Rational Gauss-Radau and rational Szegő-Lobatto quadrature on the interval and the unit circle...

Preprint August 3, 2010

Rational Gauss-Radau and rational Szego-Lobatto

quadrature on the interval and the unit circle

respectively

Karl Deckers, Adhemar Bultheel and Francisco Perdomo-Pıo

Abstract

We present a relation between rational Gauss-Radau quadrature formulas with one fixed nodein the open interval (−1, 1) that approximate integrals of the form Jµ(f) =

R

1

−1f(x)dµ(x), and

rational Szego-Lobatto quadrature formulas with two fixed nodes on the complex unit circlethat approximate integrals of the form Iµ(f) =

R π

−πf(eiθ)dµ(θ). The measures µ and µ are

assumed to be positive bounded Borel measures on the interval [−1, 1] and the complex unitcircle respectively, and are related by µ′(θ) = µ′(cos θ) |sin θ|. Further, we include some illustrativenumerical examples.

Keywords: Rational Gauss-Radau quadrature, rational Szego-Lobatto quadrature, quasi-orthogonal rational functions, para-orthogonal rational functions.

MSC: Primary 42C05; Secondary 65D32.

§1. Introduction

In this paper we shall explore the interplay between quadrature formulas for integrals of the form

Jµ(f) =∫ 1

−1f(x)dµ(x), where µ is a positive Borel measure on I = [−1, 1], and integrals of the form

Iµ(f) =∫ π

−πf(eiθ)dµ(θ), where µ is supported on [−π, π]. The integral Iµ can also be considered as

an integral over the unit circle T of the complex plane, which we shall currently do to refer to the twodifferent cases. The Joukowski transform is a well known two-to-one mapping relating the intervalI and the complex unit circle T, which can be most easily characterized by the relations z = eiθ ∈T⇔ x = cos θ ∈ I. It is used for example in Szego’s book [20] to relate orthogonal polynomials on Iwith orthogonal polynomials on T whenever the measures are related by µ′(θ) = µ′(cos θ)| sin θ|. Thecorresponding quadrature formulas can be related as well. For example in [5], Gaussian quadratureformulas on I correspond to Szego quadrature formulas on T. Both of them have distinct nodes inthe support of the measure, the weights are positive and they have a maximal domain of validity inthe space of polynomials on I and the space of Laurent polynomials on T respectively.

The classical Gauss-Radau and Gauss-Lobatto quadrature formulas correspond to prefixing oneand two nodes, respectively, in I, while Szego-Radau and Szego-Lobatto quadrature formulas corre-spond to prefixing one and two nodes, respectively, on T. The remaining nodes are freely chosen insideI, respectively T, to achieve the maximal domain of validity while maintaining positive weights. Thiswas elaborated e.g. in [4] for I and [2, 17] for T.

For integrals with an integrand having singularities close to I, respectively T, it may be advanta-geous to consider also poles for the space where the quadrature is exact. To obtain these spaces ofrational functions, one has to extend the theory of orthogonal polynomials to the theory of orthogonalrational functions with prescribed poles; see [8]. The poles are assumed to be outside the supportI, and hence, their Joukowski image is outside T. The relation between the orthogonal functions onI and T has been described for the case of real poles in [25] and more generally in [14]. Based onthis relation, precise one-to-one relations between rational Gaussian quadrature formulas and rationalSzego quadrature formulas were given in [12]. Also the cases of Radau and Lobatto, with one and twoprescribed nodes in the endpoints of the interval, respectively, were considered for the rational caseas they were in [4, 9, 7, 17].

The aim of this paper is to investigate the connection between rational Gauss-Radau quadratureformulas, this time with a prescribed node inside the open interval (−1, 1), and rational Szego-Lobatto

1

2 K. Deckers, A. Bultheel and F. Perdomo-Pıo

quadrature formulas. Both rational Gauss-Radau and rational Szego-Lobatto quadrature rules havealready been studied separately in the papers [13] and [7] respectively. Conditions were given toensure the existence of the quadrature rules, and it has been shown that the nodes and weights in thequadrature rules can be found by solving an eigenvalue problem involving respectively tridiagonal andfive-diagonal matrices. Although the rational Gauss-Radau and rational Szego-Lobatto quadraturerules were studied separately, it is still worthwhile to relate both kinds of quadrature rules becauseit can be advantageous to exploit this relation. For example: Properties as rate of convergence anderror bounds are easily transformed from one to the other (see e.g. [11]); analytic computations forthe case of the interval are often more easily performed by considering its transformation onto thecomplex unit circle (see e.g. [16]) so that more efficient numerical procedures to compute the nodesand weights can be derived (see e.g. [15, 24]); the computational effort to compute the nodes andweights for the case of the complex unit circle can be reduced by considering its transformation ontothe interval (see e.g. [3] for the polynomial case).

The outline of the paper is as follows. In Section 2 we give the necessary theoretical backgroundon orthogonal rational functions and on rational quadrature rules. Then, in Section 3 we presentthe main results regarding the relation between rational Gauss-Radau and rational Szego-Lobattoquadrature rules. We conclude in Section 4 with some illustrative numerical examples for the case ofthe Chebyshev weight function of the first kind on the interval, which corresponds to the Lebesguemeasure on the complex unit circle.

§2. Preliminaries

The field of complex numbers will be denoted by C and the Riemann sphere by C = C∪{∞}. For thereal line we use the symbol R and for the extended real line R = R∪{∞}. Further, the positive half linewill be represented by R+ = {x ∈ R : x ≥ 0}. Let a ∈ C, then ℜ{a} refers to the real part of a, whileℑ{a} refers to the imaginary part, and the imaginary unit will be denoted by i. The unit circle and theopen unit disc are denoted respectively by T = {z ∈ C : |z| = 1} and D = {z ∈ C : |z| < 1}. Wheneverthe value zero is omitted in a set X ⊆ C, this will be represented by X0. Similarly, the complementof a set Y ⊂ C with respect to a set X ⊆ C will be given by XY ; i.e., XY = {t ∈ X : t /∈ Y }.

In this paper, we will consider quadrature formulas on the unit circle T and on the intervalI = [−1, 1]. Although z and x are both complex variables, we reserve the notation z for the case ofthe unit circle and x for the case of the interval.

For any complex function f(t), with t = z or t = x, we define the involution operation or substar

conjugate by f∗(t) = f(1/t). Next, we define the super-c conjugate by fc(t) = f(t), and consequentlyfc∗ by fc

∗(t) = f(1/t). Note that, if f(t) has a pole at t = p, then f∗(t) (respectively fc(t) and fc∗(t))

has a pole at t = 1/p (respectively t = p and t = 1/p). Further, with f inv we denote the inverse ofthe function f , to avoid confusion with the notation f−1 = 1/f .

2.1. Quadrature rules on I

Let a sequence of complex poles A = {α1, α2, . . .} ⊂ CI be fixed. Unless stated otherwise, the poles inA are arbitrary complex or infinite; hence, they do not have to appear in pairs of complex conjugates.Define the basis functions

b0(x) ≡ 1, bk(x) = bk−1(x)Zk(x), with Zk(x) =x

1 − x/αk, k > 1. (2.1)

These basis functions generate the nested spaces of rational functions with poles in A defined by

L−1 = {0} and Lk := L{α1, . . . , αk} = span{b0, . . . , bk}, k > 0.

With the definition of the super-c conjugate we define Lck = {f : fc ∈ Lk}. Note that Lk and Lc

k arerational generalizations of the space Pk of polynomials of degree less than or equal to k. Indeed, if allαj = ∞, the expressions in (2.1) become Zk(x) = Zc

k(x) = x and bk(x) = bck(x) = xk.

Consider the integral

Jµ(f) =

∫ 1

−1

f(x)dµ(x),

Rational Gauss-Radau and rational Szego-Lobatto quadrature 3

where µ is a positive bounded Borel measure (in short, a measure) on I. To approximate Jµ(f), wheref is a function with singularities (possibly close to, but) outside the interval, positive rational inter-polatory quadrature rules (PRIQRs) are often preferred. An nth PRIQR is obtained by integratingan interpolating rational function in Ln−1, and is of the form

Jn(f) =

n∑

k=1

λkf(xk), {xk}nk=1 ⊂ I, xj 6= xk if j 6= k, {λk}n

k=1 ⊂ R+0 ,

so that Jµ(f) = Jn(f) for every f ∈ Rp,q = Lp · Lcq = {f · gc : f ∈ Lp, g ∈ Lq}, with 0 6 q 6 n − 1 6

p 6 n. The space Rp,q is then called the domain of validity. The existence of such a PRIQR depends,however, on the sequence of nodes {xk}n

k=1, and it is easily verified that whenever the PRIQR exists,as a consequence it should hold that Rp,q = Rc

p,q.Let ϕn ∈ Ln \Ln−1 denote an nth orthonormal rational function (ORF) with respect to the inner

product

〈f , g〉µ =

∫ 1

−1

f(x)gc(x)dµ(x);

i.e.,

〈ϕn , ϕk〉µ =

{

0, k < n1, k = n.

Whenever αn ∈ RI , the zeros xk of ϕn(x) are all distinct and in the open interval (−1, 1), and hence,can be chosen as nodes for the quadrature formula Jn(f). In this way, we obtain the n-point rationalGaussian quadrature formula, which is a PRIQR on I that has maximal domain of validity; i.e., theapproximation is exact for every function f ∈ Rn,n−1. See also [8, Chapt. 11.6], [22], [15, Thm. 2.3],and [10, Thm. 2.3.5].

