# A Note on Finite Quadrature Rules with a Kind of Freud Weight Function

Authors: Kamal Aghigh; M. Masjed-Jamei
Journal: Mathematical Problems in Engineering
 (2009)

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## Abstract

We introduce a finite class of weighted quadrature rules with the
weight function|x|−2aexp(−1/x2) on (−∞,∞) as
∫−∞∞|x|−2aexp(−1/x2)f(x)dx=∑i=1nwif(xi)+Rn[f], where xi are the zeros of polynomials orthogonal with respect to the introduced weight
function, wi are the corresponding coefficients, and Rn[f] is the error value. We show that the
above formula is valid only for the finite values of n. In other words, the condition a≥{maxn}+1/2 must always be satisfied in order that one can apply the above quadrature
rule. In this sense, some numerical and analytic examples are also given and compared.


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## Body

## 1. Introduction

Recently in [1] the differential equation
(1.1)x2(px2+q)Φn′′(x)+x(rx2+s)Φn′(x)-(n(r+(n-1)p)x2+(1-(-1)n)s2)Φn(x)=0
is introduced, and its explicit solution is shown by
(1.2)Sn(rspq|x)=∑k=0[n/2]([n2]k)(∏i=0[n/2]-(k+1)(2i+(-1)n+1+2[n/2])p+r(2i+(-1)n+1+2)q+s)xn-2k.
It is also called the generic equation of classical symmetric orthogonal polynomials [1, 2]. If this equation is written in a self-adjoint form then the first-order equation
(1.3)xddx((px2+q)W(x))=(rx2+s)W(x)
is derived. The solution of (1.3) is known as an analogue of Pearson distributions family and can be indicated as
(1.4)W(rspq|x)=exp(∫(r-2p)x2+sx(px2+q)dx).
In general, there are four main subclasses of distributions family (1.4) (as subsolutions of (1.3) whose explicit probability density functions are, respectively,
(1.5)K1W(-2a-2b-2,2a-1,1|x)=Γ(a+b+3/2)Γ(a+1/2)Γ(b+1)x2a(1-x2)b,-1≤x≤1,a+12>0,b+1>0,(1.6)K2W(-2,2a0,1|x)=1Γ(a+1/2)x2aexp(-x2),-∞<x<∞,a+12>0,(1.7)K3W(-2a-2b+2,-2a1,1|x)=Γ(b)Γ(b+a-1/2)Γ(-a+1/2)x-2a(1+x2)b,-∞<x<∞,b>0,a<12,b+a>12,(1.8)K4W(-2a+2,21,0|x)=1Γ(a-1/2)x-2aexp(-1x2),-∞<x<∞,a>12.
The values Ki;i=1,2,3,4 play the normalizing constant role in these distributions. Moreover, the value of distribution vanishes at x=0 in each four cases, that is, W(0;p,q,r,s)=0 for s≠0. Hence, (1.4) is called in [1] “The dual symmetric distributions family.”As a special case ofW(x;p,q,r,s), let us choose the values p=1,q=0,r=-2a+2, and s=2 corresponding to distribution (1.8) here and replace them in (1.1) to get
(1.9)x4Φn′′(x)+2x((1-a)x2+1)Φn′(x)-(n(n+1-2a)x2+1-(-1)n)Φn(x)=0.
