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1 Electronic Transactions on Numerical Analysis. Volume 9, 1999, pp Copyright 1999,. ISSN ETNA OTHOGONAL POLYNOMIALS AND QUADATUE WALTE GAUTSCHI Abstract. Various concepts of orthogonality on the real line are reviewed that arise in connection with quadrature rules. Orthogonality relative to a positive measure and Gauss-type quadrature rules are classical. More recent types of orthogonality include orthogonality relative to a sign-variable measure, which arises in connection with Gauss-Kronrod quadrature, and power (or implicit) orthogonality encountered in Turán-type quadratures. elevant questions of numerical computation are also considered. Key words. orthogonal polynomials, Gauss-Lobatto, Gauss-Kronrod, and Gauss-Turán rules, computation of Gauss-type quadrature rules. AMS subject classifications. 33C45, 65D32, 65F Introduction. Orthogonality concepts arise naturally in connection with numerical quadrature, when one tries to optimize the degree of precision. The classical example is the Gaussian quadrature rule, which maximizes the (polynomial) degree of exactness. Closely related quadrature rules are those of adau and Lobatto, where one or two nodes are prescribed. In Gauss-Kronrod rules about half of the nodes are prescribed, all within the support of the integration measure, which gives rise to orthogonality relative to a sign-variable measure. Quadrature rules with multiple nodes lead naturally to power (or implicit) orthogonality. The classical example here is the Turán quadrature rule. In the following, these interrelations between orthogonal polynomials and quadrature rules, as well as relevant computational algorithms, are discussed in more detail. 2. Quadrature rules and orthogonality. We begin with the simplest kind of quadrature rule, (2.1) f(t)dλ(t) = λ ν f(τ ν )+ n (f), where the integral of a function f relative to some (in general positive) measure dλ is approximated by a finite sum involving n values of f at suitably selected distinct nodes τ ν. The respective error is n (f). The support of the measure is usually a finite interval, a halfinfinite interval, or the whole real line, but could also be a finite or infinite collection of mutually distinct intervals or points. The formula (2.1) is said to have polynomial degree of exactness d if (2.2) n (f) = for all f Pd, where Pd denotes the set of polynomials of degree d. Interpolatory formulae (or Newton- Cotes formulae) are those having degree d = n 1. They are precisely the quadrature rules obtained by replacing f in (2.1) by its Lagrange polynomial of degree n 1 interpolating f at the nodes τ ν. We denote by (2.3) n ω n (t) = (t τ ν ) eceived November 1, Accepted for publicaton December 1, ecommended by F. Marcellán. Department of Computer Sciences, Purdue University, West Lafayette, IN (wxg@cs.purdue.edu). 65

2 66 W. Gautschi the node polynomial associated with the rule (2.1), i.e., the monic polynomial of degree n having the nodes τ ν as its zeros. We see that, given the n nodes τ ν, we can always achieve degree of exactness n 1. A natural question is how to select the nodes τ ν and weights λ ν to do better. This is answered by the following theorem. THEOEM 2.1. The quadrature rule (2.1) has degree of exactness d = n 1+k, k, if and only if both of the following conditions are satisfied: (a) the formula (2.1) is interpolatory; (b) the node polynomial ω n satisfies (2.4) ω n(t)p(t)dλ(t) = for all p Pk 1. It is difficult to trace the origin of this theorem, but Jacobi [11] must have been aware of it. We remark that k =corresponds to the Newton-Cotes formula, which requires no condition other than being interpolatory. The requirement (2.4), accordingly, is empty. If k>, the condition (2.4) is a condition that involves only the nodes τ ν of (2.1); it imposes exactly k nonlinear constraints on them, in fact, orthogonality (relative to the measure dλ) of the node polynomial ω n to the space of all polynomials of degree k 1. Once a set of nodes τ ν has been