In this paper, however, we will be concerned with so-called rational Gauss-type quadrature rules.These are PRIQRs with j nodes fixed in advance, where 0 < j < n, and with the remaining n − jnodes chosen to achieve the maximal possible domain of validity. Also now, the existence of thesePRIQRs depends on the nodes fixed in advance. Whenever it exists, the dimension of the domainof validity will generally decrease by one for each node that is fixed in advance. A special kind ofrational Gauss-type quadrature rules are the rational Gauss-Radau quadrature rule (j = 1) and therational Gauss-Lobatto quadrature rule (j = 2). In [13] it is proved that the nodes in an n-pointrational Gauss-Radau quadrature rule with fixed node in xα ∈ I and domain of validity Rn−1,n−1 arezeros of the so-called quasi-orthogonal rational function (qORF)

Qn,τn(x) = ϕn(x) + τn

Zn(x)

Zcn−1(x)

ϕn−1(x), τn = −Zcn−1(xα)ϕn(xα)

Zn(xα)ϕn−1(xα)∈ C. (2.2)

Moreover, it has been proved in the same reference that this qORF is orthogonal on Ln−1 with respectto measures µ supported on a subset of R.

Note that, due to the domain of validity Rn−1,n−1, the remaining nodes in a rational Gauss-Radauquadrature rule with fixed node in xα ∈ I do not depend of αn, and hence, in the remainder we mayas well assume that αn ∈ RI .

1

2.2. Quadrature rules on T

Another sequence of basis functions will be used for the unit circle. Given a sequence of complexnumbers B = {β1, β2, . . .} ⊂ D, we define the Blaschke products 2 for B as

B0(z) ≡ 1, Bk(z) = Bk−1(z)ζk(z), with ζk(z) =z − βk

1 − βkz, k > 1. (2.3)

These Blaschke products generate the nested spaces of rational functions L−1 = {0} and Lk =span{B0, . . . , Bk}, k > 0.

With the definition of the substar conjugate and the super-c conjugate we can define Lk∗ ={f : f∗ ∈ Lk}, Lc

k = {f : fc ∈ Lk} and Lck∗ = {f : fc

∗ ∈ Lk}. Note that Lk and Lck are rational

1The parameter τn in (2.2), however, does depend on αn by means of the ORF ϕn.2The products are named after Wilhelm Blaschke, who introduced these for the first time in [1].


generalizations of Pk too. Indeed, if all βj = 0 (or equivalently, 1/βj = ∞ for every j), the expressions

in (2.3) become ζk(z) = ζck(z) = z and Bk(z) = Bc

k(z) = zk.Consider now the integral

Iµ(f) =

∫ π

−π

f(z)dµ(θ), z = eiθ,

where µ is a positive bounded Borel measure (in short, a measure) on T.3 The PRIQRs to approximate

Iµ(f) are then of the form

In(f) =

n∑

k=1

λkf(zk), {zk = eiθk}nk=1 ⊂ T, zj 6= zk if j 6= k, {λk}n

k=1 ⊂ R+0 ,

so that Iµ(f) = In(f) for every f ∈ Rp = Lp · Lp∗, with (n − 1)/2 ≤ p ≤ n − 1.

Let φn ∈ Ln \ Ln−1 denote an nth ORF with respect to the inner product

⟨

f , g⟩

µ=

∫ π

−π

f(z)g∗(z)dµ(θ).

The leading coefficient κn (i.e., the coefficient of Bn(z) in the expansion of φn(z) in the basis{B0, . . . , Bn}) is then given by κn = φ∗

n(βn), where φ∗n(z) = Bn(z)φn∗(z). In the remainder we

will assume κn ∈ R+0 . Further, let δk ∈ D, ek ∈ R+

0 and ρk ∈ T be given by

δk = −

⟨(

z−βk−1

1−βkz

)

φk−1 , φj

⟩

µ⟨(

1−βk−1z

1−βk

)

φ∗k , φj

⟩

µ

, 0 6 j < k, ek =

√

(

1 − |βk|21 − |βk−1|2

)

1

(1 − |δk|2),

and ρk =δk(βk − βk−1)φk−1(βk) + (1 − βk−1βk)φ∗

k−1(βk)∣

∣δk(βk − βk−1)φk−1(βk) + (1 − βk−1βk)φ∗k−1(βk)

∣

∣

.

In [8, Chapt. 4.1] it is proved then that the ORFs φk, k > 0, with respect to the measure µ satisfy arecurrence relation of the form:

[

φk(z)φ∗

k(z)

]

= ek

(

1 − βk−1z

1 − βkz

)

[

ρk 0

0 ρk

] [

1 δk

δk 1

] [

ζk−1(z) 00 1

] [

φk−1(z)φ∗

k−1(z)

]

. (2.4)

Furthermore, it holds that

ρ2kδk =

φk(βk−1)

φ∗k(βk−1)

. (2.5)

When replacing the nth coefficients δn, ρn and en in the recurrence relation with respectively

ξn =ρnδn + τnρn

ρn + ρnτnδn

∈ T (τn ∈ T),

n =ρn + τnρnδn∣

∣ρn + τnρnδn

∣

∣

∈ T and εn = en

∣

∣ρn + τnρnδn

∣

∣ ∈ R+0 ,

we obtain the so-called para-orthogonal rational function (pORF) for µ:

Qn,τn(z) = εnn

1 − βn−1z

1 − βnz

[

ζn−1(z)φn−1(z) + ξnφ∗n−1(z)

]

= φn(z) + τnφ∗n(z) = τnQ∗

n,τn(z). (2.6)

The zeros zk of Qn,τn(z) are all distinct and on the unit circle T, and hence, can be chosen as nodes for

the quadrature formula In(f). In this way, we obtain an n-point rational Szego quadrature formula,which is a PRIQR on T that has maximal domain of validity (p = n − 1). See also [8, Chapt. 5].

Due to the presence of the parameter τn in (2.6), the nodes and weights in an n-point rationalSzego quadrature formula are (unlike in the case of the interval) not unique. For the same reason,

3The measure µ on T induces a measure on [−π, π] for which we shall use the same notation µ.


an n-point rational Szego-Radau quadrature formula (one node fixed in advance) always exists andhas maximal domain of validity too, while an n-point rational Szego-Lobatto quadrature formula (twonodes fixed in advance) has domain of validity Rn−2 whenever it exists. In [7] it is proved that thenodes in an n-point rational Szego-Lobatto quadrature rule with two distinct fixed nodes zα and zβ on

T are zeros of a pORF˜Qn,τn

(z) = φn(z)+ τnφ∗n(z) ∈ ˆLn, τn ∈ T, of degree n for which the sequence of

ORFs {φn}nk=0 is generated by means of the recurrence relation (2.4) with the sequence of parameters

{δ1, δ2, . . . , δn−2, δn−1, δn} ⊂ D,4 where δn−1 and

ξn =˜ρnδn + τn

˜ρn

˜ρn + ˜ρnτnδn

∈ T (2.7)

are uniquely determined by the nodes zα and zβ . Also now, it is easy to see that the sequence of

ORFs {φn}nk=0 form an orthonormal system in Ln with respect to ‘modified’ measures ˜µ on T; e.g.,

with respect to

d ˜µ(z) ={

1 + cα,βB(n−1)∗(z)Q2(n−1),τ2(n−1)(z)}

dµ(z), τ2(n−1) ∈ T, (2.8)

where Q2(n−1),τ2(n−1)is a pORF in Ln−1 · Ln−1 with respect to the measure µ, and cα,β is a constant

depending on zα and zβ so that cα,βB(n−1)∗(z)Q2(n−1),τ2(n−1)(z) > −1 for every z ∈ T.

2.3. Relation between I and T

We denote the Joukowski Transformation x = 12 (z + z−1) by x = J(z), mapping the open unit disc

D onto the cut Riemann sphere CI and the unit circle T onto the interval I. When z = eiθ, thenx = J(z) = cos θ. In this paper we will assume that x and z are related by this transformation. Theinverse mapping is denoted by z = J inv(x) and is chosen so that z ∈ D if x ∈ CI . With the sequenceA = {α1, α2, . . .} ⊂ CI we associate a sequence B = {β1, β2, . . .} ⊂ D, so that βk = J inv(αk), and

B = {β1, β2, . . .} ⊂ D with

β2k = β2k−1 = βk, k > 1. (2.9)

In the remainder we will consistently use the hat in our notation to refer to the sequence of num-bers (2.9). Furthermore, due to our earlier assumption that αn ∈ RI , we will assume in the remainderthat βn ∈ (−1, 1).

A connection between quadrature formulas on the unit circle and the interval I is given in e.g. [5]and [6]. If µ is a measure on I, we obtain a measure on T by setting

µ(E) = µ ({cos θ, θ ∈ E ∩ [0, π)}) + µ ({cos θ, θ ∈ E ∩ [−π, 0)}) , (2.10)

which can also be written as µ(E) =∫

E|dµ(cos θ)|. Using the Lebesgue decomposition of µ and the

change-of-variables theorem (see e.g. [19, p. 153]) it is not difficult to see that µ′(θ) = µ′(cos θ) |sin θ|(see also [20]). Clearly, this measure µ is then symmetric (i.e.; dµ(−θ) = −dµ(θ)), so that Iµ(fc

∗) =

Iµ(f) for every function f on T. In [12, Lem. 3.1] the following is proved for symmetric measures onT.

