# Extended Error Expansion of Classical Midpoint Rectangle Rule for Cauchy Principal Value Integrals on an Interval

Authors: Chunxiao Yu; Lingling Wei
Journal: Journal of Mathematics
 (2021)

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## Abstract

The classical composite midpoint rectangle rule for computing Cauchy principal value integrals on an interval is studied. By using a piecewise constant interpolant to approximate the density function, an extended error expansion and its corresponding superconvergence results are obtained. The superconvergence phenomenon shows that the convergence rate of the midpoint rectangle rule is higher than that of the general Riemann integral when the singular point coincides with some priori known points. Finally, several numerical examples are presented to demonstrate the accuracy and effectiveness of the theoretical analysis. This research is meaningful to improve the accuracy of the collocation method for singular integrals.


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## Body

## 1. Introduction

Singular integrals, especially Cauchy principal value integrals, are usually encountered in the fields of Boundary Element Method (BEM) [1–3], for example, the fluid mechanics, the elasticity and fracture mechanics, the acoustics, and the electromagnetics. In these fields (including their related physical problems), much attention has been paid to the Cauchy principal value integrals [4–8]. We now consider the following integral:(1)If;s=c.p∫abfxx−sdx=gs,s∈a,b,where fx is Holder continuous on the interval a,b, fx/x−sis the density function, c.p∫ab denotes a Cauchy principal value integral, and s is the singular point. There are many ways to define equation (1). However, these definitions can be proved to be mathematically equivalent. In this paper, we adopt the following definition:(2)c.p∫abfxx−sdx=limε⟶0∫as−εfxx−sdx+∫s+εbfxx−sdx,s∈a,b.For this level of regularity of integrand, some methods based on the Chebyshev series expansion are much effective. They are easy to include the adaptive feature and can be applied to much difficult integrands [9–11]. At the same time, there are numerous works that have been devoted to developing efficient quadrature formulas, such as the Gaussian method [12,13], the Newton–Cotes method [14–16], the spline method [17, 18], and some other methods [19–25]. Usually, Gaussian rules have good accuracy if the integrand is smooth, while Newton–Cotes rules are attractive due to their ease of implementation and flexibility of mesh. To improve the accuracy of boundary element analysis, an efficient method called general (composite) Newton–Cotes rule has been studied and used to compute Cauchy principal value integrals and Hadamard finite-part integrals [26, 27]. When the singular point s coincides with some priori known points, Newton–Cotes rules can reach a high-order convergence rate [28, 29]. This is the so-called pointwise superconvergence phenomenon of the Newton–Cotes rules.In this paper, we will focus on the superconvergence phenomenon of midpoint rectangle rules for Cauchy principal integrals with the density functionfx/x−s being replaced by the approximation function fx^i/x^i−s,i=0,1,…,n−1, where x^i is the middle point of each subinterval of a,b. Different from the idea provided by Linz in [30] to calculate the hypersingular integral on an interval, we will present a direct method to compute the Cauchy principal integral. Based on the error estimate [31–33], the error function is determined by a certain special function S0τ. We will also give the necessary and sufficient conditions to be satisfied by the superconvergence points. In addition, we will not only try to obtain the error estimate of the superconvergence phenomenon but also make some investigation about the superconvergence points.The rest of this paper is organized as follows. In Section2, some basic formulas of the midpoint rectangle rule are introduced, and our main results are presented. In Section 3, some lemmas are given and the proof of the main results is completed. In Section 4, several numerical examples are provided to validate our analysis. At last, the concluding remarks are presented.