determined that satisfies this constraint, the condition (a) then uniquely determines the weights λ ν in (2.1), for example, as the solution of the linear (Vandermonde) system of equations n λ ντ ν µ = tµ dλ(t), µ =, 1,...,n The Gaussian quadrature rule. If the measure dλ is positive, then k n, since otherwise (2.4) would have to hold for k = n +1, implying that ω n is orthogonal to all polynomials of degree n, hence, in particular, orthogonal to itself, which is impossible. Thus, k = n is optimal, in which case ω n is orthogonal to all polynomials of lower degree, i.e., ω n is the (monic) orthogonal polynomial of degree n relative to the measure dλ. We express this by writing (2.5) ω n (t) =π n (t; dλ). The interpolatory quadrature rule (2.6) f(t)dλ(t) = λ G ν f(τν G )+n G (f) corresponding to this node polynomial is precisely the Gaussian quadrature rule for the measure dλ. It was discoveredby Gauss in 1814 ([4]) in the specialcase of thelebesguemeasure dλ(t) =dt on [ 1, 1]. For general dλ, its nodes are the zeros of the orthogonal polynomial π n ( ; dλ), and its weights λ ν, the so-called Christoffel numbers, are obtainable by interpolation as above. (See, however, Theorem 3.1 below.) 2.2. The Gauss-adau quadrature rule. If the support of dλ is bounded from one side, say from the left, it is sometimes convenient to take a = inf supp dλ as one of the nodes, say τ 1 = a. According to Theorem 2.1, this reduces the degree of freedom by one, and the maximum possible value of k is k = n 1. If we write ω n (t) =ω n 1 (t)(t a), the polynomial ω n 1 having the remaining nodes as its zeros must be orthogonal to all lowerdegree polynomials relative to the measure dλ a (t) =(t a)dλ(t), i.e., (2.7) ω n 1 (t) =π n 1 (t; dλ a ).

3 Orthogonal polynomials and quadrature 67 The corresponding interpolatory quadrature rule, (2.8) f(t)dλ(t) =λ 1 f(a)+ λ ν f(τν )+n (f), ν=2 is called the Gauss-adau rule for dλ (adau [19]). There is an analogous rule in the case b = sup supp dλ <, with,say,τ n = b. Indeed, both rules make sense also in the case a<inf supp dλ resp. b>sup supp dλ. They are special cases of a quadrature rule already considered by Christoffel in 1858 ([3]) The Gauss-Lobatto quadrature rule. If the support of dλ is bounded from both sides, we may take τ 1 = a inf supp dλ and τ n = b sup supp dλ, thereby restricting the degree of freedom once more. The optimal quadrature rule becomes the Gauss-Lobatto rule for dλ ([15]), n 1 (2.9) f(t)dλ(t) =λl 1 f(a)+ λ L ν f(τ ν L )+λl n f(b)+l n (f), ν=2 whose interior nodes τ ν are the zeros of (2.1) π n 2 ( ; dλ a,b ), dλ a,b =(t a)(b t)dλ(t). It, too, is a special case of the Christoffel quadrature rule, which has an arbitrary number of prescribed nodes outside or on the boundary of the support of dλ The Gauss-Kronrod rule. This quadrature rule has also prescribed nodes, but they all are in the interior of the support of dλ, and it therefore transcends the class of Christoffel quadrature rules. In trying to estimate the error of the Gauss quadrature rule, Kronrod in 1964 ([12], [13]) indeed constructed a (2n+1)-point quadrature rule of maximum algebraic degree of exactness that has n prescribed nodes the n Gauss nodes τν G in addition to n +1free nodes. It thus has the form n+1 (2.11) f(t)dλ(t) = λ K ν f(τ ν G )+ λ K µ f(τ µ K )+K n (f). Here, the node polynomial is n+1 (2.12) ω 2n+1 (t) =π n (t; dλ)πn+1 (t), π n+1 (t) = (t τµ K ), and, according to Theorem 2.1 with n replaced by 2n +1, the quadrature rule (2.11) has degree of exactness d =2n + k if and only if (2.11) is interpolatory and π n+1 (t)p(t)π n(t; dλ)dλ(t) = for all p Pk 1. The optimal value of k is k = n +1, in which case πn+1 is orthogonal to all polynomials of lower degree, i.e., µ=1 µ=1 (2.13) π n+1(t) =π n+1 (t; π n dλ). That is, πn+1 is the (monic) polynomial of degree n +1orthogonal relative to the oscillating measure dλ n (t) =π n(t; dλ)dλ(t). While πn+1 always exists uniquely, there is no assurance that all its zeros are inside the support of dλ, or even real (cf. [5]).