Theorem 2.1. Suppose the numbers {β1, . . . , βn−1} are real and/or appear in complex conjugatepairs. Assume µ is a symmetric measure on T and let Qn,τn

(z) = φn(z) + τnφ∗n(z), τn ∈ T, be an nth

pORF with respect to µ. Then the following holds:

1. The zeros of Qn,τn(z) appear in complex conjugate pairs iff τn = ±1.

2. Qn,τn(z) has a zero in

(a) z = 1 iff τn = −1,

(b) z = −1 iff τn = (−1)n+1.

4Note that once the δk’s are fixed in the recurrence relation, so are the ρk’s and ek’s.


3. Let In(f) =∑n

k=1 λkf(zk) be an n-point rational Szego quadrature formula for Iµ(f), based on

the zeros of the pORF Qn,±1(z). Then for k = 1, . . . , n, the weight λk corresponding to the node

zk is equal to the weight λj corresponding to the node zj = zk.

Finally, note that by the Joukowski Transform, a function f(x) transforms into a function f(z) =

(f ◦ J)(z), so that f(z) = f(z−1) and Jµ(f) = 12Iµ(f). Moreover, from [16, Lem. 3.1] it follows that

every function f ∈ Lk \ Lk−1 transforms into a function f ∈ (Lck · Lk∗) \ (Lc

k−1 · L(k−1)∗); see also [10,Chapt. 3.2]. Further, the following relation between the ORFs on I and on T has been proved in [14,Thm. 4.2].

Theorem 2.2. Suppose µ is a measure on I and assume µ is the corresponding measure on T, givenby (2.10). Let ϕk be an ORF with respect to µ, and let φk be the ORF with respect to µ so thatκk ∈ R+

0 . Then there exist non-zero constants ck and dk so that

ϕk(x) = ckBk∗(z){φc2k(z) + φ∗

2k(z)} = dkBk∗(z){ζk(z)φ2k−1(z) + φc∗2k−1(z)}. (2.11)

The constant ck is explicitly given by

ck = ρk

{

1 +ℜ{φc

2k(βk)}κ2k

}−1/2

, ρk ∈ T.

§3. Relating rational Gauss-Radau with rational Szego-Lobatto

rules

In [12, Sect. 3] the n-point rational Gauss-Radau quadrature rules with fixed node in xα ∈ {±1} wererelated with (2n − 1)-point rational Szego-Radau quadrature rules with fixed node in zα ∈ {±1}. Inthis section we will consider n-point rational Gauss-Radau quadrature rules with all nodes (includingthe fixed node xα) in (−1, 1). Our aim is then to relate them with 2n-point rational Szego-Lobattoquadrature rules with fixed nodes in zα = zβ = J inv(xα) ∈ T, and to find a relation between the

parameter τn in the qORF Qn,τnand the parameter δ2n−1 in the pORF

˜Q2n,τn

∈ ˆL2n. Note that˜ρ2n ∈ {±1} and δ2n ∈ (−1, 1) due to φc

k(z) ≡ φk(z) for k ∈ {2n− 1, 2n}, while it immediately follows

from Theorem 2.1 that τn should be equal to one, so that ξ2n, defined as before in (2.7), should beequal to one too. We now have the following.

Theorem 3.1. Suppose µ is a measure on I and assume µ is the corresponding measure on T, givenby (2.10). Consider the sequence of ORFs {φk}2n

k=0, based on the sequence of numbers B2n, that aregenerated by means of the recurrence relation (2.4) with the sequence of parameters {δ1, δ2, . . . , δn−2,

δn−1, δn} ⊂ D, and let I2n(f) = λαf(zα) + λβ f(zα) +∑2n−2

k=1 λkf(zk) be a 2n-point rational Szego-

Lobatto quadrature formula with fixed nodes in {zα, zα} ⊂ T \ {−1, 1} for Iµ(f) =∫ π

−πf(z)dµ(θ),

whose nodes are the zeros zk of the pORF˜Q2n,1(z) = φ2n(z) + φ∗

2n(z). Further, suppose the zeros areordered in such a way that zn−1+k = zk for k = 1, . . . , n−1, with zk 6= zj for every 1 ≤ k < j ≤ n−1.

Then, when taking xα = J(zα), λα = λα, and xk = J(zk) and λk = λk for k = 1, . . . , n − 1, the

formula Jn(f) = λαf(xα)+∑n−1

k=1 λkf(xk) coincides with the n-point rational Gauss-Radau quadrature

formula with fixed node in xα ∈ (−1, 1) for Jµ(f) =∫ 1

−1f(x)dµ(x), based on the zeros of the qORF

Qn,τn∈ Ln \ Ln−1, given by

Qn,τn(x) = KnBn∗(z)

˜Q2n,1(z), Kn ∈ C0, with τn = −Zc

n−1(xα)ϕn(xα)

Zn(xα)ϕn−1(xα).

Proof. (The proof is similar to the proof of [12, Thm. 3.2].) From the first and second parts of

Lemma 2.1 it follows that˜Q2n,1(z) has 2n zeros, all different from 1 and −1, appearing in complex

conjugate pairs. Next, from the third part of Theorem 2.1 it then follows that λα = λβ and that

λn−1+k = λk for k = 1, . . . , n − 1. Consider now an arbitrary function f ∈ Rn−1,n−1 = Ln−1 · Lcn−1.


Clearly, its corresponding function f(z) = (f ◦J)(z) is then in (Ln−1 ·Lcn−1)

c ·(Ln−1 ·Lcn−1)∗ = R2n−2.

Consequently,

Jn(f) = λαf(xα) +

n−1∑

k=1

λkf(xk) = λαf(zα) +

n−1∑

k=1

λkf(zk) =

1

2

{

λα(zα) + λβ f(zα) +

2n−2∑

k=1

λkf(zk)

}

=1

2I2n(f) =

1

2Iµ(f) = Jµ(f).

Finally, because the equality holds for every f ∈ Rn−1,n−1, the n-point formula is the n-point rationalGauss-Radau quadrature formula with fixed node in xα ∈ (−1, 1). This concludes the proof. �

Recall from the Subsection 2.1 that the qORF Qn,τn, given by (2.2), is orthogonal with respect

to measures µ supported on a set S R. However, it has not been proved so far that (at any time)one or more of these measures µ are supported on I whenever the zeros of Qn,τn

are all in (−1, 1).So, supposing the qORF Qn,τn

has a fixed zero in xα ∈ (−1, 1), we first need to verify whether it

is possible to construct a pORF˜Q2n,1 with fixed zeros in zα = zβ = J inv(xα) before we can make

any statement in the opposite direction as in Theorem 3.1. In other words, we are interested now inthe relation between the parameters τn in the qORF and δ2n−1 in the pORF, to find out whetherδ2n−1 ∈ D for a given τn. For this, let us assume that the qORF Qn,τn

is orthogonal with respect tomodified measures µ supported on I; e.g., with respect to

dµ(x) = {1 + cαϕ2n−1(x)} dµ(x), (3.1)

where ϕ2n−1 ∈ Rn,n−1 is orthonormal on Rn−1,n−1 with respect to the measure µ, and cα is a constant

depending on xα so that cαϕ2n−1(x) > −1 on I. Further, let ˜µ be the corresponding measure on T forµ, given by (2.10), and let φ2n ∈ Ln · Ln∗ denote the ORF with respect to ˜µ so that κ2n ∈ R+

0 . Then

it follows from Theorem 2.2 that there exist constants kn ∈ C0 and cn = ρn

{

1 + φ2n(βn)κ2n

}−1/2

∈ C0

so thatknQn,τn

(x) = cnBn∗(z)˜Q2n,1(z) = dnBn∗(z){ζn(z)φ2n−1(z) + φ∗

2n−1(z)}. (3.2)

In Subsection 2.2 the parameter δn−1 in the pORF˜Qn,1(z) is just assumed to be in D. However,

in [7] it has been proved that the rational Szego-Lobatto quadrature rule with fixed nodes in zα andzβ on T can only exist whenever δn−1 lies on a circle segment in D with a certain midpoint and radius.In Lemma 3.3 we will give simple expressions for this midpoint and radius for the special case ofsymmetric measures and the sequences of complex numbers {βj}n

j=1 that are real and/or appear incomplex conjugate pairs, with βn ∈ (−1, 1). But first, we need the following lemma. The proof of allthe lemmas that follow can be found in Section 5.

Lemma 3.2. Suppose µ is a symmetric measure on T, and let φj denoted the jth ORFs with respectto µ so that their leading coefficients κj ∈ R+

0 . Assume that the numbers {βj}k−1j=1 are real and/or

appear in complex conjugate pairs, and that βk ∈ (−1, 1). Then there exist constants bk ∈ C0 and ak,with ℜ{ak} = 0, so that

φck−1(z) =

(

1 − βk−1z

z − βk−1

)

{

bkζk−1(z)φk−1(z) + akφ∗k−1(z)

}

. (3.3)

The constants satisfy the following equalities:

ak =

(

ρ2kδk − ρ

2

kδk

1 − |δk|2

)

and bk = ρ2k + δkak.

Clearly, φck−1(z) in (3.3) does not depend on the parameter δk. Therefore, also the constants ak

and bk in Lemma 3.2 have to be independent of δk; i.e., for every parameter δk ∈ D that correspondswith a modified symmetric measure ˜µ on T, it should hold that

ak =

˜ρ2k δk − ˜ρ

2

kδk

1 − |δk|2

and bk = ˜ρ2k + δkak;


or, equivalenty,

ak =bkδk − bk δk

1 + |δk|2and ˜ρ2

k = bk − δkak. (3.4)

Meanwhile, from the recurrence relation (2.4) it follows that

˜ρk =δk(βk − βk−1)φk−1(βk) + (1 − βk−1βk)φ∗

k−1(βk)∣

∣

∣δk(βk − βk−1)φk−1(βk) + (1 − βk−1βk)φ∗k−1(βk)

∣

∣

∣

, (3.5)

which leaves us with three conditions for just two parameters δk and ˜ρk. The following lemma,however, shows that these three condition are not linear independent.

