# An Algorithm for the Numerical Evaluation of Certain Finite Part Integrals

Authors: Samir A. Ashour; Hany M. Ahmed
Journal: ISRN Applied Mathematics
 (2011)

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## Abstract

Many algorithms that have been proposed for the numerical evaluation of Cauchy principal value integrals are numerically unstable. In this work we present some formulae to evaluate the known Gaussian quadrature rules for finite part integrals∫a*bw(x)(f(x)/(x-t)r)dx, r=1,2,…, and extend Clenshow's algorithm to evaluate these integrals in a stable way.


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## Body

## 1. Introduction

In  the theory of the numerical approximation of Cauchy principal value integrals,  basically two kinds of Gaussian quadrature formulae have been investigated. For the modified Gaussian formula, we refer to [1–7] and the literature cited therein. Results on the Gaussian quadrature formula in the strict sense can be found, for example, in [1, 3, 7–13]. In [14] Diethelm proposed a quadrature formula which is called Gaussian quadrature formula of the third kind. Many algorithms that have been proposed for the numerical evaluation of Cauchy principal value integrals are numerically unstable. In particular, this is true for the standard algorithm proposed in [3, Section 3] for the evaluation of the modified Gaussian formula [15, Section 4]. In this paper, we present some formulae to evaluate the known Gaussian quadrature rules for finite part integrals(1.1)∫a*bw(x)f(x)(x-t)rdx,r=1,2,…,
and extend Clenshow's algorithm [16] to evaluate these integrals in a stable way.This paper is organized as follows: in Section1, we introduce some known quadrature rules for finite part integrals and we analyze the relations among these rules. In Section 2, we extend Clenshow's algorithm [16] to evaluate any sum of derivatives of functions. In Section 3, we apply the extended algorithm to compute Lagrange interpolation polynomial and its derivatives, to evaluate some known singular integrals exactly and quadrature rules for finite part integrals in a stable way. Finally, we present a numerical example.Some hypersingular integrals can be found by successive differentiation; in some cases evaluation of these integrals becomes tedious and the formulas are lengthily. The evaluation of the such singular integrals is a necessary step for the numerical approach to the integral equations in crack problems. One of the main tasks of the proposed algorithm is evaluate these integrals. The appendix, with closed form solutions for class of these integrals, supplements the paper.


### 1.1. Guassian Quadrature Rule

Letw(x) be an integrable weight function on the interval [a,b], and let {πn(x)} be orthogonal system of polynomials associated with w(x) which satisfiesa three-term recurrence relation:(1.2)πn+1(x)=(Anx+Bn)πn(x)-lnπn-1(x),π-1(x)=0,π1(x)=0,
and hence {πn(r)(x)} satisfy a recurrence relation:(1.3)πn+1(r)(x)=(Anx+Bn)πn(r)(x)-lnπn-1(r)(x)+rAnπn(r-1)(x).
The known Gaussian quadrature rule Qn(f) for I(f):=∫abw(x)f(x)dx is defined as(1.4)Qn(f)=∑i=1nwif(xi),wi=1πn′(xi)∫abw(x)πn(x)x-xidx,πn(xi)=0,i=1,n¯.


