About a dubious proof of a correct result about closed Newton Cotes error formulas

David J. López; Jose A. Padilla; Juan Ruiz; Carlos Tapia; Juan C. Trillo

doi:10.1515/math-2023-0150

Article Open Access

About a dubious proof of a correct result about closed Newton Cotes error formulas

David J. López , Jose A. Padilla , Juan Ruiz , Carlos Tapia and Juan C. Trillo

Published/Copyright: November 24, 2023

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$Open Mathematics$

From the journal Open Mathematics Volume 21 Issue 1

Abstract

In this study, we comment about a wrong proof, at least incomplete, of the closed Newton Cotes error formulas for integration in a closed interval [ a , b ] . These error formulas appear as an intuitive generalization of the simple proof for the error formula of the trapezoidal rule, and their proofs present one controversial step, which converts the proofs in mischievous, or at least, this step needs a clear clarification that it is not easy to derive. The correct proof of such formulas comes from a technique based on the Peano kernel.

Keywords: Newton Cotes; closed; integration; error formulas; Simpson 3/8

MSC 2010: 41A05; 41A55; 65D30; 65D32

1 Introduction

Integration methods are a fundamental part of numerical analysis and constitute a way of approximating the value of integrals that cannot be calculated exactly using other methods. The numerical computation of an integral of an univariate function is normally called quadrature [1,2], and the term “cubature” is used for multiple integrals [3,4].

There are many different methods of quadrature for univariate functions. Some of them are based on a set of equally spaced points at the interval [ a , b ] , such as the closed Newton Cotes formulas and the open Newton Cotes formulas [5,6]. Other methods do not require that the knots are equally spaced, but they are interpolatory [7], and a general theory arises for these methods linking them with the Vandermonde matrices to the determination of the coefficients. When trying to locate the knots at the best places in the interval in order to reduce the approximation error, we find the Gaussian methods [8]. Many other methods try to avoid possible discontinuities in the underlying function by implementing nonlinear strategies that make them adaptive methods [9,10].

Numerical integration methods have many applications in different fields, ranging from research to various areas of engineering developments. It is used, for example, in image processing since the pixels can be interpreted as cell averages of a function [11–14], and in computational fluid dynamics where the right interpretation of the input data of the problem is again given by considering cell averages [15]. Many other examples arise in industry, for example, in naval manufacturing, just to mention a case. Computer programs, such as MaxSurf [16], for instance, offer to the naval architects many data about ships and submarines based on internal calculus where they constantly solve numerical integrals to approximate sectional areas, volumes, and inertial moments [17,18].

In many applications, the entries of the numerical integration methods are not longer the underlying functions but approximations, carried out by adequate measurements, of the function values at specific locations. Nevertheless, due to the final expressions of the formulas, the methods run equally. Maybe, the simplest integration method, but one of which its use is most extended, is the so-called trapezoidal rule. This method consists in nothing else but integrating the interpolation polynomial of degree 1 that passes through the points ( a , f ( a ) ) and ( b , f ( b ) ) instead of integrating the function. A generalization of this method to higher-order interpolation polynomials based on equally spaced points in the interval [ a , b ] constitutes the closed Newton Cotes rules.

In this article, we are going to deal with the error formulas associated with the closed Newton Cotes rules in a given interval [ a , b ] . In particular, we want to comment about a dubious proof, at least incomplete, of the Simpson 3 8 rule, i.e., the closed Newton Cotes formula using third-degree polynomials. This formula can be obtained using a modification of the simple proof for the error formula of the trapezoidal rule, although one of the crucial steps in the proof is not clear at all. In fact, it seems to be false. We explain in detail why the proof is doubtful and we indicate the already well-known technique of the Peano kernel [19–21], which gives a more elaborated but completely correct proof.

There exists the possibility of obtaining other error formulas for functions, which are fewer times differentiable, at least for Simpson rule [22]. However, the most standard error formulas for closed Newton Cotes rules, the ones that we consider in this article, assume dealing with differentiable enough functions to validate the interpolation error formulas used through the proofs.

This article is organized as follows: in Section 2, we recall the basic integration trapezoidal rule and give the proofs for its error formula. In Section 3, we present the closed Newton Cotes rules in an interval [ a , b ] . In Section 4, we present the intuitive way of deriving the error formula for the closed Newton Cotes rule with four points, i.e., the Simpson 3 8 rule, and we point out the key step that it is not well supported. In Section 5, we present the already well-known derivation of the error formula using the Peano kernel. In Section 6, we address the same situation for general close Newton Cotes rules based on polynomials of arbitrary degree n ∈ N . Finally, in Section 7, we give some conclusions.

2 The local trapezoidal rule

In this section, the main results used for the definition of the local trapezoidal rule and the derivation of the local error formula are shown, just to serve as an indicator of how someone could imagine to carry out the proofs for the attainment of the corresponding local error formulas for general closed Newton Cotes formulas that involve higher-degree polynomials.

