On the Chebyshev quadrature rule of finite part integrals

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1 Journal of Computational and Applied Mathematics 18 25) On the Chebyshev quadrature rule of finite part integrals Philsu Kim a,,1, Soyoung Ahn b, U. Jin Choi b a Major in Mathematics, Dong-A University, 84, Hadan-2 Dong, Saha-Ku, Pusan , South Korea b Department of Mathematics, KAIST, 373-1, Guseong-dong, Yuseong-gu, Daejon 35-71, South Korea Received 5 October 24; received in revised form 16 October 24 Abstract The stability and the convergence of the Chebyshev quadrature rule of one-sided finite part integrals J. Approx. Theory ) ) is considered with some numerical examples. The three-term recurrence relation is suggested for computing the quadrature weights. 24 Elsevier B.V. All rights reserved. Keywords: Finite part integrals; Chebyshev interpolation 1. Introduction The following finite part integral defined by Qf ) is considered. Qf ) := fτ) wτ) dτ, 1 τ 1.1) wτ) = 1 τ) α 1 + τ) β, β > 1, < α <, α + β = ) and fτ) is assumed to be smooth on [, 1]. The integral is to be interpreted as the Hadamard finite part. For the basic properties of it, we refer to [4] and references therein. Corresponding author. Tel.: ; fax: addresses: kimps@donga.ac.kr P. Kim), ckskan@kaist.ac.kr S. Ahn), ujinchoi@numer.kaist.ac.kr U.J. Choi). 1 This work was supported by Korea Research Foundation Grant KRF-22-3-C /$ - see front matter 24 Elsevier B.V. All rights reserved. doi:1.116/j.cam

2 148 P. Kim et al. / Journal of Computational and Applied Mathematics 18 25) In [2], Criscuolo has developed a Gaussian type quadrature rule for 1.1) based on the Lagrange interpolation polynomial at the zeros of the th orthogonal polynomial with respect to the weight w, obtaining the optimal stability result O 2α ) and rectifying an analogous result proved by Monegato in [9]. Furthermore, in [2] the more general case α < is analyzed. In [4], Elliott has analyzed two algorithms for the approximate evaluation of Qf ) α is negative, not an integer. He showed that the quadrature sums are closely related to finite-part integrals of the Bernstein polynomials. Then, Delbourgo and Elliott have extended the result of [4] to the case of integer exponents see [3]). In [6], Elliott has developed a numerical evaluation scheme for approximating 1.1) with β = using the Chebyshev interpolation polynomial at the classical node points and obtained the stability result O 2α log ), is the degree of the Chebyshev interpolation polynomial. By subtracting out the singularity, 1.1) is equivalently defined by Qf ) = If)+ f1)q, 1.2) If) = fτ) f1) wτ) dτ, q = 1 τ wτ) 1 τ dτ. Using the ewton s binomial theorem, we write 1 + τ) β = 2 β 1 1 τ ) β = 2 β 2 j= a) j denotes the Pochhammer symbol β) j j! ) 1 τ j, 1.3) 2 a) j := aa + 1) a + j 1) = Γa + j), a) Γa) = ) From the definition of Hadamard finite-part integral defined in [4, 1.3)] and 1.4), we have 1 τ) j+α dτ = 2j+α j + α = 2j+α α) j α1 + α) j, j. 1.5) Combining 1.3) and 1.5), q becomes q = 2 α+β 1 α j= β) j α) j j!1 + α) j. Hence, from the definition of hypergeometric function given in [1, ) and )], we see that q = 2 α+β 1 α Fα, β; 1 + α; 1) = 2α+β Γα)Γβ + 1) Γα + β + 1). 1.6) In [7] the author developed the Chebyshev quadrature rule based on the approximation to f by the Chebyshev interpolation polynomial p f at τ j = cos πj/, which are the zeroes of the polynomial