### 1.2. Modified Gaussian Quadrature Rules for Singular Integrals

Definition 1.1.
(1.5)f[x,y]=f(x)-f(y)x-y,f[x,x]=f′(x),f[x,t,…,t︸rtimes]=1(r-1)!dr-1dtr-1f[x,t].The finite part integral is defined as follows.Definition 1.2.
(1.6)I(f,t,r):=∫a*bw(x)f(x)(x-t)rdx=∫abw(x)f[x,t,…,t︸rtimes]dx+∑k=0r-1f(k)(t)k!I(1,t,r-k).The modified Gaussian quadrature rule forI(f,t,r) can be defined as follows.Definition 1.3.
(1.7)QnM(f,t,r):=∑i=1nwif[xi,t,…,t︸rtimes]+∑k=0r-1f(k)(t)k!I(1,t,r-k).SinceQnM(f,t,1)=∑i=1nwif[xi,t]+f(t)I(1,t,1), then by differentiating both sides (r-1)-times, using Leibniz formula and Definition 1.1, we have the following the lemma.Lemma 1.4.
(1.8)QnM(f,t,r)=1(r-1)!dr-1dtr-1QnM(f,t,1).Lemma 1.5.
Letqn(t)=I(πn,t,1); then
(1.9)I(1,t,m)=1(m-1)!dm-1dtm-1(qn(t)πn(t))+∑j=1nwj(xj-t)m,m=1,2,….Proof.
Since the quadrature rule (1.7) is Gaussian:
(1.10)I(πn,t,1)=Qn(πn,t,1)=∑i=1nwiπn[xi,t]+πn(t)I(1,t,1),
then
(1.11)qn(t)=-πn(t)∑i=1nwixi-t+πn(t)I(1,t,1).
This implies that
(1.12)I(1,t,1)=qn(t)πn(t)+∑j=1nwj(xj-t).
Differentiate both sides (m-1)-times to have (1.9).Thus the modified Gaussian quadrature rule forI(f,t,r) can be written as (1.13)QnM(f,t,r):=∑i=1nwif[xi,t,…,t︸rtimes]+∑k=0r-1f(k)(t)k!{1(r-k-1)!dr-k-1dtr-k-1(qn(t)πn(t))+∑j=1nwj(xj-t)r-k}.Special Cases
The first case:(1.14)QnM(f,t,1)=∑i=1nwif(xi)xi-t+f(t)qn(t)πn(t),
The second case:
(1.15)QnM(f,t,2)=∑i=1nwif(xi)(xi-t)2+ddt(f(t)qn(t)πn(t)),
which are the known formulae


### 1.3. The Gaussian Quadrature Formula in the Strict Sense for Singular Integrals

In the integralI(f,t,r), if we interpolate f by a polynomial Ln-1f of degree n-1 using the zeros of πn(x), that is, the nodes of the classical Gauss formula, as interpolation nodes and use I(Ln-1f,t,r) as an approximation for I(f,t,r), we obtain the Gaussian quadrature formula in the strict sense for singular integrals:(1.16)Qnstrict(f,t,1):=I(Ln-1f,t,1)=∑i=1nwiqn(xi)qn[xi,t]f(xi),(1.17)Qnstrict(f,t,r):=1(r-1)!dr-1dtr-1Qnstrict(f,t,1),
so due to (1.16), one obtains the following lemma.Lemma 1.6.
(1.18)Qnstrict(f,t,r):=∑i=1nwiqn(xi)qn[xi,t,…,t︸rtimes]f(xi).


### 1.4. Gaussian Quadrature Rule of the Third Kind

Letf∈Cr-1[a,b]. According to [14], the integral I(f,t,r) can be approximated by Qn+1L3(f,t,r):(1.19)Qn+1L3(f,t,r):=∫abw(x)Ln-1f[x,t,…,t︸rtimes]dx+∑k=0r-1f(k)(t)k!I(1,t,r-k),
which is called Gaussian quadrature rule of the third kind:forr=1, (1.20)Qn+1L3(f,t,1):=∫abw(x)Ln-1f[x,t]dx+f(t)q0(t).Lemma 1.7.
(1.21)Qn+1L3(f,t,r):=1(r-1)!dr-1dtr-1Qn+1L3(f,t,1).


### 1.5. The Connection between the Gaussian Quadrature Rules

Lemma 1.8.
(1.22)QnM(f,t,1)=Qnstrict(f,t,1)+qn(t)πn(t)(f(t)-Lnf(t)).Proof.
From (1.16),
(1.23)Qnstrict(f,t,1)=∑i=1nwiqn(xi)qn[xi,t]f(xi),wi=qn(xi)πn′(xi)=∑i=1nwif(xi)xi-t+qn(t)πn(t)∑i=1nf(xi)πn(t)πn′(xi)(t-xi)=∑i=1nwif(xi)xi-t+qn(t)πn(t)Lnf(t).
SinceQnM(f,t,1)=∑i=1nwi(f(xi)/xi-t)+f(t)(qn(t)/πn(t)), one obtains (1.22).Corollary 1.9.
Ifπn(t)=0, then
(1.24)QnM(f,t,1)=Qnstrict(f,t,1)+qn(t)πn′(t)(f(t)-Lnf(t))′.Proof.
From Definition1.3,
(1.25)QnM(f,t,1)=∑i=1nwiqn(xi)qn[xi,t]f(xi)+qn(t)∑i=1nwiqn(xi)f[xi,t]=Qnstrict(f,t,1)+qn(t)∑i=1nf(xi)-f(t)πn′(xi)(xi-t),
and since
(1.26)∑i=1nf(xi)-f(t)πn′(xi)(xi-t)=1πn′(t)(f′(t)-Ln-1[f]′(t)),
then Lemma has been proved.Lemma 1.10.
(1.27)Qn+1L3(f,t,1)=Qnstrict(f,t,1)+(f(t)-Ln-1f(t))q0(t).