Let us consider a continuous function f ( x ) , x ∈ [ a , b ] . The local trapezoidal rule uses an approximation of the underlying function in the interval [ a , b ] by a polynomial p 1 ( x ) of degree 1 , i.e., a linear approximation (see Figure 1). Either geometrically or analytically, one can easily calculate the integral of p 1 ( x ) in [ a , b ] and reach the local rule:

(1) ∫ a b f ( x ) d x ≈ ( b − a ) f ( a ) + f ( b ) 2 ,

which is nothing more than the formula of the area of the trapezoid of vertices ( a , f ( a ) ) , ( a , 0 ) , ( b , 0 ) , and ( b , f ( b ) ) according to Figure 1.

$Figure 1 Local trapezoidal rule: in blue, the integrated function f ( x ) , f\left(x), and in green, the approximation done in the interval [ a , b ] \left[a,b] by the polynomial p 1 ( x ) {p}_{1}\left(x) of degree 1 . 1.$

Figure 1

Local trapezoidal rule: in blue, the integrated function f ( x ) , and in green, the approximation done in the interval [ a , b ] by the polynomial p 1 ( x ) of degree 1 .

2.1 Error formula for the local trapezoidal rule

Assuming that f ( x ) ∈ C 2 [ a , b ] , we can first apply the Lagrange interpolation error formula to obtain

(2) E ( f , [ a , b ] ) = ∫ a b ( f ( x ) − p 1 ( x ) ) d x = 1 2 ∫ a b f ″ ( θ x ) ( x − a ) ( x − b ) d x ,

where θ x ∈ ( a , b ) depends on x , and therefore, it is not straightforward the fact of extracting the factor f ″ ( θ x ) out of the integral. In fact, it is possible to do it, but due to the well-known generalized mean value theorem for integrals, which we include here for the sake of completeness.

Theorem 1

Let f ( x ) and g ( x ) be two continuous functions in [ a , b ] with g ( x ) ≥ 0 ∀ x ∈ [ a , b ] (also for g ( x ) ≤ 0 ∀ x ∈ [ a , b ] ), and then there exist μ ∈ [ a , b ] such that

∫ a b f ( x ) g ( x ) d x = f ( μ ) ∫ a b g ( x ) d x .

Applying Theorem 1 to Expression (2) we obtain

(3) E ( f , [ a , b ] ) = f ″ ( μ ) 2 ∫ a b ( x − a ) ( x − b ) d x ,

and now, it is trivial to compute the last integral by means of the variable change x = a + ( b − a ) t , t ∈ [ 0 , 1 ] . Thus,

(4) E ( f , [ a , b ] ) = − 1 12 f ″ ( μ ) ( b − a ) 3 ,

with μ ∈ [ a , b ] .

3 Closed Newton Cotes rules

Let us consider a continuous function f ( x ) in the interval [ a , b ] and a partition X of the interval given by:

X = { x i } i = 0 n , x i = a + i h , i = 0 , … , n , and h = b − a n .

In order to estimate the integral of f ( x ) over the interval [ a , b ] , one possibility is to build the n degree interpolation polynomial based on the stencil of points S = { ( x 0 , f 0 ) , … , ( x n , f n ) } , where f i = f ( x i ) , i = 0 , … , n , with the purpose of calculating an approximated value of the integral by integrating p n ( x ) instead, i.e.,

(5) ∫ a b f ( x ) d x ≈ ∫ a b p n ( x ) d x = ∑ i = 0 n a i f i ,

where a i = ∫ a b L i ( x ) d x , and L i ( x ) , i = 0 , … , n , represent the Lagrange basis of polynomials.

The coefficients a i for the first values of n are given in Table 1.

Table 1

Coefficients a i , i = 0 , … , n for the most usual Newton Cotes local rules

n	Rule name	Formula
1	Trapezoid	h 2 ( f 0 + f 1 )
2	Simpson	h 3 ( f 0 + 4 f 1 + f 2 )
3	Simpson 3 8	3 h 8 ( f 0 + 3 f 1 + 3 f 2 + f 3 )
4	Boole	2 h 45 ( 7 f 0 + 32 f 1 + 12 f 2 + 32 f 3 + 7 f 4 )
5	Fifth degree	5 h 288 ( 19 f 0 + 75 f 1 + 50 f 2 + 50 f 3 + 75 f 4 + 19 f 5 )
6	Sixth degree	h 140 ( 41 f 0 + 216 f 1 + 27 f 2 + 272 f 3 + 27 f 4 + 216 f 5 + 41 f 6 )

4 A dubious proof of Simpson 3 8 error formula

In this section, we are going to use the same track as in the proof of the local error formula for the trapezoidal rule in order to derive the corresponding local error formula for the Simpson 3 8 local integration rule. In particular, we will be using the well-known expression for the interpolation error and the generalized mean value theorem for integrals. Let us give the following theorem, whose enunciate is true, but performing a captious proof that contains a dubious step. Despite this erroneous derivation, the error constant that appears at the end of the proof is correct. This fact is surprising since a correct result comes from a wrong conceptual application. The corrected proof using the Peano kernel strategy will be derived in the next section.

