Maohui Xia and Jin Li*

Abstract: In this paper, the classical composite middle rectangle rule for the computation of Cauchy principal value integral (the singular kernel 1/(x-s)) is discussed. With the density function approximated only while the singular kernel is calculated analysis, then the error functional of asymptotic expansion is obtained. We construct a series to approach the singular point. An extrapolation algorithm is presented and the convergence rate of extrapolation algorithm is proved. At last, some numerical results are presented to confirm the theoretical results and show the efficiency of the algorithms.

Keywords: Cauchy principal value integral, Extrapolation method, Composite rectangle rule, Superconvergence, Error expansion.

1 Introduction

In recent years, much attention has been paid to the singular integral of the form

There are many definition ofthe Cauchy principalvalue integral,in the following we adopt the definition as below:

Lotsofnumericalmethodsforsuch singularintegralshave been studied previously by many authors[Choi,Kim and Yun(2004);Ioakimidis(1985);Kim and Jin(2003);Li,Yang and Yu(2014);Yu(1992)].The classicalextrapolation method based on polynomialand rationalfunction has been widely studied.The extrapolation methods as an accelerating convergence technique has been applied to many fields in computationalmathematics[Liem,Lü and Shih(1995)].One of such extrapolation methods is Richardson extrapolation with the errorfunctionalas

here T(0)=a0,ajis constantindependentof h.

In reference Choi et al.[Choi,Kim and Yun(2004)],the asymptotic error analysis of the Euler-Maclaurin formula is obtained by using the parametric sigmoidaltransformation,with traditionalsigmoidaltransformations,a distinctimprovementon its predecessors is presented.Then in the reference Elliottetal.[Elliottand Venturino(1997)],sigmoidaltransformations to obtain better approximation to Cauchy principalvalue integrals is employed,which is also extended the Euler-Maclaurin formula to Hadamard finite-partintegrals.In the reference Sidi[Sidi(2003)]and Zeng et al.[Zeng,Lei and Huang(2014)]presented high-accuracy numerical quadrature methods for integrals of singular periodic functions which are based on the appropriate Euler-Maclaurin expansions oftrapezoidalrule approximations and their extrapolations.In recentreference,the classical Euler-Maclaurin summation formula[Sidi(2003)]expresses the difference between a definite integralover[0,1]and its approximation using the trapezoidalrule with step length h=1/m as an asymptotic expansion in powers of h togetherwith a remainderterm.

The extrapolation method for the computation of Hadamard finite-partintegrals on the interval and in a circle are studied in Li et al.[Li,Wu and Yu(2009)]and Li et al.[Li,Zhang and Yu(2013)]respectively which focus on the asymptotic expansion oferrorfunction.Based on the asymptotic expansion of the error functional,algorithm with theoretical analysis ofthe generalized extrapolation are given.In reference Zeng etal.[Zeng,Leiand Huang(2014)],quadrature formulae for hypersingular integrals and their asymptotic error expansions and the extrapolation methods for hypersingular integrals with either periodic integrand or non-periodic integrand are presented.

In this paper,we firstly obtain the errorexpansion ofthe classicalrectangle rule.Then with certain specialfunction,we presentthe explicitpartforthe firstpartofthe errorexpansion.Based on thisasymptotic expansion,we suggestan extrapolation algorithm.Aseriesof sjis selected to approximate the singular point s accompanied by the refinementof the meshes.Moreover,by means of the extrapolation technique,we notonly obtain an approximation with higher orderaccuracy butalso geta posterioriestimate of the errorfunctional.

The rest of this paper is organized as follows.In Section 2,after introducing some basic formulas ofthe classicalrectangle rule,we presentthe asymptotic error expansion ofclassicalrectangle rule for Cauchy principal value integrals.In Section 3,we finish the proof of the main theorem.In Section 4,extrapolation algorithm and a posterioriasymptotic error estimation to compute Cauchy principalvalue integrals are obtained.Finally,several numericalexamples are provided to validate our analysis.

2 Main result

Before we give ourmain results,we firstly let a=t0＜t1＜···＜tn?1＜tn=b be a uniform partition of the interval[a,b]with mesh size h=(b?a)/n.

Theorem 1 Let f(t)∈C∞[a,b]and letθ∈[0,1]be fixed.Set h=(b?a)/n for integer n and define

and

then we have

where Bk(θ)is the Bernoulli number.

Define fC(t)as the constantinterpolantfor f(t)

and also define a linear transformation

from the standard reference element[?1,1]to the subinterval[tj?1,tj].Replacing f(t)in(1)with fC(t)gives the composite middle rectangle rule:

whereωjdenotes the Cote coefficientgiven by

We also define

By lineartransformation(7),we have

where

andφ0(t),defined by

If F(τ)is the Legendre polynomialoffirstkind,φ0(t)defines the Legendre function of the second kind[Andrews(2002)].

