On the Gauss-Kronrod quadrature formula for a modified weight function of Chebyshev type

In this paper, we consider the Gauss-Kronrod quadrature formulas for a modified Chebyshev weight. Efficient estimates of the error of these Gauss–Kronrod formulae for analytic functions are obtained, using techniques of contour integration that were introduced by Gautschi and Varga (cf. Gautschi and Varga SIAM J. Numer. Anal. 20, 1170–1186 1983). Some illustrative numerical examples which show both the accuracy of the Gauss–Kronrod formulas and the sharpness of our estimations are displayed. Though for the sake of brevity we restrict ourselves to the first kind Chebyshev weight, a similar analysis may be carried out for the other three Chebyshev type weights; part of the corresponding computations are included in a final appendix.


Introduction
Consider a positive measure dσ on a real interval [a, b] having infinitely many points of increase and finite moments of all orders. It is well known that the corresponding monic orthogonal polynomials {π n } satisfy a three-term recurrence relation π −1 (t) = 0, π 0 (t) = 1, π k+1 (t) = (t − α k )π k (t) − β k π k−1 (t), k = 0, 1, 2, . . . , (1) where α k ∈ R, β k > 0, and by convention β 0 = b a dσ (t). In [11] Gautschi and Li considered a modification of the original measure, for a fixed integer n ≥ 1, given by and studied the corresponding (monic) orthogonal polynomials π m = π m,n , m = 0, 1, 2, . . . . As pointed out by the authors in [11], this kind of modifications of measures are useful, for instance, when dealing with constrained polynomial least squares approximation (see, e.g., [8]), or to provide additional interpolation points (the zeros of the induced polynomial { π n+1,n }) in the process of extending Lagrange interpolation at the zeros of π n (see [3]). Taking into account these and other applications, it seems natural to consider the numerical computation of integrals of the form by means of quadrature formulae; in particular, Gauss type rules are our main subject of interest. It is well known that the zeros and nodes of the Gauss rule can be efficiently computed by means of the eigenvalues and eigenvectors of the related tridiagonal Jacobi matrix, whose entries are given in terms of the above mentioned recursion coefficients. Then, the following problem arises in a natural way: given the recursion coefficients α k , β k for dσ , determine the recursion coefficientsα k ,β k for dσ n . Unfortunately, in general it is not feasible to get closed analytic expressions of the entries of the Jacobi matrix for the induced measure d σ n in terms of the corresponding for dσ ; in this sense, in [11] a stable numerical algorithm is given. But in the particular case of the well-known four Chebyshev weights dσ [i] , i = 1, 2, 3, 4 , where dσ [1] (t) = 1 √ 1−t 2 dt, dσ [2] (t) = √ 1 − t 2 dt, dσ [3] (t) = 1+t 1−t dt, dσ [4] (t) = 1−t 1+t dt the related induced orthogonal polynomials are easily expressible as combinations of Chebyshev polynomials of the first kind T k , i.e., orthogonal polynomials with respect to the Chebyshev weight dσ = dσ [1] (see [11, §3]). These results are very useful for the analysis of the error of the related quadrature formulas.
In the present paper, we focus on the first modified Chebyshev measure, namely dσ n (t) = dσ [1] where dσ (t) = 1 √ 1 − t 2 dt and • T n (t) = 2 1−n T n (t) , (5) with • T n denoting the corresponding nth-degree monic Chebyshev polynomial. As we said above, for this, as well as for the other modified Chebyshev weights, it is feasible to get closed expressions of the entries of the Jacobi tridiagonal matrices in terms of the corresponding for the original Chebyshev ones. These are collected in previous papers as [11, and [21,Section 2]. Such results will be useful, among other things, for computing the actual (sharp) value of the quadrature error in the numerical examples.
In this paper, we aim to obtain accurate estimates of the error of the Gauss-Kronrod quadrature formulas for analytic integrands related to this modification of the first kind Chebyshev measure; therefore, this partially completes the analysis started in [25], where those estimates were obtained for the ordinary Gauss quadrature formulas. In 1964, A. S. Kronrod, trying to estimate in a feasible way the error of the n-point Gauss-Legendre quadrature formula, developed the now called Gauss-Kronrod quadrature formula for the Legendre measure (cf. [15,16]). For a general measure dσ this formula has the form where τ ν are the zeros of π n , and the τ * μ , W ν , W * μ are chosen such that (6) has maximum degree of exactness. It turns out that a necessary and sufficient condition for this to happen is that τ * μ be the zeros of the polynomial π * n+1 (see [7,Corollary]), uniquely determined by the orthogonality relations b a π * n+1 (t)t k π n (t) dσ (t) = 0, k = 0, 1, . . . , n.