Theorem 2

Let f ( x ) be a function of class C 4 [ a , b ] . The approximation with the local Simpson 3 8 rule, using the points x 0 = a , x 1 = 2 a + b 3 , x 2 = a + 2 b 3 , x 3 = b ,

∫ a b f ( x ) d x ≈ 3 h 8 ( f 0 + 3 f 1 + 3 f 2 + f 3 ) ,

commits a local error E ( f , [ a , b ] ) = ∫ a b f ( x ) d x − ∫ a b p 3 ( x ) d x given by:

(6) E ( f , [ a , b ] ) = − 3 80 h 5 f i v ( μ ) ,

with μ ∈ [ a , b ] and h = b − a 3 .

Proof

Let us write the error E ( f , [ a , b ] ) as:

(7) E ( f , [ a , b ] ) = ∫ a b f ( x ) d x − ∫ a b p 3 ( x ) d x = ∫ a b ( f ( x ) − p 3 ( x ) ) d x .

Using the expression of the Lagrange interpolation error,

(8) f ( x ) − p 3 ( x ) = f i v ( θ x ) 4 ! ( x − x 0 ) ( x − x 1 ) ( x − x 2 ) ( x − x 3 ) ,

with θ x ∈ ( a , b ) . Plugging Expression (8) into (7), we obtain

(9) E ( f , [ a , b ] ) = ∫ a b f i v ( θ x ) 4 ! ( x − x 0 ) ( x − x 1 ) ( x − x 2 ) ( x − x 3 ) d x .

With the idea of obtaining the factor f i v ( θ x ) 4 ! out of the integral, we separate the integration interval [ a , b ] in parts. Doing this, we ensure that the function g ( x ) = ( x − x 0 ) ( x − x 1 ) ( x − x 2 ) ( x − x 3 ) will be of constant sign in each subinterval, and therefore, it will be possible to apply the generalized mean value theorem for integrals at each term. Thus,

(10) ∫ a b f i v ( θ x ) 4 ! ( x − x 0 ) ( x − x 1 ) ( x − x 2 ) ( x − x 3 ) d x = ∑ i = 0 2 ∫ x i x i + 1 f i v ( θ x ) 4 ! ( x − x 0 ) ( x − x 1 ) ( x − x 2 ) ( x − x 3 ) d x .

For each term in the sum, we can apply the mentioned mean value theorem in order to obtain

(11) ∫ x i x i + 1 f i v ( θ x ) 4 ! ( x − x 0 ) ( x − x 1 ) ( x − x 2 ) ( x − x 3 ) d x = f i v ( μ i ) 4 ! ∫ x i x i + 1 ( x − x 0 ) ( x − x 1 ) ( x − x 2 ) ( x − x 3 ) d x ,

i = 0 , 1 , 2 , with μ 0 ∈ [ a , x 1 ] , μ 1 ∈ [ x 1 , x 2 ] , μ 2 ∈ [ x 2 , x 3 ] . Then,

(12) E ( f , [ a , b ] ) = 1 4 ! ∑ i = 0 2 f i v ( μ i ) ∫ x i x i + 1 ( x − x 0 ) ( x − x 1 ) ( x − x 2 ) ( x − x 3 ) d x .

Using the variable change x = a + t ( b − a ) , we can easily compute the last integrals and we obtain

(13) ∫ a x 1 ( x − x 0 ) ( x − x 1 ) ( x − x 2 ) ( x − x 3 ) d x = − 19 7,290 ( b − a ) 5 , ∫ x 1 x 2 ( x − x 0 ) ( x − x 1 ) ( x − x 2 ) ( x − x 3 ) d x = 11 7,290 ( b − a ) 5 , ∫ x 2 b ( x − x 0 ) ( x − x 1 ) ( x − x 2 ) ( x − x 3 ) d x = − 19 7,290 ( b − a ) 5 .

Therefore,

(14) E ( f , [ a , b ] ) = − 1 4 ! h 5 19 f i v ( μ 0 ) − 11 f i v ( μ 1 ) + 19 f i v ( μ 2 ) 30 .

At this point, we are going to carry out a mischievous argument, which is not true in general, and then, it makes the proof incorrect or at least incomplete. However, the obtained result is mysteriously correct.

The last term in (14) can be written as:

(15) 19 f i v ( μ 0 ) − 11 f i v ( μ 1 ) + 19 f i v ( μ 2 ) 30 = 27 30 19 f i v ( μ 0 ) − 11 f i v ( μ 1 ) + 19 f i v ( μ 2 ) 27 ,

and then, one can wrongly interpret that the value 19 f i v ( μ 0 ) − 11 f i v ( μ 1 ) + 19 f i v ( μ 2 ) 27 is contained between the minimum and the maximum values of f i v ( x ) in the compact interval [ a , b ] , and one would follow writing

(16) 19 f i v ( μ 0 ) − 11 f i v ( μ 1 ) + 19 f i v ( μ 2 ) 27 = f i v ( μ ) ,

with μ ∈ [ a , b ] .

Using (15) and (16) in (14), we finally obtain

(17) E ( f , [ a , b ] ) = − 1 4 ! h 5 27 30 f i v ( μ ) = − 3 80 h 5 f i v ( μ ) ,

with μ ∈ [ a , b ] . □

The rest of this section is devoted to give a counterexample of the result applied in (16).