3 P. Kim et al. / Journal of Computational and Applied Mathematics 18 25) T τ) T +1 τ), commonly named the practical abscissae, T k t) is the Chebyshev polynomial of the first kind of degree k. It is defined with the remainder term R f ) as follows: Qf ) = I f ) + f1)q + R f ), 1.7) I f ) = Ip f ) = j= fτ j )w j, w j = 2 k= T k τ j )J k, J k = IT k ). 1.8) ote that the summation with primes denotes a sum with the first and last terms halved. For very restricted cases, β = and < α < 2 1, [7] shows that the above rule has the same stability result as the Gaussian quadrature rule. In this paper, the three-term recurrence relation is suggested as the method for computing the quadrature weights in Section 2. Most of all, we continue the analysis of the stability factor Λ defined by Λ = wj j= and obtain the same stability as that of the Gaussian quadrature rule of [9] without any restriction for β and α through a much simpler idea than that of [7] see Theorem 4). In Section 3, the convergence rate of the remainder R f ) is improved see Theorem 5) and some numerical examples are given in the final section, Section ) 2. The stability analysis 2.1. Computation of the quadrature weights To begin with the analysis of the stability, we first concern a practical method for computing the quadrature weights wj in 1.8). First, we note that J = IT ) =. Using the following expression for T k τ): T k τ) = k j= k) j k) j 1/2) j j!2 j 1 τ)j. We obtain J k in terms of hypergeometric function as follows. α+β Γα)Γβ + 1) J k = IT k ) = 2 Γα + β + 1) g k 1), k =, 1,..., g = 1 and g k k 1) is defined by [ ] k k) j k) j α) j k, k, α; 1 g k = = 1/2) j= j j!α + β + 1) 3 F 2 1 j 2, α + β + 1.

4 15 P. Kim et al. / Journal of Computational and Applied Mathematics 18 25) Then, using the results of [1], we see that g k satisfies the following three-term recurrence relation: β + α + k + 1)g k+1 + 2α β 1)g k + β + α k + 1)g k = 2.1) with the initial values g = 1, g 1 = β α + 1 α + β + 1. Therefore, using the three-term recurrence relation 2.1), we can easily calculate the quadrature weights wj in 1.8). For the numerical stability of the recurrence relation of 2.1), we refer to see [1] Stability analysis Throughout this paper, we assume that is an even positive integer. For estimating Λ, we need the following identity for wj of 1.8) ) wj = )j 1 1 T τ) T +1 τ) 2 τ j 1 T τ) T +1 τ) 1 τ τ j τ wτ) dτ, which is proved in [7, 2.9)]. Hence we can split w j of 1.8) as wj = w j1 + w j2 + w j3, j 1, 2.2) wj1 = )j 1 wτ j )1 + τ j ) wj2 = )j 1 τ j ) U τ) τ dτ, τ j wτ)1 τ 2 ) wτ j )1 τ j )2 ) τ j τ U τ) dτ, w j3 = )j 2 J J +1 1 τ. j Here, U k τ) denotes the second kind Chebyshev polynomial of order k. Before the estimation of Λ,we consider the following lemma, which is a basis technique of the estimation for w j2 and w j3. Lemma 1. Assume that g L 1 [, π]) is increasing or decreasing) on [, θ] and decreasing or increasing) on [ θ, π], θ [i/)π, [i + 1)/]π] for a fixed i =,..., 1. Then we have π ) gθ) sin θ dθ π max gθ) sin θ + max gπ θ) sin θ + mi g, θ [,π/] θ [,π/] {, i =, 1, m i g = 3 max gθ), i = 1,..., 2. θ [i/,i+1)/]

5 P. Kim et al. / Journal of Computational and Applied Mathematics 18 25) Proof. We first denote by A the integral in the left-hand side of the above formula and note the integrand function highly oscillates. For θ [i/)π, [i + 1)/]π], i=,..., 1, let A be split as follows: [i+1)/]π π ) [i+1)/]π A = + gθ) sin θ dθ = A i 1 + Ai 2 Ai 3. i/)π i/)π ote that { A A = 2, i =, A 1, i = 1. First, by replacing the variable θ by θ + π/ in the integral A i 1 and taking the mean value of the newand old integrals, we can find that π/) ) [i+1)/]π 2A i 1 = i + gθ) sin θ dθ + C k, i/)π k= k+1)π)/ C k = gθ) g θ + π )) sin θ dθ. kπ/ Replacing the variable θ by θ + kπ/ in each integral C k and from monotonicity conditions of g,weget i π/ i C k sin θ g θ + kπ ) g θ + k + 1 ) k= π dθ k= π/ [i+1)/]π gθ) sin θ dθ + gθ) sin θ dθ. i/)π Therefore, π/ [i+1)/]π A i 1 gθ) sin θ dθ + gθ) sin θ dθ. i/)π Similarly, by replacing the variable θ by θ π/ in A i 2,wehave π/ [i+1)/]π A i 2 gπ θ) sin θ dθ + gθ) sin θ dθ. i/)π Finally, we complete the proof using the fact sin θ sin θ. For the estimation of wj2,1 j 1, we let g α,β θ,j)= γ α,βθ) γ α,β πj/) cos θ cos πj/, γ α,βθ) = sin 2α+2 θ 2 cos2β+2 θ )