Lemma 3.3. The constants ak and bk, defined as before in Lemma 3.2, satisfy the following equality:|bk|2 = 1 + |ak|2. Further, whenever ak = 0, the equalities in (3.4)–(3.5) are satisfied with δk =δakbk, for every δa

k ∈ (−1, 1). It then holds that ˜ρ2k = ρ2

k. If, on the other hand, ak 6= 0, then the

equalities in (3.4)–(3.5) are satisfied for every δk ∈ C ∩ D, where C represents the circle with center(

ℑ{bk}ℑ{ak} , ℜ{bk}

ℑ{ak}

)

and radius |ak|−1.

The following lemma now provides an expression for Qn,τnin terms of φ2n−2 and φ∗

2n−2.

Lemma 3.4. Suppose µ is a measure on I and assume µ is the corresponding measure on T, givenby (2.10). Let Qn,τn

be a qORF with respect to µ, and let φk be the ORF with respect to µ so thatκk ∈ R+

0 . Further, assume Qn,τnis orthogonal with respect to modified measures µ supported on I,

and let Cn ∈ C0 and wn(z) be defined by

Cn = ρn˜ρ2n

√

(

1 − β2n

1 − |βn−1|2)

1

(1 − |δ2n−1|2)(1 − δ2n)and

wn(z) = ˜ρ2n−1(z − βn) + ˜ρ2n−1δ2n−1(1 − βnz),

with ρn ∈ T, ˜ρ2n ∈ {±1}, δ2n ∈ (−1, 1), and the parameters δ2n−1 and ˜ρ2n−1 satisfying

a2n−1 =b2n−1δ2n−1 − b2n−1δ2n−1

1 + |δ2n−1|2and ˜ρ2

2n−1 = b2n−1 − δ2n−1a2n−1,

where a2n−1 and b2n−1 are defined as before in Lemma 3.2. Then there exists a non-zero constant kn

so that

knQn,τn(x) = CnBn∗(z)

1 − βn−1z

(1 − βnz)2

{

wn(z)ζn−1(z)φ2n−2(z) + w∗n(z)φ∗

2n−2(z)}

. (3.6)

On the other hand, based on (2.2) we can also prove the following expression for Qn,τnin terms

of φ2n−2 and φ∗2n−2.


be a qORF with respect to µ, and let φk be the ORF with respect to µ so that

κk ∈ R+0 . Further, let Cn ∈ C0 and wn(z) be defined by

Cn = ρnˆρ2n

√

(

1 − β2n

1 − |βn−1|2)

1

(1 − |δ2n−1|2)(1 − δ2n)and

wn(z) = ˆρ2n−1(z − βn) + ˆρ2n−1δ2n−1(1 − βnz),

with ρn ∈ T, ˆρ2n ∈ {±1}, δ2n ∈ (−1, 1), and δ2n−1 and ˆρ2n−1 the coefficients in the recurrencerelation (2.4). Then it holds that

Qn,τn(x) = CnBn∗(z)

1−βn−1z

(1−βnz)2

{

[

wn(z) + τnb2n−1(1 − βn−1z)]

ζn−1(z)φ2n−2(z)+

[

w∗n(z) + τn(1 − βn−1a2n−1)

(

z − βn−1−a2n−1

1−βn−1a2n−1

)]

φ∗2n−2(z)

}

.(3.7)


where

τn = τnρn−1

ρnˆρ2n

(

1+β2n

1+β2n−1

)

√

(

1−|βn−1|21−β2

n

)

(1−|δ2n−1|2)(1−δ22n

)

(1+γ2n−2),

ρn−1 ∈ T, γ2n−2 =

δ2n−2, βn−1 ∈ (−1, 1)

ℑ{ ˆρ2n−1δ2n−1}ℑ{βn−1}

(

1−|βn−1|2

1−|δ2n−1|2)

, βn−1 /∈ (−1, 1),

(3.8)

and a2n−1 and b2n−1 are defined as before in Lemma 3.2.

In the previous two lemmas we obtained two expressions for Qn,τnin terms of φ2n−2 and φ∗

2n−2. Thesecond equality (3.7) always holds, while the first equality (3.6) only holds true under the assumptionthat Qn,τn

is orthogonal with respect to modified measures supported on I. Comparing the expressions

for Qn,τnin (3.6) and (3.7), we find that the parameters τn and δ2n−1 should satisfy the following

equalities:

Knwn(z) − wn(z) = τnb2n−1(1 − βn−1z)

Knw∗n(z) − w∗

n(z) = τn(1 − βn−1a2n−1)(

z − βn−1−a2n−1

1−βn−1a2n−1

)

,(3.9)

where wn(z), wn(z) and τn are defined as above, and

Kn =Cn

knCn

=ρn

˜ρ2n

knρnˆρ2n

√

(1 − |δ2n−1|2)(1 − δ2n)

(1 − |δ2n−1|2)(1 − δ2n)∈ C0.

We now have the following lemma.


be a qORF with respect to µ and with a fixed zero in xα ∈ (−1, 1), and assume

Qn,τnis orthogonal with respect to modified measures µ supported on I. Further, let φk be the ORF

with respect to µ so that κk ∈ R+0 , generated by means of the recurrence relation (2.4) with coefficients

δk ∈ D and ˆρk ∈ T. Suppose ρk ∈ T and ak and bk are defined as above in Theorem 2.2 and Lemma 3.2respectively, and let δ2n−1 ∈ D and ˜ρ2n−1 ∈ T denote the modified coefficients in the construction ofthe pORF Q2n,1 with fixed zeros in zα = zβ = J inv(xα), satisfying the equalities in (3.4). Then the

parameters τn in the qORF and δ2n−1 in the pORF are related by

δ2n−1 =τn

ˆρ2n−1b2n−1(1 − βn−1βn) + δ2n−1(1 − β2n)

τnˆρ2n−1[(βn − βn−1) + a2n−1(1 − βn−1βn)] + (1 − β2

n), (3.10)

where τn is given by (3.8). Or, equivalently,

τn =ˆρ2n−1(δ2n−1 − δ2n−1)

˜ρ22n−1(1 − βn−1βn) + δ2n−1(βn−1 − βn)

.

The following lemma now shows that (3.10) provides a fast and easy way to check whether for agiven τn the zeros of the qORF Qn,τn

are all in (−1, 1).

Lemma 3.7. Consider the qORF Qn,τn, with τn ∈ C, and let δ2n−1 be given by (3.10). Then the

zeros of the qORF Qn,τnare all in (−1, 1) iff δ2n−1 ∈ D.

Based on the previous two lemmas, we can now prove the following.

Theorem 3.8. Suppose µ is a measure on I and assume µ is the corresponding measure on T, givenby (2.10). Let Jn(f) = λαf(xα) +

∑n−1k=1 λkf(xk) be an n-point rational Gauss-Radau quadrature

formula with fixed node in xα ∈ (−1, 1) for Jµ(f) =∫ 1

−1f(x)dµ(x), whose nodes are the zeros of the

qORF Qn,τn∈ Ln \ Ln−1. Set xk = cos θk, and define {zk}2n−2

k=1 and {λk}2n−2k=1 by means of

zk = eiθk , λk = λk

zn−1+k = e−iθk , λn−1+k = λk

}

, k = 1, . . . , n − 1.


Further, let zα = zα = J inv(xα) and λα = λα. Then I2n(f) = λαf(zα) + λαf(zα) +∑2n−2

k=1 λkf(zk)coincides with the 2n-point rational Szego-Lobatto quadrature formula with fixed nodes in {zα, zα} ⊂T \ {−1, 1} for Iµ(f) =

∫ π

−πf(z)dµ(θ), based on the zeros of the pORF

˜Q2n,1 of the form

˜Q2n,1 = K2n

[

ζn−1(z)φ2n−1(z) + φ∗n−1(z)

]

, K2n ∈ C0,

with

φ2n−1(z) = ˜ρ2n−1e2n−11 − βn−1z

1 − βnz

[

ζn−1(z)φ2n−2(z) + δ2n−1φ∗2n−1(z)

]

,

where δ2n−1 ∈ D is given by (3.10), and e2n−1 ∈ R+0 and ˜ρ2n−1 ∈ T are the corresponding recurrence

coefficients defined as above in (2.4).

Proof. The statement directly follows from Lemmas 3.6–3.7 and (3.2). �

We conclude this section with the following remark.

Remark 3.9. In (3.1) we have given an example of a modified measure in the case of the interval.Here, the modified measure µ was assumed to be absolutely continuous with respect to the measureµ. The results in this section do not depend on this assumption made in example (3.1), but it isclear that, if the qORF Qn,τn

is orthogonal with respect to modified measures µ that are absolutelycontinuous with respect to µ, then one of these modified measures will be of the form given by (3.1).Lemma 3.7 proves the existence of modified measures that are supported on I whenever the zerosof the qORF Qn,τn

are all in (−1, 1), but it doesn’t prove that one or more of them are absolutelycontinuous with respect to µ. Hence, it may be interesting to further investigate whether the latterholds too whenever the zeros of the qORF Qn,τn

are all in (−1, 1). Of course, the same remark canbe given for example (2.8) in the case of the complex unit circle.