Remark 1

Let f ( x ) be a continuous function in [ a , b ] = [ 0 , 1 ] , and let α 0 , α 1 , and α 2 be three nonzero real numbers such that α 0 + α 1 + α 2 = 1 , with one of them having different sign than the other two. Let us suppose that α 0 > 0 , α 1 < 0 , and α 2 > 0 . We are going to give a counterexample to show that in general, the following equality is not true:

(18) α 0 g ( μ 0 ) + α 1 g ( μ 1 ) + α 2 g ( μ 2 ) = g ( μ ) ,

where μ 0 ∈ [ 0 , 1 3 ] , μ 1 ∈ [ 1 3 , 2 3 ] , μ 2 ∈ [ 2 3 , 1 ] , and μ ∈ [ 0 , 1 ] .

The counterexample can be given by:

(19) g ( x ) = 6 x , 0 ≤ x ≤ 1 6 , − 6 x + 2 , 1 6 ≤ x ≤ 1 2 , 6 x − 4 , 1 2 ≤ x ≤ 5 6 , − 6 x + 7 , 5 6 ≤ x ≤ 1 ,

with μ 0 = 1 6 , μ 1 = 1 2 , and μ 2 = 5 6 . It happens that (18) becomes

(20) α 0 g ( μ 0 ) + α 1 g ( μ 1 ) + α 2 g ( μ 2 ) = α 0 − α 1 + α 2 > 1 > g ( μ ) ,

for all μ ∈ [ 0 , 1 ] .

Note that the unique ways for (18) to be true are that ∣ α 1 ∣ ≤ ∣ α 0 ∣ and μ 1 = μ 0 or ∣ α 1 ∣ ≤ ∣ α 2 ∣ and μ 1 = μ 2 , or μ 0 = μ 1 = μ 2 .

We note also that equation (16) could not be true, as shown in the aforementioned counterexample, since it is the same kind of equation with g ( x ) = f i v ( x ) .

Let us give another explanatory example to illustrate the incompleteness of the aforementioned proof of Theorem 2. Let us consider the function f ( x ) given by:

(21) f ( x ) = x 4 24 + 55 1,296 x − 19 2,592 , 0 ≤ x ≤ 1 3 , − x 5 10 + 5 24 x 4 − 1 9 x 3 + 1 27 x 2 + 47 1,296 x − 15 2,168 , 1 3 < x ≤ 1 2 , x 5 10 − 7 24 x 4 + 7 18 x 3 − 23 108 x 2 + 8 81 x − 16 1,215 , 1 2 < x ≤ 2 3 , x 4 24 − 1 18 x 3 + 1 12 x 2 , 2 3 < x ≤ 1 .

Computing the fourth derivative of this function, we obtain

(22) f ( i v ) ( x ) = 1 , 0 ≤ x ≤ 1 3 , − 12 x + 5 , 1 3 < x ≤ 1 2 , 12 x − 7 , 1 2 < x ≤ 2 3 , 1 , 2 3 < x ≤ 1 .

Note that f ( x ) is a C 4 function and f ( i v ) ( x ) cannot satisfy equation (16) since

19 f i v ( μ 0 ) − 11 f i v ( μ 1 ) + 19 f i v ( μ 2 ) 27 = 19 − 11 f i v ( μ 1 ) + 19 27 = 38 − 11 f i v ( μ 1 ) 27 > 1 ,

unless μ 1 = 1 3 or μ 1 = 2 3 due to the fact that f ( i v ) ( x ) < 1 , ∀ x .

Therefore, in general, the given proof for Theorem 2 is wrong, unless there is a way of proving that μ 1 = μ 0 or μ 1 = μ 2 , which is not clear to be possible to be attained.

5 A correct derivation of the Simpson 3 8 error formula: the Peano kernel

In this section, and for the sake of completeness, we give a correct proof of the error formula for the Simpson 3 8 rule. This proof is based on the well-known technique of the Peano kernel [19,20]. First, we introduce the definition of the Peano kernel, and we give a lemma proving that this kernel is of constant sign for the case of Simpson 3 8 .

Given f ∈ C 4 [ a , b ] , let us consider the following linear operator:

(23) L ( f ) = ∫ a b f ( x ) d x − 3 8 h f ( a ) + 3 f 2 a + b 3 + 3 f a + 2 b 3 + f ( b ) ,

where h stands for the uniform spacing between the nodes a , 2 a + b 3 , a + 2 b 3 , and b .

The Peano kernel for the operator L is given by the following definition.

Definition 1

The Peano kernel associated with the Simpson 3 8 rule has the expression:

(24) k ( θ ) = L ( ( x − θ ) + 3 ) , ∀ θ ∈ [ a , b ] ,

where

(25) ( x − θ ) + = x − θ , x ≥ θ , 0 , otherwise.

In the following lemma, we proof that the Peano kernel in (24) maintains the constant sign.

Lemma 1

The Peano kernel k ( θ ) in (24) satisfies

k ( θ ) ≤ 0 , ∀ θ ∈ [ a , b ] .