## 1.1. Guassian Quadrature Rule

Letw(x) be an integrable weight function on the interval [a,b], and let {πn(x)} be orthogonal system of polynomials associated with w(x) which satisfiesa three-term recurrence relation:(1.2)πn+1(x)=(Anx+Bn)πn(x)-lnπn-1(x),π-1(x)=0,π1(x)=0,
and hence {πn(r)(x)} satisfy a recurrence relation:(1.3)πn+1(r)(x)=(Anx+Bn)πn(r)(x)-lnπn-1(r)(x)+rAnπn(r-1)(x).
The known Gaussian quadrature rule Qn(f) for I(f):=∫abw(x)f(x)dx is defined as(1.4)Qn(f)=∑i=1nwif(xi),wi=1πn′(xi)∫abw(x)πn(x)x-xidx,πn(xi)=0,i=1,n¯.


## 1.2. Modified Gaussian Quadrature Rules for Singular Integrals

Definition 1.1.
(1.5)f[x,y]=f(x)-f(y)x-y,f[x,x]=f′(x),f[x,t,…,t︸rtimes]=1(r-1)!dr-1dtr-1f[x,t].The finite part integral is defined as follows.Definition 1.2.
(1.6)I(f,t,r):=∫a*bw(x)f(x)(x-t)rdx=∫abw(x)f[x,t,…,t︸rtimes]dx+∑k=0r-1f(k)(t)k!I(1,t,r-k).The modified Gaussian quadrature rule forI(f,t,r) can be defined as follows.Definition 1.3.
(1.7)QnM(f,t,r):=∑i=1nwif[xi,t,…,t︸rtimes]+∑k=0r-1f(k)(t)k!I(1,t,r-k).SinceQnM(f,t,1)=∑i=1nwif[xi,t]+f(t)I(1,t,1), then by differentiating both sides (r-1)-times, using Leibniz formula and Definition 1.1, we have the following the lemma.Lemma 1.4.
(1.8)QnM(f,t,r)=1(r-1)!dr-1dtr-1QnM(f,t,1).Lemma 1.5.
Letqn(t)=I(πn,t,1); then
(1.9)I(1,t,m)=1(m-1)!dm-1dtm-1(qn(t)πn(t))+∑j=1nwj(xj-t)m,m=1,2,….Proof.
Since the quadrature rule (1.7) is Gaussian:
(1.10)I(πn,t,1)=Qn(πn,t,1)=∑i=1nwiπn[xi,t]+πn(t)I(1,t,1),
then
(1.11)qn(t)=-πn(t)∑i=1nwixi-t+πn(t)I(1,t,1).
This implies that
(1.12)I(1,t,1)=qn(t)πn(t)+∑j=1nwj(xj-t).
Differentiate both sides (m-1)-times to have (1.9).Thus the modified Gaussian quadrature rule forI(f,t,r) can be written as (1.13)QnM(f,t,r):=∑i=1nwif[xi,t,…,t︸rtimes]+∑k=0r-1f(k)(t)k!{1(r-k-1)!dr-k-1dtr-k-1(qn(t)πn(t))+∑j=1nwj(xj-t)r-k}.Special Cases
The first case:(1.14)QnM(f,t,1)=∑i=1nwif(xi)xi-t+f(t)qn(t)πn(t),
The second case:
(1.15)QnM(f,t,2)=∑i=1nwif(xi)(xi-t)2+ddt(f(t)qn(t)πn(t)),
which are the known formulae


