Error estimates for Gauss Turán quadratures and their Kronrod extensions

Transcription

1 IMA Journal of Numerical Analyi (29 29, doi:.93/imanum/drm4 Advance Acce publication on May 6, 28 Error etimate for Gau Turán quadrature and their Kronrod extenion GRADIMIR V. MILOVANOVIĆ Faculty of Electronic Engineering, Department of Mathematic, Univerity of Niš, PO Box 73, 8 Niš, Serbia MIODRAG M. SPALEVIĆ Faculty of Science, Department of Mathematic and Informatic, Univerity of Kragujevac, PO Box 6, 34 Kragujevac, Serbia AND MIROSLAV S. PRANIĆ Faculty of Science, Department of Mathematic and Informatic, Univerity of Banja Luka, M. Stojanovića 2, 5 Banja Luka, Bonia and Herzegovina [Received on 8 March 27; revied on 29 October 27] We tudy the kernel K n, (z of the remainder term R n, ( f of Gau Turán Kronrod quadrature rule with repect to one of the generalized Chebyhev weight function for analytic function. The location on the elliptic contour where the modulu of the kernel attain it maximum value i invetigated. Thi lead to effective L -error bound of Gau Turán Kronrod quadrature. Following Kronrod, uing the modulu of the difference of Gau Turán quadrature and their Kronrod extenion, we derive new error etimate for Gau Turán quadrature and compare them with the effective L -error bound derived in Milovanović & Spalević (25, BIT, 45, Keyword: Gau Turán quadrature; Kronrod extenion; -orthogonal polynomial; Stieltje polynomial; remainder term; error etimate; analytic function.. Introduction Let w be an integrable weight function on the interval (,. Gau Turán quadrature formulae of the form w(t f (tdt = n 2 λ i,ν f (i (τ ν + E n, ( f, ν= i= Downloaded from at Univerity Library Svetozar Markovic on November 9, 24 or quadrature formulae with the highet degree of algebraic preciion with multiple node, ha extenively been tudied in the lat decade from both an algebraic and a numerical point of view. Numerically table method for contructing node τ ν and coefficient λ i,ν can be found in Milovanović et al. (24 and Shi & Xu (27. Some intereting theoretical reult concerning thi theory have recently been obtained (ee Shi, 25; Kroó & Pehertorfer, 27; and reference therein. For more detail on pale@kg.ac.yu c The author 28. Publihed by Oxford Univerity Pre on behalf of the Intitute of Mathematic and it Application. All right reerved.

2 ERROR ESTIMATES FOR GAUSS TURÁN QUADRATURES AND THEIR KRONROD EXTENSIONS 487 quadrature with multiple node and correponding orthogonal polynomial, ee the book of Ghizzetti & Oicini (97 and Engel (98 and the urvey paper of Milovanović (2. We conider the error term R n, ( f of a Gau Turán Kronrod quadrature formula, firt introduced in Li (994, w(t f (tdt = n 2 σ i,ν f (i (τ ν + ν= i= n+ μ= K μ f ( ˆτ μ + R n, ( f, which i exact for all algebraic polynomial of degree le than or equal to 2( + n + n +. The node τ ν are the zero of the correponding -orthogonal polynomial π n (t π n, (t of degree n atifying the condition w(tt m [π n (t] 2+ dt =, m =,,..., n. (. The node ˆτ μ are the zero of the generalized Stieltje polynomial ˆπ n+ atifying the condition w(tt m [π n (t] 2+ ˆπ n+ (tdt =, m =,,..., n. Let Γ be a imple cloed curve in the complex plane urrounding the interval [, ] and D it interior. If the integrand f i an analytic function in D and continuou on D, then the remainder term admit the well-known contour integral repreentation The kernel i given by where K n, (z; w = R n, ( f = 2πi ρ n, (z; w = Γ K n, (z; w f (zdz. (.2 ρ n, (z; w [π n, (z; w] 2+, z / [, ], (.3 ˆπ n+ (z; w [π n, (t] 2+ ˆπ n+ (t w(tdt. (.4 z t In view of K n, (ˉz = K n, (z, the modulu of K n, i ymmetric with repect to the real axi: K n, (ˉz = K n, (z. The integral repreentation (.2 lead to a general error etimate, by uing Hölder inequality, i.e. R n, ( f = 2π Γ K n, (z; w f (zdz ( 2π where r +, /r + /r = and f r := Γ /r ( K n, (z; w r dz Γ f (z r dz /r R n, ( f 2π K n, r f r, (.5 ( Γ f (z r dz /r, max f (z, z Γ r < +, r = +., Downloaded from at Univerity Library Svetozar Markovic on November 9, 24