4 68 W. Gautschi 3. Computation of Gauss-type quadrature rules. All quadrature rules introduced in 2 can be computed via eigenvalues and eigenvectors of a symmetric tridiagonal matrix. Part or all of this matrix is made up of the coefficients in the three-term recurrence relation satisfied by the (monic) orthogonal polynomials π k ( )=π k ( ; dλ) relative to the (positive) measure dλ, (3.1) π k+1 (t) =(t α k )π k (t) β k π k 1 (t), k=, 1, 2,...,n 1, π 1 (t) =, π (t) =1, where α k = α k (dλ), β k = β k (dλ) > depend on dλ and β = dλ(t) by convention. The tridiagonal matrices involved are, respectively, the Jacobi, the Jacobi-adau, the Jacobi- Lobatto, and the Jacobi-Kronrod matrix The Gauss quadrature rule. The Jacobi matrix of order n is defined by (3.2) Jn G = J n G (dλ) = α β1 β1 α 1 β βn 1 βn 1 α n 1 The Gauss formula (2.6) can be obtained in terms of the eigenvalues and eigenvectors of Jn G according to the following theorem. THEOEM 3.1. (Golub and Welsch [1]) The Gauss nodes τν G are the eigenvalues of Jn G, and the Gauss weights λg ν are given by. (3.3) λ G ν = β [u G ν,1] 2, ν =1, 2,...,n, where u G ν is the normalized eigenvector of Jn G corresponding to the eigenvalue τν G (i.e., [u G ν ]T u G ν =1) and ug ν,1 its first component. Thus, to compute the nodes and weights of the Gauss formula, it suffices to compute the eigenvalues and first components of the eigenvectors of the Jacobi matrix Jn G. Efficient methods for this, such as the Q algorithm, are well known (see, e.g., Parlett [18]). Since the first components u G ν,1 of u G ν are easily seen to be nonzero, the positivity of the Gauss rule can be read off from (3.3) The Gauss-adau formula. We replace n by n+1in(2.8)soastohaven interior nodes, f(t)dλ(t) =λ f(a)+ λ ν f(τ ν )+ n+1 (3.4) (f), n+1 (P 2n) =. The Jacobi-adau matrix of order n +1is then defined by [ J Jn+1 = Jn+1(dλ) G = n (dλ) βn e (3.5) βn n e T n α n ], where J G n (dλ) is as in (3.2), et n =[,,...,1] is the nth coordinate vector in n,and (3.6) α n = a β n π n 1 (a) π n (a),

5 Orthogonal polynomials and quadrature 69 with π k ( )=π k ( ; dλ) as before. One then has the following theorem analogous to Theorem 3.1. THEOEM 3.2. (Golub [9]) The Gauss-adau nodes τ = a, τ 1,...,τ n in (3.4) are the eigenvalues of Jn+1, and the Gauss-adau weights λ ν are given by (3.7) λ ν = β [u ν,1] 2, ν =, 1, 2,...,n, where u ν is the normalized eigenvector of Jn+1 corresponding to the eigenvalue τν (i.e., [u ν ]T u ν =1) and u ν,1 its first component. As previously for the Gauss formula, the Q algorithm, now applied to Jn+1,isagain the method of choice to compute Gauss-adau formulae. Their positivity follows from (3.7). Theorem 3.2 remains in force if a inf supp dλ. If the right end point is the prescribed node, τn+1 = b, orifτ n+1 = b sup supp dλ, a theorem analogous to Theorem 3.2 holds with obvious changes The Gauss-Lobatto formula. We now replace n by n +2in (2.9) and write the Gauss-Lobatto formula in the form f(t)dλ(t) =λl f(a)+ λ L ν f(τ ν L )+λl n+1 f(b)+l n+2 (3.8) (f), n+2 L (P 2n+1) =. The Jacobi-Lobatto matrix of order n +2is defined by J G n (dλ) βn e n (3.9) Jn+2 L = Jn+2(dλ) L = βn e T n α n β L n+1 T β L n+1 α L n+1 with Jn G (dλ) and e n as before, and α L n+1, βn+1 L the solution of the 2 2 linear system [ ][ ] [ ] πn+1 (a) π n (a) α L n+1 aπn+1 (a) (3.1) π n+1 (b) π n (b) βn+1 L =. bπ n+1 (b) We now have THEOEM 3.3. (Golub [9]) The Gauss-Lobatto nodes τ L = a, τ 1 L,...,τL n, τ n+1 L = b in (3.8) are the eigenvalues of Jn+2 L, and the Gauss-Lobatto weights