## 1.3. The Gaussian Quadrature Formula in the Strict Sense for Singular Integrals

In the integralI(f,t,r), if we interpolate f by a polynomial Ln-1f of degree n-1 using the zeros of πn(x), that is, the nodes of the classical Gauss formula, as interpolation nodes and use I(Ln-1f,t,r) as an approximation for I(f,t,r), we obtain the Gaussian quadrature formula in the strict sense for singular integrals:(1.16)Qnstrict(f,t,1):=I(Ln-1f,t,1)=∑i=1nwiqn(xi)qn[xi,t]f(xi),(1.17)Qnstrict(f,t,r):=1(r-1)!dr-1dtr-1Qnstrict(f,t,1),
so due to (1.16), one obtains the following lemma.Lemma 1.6.
(1.18)Qnstrict(f,t,r):=∑i=1nwiqn(xi)qn[xi,t,…,t︸rtimes]f(xi).


## 1.4. Gaussian Quadrature Rule of the Third Kind

Letf∈Cr-1[a,b]. According to [14], the integral I(f,t,r) can be approximated by Qn+1L3(f,t,r):(1.19)Qn+1L3(f,t,r):=∫abw(x)Ln-1f[x,t,…,t︸rtimes]dx+∑k=0r-1f(k)(t)k!I(1,t,r-k),
which is called Gaussian quadrature rule of the third kind:forr=1, (1.20)Qn+1L3(f,t,1):=∫abw(x)Ln-1f[x,t]dx+f(t)q0(t).Lemma 1.7.
(1.21)Qn+1L3(f,t,r):=1(r-1)!dr-1dtr-1Qn+1L3(f,t,1).


## 1.5. The Connection between the Gaussian Quadrature Rules

Lemma 1.8.
(1.22)QnM(f,t,1)=Qnstrict(f,t,1)+qn(t)πn(t)(f(t)-Lnf(t)).Proof.
From (1.16),
(1.23)Qnstrict(f,t,1)=∑i=1nwiqn(xi)qn[xi,t]f(xi),wi=qn(xi)πn′(xi)=∑i=1nwif(xi)xi-t+qn(t)πn(t)∑i=1nf(xi)πn(t)πn′(xi)(t-xi)=∑i=1nwif(xi)xi-t+qn(t)πn(t)Lnf(t).
SinceQnM(f,t,1)=∑i=1nwi(f(xi)/xi-t)+f(t)(qn(t)/πn(t)), one obtains (1.22).Corollary 1.9.
Ifπn(t)=0, then
(1.24)QnM(f,t,1)=Qnstrict(f,t,1)+qn(t)πn′(t)(f(t)-Lnf(t))′.Proof.
From Definition1.3,
(1.25)QnM(f,t,1)=∑i=1nwiqn(xi)qn[xi,t]f(xi)+qn(t)∑i=1nwiqn(xi)f[xi,t]=Qnstrict(f,t,1)+qn(t)∑i=1nf(xi)-f(t)πn′(xi)(xi-t),
and since
(1.26)∑i=1nf(xi)-f(t)πn′(xi)(xi-t)=1πn′(t)(f′(t)-Ln-1[f]′(t)),
then Lemma has been proved.Lemma 1.10.
(1.27)Qn+1L3(f,t,1)=Qnstrict(f,t,1)+(f(t)-Ln-1f(t))q0(t).