3 488 G. V. MILOVANOVIĆ ET AL. The cae r = + (r = give R n, ( f ( max 2π K n,(z; w f. (.6 z Γ On the other hand, for r = (r = +, the etimate (.5 i reduced to R n, ( f ( K n, (z dz f 2π. (.7 Γ L -error bound of the type (.7 were conidered in Hunter (995 for Gau quadrature formulae and in Milovanović & Spalević (25 for Gau Turán formulae. L -error bound of the type (.7 for Gau Turán Kronrod quadrature formulae with generalized Chebyhev weight function were conidered in detail in Milovanović & Spalević (26. In thi paper, we conider L -error bound of the type (.6 for thoe quadrature. Alo, uing the modulu of the difference of Gau Turán quadrature and their Kronrod extenion, we derive new error etimate for Gau Turán quadrature and compare them with the effective L -error bound for Gau Turán quadrature derived in Milovanović & Spalević ( L -error bound for Gau Turán Kronrod quadrature A a contour Γ, we take an ellipe E ρ with foci at the point ± and a um of emi-axe ρ >, E ρ = { z C: z = } 2 (u + u, θ 2π, u = ρ e iθ. In thi ection, we tudy the magnitude of K n, (z; w on the contour E ρ for generalized Chebyhev weight function of the firt, econd and third kind, repectively. The conideration may be retricted to the upper half of E ρ, i.e. to the interval θ [, π]. The cae of the generalized Chebyhev weight function of the fourth kind i analogou to the one for the generalized Chebyhev weight function of the third kind, and it will be therefore omitted. We invetigate on the elliptic contour the location where the modulu of the kernel attain it maximum value. Thi lead to effective L -error bound for Gau Turán Kronrod quadrature. Similar reult are hard to obtain for any other weight function a no imple explicit formulae exit for both the correponding -orthogonal polynomial and the aociated Stieltje polynomial. Claical Gau Kronrod quadrature formulae received particular attention in the pat 4 year, epecially becaue of their practical ue in package for automatic integration. Concerning thi theory, ee, for intance, the recent paper by De La Calle Yern & Pehertorfer (27 and Spalević (27. Of particular importance are thoe Gau Kronrod quadrature formulae all of whoe node are real and whoe coefficient are poitive. Efficient method for their computing have been propoed recently by Laurie (997 and Calvetti et al. (2. The correponding MATLAB routine are downloadable from the webite which contain a uite of many other ueful routine, in part aembled a a companion piece to the book of Gautchi (24. However, there are ome cae when the claical real and poitive Gau Kronrod quadrature formulae do not exit. Thi i proved in Pehertorfer & Petra (2 for the Gau Kronrod quadrature formula with repect to the Gegenbauer weight function ( t 2 λ /2 for λ > 3, i.e. w 2 for > 2, and ufficiently large n. Numerical tet (uing routine from Gautchi, 24 ugget the ame for Gau Kronrod quadrature Downloaded from at Univerity Library Svetozar Markovic on November 9, 24

4 ERROR ESTIMATES FOR GAUSS TURÁN QUADRATURES AND THEIR KRONROD EXTENSIONS 489 with repect to w 3 when = 2, n 6, and when = 3, 4,...,, n 2. In all thee cae, Gau Turán Kronrod quadrature can be een a a real alternative. The particular cae of the modulu of the kernel in Gau quadrature formulae wa analyed in detail in Gautchi & Varga (983 and Gautchi et al. (99. Concerning thi approach, ee alo Hunter & Nikolov (2 and Schira (996, 997, for Gau quadrature formulae and Milovanović & Spalević (23 and Milovanović et al. (27 for Gau Turán quadrature formulae. In all thoe paper, in fact, the error etimate of Gautchi type, eaily derivable from (.6, R n, ( f l(γ 2π ( max z Γ K n,(z; w ( max f (z z Γ, (2. where l(γ i the length of the contour Γ, wa analyed. The error etimate (.7 i evidently tronger than (2. becaue of the inequality ( K n, (z dz l(γ max K n,(z; w. z Γ Γ However, error etimate of the type (.6 or (2. remain very effective and imple to ue, epecially in cae when the maximum of the modulu of the kernel i attained on the real or imaginary axi. 2. The weight function w (t = ( t 2 /2 It i well known that π n, (t; w = T n (t, for all N. T n (t i the Chebyhev polynomial of the firt kind of degree n. In Li (994 it i hown that ˆπ n+ (t; w = ( t 2 U n (t, where U n (t i the Chebyhev polynomial of the econd kind of degree n. We ue the following equalitie (ee Oicini & Roati, 975, equation 4.: ( 2 + [T n (t] 2+ = 2 2 T n(2k+ (t, k π co nθ z co θ dθ = 2π u u u n, n N, when j >, π I ( in θ in nθ co(2 j + nθ j = dθ z co θ = [ π co[(2 j + n + (n ]θ π co[(2 j + n (n ]θ dθ + dθ 4 z co θ z co θ π co[(2 j + n + (n + ]θ π ] co[(2 j + n (n + ]θ dθ dθ z co θ z co θ = [ ] 2π 4 u u u (2 j+n+n + u (2 j+n n+ u (2 j+n+n+ u (2 j+n n [ ] π u 2 = 2(u u u (2 j+n+n+ u 2 = π [ ] u (2 j+n n+ 2 u2( j+n u 2 jn, k= Downloaded from at Univerity Library Svetozar Markovic on November 9, 24 and I ( = π 2 u 2n.

5 49 G. V. MILOVANOVIĆ ET AL. Now we get ρ n, (z; w = t 2 U n (t t = 2 z t ( t 2 U n (t[t n (t] 2+ dt z t ( T n(2 j+ (t dt. j By ubtituting t = co θ, we have, in view of T n (co θ = co nθ and U n (co θ = in nθ/ in θ, ρ n, (z; w = π in nθ ( co(2 j + nθ in θ dθ z co θ j where = 2 2 = π 2 2+ ( 2 + I ( j j = π 2 2+ = π 2 2+ u ( 2 + j u 2(+n + 2n B( n, (u, u2( j+n B ( n, (u = j= ( 2 + j u 2 jn [( ( ] j j k( j u 2 jn, ( 2+ ( j 2+ j, j =,,...,, k( j =, j =. u 2( j+n According to (.3 and well-known fact T n (z = (u n + u n /2 and U n (z = (u n u n / (u u, we get where K n, (z; w = 4π B n,(u ( u 2n (u u (u n u n (u n + u n. ( Further, uing the equalitie u n + u n = [2(a 2n + co 2nθ] /2, u n u n = [2(a 2n co 2nθ] /2, a j = a j (ρ = 2 (ρ j + ρ j, j N, (2.3 Downloaded from at Univerity Library Svetozar Markovic on November 9, 24

6 ERROR ESTIMATES FOR GAUSS TURÁN QUADRATURES AND THEIR KRONROD EXTENSIONS 49 we get an explicit repreentation of K n, (z; w in the form π z B n,(ρ ( e iθ K n, (z; w = 2 /2 ρ 2n (a 2 co 2θ /2 (a 2n co 2nθ /2, (2.4 (a 2n + co 2nθ +/2 where (ee Milovanović & Spalević, 25, Lemma 4. B n, ( (ρ eiθ = ρ 2n A j co 2 jnθ with A j = A = x /2 2 x ( j/2 j /2 [k( ν] 2 x ν, x = ρ 4n, ν= k( νk( ν jx ν, j =,...,. (2.5 ν= The graph θ K n, (z; w (z E ρ for certain value of n, and ρ are diplayed in Fig.. THEOREM 2. For every fixed N and ρ >, there exit n = n (ρ, N, uch that ( max K n, (z; w = z E K n, ρ 2 (ρ + 4π B n,(ρ ( ρ ; w = ρ 2n (ρ ρ (ρ n ρ n (ρ n + ρ n, ( for every n n. When =, K n, (z; w attain it maximum modulu on the real axi for every n N. Proof. By (2.4 and (2.5, conideration may be retricted to the firt quarter of E ρ., Downloaded from at Univerity Library Svetozar Markovic on November 9, 24 FIG.. The function θ K,3 (z; w (z E.5 (left and θ K 5,2 (z; w (z E.3 (right.