6 152 P. Kim et al. / Journal of Computational and Applied Mathematics 18 25) Then for < α, β, g α,β π θ, j)= g β,α θ, j), g γ α,β θ) + sin θg α,β θ,j) α,β θ,j)= cos θ cos πj/, 2.4) the prime in g α,β θ,j) and γ α,β θ) denotes the derivative of the function about θ. We note that the equation γ α,β θ) + sin θg α,β θ,j)= has always πj/ as a solution in, π). Lemma 2. The equation γ α,β θ)+sin θg α,β θ,j)=has the other solution except πj/, say θ, only in the case β > in, π), and then there exists i=1,..., 2dependent of α, β such that θ [i/,i+1)/] and for enough large we have ) ) i 2α max θ [i/,i+1/)] g α,β θ,j) = g α,β θ,j) =O. 2.5) Proof. Putting fx)= 1 x) α x α β + β + 1)x 2 β + α + β + 2) cos 2 πj 2 + cos 2 πj ) 2α+2 πj πj β + 1) sin cos2β , we obtain the following formula for x = cos 2 θ/2: fx)= f cos 2 θ ) 2 ) x = cos θ cos πj/)γ α,β θ) + sin θg α,β θ,j)). sin θ Then we know that fx)= has a double root at x = cos 2 πj/2. From f x) = 1 x) α x β gx) cos 2 πj ) 2 x, gx) = 1 + α + β)2 + α + β)x β)1 + α + β)x + β1 + β)). It is easy to check that the equation gx) = has only one solution in, 1) only in the case β > and otherwise no solution in, 1). Hence, only in the case β >, fx) has two extreme points in, 1) and the other simple root x = cos 2 θ/2, since f)< and limx 1 fx) =+ for β >. We know γ α,β θ) + sin θg α,β θ,j) =. Therefore, there exists i = 1,..., 2 dependent of α, β such that θ [i/,[i + 1)/]]. In particular, fx) is negative for cos 2 i + 1)/2 <x<cos 2 θ/2 and positive for cos 2 θ/2 <x<cos 2 i/2. Therefore, we know that g α,β θ,j)is a local maximum,

7 P. Kim et al. / Journal of Computational and Applied Mathematics 18 25) since fcos 2 θ/2)g α,β θ,j)>. From g α,β θ,j)= 1 2α θ θ 2 sin cos2β α + β + 2) sin 2 θ ) α + 1), for enough large, we get the desired behavior 2.5). Lemma 3. Assume that < α, β and 1 j 1. Then we have max θ [,π/] sin θ g α,β θ,j) =OM α,β), 2.6) ) j 2α+1 O 1 j ) ) 2β+1, < α 1 2, < β 1 2, ) ) j 2α+1 O, < α 1 M α,β = 2, 2 < β, O 1 j ) ) 2β+1, 1 2 < α, < β 1 2, O 1), 1 2 < α, 2 < β, and max θ [,π/] sin θ g α,β π θ,j) has the same behavior by 2.4). Proof. We first note that for any j = 1, 2,..., 1 and θ [, π/], sin θ θ πj/ 1, j = 1, 2,..., 1. cos θ cos πj/ Define ζ α θ,j)= sin 2α+2 θ/2 sin 2α+2 πj/2 θ πj/, ξ β θ,j)= cos 2β+2 θ/2 cos 2β+2 πj/2 θ πj/. Then the mean value theorem yields ξ β θ,j)= β + 1) cos 2β+1 χ 2 sin χ 1 j/) 2β+1, < β 1 c 1 2, 2 1, 1 2 < β, c 1 = β + 1), χ [θ, πj/] and θ [, π/]. On the other hand, if < α 1 2, sin2α+2 θ/2is increasing and convex. So, /πj sin 2α+2 πj/2 1 2 πj/2)2α+1. Hence, again the mean value theorem gives ζ α θ,j) { 12 πj/2) 2α+1, < α 1 2, α + 1, 1 2 < α.