Observe that (7) implies that π * n+1 is a polynomial orthogonal with respect to a variable-sign measure, from which the fact that its zeros be simple and belong to the interval (a, b) is not guaranteed in advance. Polynomials of this kind were considered for the first time by T. J. Stieltjes in 1894, for the Legendre measure dσ (t) = dt on [−1, 1]. Stieltjes, in a letter to Hermite (see [1, vol 2, pp. 439-441]), conjectured that π * n+1 has n + 1 real and simple zeros, all contained in (−1, 1), and interlacing with the zeros of the nth-degree Legendre polynomial. Stieltjes' conjectures were proved by Szegő in 1935 (cf. [31]), not only for the Legendre but also for the Gegenbauer measure dσ (t) = (1 − t 2 ) λ−1/2 dt on [−1, 1], when 0 < λ ≤ 2. After that, the polynomials π * n+1 , now appropriately called Stieltjes polynomials, have apparently gone unnoticed until Kronrod's papers in 1964 (cf. [15,16]). The connection between Stieltjes polynomials and Gauss-Kronrod formulae was pointed out by Mysovskih in [20], and independently by Barrucand in [2]. A nice and detailed survey of Kronrod rules in the last 50 years is provided by Notaris [24]. Numerically stable and effective procedures for calculating Gauss-Kronrod formulas are proposed in [17] and [4].
We consider here Gauss-Kronrod quadrature formulas for the modified Chebyshev weight function of the first kind d σ n = d σ [1] n , that is, where under the assumption that all τ ν , τ * μ belong to [−1, 1]. Our analysis of the error is based on its well-known representation in terms of an integral contour of a suitable kernel; namely, if we use a Gauss-Kronrod rule I n (f ) with 2n + 1 nodes to approximate the value of the integral I σ (f ) for a certain positive measure σ (hereafter, we assume the absolute continuity of the measure σ and, hence, that dσ (t) = w(t) dt) on the real interval [−1, 1] and an analytic integrand f in a neighborhood Ω of this interval, the error of quadrature admits the following integral representation (see, e. g., [12]) where the kernel K n is given by with π n denoting, as usual, the nth degree orthogonal polynomial with respect to w, π * n+1 denoting the corresponding Stieltjes polynomial of degree n+1 for the modified Chebyshev weight, n is the commonly called 2nd kind function associated to the nodal polynomial, and Γ ⊂ Ω is any closed smooth contour surrounding the real interval [−1, 1]. Elliptic contours E ρ with foci at the points ±1 and semi-axes given by 1 2 (ρ + ρ −1 ) and 1 2 (ρ − ρ −1 ) , with ρ > 1, are often considered as contours of integration, in order to get suitable estimations of the error of quadrature; this is due to the fact that they are the level curves for the conformal function which maps the exterior of [−1, 1] onto the exterior of the unit circle |z| > 1 in the complex plane. In this sense, these elliptic level curves admit the expression where ρ > 1 and the branch of √ z 2 − 1 is taken so that |φ(z)| > 1 for |z| > 1. On the other hand, the inverse function of φ, that is, the well-known Joukowsky transform, given by will also be used in the subsequent sections.
The outline of the current paper is as follows. In Section 2 an explicit expression for the kernel (10) related to the induced Chebyshev weight d σ [1] n is provided, which will be useful to get appropriate bounds for the error of the corresponding Gauss-Kronrod rules, which represents the main contribution of the paper. In addition, the accuracy of the obtained bounds is checked by means of some illustrative numerical examples in Section 3. Finally, and for the sake of completeness, similar computations for the kernels corresponding to the other modified Chebyshev measures dσ [i] n , i = 2, 3, 4, are gathered in the final appendix. To end this introduction, let us say that the problem of estimating the quadrature error for Gauss-type rules has been thoroughly studied in the literature; see the references [12,18,19,22,23], and [26][27][28][29][30], to only cite a few. See also [5] for a very recent survey of the error estimates of Gaussian type quadrature formulae for analytic functions on ellipses.
Let us finally point out that the seemingly restricted scope of our analysis is offset, in our opinion, by the extreme sharpness of the estimations shown in Section 3.