If (1.9) is solved, the polynomial solution of monic type(1.10)S̅n(-2a+2210|x)=∏i=0[n/2]-122i+2[n/2]+(-1)n+1+2-2a×∑k=0[n/2]([n2]k)(∏i=0[n/2]-(k+1)2i+2[n/2]+(-1)n+1+2-2a2)xn-2k
is obtained. According to [1], these polynomials are finitely orthogonal with respect to a special kind of Freud weight function, that is, x-2aexp(-1/x2), on the real line (-∞,∞) if and only if a≥{maxn}+1/2; see also [3, 4]. In other words, we have
(1.11)∫-∞∞|x|-2aexp(-1x2)S̅n(-2a+2210|x)S̅m(-2a+2210|x)dx=(∏i=1n2(-1)i(i-a)+2a(2i-2a+1)(2i-2a-1))Γ(a-12)δn,m,
if and only if m,n=0,1,2,…,N=max{m,n}≤a-1/2,(-1)2a=1 and
(1.12)δn,m={0,ifn≠m,1,ifn=m.
Furthermore, the polynomials (1.10) also satisfy a three-term recurrence relation as(1.13)S̅n+1(x)=xS̅n(x)-2(-1)n(n-a)+2a(2n-2a+1)(2n-2a-1)S̅n-1(x),S̅0(x)=1,S̅1(x)=x,n∈ℕ.
But the polynomials S̅n(x;1,0,-2a+2,2) are suitable tool to finitely approximate arbitrary functions, which satisfy the Dirichlet conditions (see, e.g., [5]). For example, suppose that N=max{m,n}=3 and a>7/2 in (1.10). Then, the function f(x) can finitely be approximated as(1.14)f(x)≅C0S̅0(x;1,0,-2a+2,2)+C1S̅1(x;1,0,-2a+2,2)+C2S̅2(x;1,0,-2a+2,2)+C3S̅3(x;1,0,-2a+2,2),
where(1.15)Cm=∫-∞∞|x|-2aexp(-1/x2)S̅m(-2a+2210|x)f(x)dx(∏i=1m((2(-1)i(i-a)+2a)/(2i-2a+1)(2i-2a-1))Γ(a-1/2)),
for m=0,1,2,3.Clearly (1.14) is valid only when the general function xm|x|-2aexp(-1/x2)f(x) in (1.15) is integrable for any m=0,1,2,3. This means that the finite set {S̅i(x;1,0,-2a+2,2)}i=03 is a basis space for all polynomials of degree at most three. So if f(x)=a3x3+a2x2+a1x+a0, the approximation (1.14) is exact. By noting this, here is a good position to express an application of the mentioned polynomials in weighted quadrature rules [6, 7] by a straightforward example. Let us consider a two-point approximation as(1.16)∫-∞∞|x|-2aexp(-1x2)f(x)dx≅w1f(x1)+w2f(x2),
provided that a>5/2. According to the described themes, (1.16) must be exact for all elements of the basis f(x)={x3,x2,x,1} if and only if x1,x2 are two roots of S̅2(x;1,0,-2a+2,2). For instance, if a=3>5/2 then (1.16) should be changed to(1.17)∫-∞∞x-6exp(-1x2)f(x)dx≅w1f(23)+w2f(-23),
in which 2/3 and -2/3 are zeros of S̅2(x;1,0,-4,2), and w1,w2 are computed by solving the linear system(1.18)w1+w2=∫-∞∞x-6exp(-1x2)dx=34π,23(w1-w2)=∫-∞∞x-5exp(-1x2)dx=0.
Hence, after solving (1.18) the final form of (1.16) is known as(1.19)∫-∞∞x-6exp(-1x2)f(x)dx≅38π(f(23)+f(-23)).
This approximation is exact for all arbitrary polynomials of degree at most 3.