## 2. Sum Series Algorithm

Theorem 2.1.
The sum∑k=mnakΦk(x)(m≥0) of any series of functions which satisfy a linear recurrence relation
(2.1)Φk+1(x)+δk(x)Φk(x)+ηk(x)Φk-1(x)=skfk(x)
is given by
(2.2)∑k=mnakΦk(x)=bm(x)Φm(x)+bm+1(x){Φm+1(x)+δm(x)Φm(x)}+∑k=m+1n-1bk+1skfk(x),
where
(2.3)bn+1=bn+2=0,bk+δkbk+1+ηk+1bk+2=ak,k=n,n-1,…,m.Moreover, iffk(x) satisfies the linear recurrence relation(2.4)fk+1(x)+λk(x)fk(x)+μk(x)fk-1(x)=0,
then(2.5)∑k=mnakΦk(x)=bm(x)Φm(x)+bm+1(x){Φm+1(x)+δm(x)Φm(x)}+cm+1(x)fm+1(x)+cm+2(x){fm+2(x)+λm+1(x)fm+1(x)},
where(2.6)cn=cn+1=0,ck+λkck+1+μk+1ck+2=skbk+1,k=n-1,…,m+1.Proof.
(2.7)∑k=mnakΦk=∑k=mn(bk+δkbk+1+ηk+1bk+2)Φk(x)=(bm+δmbm+1)Φm(x)+bm+1Φm+1(x)+∑k=m+2nbk(Φk(x)+δk-1Φk-1(x)+ηk-1Φk-2(x)).
Using (2.1), we get (2.2). Moreover, if fk(x) satisfies (2.4), we can similarly deduce (2.5).In a similar way, we can prove the following theorem.Theorem 2.2.
(2.8)∑k=mnakΦk(x)=ηn+1(x)bnΦn(x)+bn-1(x){ηnΦn-1(x)+δn(x)Φn(x)}+∑k=m+1n-1bk-1skfk(x),
where
(2.9)bm-2=bm-1=0,ηk+1bk+δkbk-1+bk-2=ak,k=m,m+1,…,n.
Moreover, if fk(x) satisfies the linear recurrence relation (2.4), then
(2.10)∑k=mnakΦk(x)=ηn+1bnΦn(x)+bn-1{ηnΦn-1(x)+δnΦn(x)}+μncn-1fn-1(x)+cn-2{μn-1fn-2(x)+λn-1fn-1(x)},
where
(2.11)cm-1=cm=0,μk+1ck+λkck-1+ck-2=skbk-1,k=m+1,…,n-1.Now we extend the previous theorems to evaluate∑k=0nakΦk(r-1)(x).Suppose thatΦk(x) satisfies (2.1) with fk≡0 and that Φk(i)(x) satisfies the following recurrence relation:(2.12)Φk+1(i)(x)+δk(x)Φk(i)(x)+ηk(x)Φk-1(i)(x)=ski(x)Φk(i-1)(x).Theorem 2.3.
LetM=max{n,2r-1}; then
(2.13)∑i=0n-1aiΦi(r-1)=∑i=1r{ci-1r-iΦi-1(r-i)+cir-i[δi-1Φi-1(r-i)+Φi(r-i)]},
where