Proof

We start computing the expressions for the Peano kernel k ( θ ) . According to (24),

(26) k ( θ ) = ∫ a b ( x − θ ) + 3 d θ − 3 h 8 ( a − θ ) + 3 + 3 2 a + b 3 − θ + 3 + 3 a + 2 b 3 − θ + 3 + ( b − θ ) + 3 .

Introducing the positive part ( x − θ ) + given in (25) into Expression (26) and integrating, we obtain

(27) k ( θ ) = 1 4 ( b − θ ) 4 − 3 h 8 ( 3 2 a + b 3 − θ 3 + 3 a + 2 b 3 − θ 3 + ( b − θ ) 3 ) , a ≤ θ ≤ 2 a + b 3 , 1 4 ( b − θ ) 4 − 3 h 8 ( 3 a + 2 b 3 − θ 3 + ( b − θ ) 3 ) , 2 a + b 3 ≤ θ ≤ a + 2 b 3 , 1 4 ( b − θ ) 4 − 3 h 8 ( b − θ ) 3 , a + 2 b 3 ≤ θ ≤ b .

Let us study the sign of the previous expressions for the three cases, θ ∈ [ a , 2 a + b 3 ] , θ ∈ [ 2 a + b 3 , a + 2 b 3 ] and θ ∈ [ a + 2 b 3 , b ] .

Case 1: θ ∈ [ a , 2 a + b 3 ]
We have
k ( θ ) = 1 4 ( b − θ ) 4 − 3 h 8 ( b − θ ) 3 + 3 a + 2 b 3 − θ 3 + 3 2 a + b 3 − θ 3 , k ′ ( θ ) = − ( b − θ ) 3 + 63 h 8 ( b − θ ) 2 + 135 8 h 3 − 162 8 h 2 ( b − θ ) , k ″ ( θ ) = 3 ( b − θ ) 2 − 63 h 4 ( b − θ ) + 162 8 h 2 .
First, we see that k ″ ( θ ) < 0 , ∀ θ ∈ ( a , 2 a + b 3 ) . We observe that k ″ ( θ ) is a parabola with positive leading coefficient such that its vertex is located at v = 7 a + b 8 , where k ′ ′ ′ ( v ) = 0 and k ″ ( v ) < 0 . Moreover, k ″ ( a ) = 0 , and k ″ ( 3 a + b 4 ) = 0 . Hence, we obtain it. Second, we see that k ′ ( θ ) < 0 , ∀ θ ∈ ( a , 2 a + b 3 ) . This comes from the fact that k ′ ( a ) = 0 , and it is a strictly decreasing function since k ″ ( θ ) < 0 . Finally, we prove that k ( θ ) ≤ 0 , ∀ θ ∈ [ a , 2 a + b 3 ] . In order to see this point, we remark that k ( a ) = 0 , and k ( θ ) is a strictly decreasing function in the mentioned interval since k ′ ( θ ) < 0 , ∀ θ ∈ ( a , 2 a + b 3 ) .
Case 2: θ ∈ [ 2 a + b 3 , a + 2 b 3 ]
We have,
k ( θ ) = 1 4 ( b − θ ) 4 − 3 h 8 ( b − θ ) 3 + 3 a + 2 b 3 − θ 3 , k ′ ( θ ) = − ( b − θ ) 3 + 3 h 8 3 ( b − θ ) 2 + 9 a + 2 b 3 − θ 2 , k ″ ( θ ) = 3 ( b − θ ) 2 − 3 h 8 6 ( b − θ ) + 18 a + 2 b 3 − θ .
We note that k 2 a + b 3 = k a + 2 b 3 = − h 4 8 < 0 and k ( a + b 2 ) = − 9 h 4 64 < 0 . We also obtain that k ′ ( θ ) is a strictly increasing function, since k ″ ( θ ) > 0 , ∀ θ ∈ ( 2 a + b 3 , a + 2 b 3 ) . Moreover, k ′ 2 a + b 3 < 0 , k ′ ( a + b 2 ) = 0 , and k ′ a + 2 b 3 > 0 . Therefore, the function k ( θ ) is strictly decreasing and negative in ( 2 a + b 3 , a + b 2 ) , and it is strictly increasing, but still negative in ( a + b 2 , a + 2 b 3 ) . Thus, we obtain k ( θ ) ≤ 0 , ∀ θ ∈ [ 2 a + b 3 , a + 2 b 3 ] .
Case 3: θ ∈ [ a + 2 b 3 , b ]
This case is more straightforward:
□ k ( θ ) = 1 4 ( b − θ ) 4 − 3 h 8 ( b − θ ) 3 = 1 4 ( b − θ ) 3 b − θ − 3 h 2 ≤ − 1 4 ( b − θ ) 3 h 2 ≤ 0 .

Theorem 3

(Theorem 2 with corrected proof) Let f ( x ) be a function of class C 4 [ a , b ] . The approximation with the local Simpson 3 8 rule, using the points x 0 = a , x 1 = 2 a + b 3 , x 2 = a + 2 b 3 , x 3 = b ,

∫ a b f ( x ) d x ≈ 3 h 8 ( f 0 + 3 f 1 + 3 f 2 + f 3 ) ,

commits a local error E ( f , [ a , b ] ) given by:

(28) E ( f , [ a , b ] ) = − 3 80 h 5 f i v ( μ ) , with μ ∈ [ a , b ] , h = b − a 3 .