Now we presentthe main results below.

Theorem 2 Assume f(t)∈C∞[a,b].For the middle rectangle rule In(f;s)defined in(8),there exist certain constant ci,independent of h,such that

where s=tm?1+(1+τ)h/2,m=1,2,···,n.

Based on the theorem 2,we presentthe modify middle rectangle rule

and

Then we have the corollary

Corollary 1 Under the same assumption oftheorem 2,for the modify middle rectangle rule(15),we have

3 Proofofthe Theorem 2

Lemma 1 Assume that s∈ (tj?1,tj)for some m and let cj=2(s?tj?1)/h?1,1 ≤ j≤ n.Then,we have

Proof:By following the definition of(1)and the lineartransformation(7),we have

The case j/=m can be proved by applying the same approach to the correspondentRiemann integral.

Lemma 2 Under the same assumptions oftheorem 2,itholds that

Proof:By performing Taylorexpansion of fC(t),f(t)atthe point s,we have

Combining(22)and(23)togetherwe getthe results(20).

Lemma 3 Under the same assumptions of theorem 2,there holds Proof:By straightly calculation ofφ0(x),we easily get

Here Qn(x)be the function of the second kind associated with the Legendre polynomial Pn(x),defined by Andrews[Andrews(2002)].Then,we have,

and

where we have use the identity[Andrews(2002)],itfollows that

3.1 Proof of Theorem 1

Proof:By Lemma 2,we have

For i=m,we have

Putting(27)and(28)togetheryields

with the linear transformation from[tj?1,tj]to the identity interval[?1,1].As for the last partof

which can be considered as the error estimate of middle rectangle rule for the definite integralObviously,by the Theorem 1,itcan be expanded by the Euler-Maclaurin expansions and we have

Itis easy to see thatthere are notrelation with the singular point s which can be written as

The proof is completed.

We actually obtain the error expansion of the middle rectangle rule and moreover, get the explicit expression of the first order term. So it is easy for us to get the superconvergence point with S0(φ0,τ) = 0, which means that τ = 0 is the superconvergence point in subinterval not near the end of the interval.

4 Extrapolation method

In the above sections, we have proved that the error functional of the middle rectangle rule have the following asymptotic expansion

It is easily to see that the error functional depended on the value of ci(τ). In order to present our extrapolation algorithm, we give the Lemma below

Lemma 4 Assume f(t)∈C∞[a,b].For I(f;s)defined in(1),there holds that I(f;s)∈C∞[a,b].

The proofis similarly to the Lemma 2 in Du[Du(2001)],here we omitit.

Forthe given s,now we presentalgorithm.There exists positive integer n0such that

is a positive number. Firstly, we partition [a,b] into n0equal subinterval and get a mesh denoted by Π1with mesh size h1= (b ? a)/n0as the starting meshes. Then we refine the starting meshes Π1to get mesh Π2with mesh size h2= h1/2. In this way, a series of meshes{Πj}( j = 1,2,···) is obtained in which Πjis refined from Πj?1with mesh size denoted by hj. Then we get extrapolation scheme in Tab. 1.

Table 1: Extrapolation scheme of

Table 1: Extrapolation scheme of

T(h1)=T(1)1 T(h2)=T(2)1 T(1)2 T(h3)=T(3)1 T(2)2 T(1)3 T(h4)=T(4)1 T(3)2 T(2)3 T(1)4 T(h5)=T(5)1 T(4)2 T(3)3 T(2)4 T(1)5...............

Fora coordinateτ∈(?1,1)is given,and define

and

the following extrapolation algorithm is presented:

Step one:

Step two:

Theorem 3 Under the asymptotic expansion oftheorem 2,forτ=0 and the series ofmeshes defined by(33),we have

and a posterioriasymptotic error estimate is given by

Proof:Fora givenτ,by the asymptotic expansion of(32)we have

By the definition of Cauchy principalvalue integrals and(33),forthe firsttwo partof(36),by Taylorexpansion for I(f;sj)atthe singularpoint s,we have

Putting(37)and(36)together,yields

for a given τ,bi(s,τ)is a constant.By(38),we also have

By(38)and(40),with hj=2hj+1we also have

which implies

and

Continuing to use extrapolation process again,we can obtain accuracy O(h3).Similarly we can getthe accuracy O(h4).In this way,we continue extrapolation process and finish the proof.

5 Numerical example

In this section, computational results are reported to confirm our theoretical analysis.