## 2. The Superconvergence of the Composite Midpoint Rectangle Rule

Leta=x0<x1<⋯<xn−1<xn=b be a uniform partition of the interval a,b with a mesh size h=b−a/n. We first define the piecewise constant interpolant for fx:(3)fCx=fx^i,x^i=xi+h2,x∈xi,xi+1,and then, we define a linear transformation:(4)x=x^iτ≔τ+1xi+1−xi2+xi,τ∈−1,1,which maps the reference element −1,1 onto the subinterval xi,xi+1. By replacing fx/x−s in equation (1) with fCx/x^i−s, we can obtain the following composite midpoint rectangle rule:(5)Inf;s=c.p∫abfCxx^i−sdx=∑i=0n−1wisfx^i=c.p∫abfxx−sdx−Enf;s,where wis denotes the Cotes coefficient given by wis=h/x^i−s and Enf;s is the error function.Theorem 1 (see [5]).
Assume thatfx∈Cαa,b,α∈0,1. For the midpoint rectangle rule Inf;s defined in equation (5), assume that s∈xm,xm+1,s≠x^m,m=0,1,2,…,n−1, and there exists a positive constant C that is independent of h and s, such that(6)Enf;s≤Clnh+lnγτhα,where(7)γτ=min0≤i≤ns−xih=1−τ2.
Before presenting the main results, we first define(8)φ0x=−12xc.p∫−11ττ−xdτ,x<1,−12x∫−11ττ−xdτ,x>1,S0τ≔φ0τ+∑i=1∞φ02i+τ+φ0−2i+τ,τ∈−1,1.Theorem 2.
Assume thatfx∈Cla,b,l≥2. For the midpoint rectangle rule Inf;s defined in equation (5), assume that s=xm+1+τh/2,s≠x^m,m=0,1,2,…,n−1, and there exists a positive constant C that is independent of h and s, such that (9)Enf;s=−fsS0τ+Rns,where(10)Rns≤Clnh+γ−1τ+ηshl−1hl,where γτ is defined by equation (7) and(11)ηs=max1s−a,1b−s.Remark 1.
Under the same assumptions of Theorem2, when τ∗ is the zero of S0τ∗, we have(12)Enf;s≤Clnh+γ−1τ∗+ηsh1−lhl.


## 3. The Proof of the Main Results

In this section, we mainly complete the proof of Theorem2.