§4. Illustrative examples

In this section we will illustrate the results from the previous section by means of some numericalexperiments. For this, we consider the Chebyshev weight function of the first kind dµ(x) = dx/

√1 − x2

on I, for which the corresponding measure µ on T is the Lebesgue measure dµ(z) = dz/iz. Our purposethen is to obtain a characterization of rational Gauss-Radau rules associated with µ by giving explicitexpressions for the corresponding nodal rational functions and weights in the quadrature formulas.

Given the sequence of complex numbers Bn = {β1, β2, . . . , βn} ⊂ D, the ORFs with respect to theLebesgue measure µ are obtained by applying the recurrence relation (2.4) with δk = 0 and ρk = 1 forevery k > 0, and with initial conditions φ0(z) ≡ φ∗

0(z) ≡ 1√2π

and ζ0(z) = z (or, equivalently, β0 = 0),

so that

φk(z) =

√

1 − |βk|22π

zBk−1(z)

1 − βkz, and hence φ∗

k(z) =

√

1 − |βk|22π

1

1 − βkz, k > 0. (4.1)

Recall that for the rational Gauss-Radau quadrature rule we work with the sequence B2n ={β1, . . . , β2n} ⊂ D given by (2.9), and with βn ∈ (−1, 1), so that

B2k−1(z) = Bck(z)Bk−1(z) and B2k(z) = Bk(z)Bc

k(z). (4.2)

Further, the kth ORF with respect to the measure µ on I is given by (see also [16, 18] and [10, Chapt.3]):

ϕk(x) =

√

1 − |βk|22π

{

zBck−1(z)

1 − βkz+

B(k−1)∗(z)

z − βk

}

= Bk∗(z){φc2k(z) + φ∗

2k(z)}, k > 0;

i.e., ck = ρk = 1 for every k > 0 in Theorem 2.2. Hence, (3.6) along with (4.1)–(4.2) gives us thefollowing expression for an nth qORF with respect to the measure µ on I:

Qn,τn(x) = Cn

wn(z)Bcn−1(z) + wn∗(z)B(n−1)∗(z)

(1 − βn/z)(1 − βnz), Cn ∈ C0,


where it follows from Lemma 3.3, (3.8) and (3.10) that

wn(z) = ±[

z(1 − δ2n−1βn) − (βn − δ2n−1)]

,

with

δ2n−1 =τn(1 − βn−1βn)

τn(βn − βn−1) + (1 − β2n)

∈ (−1, 1), and τn = τn

(

1 + β2n

1 + β2

n−1

)

√

√

√

√

(

1 − |βn−1|21 − β2

n

)

.

As a result, the nodes xk = J(zk) of the n-point rational Gauss-Radau quadrature formula Jn(f)satisfy

zk

(

zk − βn

1 − βnzk

)

Bn−1(zk)Bcn−1(zk) = −1, βn :=

βn − δ2n−1

1 − δ2n−1βn

, (4.3)

where it should hold that

βn =1 + z2

αBn−1(zα)Bcn−1(zα)

zα

[

1 + Bn−1(zα)Bcn−1(zα)

] , (4.4)

in order to have a fixed node in xα = J(zα) ∈ (−1, 1). Note that for τn = 0 (and hence, δ2n−1 = 0),it holds that βn = βn. Furthermore, from (4.3) it is easily verified that βn ∈ (−1, 1) iff δ2n−1 ∈(−1, 1). Consequently, the nodes of an n-point rational Gauss-Radau quadrature formula with respectto the Chebyshev weight function of the first kind are in fact the nodes of the n-point rationalGaussian quadrature formula with respect to the same weight function, but with maximal domainof validity L{α1, . . . , αn−1, αn} · Lc

n−1 (see also [13, Sect. 6]).5 Since Ln−1 · Lcn−1 is a subspace

of L{α1, . . . , αn−1, αn} · Lcn−1, it follows that the weights λk of the n-point rational Gauss-Radau

quadrature formula Jn(f) with fixed node in xα ∈ (−1, 1) are the weights of this n-point rationalGaussian quadrature formula too; hence, they are given by (see e.g. [16]):

λk = 2π

1 +

n−1∑

j=1

[

P (zk, βj) + P (zk, βj)]

+ P (zk, βn)

−1

,

where P (z, β) denotes the Poisson kernel: P (z, β) = 1−|β|2|z−β|2 , with z ∈ T and β ∈ D.

In [24] a numerical procedure is described to compute the nodes and weights in the rational Gauss-Chebyshev quadrature rule up to machine precision in O(m · n) flops, where n denotes the numberof interpolation points and m represents the number of different poles (two poles αj and αk wereassumed to be different if αj 6= αk and αj 6= αk). This numerical procedure was implemented inMATLAB

r6. In the numerical examples that follow, we used this implementation together with theabove expressions to compute the nodes and weights in the rational Gauss-Radau quadrature rules onI. Unless stated otherwise, all the computations were done in double precision using MATLAB

r 7.

Example 4.1. Define the function f [α](x) = αx−1x−α , with α = J(β) ∈ CI . In [21, Thm. 3.2(2)] it is

proved that

Jµ(f [α]) =

∫ 1

−1

f [α](x)dx√

1 − x2= πβ.

Consider now the sequence of poles {α1, . . . , α13}, with

αk =

{

J(

1.15 − 0.2k + i sin[

(k−1)π9

])

, k 6 10

∞, k > 10.

We then approximate Jµ(f1) and Jµ(f2), with

f1(x) =1

πℜ{

10∑

k=1

f [αk](x)

}

and f2(x) = f1(x) +x

π, (4.5)

5This observation does not hold for every measure; e.g. it does not hold for the Chebyshev weight function of thesecond kind dµ(x) =

√

1 − x2dx (see [13, Sect. 6]).6MATLAB is a registered trademark of The MathWorks, Inc.


by means of an n-point rational interpolatory quadrature rule. Note that the exact solution is Jµ(f1) =Jµ(f2) = 0.5. Since the n-point rational Gaussian quadrature rule, based on the sequence of poles{α1, . . . , αn}, does not exist for 1 < n < 10, we consider an n-point rational Gauss-Radau quadraturerule instead, based on the sequence of poles

{α1, . . . , αn−1, αn}, αn = J(βn) ∈ RI , 2 6 n 6 13.

First, we consider the case in which the last pole is fixed by αn = ∞, and then we consider the casein which βn, given by (4.4), is chosen in such a way that the quadrature rule has a fixed node inxα ∈ {0, 0.1,−0.5}. Tables 1–2 then give the relative error on the approximation:

Error i =

∣

∣

∣

∣

Jn(fi) − Jµ(fi)

Jµ(fi)

∣

∣

∣

∣

, i = 1, 2.

A ’/’ means that the quadrature rule does not exists (i.e., one of the nodes of the qORF is outside Ibecause δ2n−1 /∈ D).

These tables clearly show that the approximation by means the n-point rational Gauss-Radauquadrature rule (whenever it exists) is exact for n > 11 if i = 1 and for n > 12 if i = 2, respectively.Further, in the special case of αn = ∞, the n-point rational Gauss-Radau quadrature rule withn > 10 is in fact the n-point rational Gaussian quadrature rule, which explains the exactness of thisquadrature rule for n = 11 if i = 2.

αn = ∞ xα = 0 xα = 0.1 xα = −0.5

n Error 1 Error 1 βn Error 1 βn Error 1 βn

2 1.29e + 01 2.53e + 01 5.13e − 02 2.44e + 01 1.56e − 01 1.47e + 01 −4.74e − 013 5.93e + 00 2.43e + 01 2.68e − 01 1.87e + 01 3.99e − 01 9.83e + 00 −3.71e − 014 9.17e + 00 2.37e + 01 5.66e − 01 1.24e + 01 7.63e − 01 6.26e + 00 −2.58e − 015 6.63e + 00 2.34e + 01 8.34e − 01 / 1.19e + 00 4.62e + 00 −1.93e − 016 5.36e + 00 2.80e + 01 8.78e − 01 / 1.37e + 00 3.88e + 00 −1.88e − 017 1.79e + 00 8.65e + 00 4.84e − 01 / 1.15e + 00 1.55e + 00 −1.62e − 018 1.41e + 00 1.83e + 00 9.21e − 02 2.75e + 00 7.04e − 01 2.45e + 00 3.11e − 019 6.99e − 01 4.37e − 01 −3.22e − 01 8.07e − 01 3.02e − 01 / −1.03e + 0010 1.47e − 01 5.08e − 02 −7.64e − 01 1.47e − 01 −5.10e − 03 / −2.68e + 0011 4.22e − 15 / −1.06e + 00 1.64e − 14 −1.56e − 01 / −9.24e + 0012 3.80e − 14 2.29e − 14 9.44e − 01 / 6.62e + 00 2.44e − 14 −8.92e − 0113 7.59e − 14 / −1.06e + 00 7.11e − 15 4.89e − 02 9.04e − 14 1.21e − 01

Table 1: Relative errors in the rational Gauss-Radau quadrature formulas with fixed last pole αn orwith fixed node in xα for the estimation of Jµ(f1), where f1 is given by (4.5).