(2.14)cM-i+1r-i=cM-i+2r-i=0,ckr-i+δkck+1r-i+ηk+1ck+2r-i=akr-i,k=M-i,…,i-1,1≤i≤r,akr-1={ak,0≤k≤n-1,0,n≤k≤M,akr-i=skr-i+1ck+1r-i+1,i≥2.Proof.
Let(2.15)Ym=∑i=m-1M-mair-mΦi(r-m)(1≤m≤r).
Due to Theorem 2.1, we obtain Ym=Xm+Ym+1,
(2.16)Xm=cm-1r-mΦm-1(r-m)+cmr-m[δm-1Φm-1(r-m)+Φm(r-m)],cM-m+1r-m=cM-m+2r-m=0,ckr-m+δkck+1r-m+ηk+1ck+2r-m=akr-m,k=M-m,…,m-1.
By induction, we have Y1=∑i=1r-1Xi+Yr, and by using Theorem 2.1, we obtain
(2.17)Yr=∑i=r-1M-rai0Φi=cr-10Φr-1+cr-10[δr-1Φr-1+Φr],
where
(2.18)cM-r+10=cM-r+20=0,ck0+δkck+10+ηk+1ck+20=ak0,k=M-r,…,r-1.
Finally
(2.19)Y1=∑i=1r{ci-1r-iΦi-1r-i+cir-i[δi-1Φi-1(r-i)+Φi(r-i)]},
so
(2.20)∑i=0n-1aiΦi(r-1)=Y1=∑i=1r{ci-1r-iΦi-1(r-i)+cir-i[δi-1Φi-1(r-i)+Φi(r-i)]}.The alternative algorithm that incorporateai's in an upward direction can be deduced as follows.Theorem 2.4.
LetM=max{n,2r-1}; then
(2.21)∑i=0n-1aiΦi(r-1)=∑j=1r{ηM-j+1cM-jj-1ΦM-j(r-j)+cM-j-1j-1[ηM-jΦM-j-1(r-j)+δM-jΦM-j(r-j)]},
where
(2.22)cj-3j-1=cj-2j-1=0,ηk+1ckj-1+δkck-1j-1+ck-2j-1=akj-1,k=j-1,…,M-j,1≤j≤r,ak0={ak,0≤k≤n-1,0,n≤k≤M,akj-1=skj-2ck-1j-2,k=j-1,…,M-j,j≥2.Proof.
Let(2.23)Ym=∑i=m-1M-mai(m-1)Φi(r-m)(1≤m≤r).
Due to Theorem 2.2, we obtain Ym=Xm+Ym+1,
(2.24)Xm=ηM-m+1cM-mm-1ΦM-m(r-m)+cM-m-1m-1[ηM-mΦM-m-1(r-m)+δM-mΦM-m(r-m)],cm-3m-1=cm-2m-1=0,ηk+1ckm-1+δkck-1m-1+ck-2m-1=akm-1,k=m-1,…,M-m,1≤m≤r-1.
By induction, we have Y1=∑i=1r-1Xi+Yr, and by using Theorem 2.2, we obtain
(2.25)Yr=∑i=r-1M-rair-1Φi=ηM-r+1cM-rr-1ΦM-r+cM-r-1r-1[ηM-rΦM-r-1+δM-rΦM-r],
where
(2.26)cr-3r-1=cr-2r-1=0,ηk+1ckr-1+δkck-1r-1+ck-2r-1=akr-1,k=r-1,…,M-r.
Finally
(2.27)Y1=∑j=1rηM-j+1cM-jj-1ΦM-j(r-j)+cM-j-1j-1[ηM-jΦM-j-1(r-j)+δM-jΦM-j(r-j)],
so
(2.28)∑i=0n-1aiΦi(r-1)=Y1=∑j=1rηM-j+1cM-jj-1ΦM-j(r-j)+cM-j-1j-1[ηM-jΦM-j-1(r-j)+δM-jΦM-j(r-j)].