Proof

We consider the third-degree Taylor expansion of f ( x ) in the form:

(29) f ( x ) = f ( a ) + f ′ ( a ) ( x − a ) + f ″ ( a ) 2 ! ( x − a ) 2 + f ″ ′ ( a ) 3 ! ( x − a ) 3 + 1 3 ! ∫ a x ( x − θ ) 3 f i v ( θ ) d θ ,

with θ ∈ ( a , x ) . Applying now the linear operator L defined in (23) to both sides of the equality (29), we obtain

L ( f ) = L ( p ( x ) ) + 1 3 ! L ∫ a x ( x − θ ) 3 f i v ( θ ) d θ ,

where p ( x ) = f ( a ) + f ′ ( a ) ( x − a ) + f ″ ( a ) 2 ! ( x − a ) 2 + f ″ ′ ( a ) 3 ! ( x − a ) 3 is the third-degree Taylor polynomial around the point a . Now, taking into account that L ( p ( x ) ) = 0 , since the Simpson 3 8 rule is exact for polynomials of degree 3 or less, we reach to:

(30) L ( f ) = 1 3 ! L ∫ a x ( x − θ ) 3 f i v ( θ ) d θ .

Using (25), we can rewrite Expression (30) as:

(31) L ( f ) = 1 3 ! L ∫ a b ( x − θ ) + 3 f i v ( θ ) d θ .

Let us now define g ( x ) = ∫ a b ( x − θ ) + 3 f i v ( θ ) d θ . By applying the Fubini theorem, we can see that

(32) ∫ a b g ( x ) d x = ∫ a b ∫ a b ( x − θ ) + 3 f i v ( θ ) d θ d x = ∫ a b f i v ( θ ) ∫ a b ( x − θ ) + 3 d x d θ .

Then, merging (31) and (32), we obtain

(33) L ( f ) = 1 3 ! ∫ a b f i v ( θ ) L ( ( x − θ ) + 3 ) d θ = 1 3 ! ∫ a b f i v ( θ ) k ( θ ) d θ .

At this point, we can make use of the generalized mean value theorem for integrals given in Theorem 1 and the fact that k ( θ ) ≤ 0 , ∀ θ ∈ [ a , b ] according to Lemma 1 to obtain,

(34) L ( f ) = 1 3 ! f i v ( μ ) ∫ a b k ( θ ) d θ ,

with μ ∈ [ a , b ] . From the expression of k ( θ ) in (27), we can compute

(35) ∫ a b k ( θ ) d θ = ∫ a 2 a + b 3 k ( θ ) d θ + ∫ 2 a + b 3 a + 2 b 3 k ( θ ) d θ + ∫ a + 2 b 3 b k ( θ ) d θ = 1 4 ∫ a b ( b − θ ) 4 d θ − 3 h 8 ∫ a b ( b − θ ) 3 d θ + ∫ a a + 2 b 3 3 a + 2 b 3 − θ 3 d θ + ∫ a 2 a + b 3 3 2 a + b 3 − θ 3 d θ = − 9 40 h 5 .

Finally, introducing (35) into (34) we obtain

□ E ( f , [ a , b ] ) = L ( f ) = 1 3 ! f i v ( μ ) − 9 40 h 5 = − 3 80 h 5 f i v ( μ ) , with μ ∈ [ a , b ] , h = b − a 3 .

Remark 2

The condition about the constant sign of the Peano kernel is needed in order to extract the derivative out of the integral, i.e., it is needed if we want to guarantee that for a given rule that is exact for polynomials of degree less or equal to m , the following equality holds

(36) L ( f ) = 1 m ! ∫ a b f m + 1 ( θ ) k ( θ ) d θ = 1 m ! f m + 1 ( μ ) ∫ a b k ( θ ) d θ , with μ ∈ [ a , b ] .

Note, for example, that the rule ∫ a b f ( x ) d x ≈ ( b − a ) f ( c ) , for c = 1 2 + ε , with ε such that 1 ∕ 3 − c 2 > 0 does not satisfy any equation of the type:

(37) L ( f ) = D f ′ ( μ ) , with μ ∈ [ a , b ] ,

since the general equation for a rule of approximation order m + 1 = 1 (36) is not fulfilled, i.e.,

L ( f ) = ∫ a b f ′ ( θ ) k ( θ ) d θ ≠ f ′ ( μ ) ∫ a b k ( θ ) d θ , for any μ ∈ [ a , b ] ,

due to the fact that the Peano kernel is not of constant sign in [a,b] in this case. More precisely,

k ( θ ) = L ( ( x − θ ) + 0 ) = ∫ a b ( x − θ ) + 0 d x − ( b − a ) ( c − θ ) + 0 = a − θ ≤ 0 , θ < c , b − θ ≥ 0 . θ ≥ c .