## 2. Application of Polynomials (1.10) in Weighted  Quadrature Rules: General Case

As we know, the general form of weighted quadrature rules is given by(2.1)∫αβw(x)f(x)dx=∑i=1nwif(xi)+Rn[f],
in which the weights {wi}i=1n and the nodes {xi}i=1n are unknown values, w(x) is a positive function, and [α,β] is an arbitrary interval; see, for example, [6, 7]. Moreover the residue Rn[f] is determined (see, e.g., [7]) by(2.2)Rn[f]=f(2n)(ξ)(2n)!∫αβw(x)∏i=1n(x-xi)2dx,α<ξ<β.
It can be proved in (2.1) that Rn[f]=0 for any linear combination of the sequence {1,x,x2,…,x2n-1} if and only if {xi}i=1n are the roots of orthogonal polynomials of degree n with respect to the weight function w(x) on the interval [α,β]. For more details, see [6]. Also, it is proved that to derive {wi}i=1n in (2.1), it is not required to solve the following linear system of order n×n:(2.3)∑i=1nwixij=∫αβw(x)xjdxforj=0,1,…,2n-1,
rather, one can directly use the relation(2.4)1wi=P̂02(xi)+P̂12(xi)+⋯+P̂n-12(xi)fori=1,2,…,n,
where P̂i(x) are orthonormal polynomials of Pi(x) defined as(2.5)P̂i(x)=(∫αβw(x)Pi2(x)dx)-1/2Pi(x).
In this way, as it is shown in [8, 9], P̂i(x) satisfies a particular type of three-term recurrence as(2.6)xP̂n-1(x)=αnP̂n(x)+βnP̂n-1(x)+αn-1P̂n-2(x).
Now, by noting these comments and the fact that the symmetric polynomials S̅n(x;1,0,-2a+2,2) are finitely orthogonal with respect to the weight function W(x,a)=|x|-2aexp(-1/x2) on the real line, we can define a finite class of quadrature rules as(2.7)∫-∞∞|x|-2aexp(-1x2)f(x)dx=∑j=1nwjf(xj)+Rn[f],
in which xj are the roots of S̅n(x;1,0,-2a+2,2) and wj are computed by(2.8)1wj=∑i=0n-1(S̅i*(1,0,-2a+2,2;xj))2,forj=0,1,2,…,n.
Moreover, for the residue value we have(2.9)Rn[f]=f(2n)(ξ)(2n)!∫-∞∞|x|-2aexp(-1x2)∏j=1n(x-xj)2dx,ξ∈ℝ.


### 2.1. An Important Remark

It is important to note that by applying the change of variable1/x2=t in the left-hand side of (2.7) the orthogonality interval (-∞,∞) changes to [0,∞) and subsequently(2.10)∫-∞∞|x|-2aexp(-1x2)f(x)dx=∫0∞ta-3/2e-tf(1t)dt.
As it is observed, the right-hand integral of (2.10) contains the well-known Laguerre weight function xue-x for u=a-3/2. Hence, one can use Gauss-Laguerre quadrature rules [8, 9] with the special parameter u=a-3/2. This process changes (2.7) in the form(2.11)∫-∞∞|x|-2aexp(-1x2)f(x)dx=∑j=1nwj(a-3/2)f(1xj(a-3/2))+Rn[f(1x)],
in which xj(a-3/2) are the zeros of Laguerre polynomials Ln(a-3/2)(x). But, there is a large disadvantage for formula (2.11). According to (2.2) or (2.9), the residue of integration rules generally depends on f(2n)(ξ);α<ξ<β. Thus, by noting (2.11) we should have(2.12)d2nf(1/x)dx2n=∑i=02nϕi(x)f(i)(1x),
where ϕi(x) are real functions to be computed and f(i),i=0,1,2,…,2n, are the successive derivatives of function f(x).As we observe in (2.12), f(x) cannot be in the form of an arbitrary polynomial function in order that the right-hand side of (2.12) is equal to zero. In other words, (2.11) is not exact for the basis space f(x)=xj,j=0,1,2,…,2n-1. This is the main disadvantage of using (2.11), as the examples of next section support this claim.