λl ν are given by, (3.11) λ L ν = β [u L ν,1] 2, ν =, 1, 2,...,n,n+1, where u L ν is the normalized eigenvector of Jn+2 L corresponding to the eigenvalue τν L (i.e., [u L ν ]T u L ν =1) and ul ν,1 its first component. Also Gauss-Lobatto formulae are therefore computable by the Q algorithm, now applied to Jn+2 L, and by (3.11) we still have positivity of the quadrature rule. Theorem 3.3 holds without change for a inf supp dλ and b sup supp dλ The Gauss-Kronrod formula. It has only recently been discovered by Laurie that an eigenvalue/eigenvector characterization similar to those in Theorems holds also for the Gauss-Kronrod rule (2.11). The Jacobi-Kronrod matrix of order 2n +1now has the form J G J2n+1 K = J 2n+1 K n βn e n (dλ) = (3.12) βn e T n α n βn+1 e T 1, βn+1 e 1 J n

6 7 W. Gautschi where e T 1 =[1,,...,] n and Jn is a real symmetric tridiagonal matrix which has different forms depending on whether n is odd or even: [ J G ] Jn odd = n+1:(3n 1)/2 β(3n+1)/2 e (n 1)/2 (3.13), β(3n+1)/2 e T (n 1)/2 J(3n+1)/2:2n (3.14) J Jn even = n+1:3n/2 G β (3n+2)/2 e T n/2 β (3n+2)/2 e n/2. J (3n+2)/2:2n Here, Jp:q G is the principal minor matrix of the (infinite) Jacobi matrix J G having diagonal elements α p,α p+1,...,α q,andj (3n+2)/2 :2n are symmetric tridiagonal matrices, and β(3n+2)/2 an element yet to be determined. THEOEM 3.4. (Laurie [14]) Let λ K ν > and λ K µ > in (2.11). ThenJn in (3.12) has the same eigenvalues as Jn G. Moreover, the nodes τ ν G and τ µ K are the eigenvalues of J 2n+1 K, and the Gauss-Kronrod weights are given by (3.15) λ K ν = β [u K ν,1] 2,,...,n; λ K µ = β [u K µ+n,1] 2,µ=1,...,n,n+1, where u K 1,uK 2,...,uK 2n+1 are the normalized eigenvectors of J 2n+1 K corresponding to the eigenvalues τ1 G,...,τG n,τk 1,...,τK n+1, and uk 1,1,uK 2,1,...,uK 2n+1,1 their first components. Conversely, if the eigenvalues of Jn G and J n are the same, then (2.11) exists with real nodes and positive weights. We remark that according to a result of Monegato [16] the positivity of λ K µ implies the reality of the Kronrod nodes τµ K and their interlacing with the Gauss nodes τ ν G. Once the trailing tridiagonal blocks in (3.13), (3.14), and β(3n+2)/2 if n is even, are known, the Gauss-Kronrod formula can be computed in much the same way as the Gauss formula in terms of eigenvalues and eigenvectors of the symmetric tridiagonal matrix J2n+1 K. This in fact is the way Laurie s algorithm proceeds. Note, however, that when the nodes τν G are already known, there is a certain redundancy in this algorithm in as much as these Gauss nodes are recomputed along with the Kronrod nodes τµ K. This redundancy is eliminated in a more recent algorithm of Calvetti, Golub, Gragg, and eichel, which bypasses the computation of the trailing block in (3.12) and focuses directly onto the Kronrod nodes and weights of the Gauss-Kronrod formula Laurie s algorithm. ([14]) Assume that the hypotheses of Theorem 3.4 are fulfilled. Let, as before, {π k } n k= denote the (monic) polynomials belonging to the Jacobi matrix Jn G (dλ), and thus orthogonal with respect to the measure dλ,andlet{πk }n k= be the (monic) orthogonal polynomials belonging to the matrix Jn in (3.12) and measure dµ (in general unknown). Define mixed moments by (3.16) σ k,l =(π k,π l) dµ, where (, ) dµ is the inner product relative to the measure dµ. Although the measure dµ is unknown, a few things about the moments σ k,l are known. For example, (3.17) σ k,l = for l<k, which is an immediate consequence of orthogonality. Also, (3.18) σ k,n = for k<n,