7 492 G. V. MILOVANOVIĆ ET AL. Becaue of (2.4, it uffice to prove that A j co 2 jnθ A j (a 2 co 2θ(a 2n co 2nθ(a 2n + co 2nθ 2+ (a 2 (a 2n (a 2n + 2+, for ufficiently large n, θ (, π/2] and N, where a j are given by (2.3 and A k by (2.5. When =, the lat inequality obviouly hold for each n N. The lat inequality i reduced to A j 2 A j in 2 jnθ (a 2 (a 2n (a 2n + 2+ A j j= [(a in 2 θ][(a 2n + 2 in 2 nθ][(a 2n E n, (ρ, θ in 2 nθ], where (ee Milovanović et al., 27, Thoerem ( 2 + E ρ, (n, θ = ( 2 j (a 2n + 2+ j in 2 j 2 nθ (. j Further, we have j= (a 2 (a 2n (a 2n + 2+ A j 2(a 2 (a 2n (a 2n + 2+ {(a 2 (a 2n (a 2n (a 2n (a 2 E n, (ρ, θ in 2 nθ A j in 2 jnθ + [2(a 2n in 2 θ + 2(a 2 in 2 nθ + 4 in 2 θ in 2 nθ](a 2n + co 2nθ 2+ } j= A j. If in nθ =, the lat inequality obviouly hold. After dividing thi by 2 in 2 nθ A j and denoting Q = A j, it i reduced to [ ] (a 2n + co 2nθ 2+ in 2 θ in 2 nθ (a 2n + (a in 2 θ (a 2 (a 2n E n, (ρ, θ + A Q (a 2 (a 2n (a 2n Q j=2 in 2 jnθ A j in 2 nθ (a 2 (a 2n (a 2n (2.7 In Milovanović et al. (27, Theorem 2. it i hown that ( 2 + E ρ, (n, θ 4 j (a 2n j ( j (a 2n j+. (2.8 2 j j j= Downloaded from at Univerity Library Svetozar Markovic on November 9, 24

8 ERROR ESTIMATES FOR GAUSS TURÁN QUADRATURES AND THEIR KRONROD EXTENSIONS 493 Uing (2.8 and the well-known fact in nθ/ in θ n, we conclude that the left-hand ide of (2.7 i greater than or equal to (a 2n G ρ, (n, where G ρ, (n = A Q (a 2 (a 2n n 2 (a 2n 2+ ( + (a 2 (a 2n j (a 2n j 2 j + 2 j= ( j (a 2n + 2 j 2 2 j+. Since G ρ, (n (ρ and are fixed i continuou when n and lim n + G ρ, (n = +, it follow that G ρ, (n > for each n > r, where r i the larget zero of G ρ, (n. For n we can take [r] +. The proof of Theorem 2. i not only of theoretical but alo of practical importance. We can ue the function G ρ, (n from the proof to etimate n. Numerical value of [r] + (r i the larget zero of G ρ, for certain value of ρ and are preented in Table. The mallet poible (.p. value of n are alo preented. We can ee that the.p. n i very well etimated by [r] +. A typical graph illutrating the relationhip between n and G ρ, (n i diplayed in Fig. 2 (right. THEOREM 2.2 For every fixed N and n N there exit ρ = ρ (n, >, uch that for every ρ > ρ. max K n, (z; w = z E ρ K n, ( 2 (ρ + ρ ; w, Proof. We can repeat the ame computation which led to (2.7, where we can fix n and let ρ be a variable. Since G n, (ρ i continuou when ρ >, and lim ρ + G n, (ρ = + (n and are fixed, it follow that G n, (ρ > for each ρ > t, where t i the larget zero of G n, (ρ. For ρ, we can take t. We can ue the function G n, (ρ from the proof to etimate ρ. Numerical value of t (t i the larget zero of G n, for ome value of n and are preented in Table 2. The.p. value of ρ are alo preented. We can ee that the.p. ρ i etimated by t very well. A typical graph illutrating the relationhip between ρ and G n, (ρ i diplayed in Fig. 2 (left. TABLE Numerical value of [r] + and the.p. value of n = 3 = 6 ρ [r] +.p. n [r] +.p. n Downloaded from at Univerity Library Svetozar Markovic on November 9, 24

9 494 G. V. MILOVANOVIĆ ET AL. FIG. 2. The function G 6, (ρ (left and G.2, (n (right. TABLE 2 The.p. value of ρ and numerical value of t = 2 = 5 n.p. ρ t.p. ρ t Uing the fact B n,(ρ ( B n,( ( = ( 2+, we can implify (2.6 and obtain K n, ( 2 (ρ + ρ ; w 4π ( 2 + ρ 2n (ρ ρ (ρ n ρ n (ρ n + ρ n. ( From (.6 and (2.9, we conclude that R n, ( f = O(ρ (2n(+2+, when n +. When ρ < ρ (n, or n < n (ρ,, uing the inequalitie Downloaded from at Univerity Library Svetozar Markovic on November 9, 24 (a 2 co 2θ(a 2n + co 2nθ (a 2 (a 2n +, θ π/2, and B ( n,(ρ e iθ B ( n,(ρ, we get the following crude etimate: K n, (z; w K n, ( 2 (ρ + ρ ; w ( ρ n + ρ n 2 ρ n ρ n, z E ρ.