### 3.1. Some Necessary Lemmas

Lemma 1.
Under the same assumptions of Theorem2, there holds that(13)x^i−sfx−x−sfx^i=fsx^i−x+∑k=1l−1fksk!x−skx^i−x−∑k=1l−1∑j=kl−1fjsk!j−k!x−sj−k+1x^i−xk+Rfx,where(14)Rfx=Rf1x+Rf2x+Rf3x,with(15)Rf1x=−flθ1l!x−sx^i−xl,Rf2x=flθ2l!x−slx^i−x,Rf3x=−∑k=1l−1flθ3k!l−k!x−sl−k+1x^i−x,and θ1,θ2∈xi,xi+1,θ3∈x,sors,x.Proof.
Note thatfx∈Cla,b. By applying the following Taylor expansion to fx^i at the point x, we obtain(16)fx^i=fx+∑k=1l−1fkxk!x^i−xk+flθ1l!x^i−xl,θ1∈xi,xi+1.
Similarly, we have(17)fx=fs+∑k=1l−1fksk!x−sk+flθ2l!x−sl,θ2∈xi,xi+1.
Thus,(18)x^i−sfx−x−sfx^i=fsx^i−x+∑k=1l−1fksk!x−skx^i−x−∑k=1l−1fkxk!x−sx^i−xk−flθ1l!x^i−xlx−s+flθ2l!x−slx^i−x.
On the contrary, fork=1,2,…,l−1, we have(19)fkx=fks+fk+1sx−s+⋯+flθ3l−k!x−sl−k=∑j=kl−1fjsj−k!x−sj−k+flθ3l−k!x−sl−k,where θ3∈x,sors,x. According to equations (18) and (19), we can obtain equation (13). The proof is completed.
Define the error function:(20)Emx=fx−fx^mx−sx^m−s−fsx^m−xx^m−s−∑k=1l−1fksx−skx^m−xk!x^m−s+∑k=1l−1∑j=kl−1fjsx−sj−k+1x^m−xkk!j−k!x^m−s.Lemma 2.
Assume thats∈xi,xi+1 for the integer m and let ci=2s−xi/h−1,0≤i≤n−1. Then, we have(21)φ0ci=−12c.p∫xmxm+1x^m−xx−sx^m−sdx,i=m,−12∫xixi+1x^i−xx−sx^i−sdx,i≠m.Proof.
Ifi=m, by using the definition of equation (2) and the linear transformation of equation (4), we have(22)c.p∫xmxm+1x^m−xx−sx^m−sdx=limε⟶0∫xms−εx^m−xx−sx^m−sdx+∫s+εxm+1x^m−xx−sx^m−sdx=c.p∫xmxm+1τcmτ−cmdτ=−2φ0cm.
Ifi≠m, it can be proved by applying the same approach to the corresponding Riemann integral. The proof is completed.Lemma 3.
Under the same assumptions of Theorem2, for the function Emx in equation (20), there holds that(23)c.p∫xmxm+1Emxx−sdx≤Chllnγτ,where γτ is defined in equation (7).Proof.
Sincefx∈Cla,b, by using Taylor expansion, we have(24)Emix≤Chl−i,i=0,1.
From the following defined relationship,(25)c.p∫abfxx−sdx=∫abfx−fsx−sdx+fslnb−ss−a,we have(26)c.p∫xmxm+1Emxx−sdx=∫xmxm+1Emx−Emsx−sdx+Emslnxm+1−ss−xm.
Nowm we estimate equation (26) term by term:(27)∫xmxm+1Emx−Emsx−sdx=∫xmxm+1Em′ξmdx≤Chl,ξm∈xm,sors,xm+1,(28)Emslnxm+1−ss−xm≤Chllnγτ.
From equations (26)–(28), we can obtain equation (23). The proof is completed.Lemma 4 (see [5]).
Forτ∈−1,1 and m≥1, we have(29)∑i=m∞φ02i+τ+∑i=n−m+1∞φ0−2i+τ≤Chηs.