Example 1 We consider the Cauchy principal value integrals with f (t) = t3, a = 0,b = 1.Obviously the integrand function f (t) is smooth enough and by (1), we examine the dynamic point s = t[n/4]+ (τ + 1)h/2 with τ = 0,±2/3,1/2.

Table 2: The error of the rectangle rule to s = t[n/4] + (τ + 1)h/2

From Tab.2,we know that the convergence rate is O(h)with the coordinate location of singularpointequalzero,while forthe localcoordinate ofsingularpointdo notequalzero,itis notconvergence in generalwhich coincide with our analysis.

For the modify classicalrectangle rule,from Tab.3,we can see thatthe convergence rate can reach O(h)for the localcoordinate of singular pointequalzero or not,which is also coincide with our corollary.

Example 2 We consider the Cauchy principal value integrals with f(t)=t3a=0,b=1.Obviously the integrand function f(t)is smooth enough and by(1),with s=0.25 and the exact value is 5.379991503437726e-01,we use s=t[n/4]+(τ+1)h/2 with τ=0,to approximation s=0.25.

From Tab.4,we know thatthe convergence rate oferrorestimate ofthe classicalrectangle rule is O(h)for the firstcolumn,and the convergence rate of second column,third column and fourth column are O(h2),O(h3)and O(h4)respectively.From Tab.5,we know thatthe convergence rate ofposterioriestimate ofthe classicalrectangle rule rule is the same as the the convergence rate of error estimate of the classicalrectangle rule which agree with our theorem.

Table 4:Errorestimate ofthe classicalrectangle rule sj=s+(τ+1)h j/2

Table 5:A posterioriestimate of the classicalrectangle rule rule sj=s+(τ+1)h j/2

Forthe case the singular pointis nearthe end ofthe intervalwith s=1/1024,and the exact value is 3.338225747121760e-01.We choose the starting meshes n0=1024,the convergence rate is also O(h)for the first column,and the convergence rate of second column,third column and fourth column are O(h2),O(h3)and O(h4)respectively in Tab.6 and Tab.7 which agree with ourtheorem.

Forthe case the singularpointis notlocated as the mesh-point,we can notfind the proper starting meshes.We have lots of methods to solve the problem,we adoptthe methods by moving the starting meshes a little to make the singularpointbe located atthe mesh point.In fact,itis notdifficultto extend our methods to the quasi-uniform meshes and the proof is similarly to Theorem 2.

Example 3 Let f(t)=t3,a=0,b=1 for the case of quasi-uniform meshes,we consider the case of s=1/and make sure s is located at the meshes point by moving the starting meshes a little and refine the meshes each time.

Table 6:Error estimate of the classicalrectangle rule s=1/1024,sj=s+(τ+1)h j/2

Table 7:A posterioriestimate of the classicalrectangle rule sj=s+(τ+1)h j/2

From Tab.8,for the singular pointwe know thatthe convergence rate of error estimate ofthe classicalrectangle rule is O(h)forthe firstcolumn,and the convergence rate ofsecond column,third column and fourth column are O(h2),O(h3)and O(h4)respectively.From Tab.9,we know that the convergence rate of posteriori estimate of the classical rectangle rule is the same as the the convergence rate of error estimate of the classical rectangle rule which agree with our theorem.

Table 8:Errorestimate ofthe classicalrectangle rule with s=

Table 8:Errorestimate ofthe classicalrectangle rule with s=

0 h2?extra h3?extra h4?extra 32 7.5514e-02 64 3.6402e-02 -2.7097e-03 128 1.7875e-02 -6.5214e-04 3.3719e-05 256 8.8575e-03 -1.6001e-04 4.0393e-06 -2.0067e-07 512 4.4089e-03 -3.9631e-05 4.9451e-07 -1.1884e-08 1024 2.1995e-03 -9.8618e-06 6.1181e-08 -7.2315e-10

Table 9:A posterioriestimate ofthe classicalrectangle rule with s=

Table 9:A posterioriestimate ofthe classicalrectangle rule with s=

0 h2?extra h3?extra h4?extra 32 64 3.9112e-02 128 1.8527e-02 -6.8586e-04 256 9.0175e-03 -1.6405e-04 4.2400e-06 512 4.4486e-03 -4.0125e-05 5.0640e-07 -1.2586e-08 1024 2.2094e-03 -9.9230e-06 6.1904e-08 -7.4403e-10

Acknowledgment:The work of Jin Li was supported by National Natural Science Foundation of China (Grant No. 11471195 ), China Postdoctoral Science Foundation(Grant No. 2015T80703), Shan-dong Provincial Natural Science Foundation of China(Grant No. ZR2016JL006) and Na-tional Natural Science Foundation of China (Grant No.11771398).