10 ERROR ESTIMATES FOR GAUSS TURÁN QUADRATURES AND THEIR KRONROD EXTENSIONS The weight function w 2 (t = ( t 2 +/2 It i well known that π n, (t; w 2 = U n (t. In Milovanović & Spalević (26 it i hown that ˆπ n+ (t; w 2 = T n+ (t. We ue the following equalitie (ee Oicini & Roati, 975, equation 4.2: ( 2 + ( t 2 [U n (t] 2+ = 2 2 ( k U n(2k++2k (t, k when j >, I (2 j = π and I (2 = π 2 u 2(n+. Now we get π k= in θ in (n + θ dθ = π z co θ u n+, n N, (2. in θ co(n + θ in(2 j + (n + θ dθ z co θ = [ π in θ in[2( j + (n + ]θ π dθ + 2 z co θ = π [ ] 2 u 2( j+(n+ + u 2 j (n+, ρ n, (z; w 2 = = = 2 2 ] in θ in[2 j (n + ]θ dθ z co θ ( t 2 +/2 [U n(t] 2+ T n+ (t dt z t t 2 {( t 2 [U n (t] 2+ } T n+(t dt z t t 2 T n+(t ( 2 + ( j U n(2 j++2 j (t dt. z t j By ubtituting t = co θ, we have, in view of T n (co θ = co nθ and U n (co θ = in nθ/ in θ, ρ n, (z; w 2 = π co(n + θ ( ( j in(2 j + (n + θ in θ dθ z co θ j = ( ( j I (2 j j = π ( ( j ( ( j j u2( j+(n+ j u 2 j (n+ j= = π ( 2 2+ u 2(+(n+ + [( ( ] ( j j j u2( j+(n+ = π 2 2+ u 2(n+ B(2 n, (u, Downloaded from at Univerity Library Svetozar Markovic on November 9, 24

11 496 G. V. MILOVANOVIĆ ET AL. where B (2 n, (u = ( j k( j u 2 j (n+. According to (.3, we get [ K n, (z; w 2 = π 2 2 u u u n+ u (n+ ] 2+ B (2 n,(u u 2(n+ (u n+ + u (n+. (2. In the ame way a in the previou cae, we get the explicit repreentation of K n, (z; w 2, K n, (z; w 2 = [ π 2 2+/2 ρ 2(n+ a 2 co 2θ a 2n+2 co(2n + 2θ ] +/2 B n,(ρ (2 e iθ. (2.2 [a 2n+2 + co(2n + 2θ] /2 In the proof of the next theorem, we ue the following two lemma and the function l B n,,l (u = ( j k( j + lu 2(n+ j, N, l =,,...,. (2.3 LEMMA 2. For N, l =,,...,, n odd and r >, we have that B n,,l (r = B n,,l (ri >. Proof. We prove thi by induction. For =, we have B n,, (u =. To pa from to, we ue the following equalitie: B n,, (u = (2 u 2(n+ B n,, (u + B n,, (u, B n,,l (u = B n,,l (u + 2B n,,l (u + B n,,l+ (u, l =,..., 2, B n,, (u = B n,, 2 (u + 2, B n,, (u =, which follow from the well-known identity ( ( = + 2 k k ( 2 k + ( 2. k 2 LEMMA 2.2 For N, n odd, ρ > and l =,,...,, we have that (2.4 Downloaded from at Univerity Library Svetozar Markovic on November 9, 24 B n,,l (ρ e θi (a 2n+2 co(2n + 2θ /2 B n,,l(ρi (a 2n+2 /2. Proof. Thi lemma can alo be proved by induction. The tatement for = i obviou becaue of B n,, (u =. We demontrate the induction tep for the cae l =. The other cae (l =, 2,..., 2 and l = are imilar.

12 ERROR ESTIMATES FOR GAUSS TURÁN QUADRATURES AND THEIR KRONROD EXTENSIONS 497 Firt, we prove 2(ρ e θi 2n+2 (a 2n+2 co(2n + 2θ /2 2(ρi2n+2. (2.5 (a 2n+2 /2 Uing the notation r = ρ 2n+2 and φ = (2n + 2θ we get 2(ρ e θi 2n+2 (a 2n+2 co(2n + 2θ /2 = 4r 2 4r co φ + /2(r + r co φ = 2r 4r + 2 /2(r + r co φ, from which we ee that the left-hand ide of (2.5 attain it maximum when θ = π/2. Uing (2.4 and (2.5 we get B n,, (ρ e θi (a 2n+2 co(2n + 2θ /2 B n,, (ρ e θi (a 2n+2 co(2n + 2θ ( /2 2(ρ e θi 2n+2 ρ 2n+2 (a 2n+2 co(2n + 2θ /2 + B n,, (ρ e θi (a 2n+2 co(2n + 2θ ( /2 (a 2n+2 co(2n + 2θ /2 B n,, (ρi (a 2n+2 ( /2 2(ρi 2n+2 (ρi 2n+2 (a 2n+2 /2 + B n,,(ρi (a 2n+2 ( /2 (a 2n+2 /2 = B n,,(ρi (a 2n+2 /2. THEOREM 2.3 K n, (z; w 2 attain it maximum on the imaginary axi when n i odd, i.e. ( max K n, (z; w 2 = i z E K n, ρ 2 (ρ π B n,(ρi (ρ (2 + ρ 2+ ρ ; w 2 = 4 ρ 2(n+ (ρ n+ + ρ (n+ (ρ n+ ρ (n+ 2+. (2.6 When n i even, for all N and ρ >, there exit an even n = n (ρ, N, uch that K n, (z; w 2 attain it maximum on the imaginary axi for each n n. The maximum i given by π B n,(ρi (ρ (2 + ρ 2+ max K n, (z; w 2 = z E ρ 4 ρ 2(n+ (ρ n+ ρ (n+ (ρ n+ + ρ (n+. ( Proof. For n odd, note that B (2 n, = B n,,. By (2.2 and Lemma 2.2, it uffice to prove that the function (a 2 co 2θ +/2 H(θ = [a 2n+2 co(2n + 2θ] /2 [a2n+2 2 co2 (2n + 2θ] attain it maximum at θ = π/2, which i obviou. The cae when n i even can be proved in the ame way a Theorem 2.4 from Milovanović et al. (27. All we have to change i the parameter α. The new value of α are α = k(2ν +, α >. k(2ν In the ame way a Theorem 2.2 we can prove the next theorem. Downloaded from at Univerity Library Svetozar Markovic on November 9, 24