### 3.2. Proof of Theorem2

Proof.
By Lemma1, we have(30)∫axm+∫xm+1bfxx−sdx−∑i=0,i≠mn−1fx^ihx^i−s=∑i=0,i≠mn−1∫xixi+1fxx−s−fx^ix^i−sdx=∑i=0,i≠mn−1∫xixi+1x^i−sfx−x−sfx^ix−sx^i−sdx=fs∑i=0,i≠mn−1∫xixi+1x^i−xx−sx^i−sdx+∑k=1l−1fksk!∑i=0,i≠mn−1∫xixi+1x−sk−1x^i−xx^i−sdx−∑k=1l−1∑j=kl−1fjsk!j−k!∑i=0,i≠mn−1∫xixi+1x−sj−kx^i−xkx^i−sdx−∑i=0,i≠mn−1∫xixi+1flθ1x^i−xll!x^i−sdx+∑i=0,i≠mn−1∫xixi+1flθ2x−sl−1x^i−xl!x^i−sdx−∑k=1l−1∑i=0,i≠mn−1∫xixi+1flθ3x−sl−kx^i−xkk!l−k!x^i−sdx.
From the definition ofEmx in equation (20), we have(31)c.p∫xmxm+1fxx−s−fx^mx^m−sdx=c.p∫xmxm+1Emxx−sdx+fsc.p∫xmxm+1x^m−xx−sx^m−sdx+∑k=1l−1fksk!c.p∫xmxm+1x−sk−1x^m−xx^m−sdx−∑k=1l−1∑j=kl−1fjsk!j−k!c.p∫xmxm+1x−sj−kx^m−xkx^m−sdx.
According to equations (30) and (31), we can obtain(32)c.p∫abfxx−sdx−∑i=0n−1fx^ihx^i−s=∑i=0n−1c.p∫xixi+1fxx−s−fx^ix^i−sdx=∑i=0,i≠mn−1∫xixi+1fxx−s−fx^ix^i−sdx+c.p∫xmxm+1fxx−s−fx^mx^m−sdx=−fsS0τ+Rns,where(33)Rns=Rn1s+Rn2s+Rn3s+Rn4s,with(34)Rn1s=∑k=1l−1fksk!∑i=0,i≠mn−1∫xixi+1x−sk−1x^i−xx^i−sdx+c.p∫xmxm+1x−sk−1x^m−xx^m−sdx−∑k=1l−1∑j=kl−1fjsk!j−k!∑i=0,i≠mn−1∫xixi+1x−sj−kx^i−xkx^i−sdx+c.p∫xmxm+1x−sj−kx^m−xkx^m−sdx,(35)Rn2s=−∑i=0,i≠mn−1∫xixi+1flθ1x^i−xll!x^i−sdx+∑i=0,i≠mn−1∫xixi+1flθ2x−sl−1x^i−xl!x^i−sdx−∑k=1l−1∑i=0,i≠mn−1∫xixi+1flθ3x−sl−kx^i−xkk!l−k!x^i−sdx,(36)Rn3s=c.p∫xmxm+1Emxx−sdx,(37)Rn4s=fs∑i=m∞φ02i+τ+∑i=n−m+1∞φ0−2i+τ.