7 Orthogonal polynomials and quadrature 71 again by orthogonality, since π n = πn by Theorem 3.4. It is easy, moreover, to derive the recurrence relation (3.19) σ k,l+1 σ k+1,l (α k α l)σ k,l β k σ k 1,l β l σ k,l 1 =, where {α k }n 1 k= are the diagonal elements of J n and { βk }n 1 k=1 the elements on the side diagonals, while {α l } n 1 l=, { β l } n 1 l=1 are the analogous elements of J n G. Some of the elements α k, β k are known according to the structure of the matrices in (3.13) and (3.14). Indeed, if we assume for definiteness that n is odd, then (3.2) α k = α n+1+k for k (n 3)/2, β k = β n+1+k for k (n 1)/2. Using the facts that σ, = dµ (t) =β = β n+1 and σ 1,l =for l =, 1,...,n 1, σ, 1 =,andσ k,k 2 = σ k,k 1 =for k =1, 2,...,(n 1)/2, one can solve (3.19) for σ k,l+1 and compute the entries σ k,l in the triangular array indicated by black dots in Fig. 3.1 (drawn for n =7), since the α k and β k required are known by (3.2) except for the top element in the triangle. For this element the α k for k =(n 1)/2 is not yet known, but, by good fortune, it multiplies the element σ (n 1)/2,(n 3)/2, which is zero by (3.17). This mode of recursion from left to right has already been used by Salzer [21] in another context. k n -1 ( n-3)/2 l n ( =7) n FIG.3.1.Computation of the mixed moments At this point, one can switch to a recursion from bottom up, using the recurrence relation (3.19) solved for σ k+1,l. This computes all entries indicated by a cross in Fig. 3.1, proceeding from the very bottom to the very top of the array. For each k with (n 1)/2 k n 1, the entries in Fig. 3.1 surrounded by boxes are those used to compute the as yet unknown α k, βk according to α k = α k + σ k,k+1 σ k 1,k, βk = σ k,k (3.21). σ k,k σ k 1,k 1 σ k 1,k 1 This is the way Sack and Donovan [2] proceeded to generate modified moments in the modified Chebyshev algorithm (cf. [6, 5.2]). In the present case, crucial use is made of the

8 72 W. Gautschi property (3.18) of mixed moments, which provides the zero boundary values at the right edge of the σ-tableau. Once in possession of all the α k, β k required to compute J n, one proceeds as described earlier The algorithm of Calvetti, Golub, Gragg, and eichel. ([2]) Here we assume for simplicity that n is even. There is a similar, though slightly more complicated, algorithm when n is odd. The symmetric tridiagonal matrix Jn in (3.12) determines its own set of orthogonal polynomials, in particular, an n-point Gauss quadrature rule relative to some (unknown) measure dµ. Since by Theorem 3.4 the eigenvalues of Jn are the same as those of Jn G, the Gauss rule in question has nodes τν G and certain positive weights µ ν, ν =1, 2,...,n. Likewise, the (known) matrix Jn+1:3n/2 G of order n/2 in (3.14) determines another Gauss rule whose nodes and weights we denote respectively by τ κ and λ κ, κ =1, 2,...,n/2. Letv =[v 1,v 2,...,v n ] and v =[v1,v 2,...,v n ] be the matrices of normalized eigenvectors of J n G and J n, respectively. The algorithm requires the last components v 1,n,v 2,n,...,v n,n of the eigenvectors v 1,v 2,...,v n and the first components v1,1,v 2,1,...,v n,1 of the eigenvectors v1,v 2,...,v n. The latter, according to Theorem 3.1, are related to the Gauss weights µ ν by means of the relation (3.22) β [v ν,1] 2 = µ ν, ν =1, 2,...,n, and can therefore be computed as the positive square roots of µ ν/β. It can be easily shown, on the other hand, that the weights µ ν are computable with the help of the second Gauss rule above as (3.23) n/2 µ ν = l ν ( τ κ ) λ κ, κ=1 ν =1, 2,...,n, where l ν are the elementary Lagrange polynomials associated with the nodes τ G 1,τG 2,...,τ G n. With these auxiliary quantities computed, the remainder of the algorithm consists of the consolidation phase of a divide-and-conquer algorithm due to Borges and Gragg ([1]). The spectral decomposition of J n is (3.24) Define (3.25) J n = v D τ [v ] T, D τ = diag (τ G 1,τ G 2,...,τ G n ). V := v 1, v a matrix of order 2n +1. We then have from (3.12) that D τ λi βn v T e n V T (J2n+1 K λi)v = βn e T n v α (3.26) n λ βn+1 e T 1 v βn+1 [v ] T e 1 D τ λi, where the matrix on the right is a diagonal matrix plus a Swiss cross containing the elements of e T n v and et 1 v previously computed. By orthogonal similarity transformations involving

9 Orthogonal polynomials and quadrature 73 a permutation and a sequence of Givens rotations, the matrix in (3.26) can be reduced to the form D τ λi (3.27) Ṽ T (J2n+1 K λi)ṽ = D τ λi c, c T α n λ where Ṽ is the transformed matrix V and c a vector containing the entries in positions n +1 to 2n of the transformed vector [ β n e T n v, β n+1 e T 1 v,α n ]. It is now evident from (3.27) that one set of eigenvalues of J2n+1 K is {τ1 G,τ2 G,...,τn G }. The remaining eigenvalues are those of the trailing block in (3.27). To compute them, observe that [ ] [ Dτ λi c = α n λ c T where in terms of the components c ν of c, I c T (D τ λi) 1 1 ][ Dτ λi c T f(λ) ], (3.28) f(λ) =λ α n + c 2 ν τ G ν λ. Thus, the remaining eigenvalues of J2n+1 K are the zeros of f(λ), which can be seen to interlace with the nodes τν G. The first components of the normalized eigenvectors u K 1,uK 2,...,uK 2n+1 needed according to Theorem 3.4 are also computable from the columns of V by keeping track of the orthogonal transformations. 4. Quadrature rules with multiple nodes. We now consider quadrature rules of the form f(t)dλ(t) = r 1 (4.1) λ (ρ) ν f (ρ) (τ ν )+ n (f), ρ= where each τ ν is a node of multiplicity r. The underlying interpolation process is the one of Hermite, which is exact for polynomials of degree r n 1. We therefore call (4.1) interpolatory if it has degree of exactness d = r n 1. As before, the node polynomial is defined by (4.2) n ω n (t) = (t τ ν ). In analogy to Theorem 2.1, we now have the following theorem. THEOEM 4.1. The quadrature rule (4.1) has degree of exactness d = r n 1+k, k, if and only if both of the following conditions are satisfied: (a) the formula (4.1) is interpolatory; (b) the node polynomial ω n satisfies (4.3) [ω n(t)] r p(t)dλ(t) = for all p Pk 1. The case k =corresponds to the Newton-Cotes-Hermite formula, which requires no extra condition beyond that of being interpolatory. If k >, the condition (b) is again a condition involving only the nodes τ ν, namely orthogonality of the rth power of ω n to all

10 74 W. Gautschi polynomials of degree k 1. It is immediately clear from (4.3) that for positive measures dλ the multiplicity r must be an oddinteger, since otherwise we couldnot have orthogonality of ω r n to a constant, let alone to P k 1. It is customary, therefore, to write (4.4) r =2s +1, s. It then follows that k n, since otherwise k = n +1would imply orthogonality of ωn r to all polynomials of degree n, in particular, orthogonality to ω n,which,r being odd, is impossible. Thus, k = n is optimal; the corresponding interpolatory quadrature rule is called is orthogonal to all polynomials of degree n 1, the polynomial ω n thus the so-called s-orthogonal polynomial of degree n. There is an extensive theory of these