Let us consider the interval [ a , b ] = [ 0 , 1 ] , and let us suppose that L ( f ) satisfies (37). Then, applying the error formula for the function f ( x ) = x , we obtain that D = 1 2 − c < 0 . And applying again the error formula, but this time for the function f ( x ) = x 2 , we obtain 1 3 − c 2 = 2 D μ , with μ ∈ [ 0 , 1 ] . Since D < 0 and μ ≥ 0 , we obtain that 1 3 − c 2 ≤ 0 , which immediately gives a contradiction by hypothesis with the chosen value of c .

6 About the dubious proof for general closed Newton Cotes error formulas

As we have pointed out in the previous section dedicated to the correct derivation of the error formula for Simpson 3 8 rule, more advanced techniques based on the Peano kernel are needed. These proofs are quite technical, and require to prove that the Peano kernel satisfies certain positivity requirements. In [21], pages 308 to 313, the interested reader can consult the correct derivation for the general case of the closed Newton Cotes error formulas. Previous studies [5,23,24] also give some interesting insights on the topic. In this section we are going to connect this known result with the fact that the dubious proof carried out for Simpson 3 8 can be also reproduced for the general case, and in the same way, it gives the correct error formulas. The dubious step is validated just because of the existence of the complete proofs using the positivity of the Peano kernel. Let us explain the situation with more detail by considering two different cases, the case of the closed Newton Cotes formulas coming from a polynomial p n ( x ) with n an odd integer and the case where n is an even integer.

6.1 Closed Newton Cotes error formulas for n odd

Let us consider n an odd integer. When using n + 1 equally spaced points x i , i = 0 , … , n in the interval [ a , b ] to build the closed Newton Cotes rule, one builds a polynomial p n ( x ) by the Lagrange interpolation, and according to [21], the error can be expressed as:

(38) E ( f , [ a , b ] ) = ∫ a b f ( x ) d x − ∫ a b p n ( x ) d x = C h n + 2 ( n + 1 ) ! f ( n + 1 ) ( μ ) ,

where μ ∈ ( a , b ) , and C is a constant given by:

(39) C = ∫ a b q n + 1 ( x ) d x h n + 2 ,

where q n + 1 ( x ) = ( x − a ) ( x − ( n − 1 ) a + b n ) … ( x − a + ( n − 1 ) b n ) ( x − b ) . This expression amounts to the error committed by the integration rule for the function f ( x ) = x n + 1 .

On the other hand, if we carry out the right-forward steps of the dubious proof, we have that

(40) E ( f , [ a , b ] ) = ∫ a b f ( n + 1 ) ( μ x ) ( n + 1 ) ! q n + 1 ( x ) d x = ∑ i = 1 n ∫ x i − 1 x i f ( n + 1 ) ( μ x ) ( n + 1 ) ! q n + 1 ( x ) d x , = 1 ( n + 1 ) ! ∑ i = 1 n f ( n + 1 ) ( μ i ) ∫ x i − 1 x i q n + 1 ( x ) d x ,

where we have applied that the function q n + 1 ( x ) has constant sign in each [ x i , x i + 1 ] . Denoting now σ i ≔ ∫ x i − 1 x i q n + 1 ( x ) d x , we obtain,

(41) E ( f , [ a , b ] ) = ∑ i = 1 n σ i ( n + 1 ) ! ∑ i = 1 n ( f ( n + 1 ) ( μ i ) σ i ) ∑ i = 1 n σ i , = ∫ a b q n + 1 ( x ) d x ( n + 1 ) ! ∑ i = 1 n ( f ( n + 1 ) ( μ i ) σ i ) ∑ i = 1 n σ i ,

since ∫ a b q n + 1 ( x ) d x = ∑ i = 1 n σ i .

Comparing the expressions for the error (41) and (38), we observe that they read exactly the same if we assume that there exists μ in [ a , b ] such that

(42) ∑ i = 1 n ( f ( n + 1 ) ( μ i ) σ i ) ∑ i = 1 n σ i = f ( n + 1 ) ( μ ) .

Therefore, the correctness of Expression (38), which was derived using the Peano kernel, explains why the dubious derivation gives mysteriously the correct error formula. Note that this assumption in (42) is not in general valid as pointed out in the counterexample given in (21), since σ 1 < 0 , σ 2 > 0 , … , σ n < 0 .

But in this particular case of the proof, we have the guarantee that it is true due to the Peano kernel.

6.2 Closed Newton Cotes error formulas for n even

Let us consider now n an even integer. We again use n + 1 equally spaced points x i , i = 0 , … , n in the interval [ a , b ] to build the closed Newton Cotes rule, i.e., we construct a polynomial p n ( x ) by Lagrange interpolation, and according to [21], since the rule is this time exact of degree n + 1 , the error can be expressed in this case as:

(43) E ( f , [ a , b ] ) = ∫ a b f ( x ) d x − ∫ a b p n ( x ) d x = C h n + 3 ( n + 2 ) ! f ( n + 2 ) ( μ ) ,

where μ ∈ ( a , b ) , and C is a constant given by:

(44) C = ∫ a b q n + 2 ( x ) d x h n + 3 ,

where q n + 2 ( x ) = ( x − a ) … x − ( n 2 + 1 ) a + ( n 2 − 1 ) b n x − a + b 2 2 x − ( n 2 − 1 ) a + ( n 2 + 1 ) b n … ( x − b ) . This expression amounts to the error committed by the integration rule for the function f ( x ) = x n + 2 .