8 154 P. Kim et al. / Journal of Computational and Applied Mathematics 18 25) Since sin θ g α,β θ,j) sin 2α+2 θ/2 cos 2β+2 θ/2 sin 2α+2 πj/2 cos 2β+2 πj/2 θ πj/ ζ πj α θ,j)cos2β πj ξ β θ,j)sin2α ζ α θ,j)ξ β θ,j) ζ π α θ,j) 2 πj ) 2β+2 πj + ξ β 2 θ,j) 2 ) πj θ ) 2α+2 + ζ α θ,j)ξ β θ,j)πj, summarizing the above inequalities, we obtain 2.6). Based on the above Lemmas 2 and 3, finally we prove the following asymptotic behavior of the stability Λ. Theorem 4. Let Λ be the stability factor defined in 1.9) of the quadrature rule 1.7). Then we have that Λ = O 2α ), < α <. 2.7) Proof. By using the facts [8, 2.2),2.8),2.1)] and the change of variable τ = cos θ, we first rewrite w j1 as follows: for 1 j 1, Since w j1 = )j 2 α+β+1 = 2α+β+2 2α πj πj sin cos2β πj πj sin2α cos2β π /2 k= sin θ cos πj/ cos θ dθ sin2k + 1)πj/). 2k + 1 /2 k= sin2k + 1)πj/) 2k + 1 πj/ = sin θ 2 sin θ dθ has maximum value at j = 1, we obtain the following estimation: j=1 wj1 2α+β+1 π/ = O 1 /2 j=1 sin θ sin θ dθ 2α πj πj sin cos2β j=1 ) 2β+1 ) j 2α + j=/2+1 1 j = O 2α ), < α <, 2.8) we used the Riemann lower sum.

9 P. Kim et al. / Journal of Computational and Applied Mathematics 18 25) For the estimation of wj2, by the change of variable τ = cos θ in w j2, we note that wj2 = 1 2 α+β+2 π 1 cos πj/) g α,β θ,j)sin θ dθ = O 1 j ) 2 π ) g α,β θ,j)sin θ dθ, g α,β θ,j)is defined in 2.3). Then using Lemma 1 and the asymptotic behaviors 2.5) and 2.6), we obtain j=1 w j2 =O 2α ). 2.9) For estimating w j3, we use the identity T τ) T +1 τ) = 2U τ)1 τ 2 ). Then J J +1 = 2 U τ)1 + τ)wτ) dτ = 2 π gθ) sin θ dθ, gθ) = 1 + cos θ) β+1 1 cos θ) α. ote that gθ) is a decreasing function. Thus, the result of Lemma 1 yields J J +1 =O 2α ). From this asymptotic behavior, we obtain wj3 = 1 J J τ = O 2α ). 2.1) j j=1 j=1 For estimating w, we need the following asymptotic behavior for J k: J k =Ok 2α ), k 1, 2.11) which is derived from the asymptotic analysis of [1] and the three-term recurrence relation 2.1). From 1.8) and 2.11), we get ) w 2 J k + J = O 1 k 2α = O 2α ) 2.12) and also, w = 1 k=1 k=1 U τ)wτ) dτ wτ) dτ = O1). 2.13) Hence, by summarizing the asymptotic behaviors 2.8) 2.1), 2.12) and 2.13), we complete the asymptotic behavior 2.7).

10 156 P. Kim et al. / Journal of Computational and Applied Mathematics 18 25) The convergence rate Using this Theorem 4 and recalling [7, Lemma 4.1], we improve the convergence rate of the rule as follows: Theorem 5. Let us consider the quadrature rule 1.7). Suppose the function fτ) possesses continuous derivatives up to order p 1 and the derivative f p) τ) satisfies Hölder continuity of order ρ. Then the remainder term R f ) satisfies R f ) =O p ρ 2α ). Proof. Let p τ) be a polynomial of degree satisfying max τ [,1] fτ) p τ) M p ρ with a constant M see [7, Lemma 4.1]). Putting r τ) = fτ) p τ), [7, Theorem 4 and Lemma 4.1] showthat R f ) Ir ) +C 1 2α p ρ, Ir ) = wτ) r τ) r 1) dτ 3.1) 1 τ for some constant C 1. For the estimation of Ir ), we use the following result of [5]: inf p sup x<y 1 r x) r y) x y λ = O p ρ+λ ) for < λ 1 and λ p + ρ. Thus, if we take λ > α, then Ir ) can be estimated as follows: since inf p Ir ) inf 1 τ) α+λ 1 + τ) β r τ) r 1) p 1 τ λ dτ = O p ρ+λ ), 1 τ) α+λ 1 + τ) β dτ <. Then we choose λ as small as we can, viz. λ = α + ε for any sufficiently small ε >. Therefore, we obtain R f ) =O p ρ 2α + α+ε )) = O p ρ 2α ). Remark6. We remark that in [2] Criscuolo proved that the Gaussian type quadrature converges with order O p ρ ) under the same assumptions of Theorem 5, with ρ = 1 when < α 2 1. So that, we deduce that the method in [2] performs better than the present method. 4. Some numerical examples In order to test estimation 2.7), according to formulas 1.8) and 1.9), we calculate the values of Λ / 2α with α =.3 varying the node number from 8 to 2 by step size 2 and increasing the index β from.6 to.6 by step size.2. In Table 1, we list these values. For each fixed index β, Table 1 shows