Now, we estimate these four terms one by one. ForRn1s, by the definition of equation (2) and the linear transformation of equation (4), we have(38)∑k=1l−1∑j=kl−1fjsk!j−k!∑i=0,i≠mn−1∫xixi+1x−sj−kx^i−xkx^i−sdx+c.p∫xmxm+1x−sj−kx^m−xkx^m−sdx≤C∑k=1l−1∑j=kl−1fjsk!j−k!h2j∑i=0n−11ci≤Chlγ−1τ.
Then, we obtain(39)c.p∫xmxm+1x−sj−kx^m−xkx^m−sdx≤C1cmh2j.
Similarly, by applying the same approach to the corresponding Riemann integral, we obtain(40)∫xixi+1x−sj−kx^i−xkx^i−sdx≤C1cih2j.
By substituting equations (39) and (40) into the second part of Rn1s, we can obtain(41)∑k=1l−1∑j=kl−1fjsk!j−k!∑i=0,i≠mn−1∫xixi+1x−sj−kx^i−xkx^i−sdx+c.p∫xmxm+1x−sj−kx^m−xkx^m−sdx≤C∑k=1l−1∑j=kl−1fjsk!j−k!h2j∑i=0n−11ci≤Chlγ−1τ.
In the same way, we can obtain(42)∑k=1l−1fksk!∑i=0,i≠mn−1∫xixi+1x−sk−1x^i−xx^i−sdx+c.p∫xmxm+1x−sk−1x^m−xx^m−sdx≤C∑k=1l−1fksk!h2k∑i=0n−11ci≤Chlγ−1τ.
As forRn2s, we have(43)∑i=0,i≠mn−1∫xixi+1flθ1x^i−xll!x^i−sdx≤Chl∑i=0,i≠mn−1∫xixi+11x^i−sdx≤Chllnh+1.
From equation (40), we can obtain(44)∑i=0,i≠mn−1∫xixi+1flθ2x−sl−1x^i−xl!x^i−sdx≤Ch2l∑i=0,i≠mn−11ci≤Chlγ−1τ,(45)∑k=1l−1∑i=0,i≠mn−1∫xixi+1flθ3x−sl−kx^i−xkk!l−k!x^i−sdx≤Ch2l∑i=0,i≠mn−11ci≤Chlγ−1τ.
According to Lemmas3 and 4, we know(46)Rn3s≤Chllnγτ,(47)Rn4s≤Chηs.
From the above estimates, equation (10) can be obtained. The proof is completed.
In addition, from [5], we know S0φ0,τ=−π/2tanπτ+1/2.