The proof of error Formula (43) follows a similar path as in (38), but taking into account as a first step that,

E ( f , [ a , b ] ) = ∫ a b f ( x ) d x − ∫ a b p ˜ n + 1 ( x ) d x + ∫ a b p ˜ n + 1 ( x ) d x − ∫ a b p n ( x ) d x = ∫ a b f ( x ) d x − ∫ a b p ˜ n + 1 ( x ) d x ,

where p ˜ n + 1 ( x ) is the Hermite interpolation polynomial that satisfies the same interpolation conditions than p n ( x ) plus the extra condition p ˜ ′ ( a + b 2 ) = f ′ ( a + b 2 ) . In this case, it is clear that the interpolation error is given by:

f ( x ) − p ˜ n + 1 ( x ) = f ( n + 2 ) ( μ x ) ( n + 2 ) ! q n + 2 ( x ) d x ,

with μ x an intermediate point located in ( a , b ) and depending on the value of x ∈ [ a , b ] .

On the other hand, if we carry out again the right-forward steps of the dubious proof, we have that,

(45) E ( f , [ a , b ] ) = ∫ a b f ( n + 2 ) ( μ x ) ( n + 2 ) ! q n + 2 ( x ) d x = ∑ i = 1 n ∫ x i − 1 x i f ( n + 2 ) ( μ x ) ( n + 2 ) ! q n + 2 ( x ) d x , = 1 ( n + 2 ) ! ∑ i = 1 n f ( n + 2 ) ( μ i ) ∫ x i − 1 x i q n + 2 ( x ) d x ,

where we have applied that the function q n + 2 ( x ) has constant sign in each [ x i , x i + 1 ] . Denoting now σ i ≔ ∫ x i − 1 x i q n + 2 ( x ) d x , we obtain

(46) E ( f , [ a , b ] ) = ∑ i = 1 n σ i ( n + 2 ) ! ∑ i = 1 n ( f ( n + 2 ) ( μ i ) σ i ) ∑ i = 1 n σ i , = ∫ a b q n + 2 ( x ) d x ( n + 2 ) ! ∑ i = 1 n ( f ( n + 2 ) ( μ i ) σ i ) ∑ i = 1 n σ i ,

since ∫ a b q n + 2 ( x ) d x = ∑ i = 1 n σ i .

Comparing the expressions for the error (46) and (43), we observe that they read exactly the same if we assume that there exists μ in [ a , b ] such that

(47) ∑ i = 1 n ( f ( n + 2 ) ( μ i ) σ i ) ∑ i = 1 n σ i = f ( n + 2 ) ( μ ) .

Therefore, also in this case, the correctness of Expression (43), which was derived using the Peano kernel, explains why the dubious derivation gives mysteriously the correct error formula. We need to mention and emphasize that this assumption in (47) is not in general valid either for μ i and σ i whatever. It is just valid in this particular case as it is proved by the reasoning we have shown.

7 Conclusions

In this article, we have commented about the incompleteness or the incorrectness of an extended proof of the closed Newton Cotes error formulas for numerical integration in a closed interval [ a , b ] . Albeit these formulas, the ones for Simpson and Simpson 3 8 and the rest of the closed Newton Cotes error formulas for higher degrees of approximation, come as an intuitive generalization of the simple proof for the error formula of the Trapezoidal rule, they contain a controversial step that it is not easy to overcome with simple arguments. In fact, one needs to be careful about using apparently wrong logical arguments, which, however, obtain correct results. One example of this is just the mentioned proof that includes a particular step without the needed rigor; it results to be correct, but it is not easy to see. Just for completeness and to indicate that the appropriate proof is more intricate, we have added the correct proof for the error formula of Simpson 3 8 based on the Peano kernel. For higher degrees, the correct derivation can be consulted in [21] through pages 308 to 313 . For the cases of dealing with similar formulas to Newton Cotes, or with weaker assumptions with respect to the regularity of the integrated function, one can consult other related techniques in [19,20,22].

Funding information: This research has been partially supported through the Programa de Apoyo a la investigación de la fundación Séneca-Agencia de Ciencia y Tecnología de la Región de Murcia 19374/PI714 and through the national research project MTM2015-64382-P (MINECO/FEDER).
Author contributions: All authors have accepted responsibility for the entire content of this manuscript and approved its submission.
Conflict of interest: The authors state no conflicts of interest.
Data availability statement: Data sharing is not applicable to this article as no data sets were generated or analyzed during the current study.

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Received: 2022-11-17

Revised: 2023-10-24

Accepted: 2023-10-24

Published Online: 2023-11-24

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

https://doi.org/10.1515/math-2023-0150

Keywords for this article

Newton Cotes; closed; integration; error formulas; Simpson 3/8

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