## 2.1. An Important Remark

It is important to note that by applying the change of variable1/x2=t in the left-hand side of (2.7) the orthogonality interval (-∞,∞) changes to [0,∞) and subsequently(2.10)∫-∞∞|x|-2aexp(-1x2)f(x)dx=∫0∞ta-3/2e-tf(1t)dt.
As it is observed, the right-hand integral of (2.10) contains the well-known Laguerre weight function xue-x for u=a-3/2. Hence, one can use Gauss-Laguerre quadrature rules [8, 9] with the special parameter u=a-3/2. This process changes (2.7) in the form(2.11)∫-∞∞|x|-2aexp(-1x2)f(x)dx=∑j=1nwj(a-3/2)f(1xj(a-3/2))+Rn[f(1x)],
in which xj(a-3/2) are the zeros of Laguerre polynomials Ln(a-3/2)(x). But, there is a large disadvantage for formula (2.11). According to (2.2) or (2.9), the residue of integration rules generally depends on f(2n)(ξ);α<ξ<β. Thus, by noting (2.11) we should have(2.12)d2nf(1/x)dx2n=∑i=02nϕi(x)f(i)(1x),
where ϕi(x) are real functions to be computed and f(i),i=0,1,2,…,2n, are the successive derivatives of function f(x).As we observe in (2.12), f(x) cannot be in the form of an arbitrary polynomial function in order that the right-hand side of (2.12) is equal to zero. In other words, (2.11) is not exact for the basis space f(x)=xj,j=0,1,2,…,2n-1. This is the main disadvantage of using (2.11), as the examples of next section support this claim.


## 3. Examples

Example 3.1.
Since a 2-point formula was presented in (1.19), in this example we consider a 3-point integration formula. For this purpose, we should first note that according to (1.11) the condition a>7/2 is necessary. Hence, let us, for instance, assume that a=4. After some computations the related quadrature rule would take the form
(3.1)∫-∞∞x-8exp(-1x2)f(x)dx=316π(3f(23)+4f(0)+3f(-23))+R3[f],
where
(3.2)R3[f]=f(6)(ξ)6!∫-∞∞x-8exp(-1x2)(S̅3(-6210|x))2dx=π1080f(6)(ξ),ξ∈R,
and x1=2/3, x2=0, and x3=-2/3 are the roots of S̅3(x;1,0,-6,2)=x3-(2/3)x. Moreover, w1,w2,w3 can be computed by
(3.3)1wj=∑i=02(S̅i*(xj;1,0,-6,2))2,j=1,2,3,
in which
(3.4)S̅i*(xj;1,0,-6,2)=S̅i(xj;1,0,-6,2)〈S̅i(xj;1,0,-6,2),S̅i(xj;1,0,-6,2)〉1/2.Example 3.2.
To have a 4-point formula, we should again note thata>9/2 is a necessary condition. In this sense, if, for example, a=5 then we eventually get
(3.5)∫-∞∞x-10exp(-1x2)f(x)dx=1564π(7-210)(f(10+21015)+f(-10+21015))+1564π(7+210)(f(10-21015)+f(-10-21015))+R4[f],
where
(3.6)R4[f]=f(8)(ξ)8!∫-∞∞x-10exp(-1x2)(S̅4(-8210|x))2dx=π75600f(8)(ξ),ξ∈R.
Clearly this formula is exact for the basis elements f(x)=xj,j=0,1,2,…,7, and the nodes of quadrature (3.5) are the roots of S̅4(x;1,0,-8,2)=x4-(4/3)x2+4/15.


## 4. Numerical results

In this section, some numerical examples are given and compared. The numerical results related to the 2-point formula (1.19) are presented in Table 1, the results related to 3-point formula (3.1) are given in Table 2, and finally the results related to 4-point formula (3.5) are presented in Table 3.Table 1
∫-∞+∞x-6exp(-1/x2)f(x)dx.
f(x)Approx. value (2-point)Exact valueErrorcosx0.91030375120.93825391410.0279501629exp(-2/x2)0.06618396080.08527722570.0190932649exp(-cosx)0.67025592970.68126453980.0110086101Table 2
∫-∞+∞x-8exp(-1/x2)f(x)dx.
f(x)Approx. value (3-point)Exact valueErrorexp(-cosx)1.4944208941.4928418210.0015790731+sinx23.8660242283.8667005600.0006763321+cosx24.5447089794.5612667610.016557782Table 3
∫-∞+∞x-10e(-1/x2)f(x)dx.
f(x)Approx. value (4-point)Exact valueError1+cosx216.2177693616.219785390.002016030(1+x2)-1/210.3098775310.317047400.007116987exp(-x2-2)1.1982190381.1991251360.000906098



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*Source: 421546-2009-07-29.xml*