13 498 G. V. MILOVANOVIĆ ET AL. THEOREM 2.4 For every even n and every N, there exit ρ = ρ (n, >, uch that K n, (z; w 2 attain it maximum on the imaginary axi for each ρ > ρ. Similarly a in the Section 2., we can implify (2.6 and (2.7. We obtain ( K n, 2 (ρ + ρ ; w 2 π ( [ ] ρ + ρ 4 ρ 2(n+ (ρ n+ + ρ (n+ ρ n+ ρ (n+ (2.8 and conclude that R n, ( f = O(ρ (2n(+2+3, when n +. When the maximum point i not on the imaginary axi (n i even, ρ < ρ (n, or n < n (ρ,, uing the inequalitie (a 2n+2 co(2n + 2θ(a 2n+2 + co(2n + 2θ (a 2n+2 (a 2n+2 +, and B (2 n,(ρ e iθ B (2 n,(ρi, we get the following crude etimate: K n, (z; w 2 K n, ( i 2 (ρ ρ ; w The weight function w 3 (t = ( t /2 ( + t +/2 2 (ρ n+ + ρ (n+ ρ n+ ρ (n+, z E ρ. Let V n (t and W n (t repreent Chebyhev polynomial of degree n of the third and fourth kind, repectively. It i well known that π n, (t; w 3 = V n (t. In Milovanović & Spalević (26 it i hown that ˆπ n+ (t; w 3 = ( tw n (t. We ue the following equality (ee Oicini & Roati, 975, equation 4.6, and (2., when j >, I (3 j = π and I (3 = π 2 Now we get u (2 j+n+ j+n+ ( + t [V n (t] 2+ = 2 ( 2 + j in θ co[(2 j + n + j + /2]θ in(n + /2θ dθ z co θ = [ π in θ in[(2 j + n + j + n + ]θ π dθ 2 z co θ = π [ ] 2 u (2 j+n+ j+n+, u (2 j+n+ j n [ ]. V n(2 j++ j (t, ] in θ in[(2 j + n + j n]θ dθ z co θ ( + t +/2 ( t[v n (t] 2+ W n (t ρ n, (z; w 3 = ( t /2 dt z t = t 2 W n(t ( 2 + z t 2 V n(2 j++ j (t dt. j Downloaded from at Univerity Library Svetozar Markovic on November 9, 24

14 ERROR ESTIMATES FOR GAUSS TURÁN QUADRATURES AND THEIR KRONROD EXTENSIONS 499 By ubtituting t = co θ, in view of we have where we get ρ n, (z; w 3 = 2 π V n (co θ = = 2 = π 2 co(n + /2θ, W n (co θ = co θ/2 in 2 θ in(n + /2θ z co θ in θ/2 ( 2 + I (3 j j ( 2 + j u (2 j+n+ j+n+ = π 2 u (2+n++n+ + = π 2 B(3 u2n+ n, (u, B (3 n, (u = According to (.3 and the well-known fact V n (z = un+ + u n u + K n, (z; w 3 = π ( 2 ( 2 + j j= in(n + /2θ, in θ/2 co[(2 j + n + j + /2]θ dθ co θ/2 ( 2 + j u [( ( ] j j k( j u (2n+ j. u + u n+ + u n, W n (z = un+ u n, u (2 j+n+ j n u (2 j+n+ j+n+ 2+ B (3 n,(u u 2n (u (u n+ u n. (2.9 A in the two previou cae, we get an explicit repreentation of K n, (z; w 3 : 2π (a + co θ + B n,(ρ (3 e iθ K n, (z; w 3 = 2 ρ 2n+ (a 2 co 2θ /2 (a 2n+ co(2n + θ /2 (a 2n+ + co(2n + θ +/2. (2.2 THEOREM 2.5 For every fixed ρ > and N, there exit n = n (ρ, N, uch that ( max K n, (z; w 3 = z E ρ K n, 2 (ρ + ρ ; w 3 = π B (3 ( n,(ρ ρ ρ 2n (ρ (ρ n+ ρ n ρ n+ + ρ n, (2.2 Downloaded from at Univerity Library Svetozar Markovic on November 9, 24

15 5 G. V. MILOVANOVIĆ ET AL. for every n n. When =, K n, (z; w 3 attain it maximum modulu on the poitive real axi for every n N. Proof. Becaue of (2.2, it uffice to prove that (a + co θ + B (3 n,(ρ e iθ (a 2 co 2θ /2 (a 2n+ co(2n + θ /2 (a 2n+ + co (2n + θ /2+ (a + + B (3 n,(ρ (a 2 /2 (a 2n+ /2 (a 2n+ + /2+, for ufficiently large n (n n (n, and θ (, π] (for each n N when =. It i obviou that (a + co θ + (a + +. (2.22 On the bai of the reult from the proof of Theorem 2., we have B (3 n,(ρ e iθ (a 2 co 2θ /2 (a 2n+ co(2n + θ /2 (a 2n+ + co (2n + θ /2+ = = B ( n+/2, (ρ eiθ (a 2 co 2θ /2 [a 2(n+/2 co(2(n + /2θ] /2 [a 2(n+/2 + co (2(n + /2θ] /2+ B ( n+/2, (ρ (a 2 /2 (a 2(n+/2 /2 (a 2(n+/2 + /2+ B (3 n,(ρ (a 2 /2 (a 2n+ /2 (a 2n+ + /2+. The lat inequality hold for ufficiently large n when > and each n ( N when =, and with (2.22 that complete the proof. For n, we can take [(2r /2] +, where r i the larget zero of G ρ, (n. In the ame way a Theorem 2.2, we can prove the following reult. THEOREM 2.6 For every fixed N and n N, there exit ρ = ρ (n, >, uch that K n, (z; w 3 attain it maximum on the poitive real axi for all ρ > ρ. In the ame way a in the Section 2. and 2.2, we can how that R n, ( f = O(ρ (2n(+2+2, when n +. In the cae when the maximum point i not on the poitive real axi (ρ < ρ (n, or n < n (ρ,, uing the inequalitie a + co θ a2 co 2θ a 2n+ + co(2n + θ < a + a2, < θ π, a 2n+ + Downloaded from at Univerity Library Svetozar Markovic on November 9, 24 and B n,(ρ (3 e iθ B n,(ρ, (3 we get the following crude etimate: ( 2 K n, (z; w 3 (ρ n+/2 K n, 2 (ρ + + ρ (n+/2 ρ ; w 3 ρ n+/2 ρ (n+/2, z E ρ.