Example 4.2. Finally, consider the function

f3(x) =1

πsin

(

1

(x − υ)2 + ω2

)

, υ ∈ I, ω ∈ R+0 , (4.6)

which is similar to the function considered in [23, Ex. 5.7]. This function has essential singularities inx = υ ± ωi. For ω very close to 0, this function is extremely oscillating near these singularities. Sincean essential singularity can be viewed as a pole of infinity multiplicity, this suggests taking

αk = υ + (−1)kωi, k = 1, . . . , n. (4.7)

For υ = ℜ{J(0.20 + 0.75i)} = 6412410 and ω = |ℑ {J(0.20 + 0.75i)}| = 477

1928 we obtain with the aid of

MAPLEr7 9.5 that Jµ(f3) ≈ 0.4259620458467829.

7MAPLE is a registered trademark of Waterloo Maple, Inc.


αn = ∞ xα = 0 xα = 0.1 xα = −0.5


2 1.29e + 01 2.52e + 01 5.13e − 02 2.47e + 01 1.56e − 01 1.39e + 01 −4.74e − 013 5.93e + 00 2.40e + 01 2.68e − 01 1.92e + 01 3.99e − 01 9.38e + 00 −3.71e − 014 9.17e + 00 2.32e + 01 5.66e − 01 1.31e + 01 7.63e − 01 6.04e + 00 −2.58e − 015 6.63e + 00 2.28e + 01 8.34e − 01 / 1.19e + 00 4.47e + 00 −1.93e − 016 5.36e + 00 2.73e + 01 8.78e − 01 / 1.37e + 00 3.74e + 00 −1.88e − 017 1.79e + 00 8.29e + 00 4.84e − 01 / 1.15e + 00 1.43e + 00 −1.62e − 018 1.41e + 00 1.77e + 00 9.21e − 02 2.32e + 00 7.04e − 01 2.26e + 00 3.11e − 019 6.99e − 01 5.56e − 01 −3.22e − 01 6.95e − 01 3.02e − 01 / −1.03e + 0010 1.47e − 01 2.03e − 01 −7.64e − 01 1.48e − 01 −5.10e − 03 / −2.68e + 0011 4.44e − 15 / −1.06e + 00 2.25e − 02 −1.56e − 01 / −9.24e + 0012 3.97e − 14 2.25e − 14 9.44e − 01 / 6.62e + 00 2.25e − 14 −8.92e − 0113 7.62e − 14 / −1.06e + 00 6.22e − 15 4.89e − 02 8.90e − 14 1.21e − 01

Table 2: Relative errors in the rational Gauss-Radau quadrature formulas with fixed last pole αn orwith fixed node in xα for the estimation of Jµ(f2), where f2 is given by (4.5).

Because αk /∈ R for every k > 0, the n-point rational Gaussian quadrature rule, based on thesequence of poles {α1, . . . , αn}, does not exist for every n > 0. For this reason we consider an n-pointrational Gauss-Radau quadrature rule instead, based on the sequence of poles

{α1, . . . , αn−1, αn}, αn = J(βn) ∈ RI , 2 6 n 6 33.

First, we consider the case in which the last pole is fixed by αn = ∞, and then we consider the casein which βn, given by (4.4), is chosen in such a way that the quadrature rule has a fixed node inxα ∈ {0, 0.1, 0.2}. Figure 1 then graphically shows the relative error

Error 3 =

∣

∣

∣

∣

Jn(f3) − Jµ(f3)

Jµ(f3)

∣

∣

∣

∣

as a function of the number of nodes in the quadrature rules, while the numerical results, togetherwith the parameter βn, are given in Table 3 for some values of n. The graphs clearly show that thequadrature rules globally perform equally well, but that locally one rule may perform better over theother or some of the rules may not exist. Note that the rational Gauss-Radau quadrature does notexist for

n ∈

{3, 4, 7, 8, 11, 12, 15, 16, 19, 20, 23, 24, 27, 28, 31, 32} if xα = 0{2, 3, 5, 6, 8, 9, 11, 12, 14, 15, 18, 21, 24, 27, 30, 33} if xα = 0.1{2, 4, 7, 9, 11, 13, 14, 16, 18, 20, 23, 25, 27, 30, 32} if xα = 0.2.

Thus, to ensure the existence of the quadrature rule for every n, it may be better to fix the last poleαn rather than to fix the node xα.

§5. Proofs

Proof of Lemma 3.2

Note that under the conditions given on the measure µ and the numbers {βj}kj=1, it holds that

φck(z) ≡ φk(z). Thus, it follows from the recurrence relation (2.4) that

φk(z) = ρkek1−β

k−1z

1−βkz

[

ζk−1(z)φk−1(z) + δkφ∗k−1(z)

]

=

ρkek1−βk−1z1−βkz

[

ζck−1(z)φc

k−1(z) + δkφc∗k−1(z)

]

= φck(z)

φ∗k(z) = ρkek

1−βk−1z

1−βkz

[

δkζk−1(z)φk−1(z) + φ∗k−1(z)

]

=

ρkek1−βk−1z1−βkz

[

δkζck−1(z)φc

k−1(z) + φc∗k−1(z)

]

= φc∗k (z),


0 5 10 15 20 25 30 3510

−16

10−14

10−12

10−10

10−8

10−6

10−4

10−2

100

102

Number of interpolation points

Rela

tive

err

or

αn = ∞xα = 0xα = 0.1xα = 0.2

Figure 1: Relative errors in the rational Gauss-Radau quadrature formulas with fixed last pole in αn

or with fixed node in xα for the estimation of Jµ(f3), where f3 is given by (4.6). (For n = 31, weobtained with fixed last pole in αn = ∞ that Error 3 = 0.)

αn = ∞ xα = 0 xα = 0.1 xα = 0.2


2 6.32e − 01 1.30e + 00 9.94e − 01 / 1.74e + 00 / 5.00e + 003 4.85e − 01 / 1.60e + 02 / −1.81e + 00 4.92e − 01 −2.17e − 014 3.87e − 02 / −1.02e + 00 3.13e − 02 3.62e − 02 / 1.62e + 005 2.12e − 01 2.04e − 01 −1.25e − 02 / 1.53e + 00 6.22e − 01 −8.20e − 018 3.15e − 03 / −1.04e + 00 / 1.35e + 00 3.46e − 01 3.59e − 019 1.37e − 02 1.71e − 02 −2.51e − 02 / −2.60e + 00 / 2.46e + 0116 8.41e − 05 / −1.10e + 00 1.05e − 04 −2.30e − 01 / −7.84e + 0017 1.44e − 05 1.49e − 05 −5.02e − 02 7.99e − 06 9.48e − 01 1.28e − 05 1.21e − 0132 2.61e − 16 / −1.22e + 00 1.30e − 16 5.03e − 01 / −2.36e + 0133 7.82e − 16 3.91e − 16 −1.01e − 01 / 6.28e + 00 2.61e − 16 4.16e − 02

Table 3: Relative errors in the rational Gauss-Radau quadrature formulas with fixed last pole in αn

or with fixed node in xα for the estimation of Jµ(f3), where f3 is given by (4.6).

and hence, that

ek1 − βk−1z

1 − βkz(1 − |δk|2)ζc

k−1(z)φck−1(z) = ρkφk(z) − ρkδkφ∗

k(z) =

ek1 − βk−1z

1 − βkz

[

(ρ2k − ρ

2

kδ2

k)ζk−1(z)φk−1(z) + (ρ2kδk − ρ

2

kδk)φ∗k−1(z)

]

.

The equality in (3.3) now easily follows.


Proof of Lemma 3.3

Suppose δk ∈ D satisfies the first equality in (3.4). First, we prove that if ˜ρk ∈ T is given by (3.5),

then δk and ˜ρk satisfy the second equality in (3.4) too. For this, let us assume that σ2k = bk − δkak,

and define

φk(z) = σkek1 − βk−1z

1 − βkz

[

ζk−1(z)φk−1(z) + δkφ∗k−1(z)

]

, βk ∈ (−1, 1), ek ∈ R+0 .

Then it follows from Lemma 3.2 that

σkφck(z) − σk δkφc∗

k (z) = ek1 − βk−1z

1 − βkz(1 − |δk|2)ζc

k−1(z)φck−1(z)

= ek1 − βk−1z

1 − βkz

[

(σ2k − σ2

k δ2

k)ζk−1(z)φk−1(z) + (σ2k δk − σ2

k δk)φ∗k−1(z)

]

= σkφk(z) − σk δkφ∗k(z).

Thus, setting fk(z) = φk(z) − φck(z) and Ck = σk

σk

δk ∈ D, we obtain that

fk(z) = Ckf∗k (z) ⇔ f∗

k (z) = Ck[f∗k (z)]∗ = Ckfk(z),

and hence,fk(z) = |Ck|2 fk(z).

Since Ck ∈ D, it follows that fk(z) ≡ 0. Consequently, φk(z) ≡ φck(z) and φ∗

k(βk) ∈ R, which means

that either σk = ˜ρk or σk = − ˜ρk, where ˜ρk is given by (3.5).Next, assuming that ak = 0, it follows that ℑ{bk δk} = 0 which holds true for every δk = δa

kbk with

δak ∈ R. Since δk ∈ D, we find that δa

k ∈ (− |bk|−1, |bk|−1

), where ˜ρ2k = bk = ρ2

k ∈ T.Finally, for ak 6= 0 we have that

ak =bkδk − bk δk

1 + |δk|2⇔∣

∣

∣

∣

δk +bk

ak

∣

∣

∣

∣

2

=|bk|2 − |ak|2

|ak|2,

where we used the fact that ak = −ak. On the other hand we have that

˜ρ2k = bk − δkak ⇒

∣

∣

∣

∣

δk +bk

ak

∣

∣

∣

∣

2

=1

|ak|2.

Thus, we obtain two circles with the same center(

ℑ{bk}ℑ{ak} , ℜ{bk}

ℑ{ak}

)

and at least one point δk in common.