## 3.1. Some Necessary Lemmas

Lemma 1.
Under the same assumptions of Theorem2, there holds that(13)x^i−sfx−x−sfx^i=fsx^i−x+∑k=1l−1fksk!x−skx^i−x−∑k=1l−1∑j=kl−1fjsk!j−k!x−sj−k+1x^i−xk+Rfx,where(14)Rfx=Rf1x+Rf2x+Rf3x,with(15)Rf1x=−flθ1l!x−sx^i−xl,Rf2x=flθ2l!x−slx^i−x,Rf3x=−∑k=1l−1flθ3k!l−k!x−sl−k+1x^i−x,and θ1,θ2∈xi,xi+1,θ3∈x,sors,x.Proof.
Note thatfx∈Cla,b. By applying the following Taylor expansion to fx^i at the point x, we obtain(16)fx^i=fx+∑k=1l−1fkxk!x^i−xk+flθ1l!x^i−xl,θ1∈xi,xi+1.
Similarly, we have(17)fx=fs+∑k=1l−1fksk!x−sk+flθ2l!x−sl,θ2∈xi,xi+1.
Thus,(18)x^i−sfx−x−sfx^i=fsx^i−x+∑k=1l−1fksk!x−skx^i−x−∑k=1l−1fkxk!x−sx^i−xk−flθ1l!x^i−xlx−s+flθ2l!x−slx^i−x.
On the contrary, fork=1,2,…,l−1, we have(19)fkx=fks+fk+1sx−s+⋯+flθ3l−k!x−sl−k=∑j=kl−1fjsj−k!x−sj−k+flθ3l−k!x−sl−k,where θ3∈x,sors,x. According to equations (18) and (19), we can obtain equation (13). The proof is completed.
Define the error function:(20)Emx=fx−fx^mx−sx^m−s−fsx^m−xx^m−s−∑k=1l−1fksx−skx^m−xk!x^m−s+∑k=1l−1∑j=kl−1fjsx−sj−k+1x^m−xkk!j−k!x^m−s.Lemma 2.
Assume thats∈xi,xi+1 for the integer m and let ci=2s−xi/h−1,0≤i≤n−1. Then, we have(21)φ0ci=−12c.p∫xmxm+1x^m−xx−sx^m−sdx,i=m,−12∫xixi+1x^i−xx−sx^i−sdx,i≠m.Proof.
Ifi=m, by using the definition of equation (2) and the linear transformation of equation (4), we have(22)c.p∫xmxm+1x^m−xx−sx^m−sdx=limε⟶0∫xms−εx^m−xx−sx^m−sdx+∫s+εxm+1x^m−xx−sx^m−sdx=c.p∫xmxm+1τcmτ−cmdτ=−2φ0cm.
Ifi≠m, it can be proved by applying the same approach to the corresponding Riemann integral. The proof is completed.Lemma 3.
Under the same assumptions of Theorem2, for the function Emx in equation (20), there holds that(23)c.p∫xmxm+1Emxx−sdx≤Chllnγτ,where γτ is defined in equation (7).Proof.
Sincefx∈Cla,b, by using Taylor expansion, we have(24)Emix≤Chl−i,i=0,1.
From the following defined relationship,(25)c.p∫abfxx−sdx=∫abfx−fsx−sdx+fslnb−ss−a,we have(26)c.p∫xmxm+1Emxx−sdx=∫xmxm+1Emx−Emsx−sdx+Emslnxm+1−ss−xm.
Nowm we estimate equation (26) term by term:(27)∫xmxm+1Emx−Emsx−sdx=∫xmxm+1Em′ξmdx≤Chl,ξm∈xm,sors,xm+1,(28)Emslnxm+1−ss−xm≤Chllnγτ.
From equations (26)–(28), we can obtain equation (23). The proof is completed.Lemma 4 (see [5]).
Forτ∈−1,1 and m≥1, we have(29)∑i=m∞φ02i+τ+∑i=n−m+1∞φ0−2i+τ≤Chηs.