Error bounds of Gauss-Kronrod rules for the measure d σ [1]
n Hereafter, the (monic) orthogonal polynomials relative to the positive measure d σ [1] n (t) = [π n (t)] 2 dσ [1] (t), defined in (2), will be denoted simply by π m,n , m = 0, 1, 2, . . . . For simplicity, here and in the next section we only consider what may be referred to as the "diagonal" setting, that is, the case where m = n and we simply denote π n = π n,n ; on the other hand, π * n+1 will denote the corresponding Stieltjes polynomial. Our first result gives the explicit expression of the kernel K n in this case.

Lemma 1 The kernel K n is given by
with ξ given by (12).
Proof By (10) the corresponding kernel is given by with U k denoting as usual the Chebyshev polynomial of the second kind of degree k, andŮ k being the monic one, and that is, Thus, by using the identity (see [12, p. 1176 we get Next, to compute the denominator of the kernel the following representation will be used (see [12, pp. 1176-1177]) Then, the proof of (13) easily follows. Now, we are in a position to obtain bounds for the error of the Gauss-Kronrod quadrature formula using (9) and (13). To do it, several methods will be employed.

L ∞ -bound for the error
On the sequel, for a function g and a compact subset E of the complex plane, the L ∞ -norm of g on E will be denoted by Now, from (9) and taking Γ = E ρ for certain ρ > 1, we easily get that if f is analytic on E ρ and its interior, where l(E ρ ) represents the length of the ellipse E ρ . If we denote by D ρ the closed interior of E ρ , define Next, we have the following L ∞ -bound for the error of the Gauss-Kronrod quadrature formula.

Theorem 1
The error of the Gauss-Kronrod quadrature formula for d σ [1] n is bounded by where the expression of a j is given in (18).
Proof From (13) and using polar coordinates and the Joukowsky transform (12), the modulus of the kernel in this case may be expressed in the form with the a j given by (18), because |2ξ 2n + 1| = 4ρ 4n + 4ρ 2n cos 2nθ + 1.
Since the numerator and denominator of this expression obviously reach its maximum and minimum, respectively, at θ = 0 for all ρ > 1, we can directly state that max On the other hand, the length of the ellipse can be estimated by (cf. [28]) , and thus, (17) yields the bound (19).

Error bounds based on an expansion of the remainder
If f is an analytic function in the interior of E ρ , for some ρ > 1, it admits the expansion where α k are given by The prime in the corresponding sum denotes that the first term is taken with the factor 1/2. The series converges for each z in the interior of E ρ . In general, the Chebyshev-Fourier coefficients α k in the expansion are unknown; however, Elliott [6] described a number of ways to estimate or bound them. In particular, under our assumptions the following upper bound will be useful, The following result provides the desired bound for the error of quadrature.

Theorem 2
The following bound for the error of the Gauss-Kronrod quadrature formula based on the expansion of the remainder is obtained for d σ [1] n : For the proof of this theorem, we need a result by D. B. Hunter [14,Lemma 5], which is included below, to make the paper self-contained.

Lemma 2
With ξ and z as in (12), we have: otherwise .
The remainder term R n (f ) can be represented in the form where the coefficients n,k are independent on f . Namely, using (21) and (24) in (9) we obtain Applying Lemma 2, this reduces to (25)  Now we easily reach the following expression for the error of quadrature Then, inequality (22) yields Finally, the bound (23) is easily attained.

L 1 -bound for the error
From the integral expression (9), the error of quadrature may be bounded in the form where Now, we shall prove the following result.

Numerical results
Throughout this section, several numerical experiments are displayed to illustrate the results in previous section. In this sense, the obtained error bounds r 1 (f ) , r 2 (f ) and r 3 (f ) have been tested for the three following characteristic examples (commonly used in the literature on numerical integration): where a < −1, c ≤ b ≤ a and k ∈ N, l, m ∈ N 0 , and it is easy to check that the following properties are satisfied, It is clear that the functions f 0 (z) and f 1 (z) are entire, so ρ max = ∞ in both cases. Otherwise, for f 2 (z) the condition a < −1, c ≤ b ≤ a means that the function f is analytic inside the elliptical contour E ρ max , for a certain ρ max > 1, where |a| = 1 2 (ρ max + ρ −1 max ). We use some values of parameters a, b, c which have been used in literature (see, e. g., [30]); in particular, a = −1.408333333333333, b = −1.892857142857143, c = −2.408695652173913, k = 1, l = 5, m = 10, which means that ρ max = 2.4.
First of all and though it is a well-known fact that the Gauss-Kronrod quadrature formula is a refinement of the classical Gauss rule (up to the extent that the value given by the former is commonly used to estimate the error of the latter), let us previously include a small table (Table 1) comparing the actual estimations of the error of both quadrature rules for some examples corresponding to the integrand f 0 . We denote by "Error GF" and "Error GKF" the errors of the classical Gauss formula and the Gauss-Kronrod rule, respectively. The numerical examples displayed below clearly show that the number of precision digits of the latter is approximately the  Tables 2, 3, and 4, where "Error" means the actual (sharp) error and I σ (f ) represents the exact value of the integral 1

−1 f (t) d σ n (t).
Tables 2-4 above show how sharp the bounds of the quadrature error obtained in Section 2 are; namely, the average deviation from the actual value of the error does not exceed one precision digit. At the same time, the high accuracy of the Gauss-Kronrod rules, especially in the case of the entire integrands f 0 and f 1 , is clearly shown. Table 2 The values of the derived bounds r 1 (f 0 ), r 2 (f 0 ), r 3 (f 0 ), compared with the actual error for some values of n, ω n, ω  Table 3 The values of the derived bounds r 1 (f 1 ), r 2 (f 1 ), r 3 (f 1 ), compared with the actual error for some values of n, ω n, ω
Let us mention that throughout this appendix we continue dealing with the diagonal setting m = n and using the abbreviate notation for the corresponding orthogonal and Stieltjes polynomials for the modified measures, that is, π [i] n and π [i] * n+1 , respectively. As we will see below, after computing the kernels our main finding will be that the argument θ of the point where each of these kernels attains its maximum modulus remains constant for ρ is big enough; e.g., there exists some ρ * > 0 such that for ρ ≥ ρ * , the argument of the extremum will be θ = θ 0 = const.