For this reason, their radii have to be the same, which concludes the proof.

Proof of Lemma 3.4

Before we can prove Lemma 3.4, we first need the following.

Lemma 5.1. In the special case in which βk ∈ (−1, 1), the constants ck and dk in Theorem 2.2 aregiven by

ck = ρk

{

1 + δ2k

}−1/2

and dk = ρkˆρ2k

{

1 − δ2k

}−1/2

, ˆρ2k ∈ {±1}.

Proof. Since the measure µ in Theorem 2.2 is symmetric, and β2k = βk = β2k−1 ∈ (−1, 1), it follows

from (2.5) that δ2k = ˆρ22k δ2k = φ2k(βk)

κ2k

=φc

2k(βk)

κ2k

∈ (−1, 1). As a result

ck = ρk

{

1 + δ2k

}−1/2

.

Next, by means of the recurrence relation (2.4) we obtain, with φc2k(z) ≡ φ2k(z) that

ckBk∗(z){φ2k(z) + φ∗2k(z)} = cke2k

ˆρ2k(1 + δ2k)Bk∗(z){ζk(z)φ2k−1(z) + φc∗2k−1(z)},


and hence, that

dk = cke2kˆρ2k(1 + δ2k) = ρk

ˆρ2k

√

√

√

√

√

1 + δ2k

1 −∣

∣

∣δ2k

∣

∣

∣

2 = ρkˆρ2k

{

1 − δ2k

}−1/2

.

�

From (3.2) and Lemma 5.1 it follows that

knQn,τn(x) = ρn

˜ρ2n

{

1 − δ2n

}−1/2

Bn∗(z){ζn(z)φ2n−1(z)+φ∗2n−1(z)}, ˜ρ2n ∈ {±1}, δ2n ∈ (−1, 1).

Next, applying the recurrence relation (2.4) on φ2n−1 and φ∗2n−1 we obtain that

knQn,τn(x) = ρn

˜ρ2n

{

1 − δ2n

}−1/2

e2n−1Bn∗(z)(1 − βn−1z)

(1 − βnz)2×

{

[

˜ρ2n−1(z − βn) + ˜ρ2n−1δ2n−1(1 − βnz)]

ζn−1(z)φ2n−2(z)+

[

˜ρ2n−1δ2n−1(z − βn) + ˜ρ2n−1(1 − βnz)]

φ∗2n−2(z)

}

,

where

ρn˜ρ2n

{

1 − δ2n

}−1/2

e2n−1 = Cn.

Finally, with wn(z) as defined above, we have that

w∗n(z) = ˜ρ2n−1(1 − βnz) + ˜ρ2n−1δ2n−1(z − βn),

which ends the proof.

Proof of Lemma 3.5

First, note that with x = J(z) and αk = J(βk) we have that

Zn(x)

Zcn−1(x)

=

(

1 + β2n

1 + β2

n−1

)

(1 − βn−1z)(z − βn−1)

(1 − βnz)(z − βn).

Hence, from (2.2) and (2.11) it follows that

Qn,τn(x) = dnBn∗(z){ζn(z)φ2n−1(z) + φ∗

2n−1(z)}+

τn

(

1 + β2n

1 + β2

n−1

)

(1 − βn−1z)(z − βn−1)

(1 − βnz)2cn−1Bn∗(z){φc

2n−2(z) + φ∗2n−2(z)}.

Next, using the recurrence relation (2.4) for φ2n−1 and φ∗2n−1, together with Lemma 3.2 for φc

2n−2,

and setting Cn = dne2n−1 and

τn = τn

(

1 + β2n

1 + β2

n−1

)

cn−1

e2n−1dn

then yields

Qn,τn(x) = CnBn∗(z)×

{[

(

1 − βn−1z

1 − βnz

)

(ˆρ2n−1ζn(z) + ˆρ2n−1δ2n−1) + τnb2n−1

(

1 − βn−1z

1 − βnz

)2]

ζn−1(z)φ2n−2(z)+

[

(

1 − βn−1z

1 − βnz

)

(ˆρ2n−1δ2n−1ζn(z)+ˆρ2n−1)+τn1 − βn−1z

(1 − βnz)2[(1−βn−1z)a2n−1+(z−βn−1)]

]

φ∗2n−2(z)

}

.


The equality in (3.7) now easily follows.Finally, from (3.3) we deduce that

ℜ{φc2n−2(βn−1)} =

{

φ2n−2(βn−1), βn−1 ∈ (−1, 1)

a2n−1

(

1−|βn−1|2βn−1−β

n−1

)

κ2n−2, βn−1 /∈ (−1, 1).

This concludes the proof.

Proof of Lemma 3.6

Note that (3.9) is equivalent with

Kn˜ρ2n−1(δ2n−1 − βn

˜ρ22n−1) − ˆρ2n−1(δ2n−1 − βn

ˆρ22n−1) = τnb2n−1

Kn˜ρ2n−1(δ2n−1βn − ˜ρ2

2n−1) − ˆρ2n−1(δ2n−1βn − ˆρ22n−1) = τnb2n−1βn−1

Kn˜ρ2n−1(

˜ρ22n−1δ2n−1 − βn) − ˆρ2n−1(

ˆρ22n−1δ2n−1 − βn) = τn(1 − βn−1a2n−1)

Kn˜ρ2n−1(

˜ρ22n−1δ2n−1βn − 1) − ˆρ2n−1(

ˆρ22n−1δ2n−1βn − 1) = τn(βn−1 − a2n−1) .

(5.1)

Hence, we obtain in this way four equalities in two unknowns δ2n−1 and Kn. With

˜ρ22n−1 = b2n−1 − δ2n−1a2n−1 =

[

b2n−1 + δ2n−1a2n−1

]−1

∈ T

andˆρ22n−1 = b2n−1 − δ2n−1a2n−1 =

[

b2n−1 + δ2n−1a2n−1

]−1

∈ T,

the first two equalities in (5.1) become{

Kn˜ρ2n−1[δ2n−1(1 + βna2n−1) − βnb2n−1] − ˆρ2n−1[δ2n−1(1 + βna2n−1) − βnb2n−1] = τnb2n−1

Kn˜ρ2n−1[δ2n−1(βn + a2n−1) − b2n−1] − ˆρ2n−1[δ2n−1(βn + a2n−1) − b2n−1] = τnb2n−1βn−1 ,

(5.2)and the last two equalities in (5.1) become

Kn˜ρ2n−1

[

δ2n−1(1−βna2n−1)−βnb2n−1

b2n−1+δ2n−1a2n−1

]

− ˆρ2n−1

[

δ2n−1(1−βna2n−1)−βnb2n−1

b2n−1+δ2n−1a2n−1

]

= τn(1 − βn−1a2n−1)

Kn˜ρ2n−1

[

δ2n−1(βn−a2n−1)−b2n−1

b2n−1+δ2n−1a2n−1

]

− ˆρ2n−1

[

δ2n−1(βn−a2n−1)−b2n−1

b2n−1+δ2n−1a2n−1

]

= τn(βn−1 − a2n−1) .

(5.3)

Solving the system of equations (5.2) for Kn˜ρ2n−1 and δ2n−1, eventually leads to

Kn˜ρ2n−1 = ˆρ2n−1

{

1 + τnˆρ2n−1

(

(βn − βn−1) + a2n−1(1 − βn−1βn)

1 − β2n

)}

, (5.4)

and δ2n−1 given by (3.10). On the other hand, solving the system of equations (5.3) for Kn˜ρ2n−1 and

δ2n−1, we obtain that

δ2n−1 = b2n−1τn

ˆρ2n−1[(βn−1 − βn)a2n−1 − (1 − βn−1βn)] − δ2n−1(1 − β2n)

τnˆρ2n−1(βn−1 − βn)(1 − a2

2n−1) − b2n−1(1 − β2n)

=τn

ˆρ2n−1[(βn−1 − βn)a2n−1 − (1 − βn−1βn)] − δ2n−1(1 − β2n)

τnˆρ2n−1(βn−1 − βn)b2n−1 − (1 − β2

n), (5.5)

while Kn˜ρ2n−1 again is given by (5.4). So, it remains to prove that the equalities given by (3.10)

and (5.5) are equivalent. Eliminating τn in (3.10) and (5.5), we find that it should hold that

ˆρ2n−1(δ2n−1 − δ2n−1)

(b2n−1 − δ2n−1a2n−1)(1 − βn−1βn) + δ2n−1(βn−1 − βn)=

ˆρ2n−1(δ2n−1 − δ2n−1)

(a2n−1 − δ2n−1b2n−1)(βn−1 − βn) − (1 − βn−1βn),


or, equivalently,

{

ˆρ2

2n−1(δ2n−1 − δ2n−1) − ˜ρ22n−1(δ2n−1 − δ2n−1)

}{

(1 − βn−1βn) + ˜ρ2

2n−1δ2n−1(βn−1 − βn)

}

= 0.

We now have that

ˆρ2

2n−1(δ2n−1 − δ2n−1) − ˜ρ22n−1(δ2n−1 − δ2n−1) =

(b2n−1 + δ2n−1a2n−1)(δ2n−1 − δ2n−1) − (b2n−1 − δ2n−1a2n−1)(δ2n−1 − δ2n−1) =

− (b2n−1δ2n−1 − b2n−1δ2n−1) + (b2n−1δ2n−1 − b2n−1δ2n−1) + a2n−1

(

∣

∣

∣δ2n−1

∣

∣

∣

2

−∣

∣

∣δ2n−1

∣

∣

∣

2)

= 0,

which ends the proof.