polynomials, for which we refer to the book of Ghizzetti and Ossicini [8, 3.9,4.13]. We mention here only the elegant result of Turán [22], according to which ω n is the extremal polynomial of the Gauss-Turán rule for the measure dλ ([22]). For it, the polynomial ω 2s+1 n (4.5) [ω(t)]2s+2 dλ(t) = min, where the minimum is sought among all monic polynomials ω of degree n. Fromthis,there follows in particular the existence and uniqueness of real Gauss-Turán formulae, but not necessarily their positivity. Nevertheless, it has been shown by Ossicini and osati [17] that >, ν =1, 2,...,n,ifρ is even. λ (ρ) ν 5. Computation of Gauss-Turán quadrature rules. (Gautschi and Milovanović[7]) The computation of the Gauss-Turán formula f(t)dλ(t) = 2s λ ν (σ)t f (σ) (τν T )+n T (f), (5.1) σ= n T (P 2(s+1)n 1) =, involves two stages: (i) The generation of the s-orthogonal polynomial π n = π n,s, i.e., the polynomial π n satisfying the orthogonality relation (5.2) [π n(t)] 2s+1 p(t)dλ(t) = for all p Pn 1; the zeros of π n are the desired nodes τν T in (5.1). (ii) The computation of the weights λ ν (σ)t. Since the latter stage requires knowledge of the nodes τν T, we begin with the computation of the s-orthogonal polynomial π n,s. We reinterpret power orthogonality in (5.2) as ordinary orthogonality with respect to the measure (5.3) dλ n,s (t) =[π n (t)] 2s dλ(t), which, like dλ, is also positive, (5.4) π n(t)p(t)dλ n,s (t) = for all p Pn 1. Since the measure dλ n,s involves the unknown π n, one also talks about implicit orthogonality.

11 Orthogonal polynomials and quadrature 75 The measure dλ n,s being positive, it defines a sequence π k (t) = π k (t; dλ n,s ), k =, 1,...,n, of orthogonal polynomials, of which only the last one for k = n is of interest to us. They satisfy a three-term recurrence relation (3.1), with coefficients α k, β k given by the well-known inner product formulae α k = tπ2 k (t)dλ n,s(t) π2 k (t)dλ, k =, 1,...,n 1, n,s(t) (5.5) β = dλ n,s(t), β k = π2 k (t)dλ n,s(t) π2 k 1 (t)dλ n,s(t), k =1,...,n 1. This constitutes a system of 2n nonlinear equations for the 2n coefficients α,α 1,..., α n 1 ; β,β 1,...,β n 1, which can be written in the form (5.6) φ =, φ T =[φ,φ 1,...,φ 2n 1 ], where, in view of (5.3), φ := β π2s n (t)dλ(t), (5.7) φ 2ν := [β νπν 1(t) 2 πν(t)]π 2 n 2s (t)dλ(t),,...,n 1, φ 2ν+1 := (α ν t)πν 2 (t)π2s n (t)dλ(t), ν=, 1,...,n 1. Here, each π k is to be considered a function of α,...,α k 1 ; β 1,...,β k 1 by virtue of the three-term recurrence relation (3.1) which it satisfies. Since all integrands in (5.7) are polynomials of degree 2(s +1)n 1, the integrals in (5.7) can be computed exactly by an (s +1)n-point Gaussian quadrature rule relative to the measure dλ. Any method for solving systems of nonlinear equations, such as Newton s method or quasi-newton methods, can now be applied to solve (5.6). (For Newton s method, in this context, see [7].) We now turn to the computation of the quadrature weights. Here we note that (5.8) ω ρ,ν (t) =(t τ T ν ) ρ µ ν(t τ T µ ) 2s+1, ρ =, 1,...,2s; ν =1,...,n, are polynomials of degree (2s +1)n 1, hence, by (5.1), n T (ω ρ,ν) =. This can be written as 2s (5.9) λ κ (σ)t ω ρ,ν (σ) (τκ T )=b ρ,ν, κ=1 σ= with (5.1) b ρ,ν = ω ρ,ν(t)dλ(t). By virtue of the definition (5.8) of ω ρ,ν,wehaveω ρ,ν (σ) (τκ T )=for κ ν, so that the system (5.9) breaks up into n separate linear systems for the weights λ ν (σ)t, σ =, 1,...,2s, (5.11) 2s σ= λ ν (σ)t ω ρ,ν (σ) (τ ν T )=b ρ,ν, ρ =, 1,...,2s.

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