11 P. Kim et al. / Journal of Computational and Applied Mathematics 18 25) Table 1 Evaluation of Λ / 2α with α =.3 β = 8 = 1 = 12 = 14 = 16 = 18 = Table R f ) with α =.9 β Present method Elliott [6] the 2nd the 3rd = = = = = = = that the quantities Λ / 2α are almost constant independent of = 2k, k = 4, 5,...,1. On the other hand, it also shows that the parameter index β hardly influences the behavior of Λ / 2α. We nowtest the rule Q f ) = I f ) + f1)q with the improper integral Qf ) = 1 τ) τ) 1.1 dτ = 2Γ2.1)Γ.9). To do this, we set wτ)=1 τ).9 1+τ) β,fτ)=1 τ) τ) 1.1 β with β=.5+.2k, k=, 1, 3, 4. In Table 2, we display 1 2 R f ) :=1 2 Qf ) Q f ), varying both and β. We also list the corresponding errors to the second and third quadrature rules in [6]. For these rules in [6], we take fτ) = 1 τ) τ) 1.1, wτ) = 1 τ).9. From Table 2, it appears that the present method is converging better than the second rule of [6], which is similar to our method and based on the Chebyshev interpolation polynomial with the classical node points cos2j )π/2), j =1,...,. The last column of Table 2 is the result obtained by the third rule of [6], which is based on the Muneer s approximation and the Chebyshev interpolation. The results show

12 158 P. Kim et al. / Journal of Computational and Applied Mathematics 18 25) Table R f ) with α =.9 Present method The 2nd rule of [6] 1 5 R f ) 1 5 R f ) log 1 5 R f ) = = = = = that this method is converging a little faster than the present method. We also see that the convergence of the present method does not depend on the parameter β. Finally, we conclude this section by the test of the rule Q f ) = I f ) + f1)q with the finite part integral Qf ) = 1 + τ) 2.1 Γ.9)Γ3.1) dτ = τ) Γ2.2) We let wτ) = 1 τ).9,fτ) = 1 + τ) 2.1.InTable 3, we display the error 1 5 R f ) :=1 5 Qf ) Q f ) and also list the corresponding errors to the second rule in [6]. Table 3 shows that our method s results are better than that of the method in [6] by log. Aswe remarked, it is caused from the fact that our method makes the growth of the stability factor in [6] to slow by log, and it means our method is more stable. Acknowledgements The authors would like to express their sincere gratitude to the reviewers for many helpful comments and valuable suggestions on the first draft of this paper. References [1] M. Abramowitz, I.A. Stegun, Handbook of Mathematical Functions, Applied Mathematics Series, vol. 55, ational Bureau of Standard, Washington, DC, [2] G. Criscuolo, On the numerical evaluation of certain Hadamard finite-part integrals, Suppl. Rend. Circ. Matem. Palermo Ser. II ) [3] D. Delbourgo, D. Elliott, On the approximate evaluation of Hadamard finite-part integrals, IMA J. umer. Anal ) [4] D. Elliott, An asymptotic analysis of two algorithms for certain Hadamard finite-part integrals, IMA J. umer. Anal ) [5] D. Elliott, On the Hölder semi-norm of the remainder in polynomial approximation, Bull. Austral. Math. Soc ) [6] D. Elliott, Three algorithms for Hadamard finite-part integrals and fractional derivatives, J. Comp. Appl. Math )

13 P. Kim et al. / Journal of Computational and Applied Mathematics 18 25) [7] P. Kim, A Chebyshev quadrature rule for one sided finite part integrals, J. Approx. Theory ) [8] P. Kim, U.J. Choi, A quadrature rule of interpolatory type for Cauchy integrals, J. Comp. Appl. Math ) [9] G. Monegato, The numerical evaluation of a 2-D Cauchy principal value integral arising in boundary integral equation methods, Math. Comp ) [1] R. Piessens, M. Branders, The evaluation and application of some modified moments, BIT )