Proof of Lemma 3.7

Note that, due to the equality in (3.2), it suffice to prove that one of the zeros of r2n−1(z) :=1

e2n−1{ζn(z)φ2n−1(z) + φ∗

2n−1(z)} tends to either 1 or −1 whenever∣

∣

∣δ2n−1

∣

∣

∣tends to one. Or, equiva-

lently, either r2n−1(1) = 0 or r2n−1(−1) = 0 if δ2n−1 ∈ T.So, let ν ∈ {±1}. Then we have that r2n−1(ν) = 0 iff

1

e2n−1

{

ζn(ν)φ2n−1(ν) + B2n−1(ν)φ(2n−1)∗(ν)}

= 0

⇔ ζn(ν)

e2n−1

{

φ2n−1(ν) + B2n−2(ν)φ(2n−1)∗(ν)}

= 0

⇔ 1

e2n−1

{

φ2n−1(ν) +∣

∣

∣Bn−1(ν)∣

∣

∣

2

φc2n−1(ν)

}

= 0

⇔ φ2n−1(ν)

e2n−1= 0

⇔ ˜ρ2n−11 − βn−1ν

1 − βnν

[

ζn−1(ν)φ2n−2(ν) + δ2n−1φ∗2n−2(ν)

]

= 0, (5.6)

which implies that φ2n−1(z)e2n−1

is (up to a nonzero multiplicative factor) a pORF. Consequently, from (2.6)

together with (5.6) and Theorem 2.1 we deduce that r2n−1(ν) = 0 iff

δ2n−1 = ξ2n−1 :=ˆρ2n−1δ2n−1 − ν ˆρ2n−1

ˆρ2n−1 − ν ˆρ2n−1δ2n−1

∈ T. (5.7)

Since δ2n−1 and ˜ρ2n−1 satisfy the equalities in (3.4), it follows from Lemma 3.3 that the equalityin (5.7) can only hold if

ξ2n−1 ∈ {±1} for a2n−1 = 0 or ξ2n−1 ∈ C ∩ T for a2n−1 6= 0,

where C is defined as before in Lemma 3.3. If a2n−1 = 0 it follows from (5.7) that ξ2n−1 = −ν, whichindeed is in {±1}. This proves the statement for the case of a2n−1 = 0.

Finally, consider the case in which a2n−1 6= 0. From the first equality in (3.4) we then deduce that

ξ2n−1 ∈ C ∩ T iff

b2n−1ξ2n−1 − b2n−1ξ2n−1 = 2a2n−1. (5.8)

Taking into account that

a2n−1 =

ˆρ22n−1δ2n−1 − ˆρ

2

2n−1δ2n−1

1 − |δ2n−1|2

and b2n−1 = ˆρ22n−1 + δ2n−1a2n−1,


we find that (5.8) is equivalent with

{

ˆρ22n−1ξ2n−1 − ˆρ

2

2n−1ξ2n−1

}

(1 − |δ2n−1|2) =

(2 − δ2n−1ξ2n−1 − δ2n−1ξ2n−1)

{

ˆρ22n−1δ2n−1 − ˆρ

2

2n−1δ2n−1

}

.(5.9)

After replacing ξ2n−1 with the expression on the right hand side of (5.7) and simplifying, it becomes

clear that the equality in (5.9) indeed holds true for every δ2n−1 ∈ D and ˆρ2n−1 ∈ T, which ends theproof.

Acknowledgements

The first author is a Postdoctoral Fellow of the Research Foundation - Flanders (FWO). The workof the first and second author is partially supported by the Belgian Network DYSCO (DynamicalSystems, Control, and Optimization), funded by the Interuniversity Attraction Poles Programme,initiated by the Belgian State, Science Policy Office. The scientific responsibility rests with its au-thors. The work of the third author has been partially supported by a Grant of Agencia Canaria deInvestigacion, Innovacion y Sociedad de la Informacion del Gobierno de Canarias.

References

[1] Blaschke W. (1915)Erweiterung des Satzes von Vitali uber Folgen analytischer Funktionen, Berichte uber die Ver-handlungen der Koniglich-Sachsische Gesellschaft der Wissenschaften zu Leipzig, Mathematisch-Physische Klasse 67, 194–200.

[2] Bultheel A., R. Cruz-Barroso, K. Deckers and P. Gonzalez-Vera (2008)Rational Szego quadratures associated with Chebyshev weight functions, Mathematics of Com-putation 78(266), 1031–1059.

[3] Bultheel A., R. Cruz-Barroso, P. Gonzalez-Vera and F. Perdomo-Pıo (2010)Computation of Gauss-type quadrature formulas with some preassigned nodes, Jaen Journal onApproximation (Accepted).

[4] Bultheel A., R. Cruz-Barroso and M. Van Barel (2010)On Gauss-type quadrature formulas with prescribed nodes anywhere on the real line, Calcolo47(1), 28–41.

[5] Bultheel A., L. Daruis and P. Gonzalez-Vera (2001)A connection between quadrature formulas on the unit circle and the interval [−1, 1], Journal ofComputational and Applied Mathematics 132(1), 1–14.

[6] Bultheel A., L. Daruis and P. Gonzalez-Vera (2005)Positive interpolatory quadrature formulas and para-orthogonal polynomials, Journal of Com-putational and Applied Mathematics 179(1–2), 97–119.

[7] Bultheel A., P. Gonzalez-Vera, E. Hendriksen and O. Njastad (2010)Computation of rational Szego-Lobatto quadrature formulas, Applied Numerical Mathematics(Published online 25 May 2010).

[8] Bultheel A., P. Gonzalez-Vera, E. Hendriksen and O. Njastad (1999)Orthogonal Rational Functions, volume 5 of Cambridge Monographs on Applied and Computa-tional Mathematics, Cambridge University Press, Cambridge.

[9] Bultheel A., P. Gonzalez-Vera, E. Hendriksen and O. Njastad (2009)Rational quadrature formulas on the unit circle with prescribed nodes and maximal domain ofvalidity, IMA Journal Numererical Analysis (Published online 21 August 2009).


[10] Deckers K. (2009)Orthogonal Rational Functions: Quadrature, Recurrence and Rational Krylov, PhD thesis, 4February 2009, Department of Computer Science, K.U.Leuven.

[11] Deckers K. and A. Bultheel (2010)Rational interpolation and quadrature on the interval and on the unit circle (Submitted).

[12] Deckers K., A. Bultheel, R. Cruz-Barroso and F. Perdomo-Pıo (2010)Positive rational interpolatory quadrature formulas on the unit circle and the interval, AppliedNumerical Mathematics (Published online 30 March 2010).

[13] Deckers K., A. Bultheel and J. Van Deun (2010)A generalized eigenvalue problem for quasi-orthogonal rational functions (Submitted).

[14] Deckers K., J. Van Deun and A. Bultheel (2007)An extended relation between orthogonal rational functions on the unit circle and the interval[−1, 1], Journal of Mathematical Analysis and Applications 334(2), 1260–1275.

[15] Deckers K., J. Van Deun and A. Bultheel (2009)Computing rational Gauss-Chebyshev quadrature formulas with complex poles: the algorithm,Advances in Engineering Software 40(8), 707–717.

[16] Deckers K., J. Van Deun and A. Bultheel (2008)Rational Gauss-Chebyshev quadrature formulas for complex poles outside [−1, 1], Mathematicsof Computation 77(262), 967–983.

[17] Jagels C. and L. Reichel (2007)Szego-Lobatto quadrature rules, Journal of Computational and Applied Mathematics 200(1),116–126.

[18] Rovba E. A. (1999)Orthogonal systems of rational functions on the segment and quadratures of Gauss-type, Math-ematica Balkanica (N.S.) 13(1–2), 187–198.

[19] Rudin W. (1987)Real and Complex Analysis (3rd edition), McGraw-Hill, New York.

[20] Szego G. (1975)Orthogonal Polynomials (4th edition), volume 33 of Amer. Math. Soc. Colloq. Publ., AmericanMathematical Society, Providence, Rhode Island.

[21] Van Deun J. and A. Bultheel (2006)A quadrature formula based on Chebyshev rational functions, IMA Journal of Numerical Anal-ysis 26(4), 641–656.

[22] Van Deun J. and A. Bultheel (2003)Orthogonal rational functions and quadrature on an interval, Journal of Computational andApplied Mathematics 153(1–2), 487–495.

[23] Van Deun J., A. Bultheel and P. Gonzalez-Vera (2006)On computing rational Gauss-Chebyshev quadrature formulas, Mathematics of Computation75(253), 307–326.

[24] Van Deun J., K. Deckers, A. Bultheel and J. A. C. Weideman (2008)Algorithm 882: Near best fixed pole rational interpolation with applications in spectral methods,ACM Transactions on Mathematical Software 35(2), 14:1–14:21.

[25] Van gucht P. and A. Bultheel (2000)A relation between orthogonal rational functions on the unit circle and the interval [−1, 1],Communications in the Analytic Theory of Continued Fractions 8, 170–182.


Karl Deckers, Adhemar Bultheel,Department of Computer Science, K.U.Leuven, Heverlee, [email protected], [email protected]

Francisco Perdomo-Pıo,Department of Mathematical Analysis, La Laguna University, Tenerife, [email protected]

Rational Gauss-Radau and rational Szegő-Lobatto quadrature on the interval and the unit circle...

Documents

Transcript of Rational Gauss-Radau and rational Szegő-Lobatto quadrature on the interval and the unit circle...