table 2. In summary(i)for , we have ;(ii)for , we have .
Remark 8. The function considered in [3] corresponds to the Peano kernel . Also the constants obtained in [3] correspond to the constants we obtained here.
6.3. First Simpson’s Rule
Let and ; then the quadrature formula is where with and with . For , the kernel is given by
For this method and , so . We obtain , and for any we have . We have only computed , for (or ), to compare with the midpoint and trapezoidal rules (see Table 3). Identifying is more complicated.
6.4. Second Simpson’s Rule
Let and ; then the quadrature formula is where with and with . For the kernel is given by
For this method and , so . We obtain , and for any we have . We have only computed , for (or ), to compare with the midpoint and trapezoidal rules (see Table 3). Identifying is more complicated.
6.5. Discussion
Let us observe that the constants of Table 1 (for the midpoint rule) are all equal or less than the constants of Table 2 (for the trapezoidal rule). Moreover, for the second (noncorrected, ) Simpson rule are all less than the of the first (noncorrected, ) Simpson rule which are themselves less than the of the corrected midpoint rule.
7. Corrected Composite Quadrature Rules
7.1. The Method
For an integral , where , a composite rule uses a partition of in subintervals and applies a formula on each subinterval. To simplify here we consider subintervals of equal length . Let us set for and for . Then The corrected composite rule is then defined by since there are cancellations of the first derivatives at the endpoints of the subintervals. The truncation error is But It follows that for and .
7.2. Numerical Examples
To illustrate the results, we approximate the following integral: using (55) for and for . Since for the absolute error, noted as , we have from (58) the order is estimated by the formula for , in Tables 4, 5, 6, and 7. Because the function we integrate is regular, we can identify the maximal orders and which correspond to the theoretical results.
Conflict of Interests
The author declares that there is no conflict of interests regarding the publication of this paper.