16 ERROR ESTIMATES FOR GAUSS TURÁN QUADRATURES AND THEIR KRONROD EXTENSIONS 5 3. Error etimate for Gau Turán quadrature A very popular method for obtaining a practical error etimate in numerical integration i to ue two quadrature formulae A and B, where the node ued by formula B form a proper ubet of thoe ued by formula A, and where rule A i alo of higher degree of preciion. Kronrod (964a,b originated thi method, which ha been ued many time to date. For more detail, ee, e.g. Monegato (982, 2, Laurie (996 and Spalević (27. The difference A( f B( f, i.e. R (A ( f R (B ( f where f i the integrand, i uually quite a good etimate of the error for the rule B. Following thi idea, taking Kronrod extenion of Gau Turán quadrature (K a rule A and Gau Turán quadrature (GT a rule B, we derive new error etimate for Gau Turán quadrature which have the ame form (cont f a the effective L -error bound from Milovanović & Spalević (25. The following integral for k N and a > will be ueful in thi ection: π co kθ J k (a = (a + co θ 2+ (a co θ dθ, M k(ρ = π (a ± co θ k dθ. π In Milovanović & Spalević (26, Lemma 3. it i hown that [ J k (a = π x+ x k/2 + x k/2 x 2 (x x 2 ν/2 h( ν + h (2 ν (x 2 x 2 ν (2 ν! ν= ν ( 2 + l ν l ( 2 (x l l 2 p (x + p, l where and where l= p= a = x + 2 ν+k+ (2 + + k!, x >, h( ν = ( x (ν + k +! x (ν+k+/2 h (2 ν = In the ret of the paper, we ue ψ(, x = J (a = ( ν ν= 2 ν { ( ν k+ (2+ k! (ν k+! x (ν k+/2, k ν +,, k > ν +. 2π x+ (x 2 [ x (x ( 2 2 ψ(, x], x ( ν ( ( ( l 2 l ( 2 ν l+ ( l. ν + l x x + l= From Gradhteyn & Ryzhik (2, equation and 3.66., we have that ( k ( ρ ρ ρ + ρ k ( k 2 M k (ρ = P k 2 ρ ρ = (2ρ k ρ 2ν, ν where P k i the Legendre polynomial of degree k. ν= Downloaded from at Univerity Library Svetozar Markovic on November 9, 24

17 52 G. V. MILOVANOVIĆ ET AL. 3. The weight function w (t = ( t 2 /2 Firt, we derive a connection between the kernel K n, (GT (z; w from the Gau Turán quadrature formula and the kernel K n, (K (z; w from the Kronrod extenion of the Gau Turán quadrature formula. The term ρ n, (z; w = ρ (K n, (z; w can be written in another form a [ u n u n ( 2 + ρ n, (z; w = u n Z n, ( (u + π 2 2+ where (ee Milovanović & Spalević, 23 Z ( n, (u = ( 2 + j u 2 jn. ], According to (.3, we get ( π 2+ K n, (K (z; w = K n, (GT 2 (z; w + 2+ ( z 2 [T n (z] 2+ U n (z. (3. THEOREM 3. For the remainder term of the Gau Turán quadrature formula R (GT n, ( f ; w = R n, ( f ; w and the remainder term of the Kronrod extenion of the Gau Turán quadrature formula R (K n, ( f ; w = E n, ( f ; w, we have that R (K n, ( f ; w R (GT n, ( f ; w V ( n,,ρ f, where V ( n,,ρ = π 2 ( 2 + Proof. According to (.2, we obtain Further, we get R (K n, ( f ; w R (GT n, ( f ; w = 2π E ρ K (K ( 2 + J (a 2n, W n,,ρ ( = 2π R (K n, ( f ; w R (GT n, ( f ; w W ( n,,ρ f, [K (K Γ n, (z; w K n, (GT (z; w dz = π ( (ρ n + ρ n (ρ n ρ n 2+. n, (z; w K n, (GT (z; w ] f (zdz (K K n, 2π (z; w K n, (GT (z; w f. E ρ z 2 T n (z 2+ U n (z dz = π ( 2 + 2π 2 (a 2n co 2nθ /2 (a 2n + co 2nθ = π ( 2 + π 2 dθ. (a 2n co θ(a 2n + co θ 2+ +/2 dθ Downloaded from at Univerity Library Svetozar Markovic on November 9, 24