## 3. Computational Aspects and Numerical Examples

In this section, we use the previous algorithm to evaluate the known Gaussian quadrature rules for the integrals in (1.6).Lemma 3.1.
LetLn-1(f)=∑i=0n-1ai[f]πi; then(3.1)ai[f]=1hi∑j=1nwjπi(xj)f(xj),
where wi is the weight of the classical Gaussian quadrature formula corresponding to the node xi.Proof.
Since(3.2)∑i=0n-1ai[f]πi=∑i=1nf(xi)πn(x)πn′(xi)(xi-x),
then multiply both sides by w(x)πj(x)(0≤j≤n-1), and integrate over [a,b], we derive
(3.3)aj[f]=1hj∑i=1nf(xi)πn′(xi)I(πnπj,xi,1),
by using the orthogonality of πn, we get I(πnπj,xi,1)=πj(xi)qn(xi) and by knowing that wi=qn(xi)/πn′(xi), we obtain (3.1).Now, sinceπn(x) satisfy (2.1) which fr=0and ηn=ln, the following statement follows from Theorem 2.3.Lemma 3.2.
(3.4)Ln-1[f]=c0+c1(α0+π1),
where (3.5)cn+1=cn+2=0,ck+αkck+1+lk+1ck+2=ak,k=n-1,…,0,(3.6)αk=-(Akt+Bk).
Since πn′ satisfies recurrence relation (2.12), so we can apply Theorem 2.3 to have the following lemma.Lemma 3.3.
(3.7)Ln-1′[f]=c11π1′+c10π1+c20(α1π1+π2),
where c11, c10, c20 are given in Theorem 2.3.Sinceqi(r-1)(t) obeys recurrence relation (2.12), so we can evaluate the pervious Gaussian quadrature rules introduced in Sections 1.2–1.4 by the following theorems.Theorem 3.4.
(3.8)Qnstrict(f,t,r)=1(r-1)!∑i=0n-1ai[f]qi(r-1)(t),
where
(3.9)∑i=0n-1ai[f]qi(r-1)(t)=∑i=1r{ci-1r-iqi-1(r-i)(t)+cir-i[αi-1qi-1(r-i)(t)+qi(r-i)]},
where cik’s are given by Theorem 2.3.Theorem 3.5.
Ifπn(t)≠0, then
(3.10)QnM(f,t,r)=1(r-1)!∑i=0n-1ai[f]ρi(r-1)(t)+1(r-1)!dr-1dtr-1{qn(t)πn(t)f(t)},
where (using Theorem 2.3)
(3.11)∑i=0n-1ai[f]ρi(r-1)(t)=∑i=1r{ci-1r-iρi(r-i)(t)+cir-i[αi-1ρi-1(r-1)(t)+ρi(r-1)(t)]},
where ρi(t)=qi(t)-qn(t)πi(t)/πn(t), which fulfill a three-term recurrence (1.2), and cik are given by Theorem 2.3.Proof.
Using Lemma1.8, we have
(3.12)QnM(f,t,1)=∑i=0n-1ai[f]I(πi,t,1)+qn(t)πn(t)(f(t)-Ln-1[f](t))=∑i=0n-1ai[f]qi(t)+qn(t)πn(t)(f(t)-Ln-1[f](t))=∑i=0n-1ai[f]ρi(t)+qn(t)πn(t)f(t).
Then differentiating both sides (r-1)-times, we obtain relation (3.10).Theorem 3.6.
Ifπn(t)=0, then
(3.13)QnM(f,t,r)=1(r-1)!∑i=0n-1ai[f]qi(r-1)(t)+1(r-1)!dr-1dtr-1{qn(t)πn(t)(f′(t)-Ln-1′[f](t))}.Proof.
Using Corollary1.9, we have
(3.14)QnM(f,t,1)=∑i=0n-1ai[f]I(πi,t,1)+qn(t)πn(t)(f′(t)-Ln-1′[f](t))=∑i=0n-1ai[f]qi(t)+qn(t)πn(t)(f′(t)-Ln-1′[f](t)),
and by differentiating both sides (r-1)-times, we obtain relation (3.13).A direct result of Lemma1.10 and Theorem 2.3 is the following theorem.Theorem 3.7.
(3.15)Qn+1L3(f,t,r)=1(r-1)!∑i=0n-1ai[f]σi(r-1)(t)+1(r-1)!dr-1dtr-1(f(t)q0(t)),Qn+1L3(f,t,r)=1(r-1)!∑i=0r-1{cir-i-1σi(r-i-1)+ci+1r-i-1[αiσi(r-i-1)+σi(r-i-1)]}+1(r-1)!dr-1dtr-1(f(t)q0(t)),
where σi=qi(t)-q0(t)πi(t) and ci(k) are given by Theorem 2.3.


## 4. A Numerical Example

Hui* and Shia [17] consider the modified Gaussian quadrature(4.1)∫-1*11-x2f(x)(x-t)2dx≅-πf′(t)Un-1(t)Tn(t)-πf(t)W(Un-1,Tn)Un-1(t)2+∑i=1nwif(xi)(xi-t)2,
where W(Un-1,Tn)=Un-1Tn′-Un-1′Tn.In Table1, numerical results are obtained for(4.2)I(f,0,2)=∫-1*11-x2exx2dx≅-2.3395562533389707,
by using the quadrature rule QnM(f,0,2); in this case, w(x)=1-x2; and xi,n are the zeros of Chebyshev polynomial of the second kind Un; then we have(4.3)xi,n=cos⁡(i+1n+1π),wi,n=πn+1(1-xi,n2),qn(t)=-πTn+1(x).
Then applying the algorithm in Theorem 3.6 for r=2, li=1, αi=-2t, we obtain the following results comparing with these of Hui* and Shia [17].Table 1nEquation (4.1)Equation (3.13)4−2.33956−2.339565NA−2.339426−2.33956−2.339567NA−2.339568−2.33956−2.33956The results indicate that the algorithm is indeed stable even whent coincides with one of the nodes of the quadrature formula (in this case, this node is located at the origin). However, in deriving the modified method [17] there are no numerical values presented for n=5 and 7 that is why it is assumed that the roots of the orthogonal polynomials are distinct from t.



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*Source: 571583-2011-06-22.xml*