## 3.2. Proof of Theorem2

Proof.
By Lemma1, we have(30)∫axm+∫xm+1bfxx−sdx−∑i=0,i≠mn−1fx^ihx^i−s=∑i=0,i≠mn−1∫xixi+1fxx−s−fx^ix^i−sdx=∑i=0,i≠mn−1∫xixi+1x^i−sfx−x−sfx^ix−sx^i−sdx=fs∑i=0,i≠mn−1∫xixi+1x^i−xx−sx^i−sdx+∑k=1l−1fksk!∑i=0,i≠mn−1∫xixi+1x−sk−1x^i−xx^i−sdx−∑k=1l−1∑j=kl−1fjsk!j−k!∑i=0,i≠mn−1∫xixi+1x−sj−kx^i−xkx^i−sdx−∑i=0,i≠mn−1∫xixi+1flθ1x^i−xll!x^i−sdx+∑i=0,i≠mn−1∫xixi+1flθ2x−sl−1x^i−xl!x^i−sdx−∑k=1l−1∑i=0,i≠mn−1∫xixi+1flθ3x−sl−kx^i−xkk!l−k!x^i−sdx.
From the definition ofEmx in equation (20), we have(31)c.p∫xmxm+1fxx−s−fx^mx^m−sdx=c.p∫xmxm+1Emxx−sdx+fsc.p∫xmxm+1x^m−xx−sx^m−sdx+∑k=1l−1fksk!c.p∫xmxm+1x−sk−1x^m−xx^m−sdx−∑k=1l−1∑j=kl−1fjsk!j−k!c.p∫xmxm+1x−sj−kx^m−xkx^m−sdx.
According to equations (30) and (31), we can obtain(32)c.p∫abfxx−sdx−∑i=0n−1fx^ihx^i−s=∑i=0n−1c.p∫xixi+1fxx−s−fx^ix^i−sdx=∑i=0,i≠mn−1∫xixi+1fxx−s−fx^ix^i−sdx+c.p∫xmxm+1fxx−s−fx^mx^m−sdx=−fsS0τ+Rns,where(33)Rns=Rn1s+Rn2s+Rn3s+Rn4s,with(34)Rn1s=∑k=1l−1fksk!∑i=0,i≠mn−1∫xixi+1x−sk−1x^i−xx^i−sdx+c.p∫xmxm+1x−sk−1x^m−xx^m−sdx−∑k=1l−1∑j=kl−1fjsk!j−k!∑i=0,i≠mn−1∫xixi+1x−sj−kx^i−xkx^i−sdx+c.p∫xmxm+1x−sj−kx^m−xkx^m−sdx,(35)Rn2s=−∑i=0,i≠mn−1∫xixi+1flθ1x^i−xll!x^i−sdx+∑i=0,i≠mn−1∫xixi+1flθ2x−sl−1x^i−xl!x^i−sdx−∑k=1l−1∑i=0,i≠mn−1∫xixi+1flθ3x−sl−kx^i−xkk!l−k!x^i−sdx,(36)Rn3s=c.p∫xmxm+1Emxx−sdx,(37)Rn4s=fs∑i=m∞φ02i+τ+∑i=n−m+1∞φ0−2i+τ.
Now, we estimate these four terms one by one. ForRn1s, by the definition of equation (2) and the linear transformation of equation (4), we have(38)∑k=1l−1∑j=kl−1fjsk!j−k!∑i=0,i≠mn−1∫xixi+1x−sj−kx^i−xkx^i−sdx+c.p∫xmxm+1x−sj−kx^m−xkx^m−sdx≤C∑k=1l−1∑j=kl−1fjsk!j−k!h2j∑i=0n−11ci≤Chlγ−1τ.
Then, we obtain(39)c.p∫xmxm+1x−sj−kx^m−xkx^m−sdx≤C1cmh2j.
Similarly, by applying the same approach to the corresponding Riemann integral, we obtain(40)∫xixi+1x−sj−kx^i−xkx^i−sdx≤C1cih2j.
By substituting equations (39) and (40) into the second part of Rn1s, we can obtain(41)∑k=1l−1∑j=kl−1fjsk!j−k!∑i=0,i≠mn−1∫xixi+1x−sj−kx^i−xkx^i−sdx+c.p∫xmxm+1x−sj−kx^m−xkx^m−sdx≤C∑k=1l−1∑j=kl−1fjsk!j−k!h2j∑i=0n−11ci≤Chlγ−1τ.
In the same way, we can obtain(42)∑k=1l−1fksk!∑i=0,i≠mn−1∫xixi+1x−sk−1x^i−xx^i−sdx+c.p∫xmxm+1x−sk−1x^m−xx^m−sdx≤C∑k=1l−1fksk!h2k∑i=0n−11ci≤Chlγ−1τ.
As forRn2s, we have(43)∑i=0,i≠mn−1∫xixi+1flθ1x^i−xll!x^i−sdx≤Chl∑i=0,i≠mn−1∫xixi+11x^i−sdx≤Chllnh+1.
From equation (40), we can obtain(44)∑i=0,i≠mn−1∫xixi+1flθ2x−sl−1x^i−xl!x^i−sdx≤Ch2l∑i=0,i≠mn−11ci≤Chlγ−1τ,(45)∑k=1l−1∑i=0,i≠mn−1∫xixi+1flθ3x−sl−kx^i−xkk!l−k!x^i−sdx≤Ch2l∑i=0,i≠mn−11ci≤Chlγ−1τ.
According to Lemmas3 and 4, we know(46)Rn3s≤Chllnγτ,(47)Rn4s≤Chηs.
From the above estimates, equation (10) can be obtained. The proof is completed.
In addition, from [5], we know S0φ0,τ=−π/2tanπτ+1/2.