18 ERROR ESTIMATES FOR GAUSS TURÁN QUADRATURES AND THEIR KRONROD EXTENSIONS 53 We can continue applying Hölder inequality with different choice of the parameter r and r, uch that /r + /r =. The cae r = r = 2 give the bound with the factor V n,,ρ, ( wherea the cae r = and r = give the bound with the factor W n,,ρ. ( The bound with V ( n,,ρ i harper than the bound with W ( n,,ρ, wherea the factor W ( n,,ρ i impler and exhibit more clearly it behaviour with repect to ρ and n. A i een from Fig. 3, the factor V ( n,,ρ and W ( n,,ρ (a function of ρ very quickly become the ame a the upper bound on L n, (E ρ in (4.8 from Theorem 4.3 in Milovanović & Spalević (25. Numerical tet alo ugget that V ( n,,ρ i greater than or equal to L n, (E ρ, but le than or equal to the upper bound on L n, (E ρ mentioned above, i.e. V ( n,,ρ can be een a a new upper bound on L n, (E ρ which i harper than the bound from Milovanović & Spalević (25. The ame can be aid in the Section 3.2 and The weight function w 2 (t = ( t 2 +/2 We derive a connection between the kernel K n, (GT (z; w 2 from the Gau Turán quadrature formula and the kernel K n, (K (z; w 2 from the Kronrod extenion of the Gau Turán quadrature formula. The term ρ n, (z; w 2 can be written in another form a ρ n, (z; w 2 = π 2 2+ [ u n+ + u (n+ ( ] 2 + u n+ Z n, (2 (u, where (ee Milovanović & Spalević, 23 ( 2 + Z n, (2 (u = ( j j u 2 j (n+. According to (.3, we get ( π 2+ K n, (K (z; w 2 = K n, (GT 2 (z; w 2 2+ [U n (z] 2+ T n+ (z. (3.2 Downloaded from at Univerity Library Svetozar Markovic on November 9, 24 FIG. 3. The upper bound on L n, (E ρ (dahed line and V n,,ρ ( (olid line, left (W n,,ρ ( right a function of ρ when n = and = 3.

19 54 G. V. MILOVANOVIĆ ET AL. THEOREM 3.2 For the remainder term of the Gau Turán quadrature formula R n, (GT ( f ; w 2 and the remainder term of the Kronrod extenion of the Gau Turán quadrature formula R n, (K ( f ; w 2, we have that R (K n, ( f ; w 2 R (GT n, ( f ; w 2 V (2 n,,ρ f, R (K n, ( f ; w 2 R (GT n, ( f ; w 2 W (2 n,,ρ f, where V (2 n,,ρ = π 2 2+ W (2 n,,ρ = π 2 ( 2 + Proof. According to (.2, we get Further, we get E ρ K (K ( 2 + R (K n, ( f ; w 2 R (GT n, ( f ; w 2 = 2π n, (z; w 2 K n, (GT (z; w 2 dz = π ( = π ( M 2+2 (ρ 2 J (a 2n+2, M + (ρ 2 (ρ n+ + ρ (n+ (ρ n+ ρ (n+ 2+. E ρ U n (z 2+ T n+ (z dz [K (K Γ n, (z; w 2 K n, (GT (z; w 2 ] f (zdz (K K n, 2π (z; w 2 K n, (GT (z; w 2 f. 2π (a 2 co 2θ + (a 2n+2 co(2n + 2θ +/2 (a 2n+2 + co(2n + 2θ π ( ( 2 + 2π 2 2+ (a 2 co 2θ 2+2 dθ = π 2 2 = π 3/2 2 2 ( 2π /2 /2 (a 2n+2 co(2n + 2θ 2+ (a 2n+2 + co(2n + 2θ dθ ( ( 2 + π ( 2 + /2 ( π (a 2 co θ 2+2 dθ M 2+2 (ρ 2 J (a 2n+2. /2 dθ /2 (a 2n+2 co θ 2+ (a 2n+2 + co θ dθ Downloaded from at Univerity Library Svetozar Markovic on November 9, 24 Similarly a in the proof of the previou theorem, applying Hölder inequality with r = and r =, we obtain the bound with the factor W n,,ρ. (2

20 ERROR ESTIMATES FOR GAUSS TURÁN QUADRATURES AND THEIR KRONROD EXTENSIONS The weight function w 3 (t = ( t /2 ( + t +/2 We derive a connection between the kernel K n, (GT (z; w 3 from the Gau Turán quadrature formula and the kernel K n, (K (z; w 3 from the Kronrod extenion of the Gau Turán quadrature formula. The term ρ n, (z; w 3 can be written in another form a [ ρ n, (z; w 3 = π u 2n+ ( ] u 2n+ Z n, (3 (u +, where (ee Milovanović & Spalević, 23 According to (.3, we get Z (3 n, (u = ( 2 + j u j (2n+. ( π 2+ K n, (K (z; w 3 = K n, (GT 2 (z; w 3 + ( z[v n (z] 2+ W n+ (z. (3.3 THEOREM 3.3 For the remainder term of the Gau Turán quadrature formula R n, (GT ( f ; w 3 and the remainder term of the Kronrod extenion of the Gau Turán quadrature formula R n, (K ( f ; w 3, we have that R (K n, ( f ; w 3 R (GT n, ( f ; w 3 V (3 n,,ρ f, where V (3 n,,ρ = π W (3 n,,ρ = 2π ( 2 + Proof. According to (.2, we get 2 R (K n, ( f ; w 3 R (GT n, ( f ; w 3 = 2π R (K n, ( f ; w 3 R (GT n, ( f ; w 3 W (3 n,,ρ f, ( 2 + M2+2(ρ J (a 2n+, M + (ρ (ρ n+/2 + ρ (n+/2 (ρ n+/2 ρ (n+/2 2+. [K (K Γ Further, we get K n, (K (z; w 3 K n, (GT (z; w 3 dz E ρ = π ( E ρ z V n (z 2+ W n (z dz n, (z; w 3 K n, (GT (z; w 3 ] f (zdz (K K n, 2π (z; w 3 K n, (GT (z; w 3 f. Downloaded from at Univerity Library Svetozar Markovic on November 9, 24