## 4. Numerical Examples

Example 1.
We first consider the Cauchy principal value integral withfx=x6,a=−1,andb=1. The exact value can be expressed as 2s5+2/3s3+2/5s+s6log1−s/1+s.
We adopt the uniform mesh method to examine the convergence rate of the midpoint rectangle ruleInx6,s with the dynamic points s=xn/4+1+τh/2 and s=a+1+τh/2, respectively. For different values of n, the error distributions are shown in Figure 1 and Figure 2, respectively.
Whens=xn/4+1+τh/2, the convergence rate of the midpoint rectangle rule is Oh2 at the superconvergence points, and there is no convergence rate at the non-superconvergence points, as shown in Table 1. When s=a+1+τh/2, because of the influence of ηs, no superconvergence phenomenon occurs at both the superconvergence points and the non-superconvergence points, which coincides with the theoretical analysis, as shown in Table 2. For the case of l=2, these numerical results agree quite well with the theoretical results in Theorem 2.Figure 1
Error distributions for the midpoint rectangle rule withs=xn/4+1+τh/2.
(a)(b)(c)(d)Figure 2
Error distributions for the midpoint rectangle rule withs=a+1+τh/2.
(a)(b)(c)(d)Table 1
Errors and convergence for the midpoint rectangle rule withs=xn/4+1+τh/2.
nτ=1τ=−1τ=1/2τ=2/3326.2074e − 0047.2296e − 0042.6547e − 0021.4010e − 002641.6800e − 0041.8082e − 0043.6631e − 0022.0390e − 0021284.3607e − 0054.5210e − 0054.2533e − 0022.4147e − 0022561.1102e − 0051.1303e − 0054.5724e − 0022.6186e − 0025122.8007e − 0062.8257e − 0064.7384e − 0022.7249e − 00210247.0330e − 0077.0643e − 0074.8230e − 0022.7791e − 002Convergence ratio1.95711.9998——Table 2
Errors and convergence for the midpoint rectangle rule withs=a+1+τh/2.
nτ=1τ=−1τ=1/2τ=2/3322.2989e − 002—2.3986e − 0001.3505e − 000642.9689e − 002—2.7767e − 0001.5907e − 0001283.3079e − 002—2.9831e − 0001.7230e − 0002563.4782e − 002—3.0909e − 0001.7924e − 0005123.5635e − 002—3.1459e − 0001.8280e − 00010243.6062e − 002—3.1738e − 0001.8460e − 000Convergence ratio————Example 2.
In the second example, we further study the accuracy of the midpoint rectangle rule. We next consider the Cauchy principal value integral withfx=x2−1,a=−1,b=1. The exact value can be expressed as 2s+1−s2log1+s/1−s.
We adopt the uniform mesh method to examine the convergence rate of the midpoint rectangle ruleInx2−1,s with the dynamic points s=xn/4+1+τh/2 and s=a+1+τh/2, respectively. For different values of n, the error distributions are shown in Figure 3 and Figure 4, respectively.
For the case ofs=xn/4+1+τh/2, when the local coordinate of singular point τ=±1, the convergence rate of the midpoint rectangle rule is Oh2 at the superconvergence points, and there is no convergence rate at the non-superconvergence points, as shown in Table 3. For the case of s=a+1+τh/2 because of the limitation of the boundary conditionfa=fb=0, the convergence rate is Oh at both the superconvergence points and the non-superconvergence points, as shown in Table 4. These numerical results are consistent with the theoretical results of l=2 in Theorem 2.Figure 3
Error distributions for the midpoint rectangle rule withs=xn/4+1+τh/2.
(a)(b)(c)(d)Figure 4
Error distributions for the midpoint rectangle rule withs=a+1+τh/2.
(a)(b)(c)(d)Table 3
Errors and convergence for the midpoint rectangle rule withs=xn/4+1+τh/2.
nτ=1τ=−1τ=1/2τ=2/3323.5138e − 0044.3272e − 0042.4969e + 0001.4503e + 000649.7712e − 0051.0842e − 0042.4282e + 0001.4065e + 0001282.5746e − 0052.7122e − 0052.3926e + 0001.3837e + 0002566.6071e − 0066.7814e − e − 0062.3745e + 0001.3721e + 0005121.6735e − 0061.6954e − 0062.3654e + 0001.3662e + 00010244.2110e − 0074.2385e − 0072.3608e + 0001.3633e + 000Convergence ratio1.94091.9991——Table 4
Errors and convergence for the midpoint rectangle rule withs=a+1+τh/2.
nτ=1τ=−1τ=1/2τ=2/3324.4135e − 003—2.9313e − 0011.8911e − 001642.2443e − 003—1.4833e − 0019.5823e − 0021281.1313e − 003—7.4603e − 0024.8228e − 0022565.6792e − 004—3.7411e − 0022.4193e − 0025122.8452e − 004—1.8733e − 0021.2116e − 00210241.4240e − 004—9.3735e − 0036.0631e − 003Convergence ratio0.9908—0.99340.9926


## 5. Conclusions

In this paper, the interpolation method and Taylor expansion method are used to obtain the extended error expansion of classical composite midpoint rectangle rule for the computation of Cauchy principal value integrals. For the case offx∈Cla,b,l≥2, the error expansion and its accuracy are analyzed by theoretical proofs and numerical experiments. Based on the expansion of the error function, some superconvergence results are obtained. It shows that the increased rate of convergence occurs at the singular points whose location changes are allowed. This kind of Cauchy principal value integral can be widely used in many engineering areas, and the positions of singular points are fixed in real applications. Moreover, it is very possible to extend the above presented results to improve the accuracy of the collocation method for singular integrals by choosing the superconvergence points to be the collocation points.



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