21 56 G. V. MILOVANOVIĆ ET AL. = π ( 2 + 2π 2 (a + co θ + (a 2n+ + co(2n + θ +/2 (a 2n+ co(2n + θ π ( ( 2 + 2π 2 (a + co θ 2+2 dθ /2 /2 dθ ( 2π = π ( = π 3/2 2 /2 (a 2n+ + co(2n + θ 2+ (a 2n+ co(2n + θ dθ ( π /2 ( π (a + co θ 2+2 dθ ( 2 + M2+2(ρ J (a 2n+. /2 (a 2n+ + co θ 2+ (a 2n+ co θ dθ Similarly a in the proof of the previou two theorem, applying Hölder inequality with r = and r =, we obtain the bound with the factor W n,,ρ. (2 REMARK 3. Concerning ome intereting hitorical detail on Gau Kronrod quadrature rule, ee Gautchi (25. Acknowledgment We are grateful to the referee for uggetion, which have improved the firt verion of thi paper. Funding Swi National Science Foundation (SCOPES Joint Reearch Project No. IB New Method for Quadrature ; Serbian Minitry of Science (Reearch Project: Approximation of linear operator (No. #445 & Orthogonal ytem and application (No. #444C. REFERENCES CALVETTI, D., GOLUB, G. H., GRAGG, W. B. & REICHEL, L. (2 Computation of Gau-Kronrod rule. Math. Comput., 69, DE LA CALLE YSERN, B. & PEHERSTORFER, F. (27 Ultrapherical Stieltje polynomial and Gau-Kronrod quadrature behave nicely for λ <. SIAM J. Numer. Anal., 45, ENGELS, H. (98 Numerical Quadrature and Cubature. London: Academic Pre. GAUTSCHI, W. (24 Orthogonal polynomial: computation and approximation. Numerical Mathematic and Scientific Computation (G. H. Golub, Ch. Schwab & E. Süli ed. Oxford: Oxford Univerity Pre. GAUTSCHI, W. (25 A hitorical note on Gau-Kronrod quadrature. Numer. Math.,, GAUTSCHI, W., TYCHOPOULOS, E. & VARGA, R. S. (99 A note on the contour integral repreentation of the remainder term for a Gau-Chebyhev quadrature rule. SIAM J. Numer. Anal., 27, GAUTSCHI, W. & VARGA, R. S. (983 Error bound for Gauian quadrature of analytic function. SIAM J. Numer. Anal., 2, GHIZZETTI, A. & OSSICINI, A. (97 Quadrature Formulae. Berlin: Akademie. GRADSHTEYN, I. S. & RYZHIK, I. M. (2 Table of Integral, Serie, and Product. San Diego: Academic Pre. Downloaded from at Univerity Library Svetozar Markovic on November 9, 24

22 ERROR ESTIMATES FOR GAUSS TURÁN QUADRATURES AND THEIR KRONROD EXTENSIONS 57 HUNTER, D. B. (995 Some error expanion for Gauian quadrature. BIT, 35, HUNTER, D. B. & NIKOLOV, G. (2 On the error term of ymmetric Gau-Lobatto quadrature formulae for analytic function. Math. Comput., 69, KRONROD, A. S. (964a Integration with control of accuracy. Sov. Phy. Dokl., 9, 7 9. KRONROD, A. S. (964b Node and Weight for Quadrature Formulae. Sixteen Place Table. Mocow: Nauka. (Tranlation by Conultant Bureau, New York, 965. KROÓ, A. & PEHESTORFER, F. (27 Aymptotic repreentation of L p -minimal polynomial, < p <. Contr. Approx., 25, LAURIE, D. P. (996 Anti-Gauian quadrature formula. Math. Comput., 65, LAURIE, D. P. (997 Calculation of Gau-Kronrod quadrature rule. Math. Comput., 66, LI, S. (994 Kronrod extenion of Turán formula. Stud. Sci. Math. Hung., 29, MILOVANOVIĆ, G. V. (2 Quadrature with multiple node, power orthogonality, and moment-preerving pline approximation. J. Comput. Appl. Math., 27, MILOVANOVIĆ, G. V. & SPALEVIĆ, M. M. (23 Error bound for Gau-Turán quadrature formulae of analytic function. Math. Comput., 72, MILOVANOVIĆ, G. V. & SPALEVIĆ, M. M. (25 An error expanion for Gau Turán quadrature and L -etimate of the remainder term. BIT, 45, MILOVANOVIĆ, G. V. & SPALEVIĆ, M. M. (26 Gau-Turán quadrature of Kronrod type for generalized Chebyhev weight function. Calcolo, 43, MILOVANOVIĆ, G. V., SPALEVIĆ, M. M. & CVETKOVIĆ, A. S. (24 Calculation of Gauian type quadrature with multiple node. Math. Comput. Model., 39, MILOVANOVIĆ, G. V., SPALEVIĆ, M. M. & PRANIĆ, M. S. (27 Maximum of the modulu of kernel in Gau- Turán quadrature. Math. Comput., DOI:.9/S PII: S ( MONEGATO, G. (982 Stieltje polynomial and related quadrature rule. SIAM Rev., 24, MONEGATO, G. (2 An overview of the computational apect of Kronrod quadrature rule. Numer. Algorithm, 26, OSSICINI, A. & ROSATI, F. (975 Funzioni caratteritiche nelle formule di quadratura gauiane con nodi multipli. Boll. Union Mat. Ital.,, PEHERSTORFER, F. & PETRAS, K. (2 Ultrapherical Gau-Kronrod quadrature i not poible for λ > 3. SIAM J. Numer. Anal., 37, SCHIRA, T. (996 The remainder term for analytic function of Gau-Lobatto quadrature. J. Comput. Appl. Math., 76, SCHIRA, T. (997 The remainder term for analytic function of ymmetric Gauian quadrature. Math. Comput., 66, SHI, Y. G. (25 Chritoffel type function for m-orthogonal polynomial. J. Approx. Theory, 37, SHI, Y. G. & XU, G. (27 Contruction of σ -orthogonal polynomial and Gauian quadrature formula. Adv. Comput. Math., 27, SPALEVIĆ, M. M. (27 On generalized averaged Gauian formula. Math. Comput., 76, Downloaded from at Univerity Library Svetozar Markovic on November 9, 24