Numerische Mathematik

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1 Numer. Math. (28) 9:43 65 DOI.7/s Numerische Mathematik The superconvergence of Newton Cotes rules for the Hadamard finite-part integral on an interval Jiming Wu Weiwei Sun Received: 2 May 27 / Revised: 5 October 27 / Published online: 2 December 27 Springer-Verlag 27 Abstract We study the general (composite) Newton Cotes rules for the computation of Hadamard finite-part integral with the second-order singularity and focus on their pointwise superconvergence phenomenon, i.e., when the singular point coincides with some a priori known point, the convergence rate is higher than what is globally possible. We show that the superconvergence rate of the (composite) Newton Cotes rules occurs at the zeros of a special function and prove the existence of the superconvergence points. Several numerical examples are provided to validate the theoretical analysis. Mathematics Subject Classification (2) 65D3 65D32 Introduction We consider the Hadamard finite-part integral of the form (see e.g., [3,6,25]) The work of J. Wu was partially supported by the National Natural Science Foundation of China (No. 6725) and a grant from the Research Grants Council of the Hong Kong Special Administrative Region, China (No. CityU 257). The work of W. Sun was supported in part by a grant from the Research Grants Council of the Hong Kong Special Administrative Region, China (No. City U 257) and the National Natural Science Foundation of China (No. 6777). J. Wu Laboratory of Computational Physics, Institute of Applied Physics and Computational Mathematics, P. O. Box 89, Beijing 88, People s Republic of China wu_jiming@iapcm.ac.cn W. Sun (B) Department of Mathematics, City University of Hong Kong, Kowloon, Hong Kong maweiw@math.cityu.edu.hk 23

2 44 J.Wu,W.Sun a f (x) dx : lim (x s) 2 ε s ε a f (x) (x s) 2 dx + s+ε f (x) 2 f (s) dx, s (a, b). (x s) 2 ε (.) f (x) is said to be finite-part integrable with respect to the weight (x s) 2 if the limit on the right hand side of the above equation exists. A sufficient condition for f (x) to be finite-part integrable is that its first derivative f (x) is Hölder continuous. Throughout this paper, denotes an integral in the Hadamard finite-part sense and by contrast, a Cauchy principal value integral or a finite Hilbert transform. The Hadamard finite-part integral is related to the usual Cauchy principal value integral by a f (x) (x s) 2 dx d ds a f (x) x s dx. (.2) In many occasions, this identity has been used as an alternative definition of the Hadamard finite-part integral (cf. [5,9,7,]). Integrals of the form (.) appear frequently in boundary element methods (BEMs) and other numerical computations [2,3,,2]. The efficiency of BEMs often depends upon the efficiency of numerical evaluation of such Hadamard finite-part integrals. Numerous work has been devoted in developing efficient quadrature formulas, such as the Gaussian method [,,5,7,22], the (composite) Newton Cotes method [6,3,9,23,25], the transformation method [5,7] and some other methods [9,2]. The Newton Cotes rule is a commonly used one in many areas due to its ease of implementation and flexibility of mesh. Newton Cotes rules for Riemann integrals have been well studied. The accuracy of the (composite) Newton Cotes rules for Riemann integrals is O(h k+ ) for odd k and O(h k+2 ) for even k. However, the rules are less accurate for Hadamard finitepart integrals due to the hypersingularity of the integrand at the singular point s. The (composite) Newton Cotes rules for Hadamard finite-part integrals were first studied in [3], where the error estimates obtained for the trapezoidal rule and Simpson s rule are much lower than their counterparts for Riemann integrals. Our numerical experiments show that, when the singular point s coincides with some a priori known point, Newton Cotes rules can reach a higher-order convergence rate. We refer to this as the pointwise superconvergence phenomenon of the (composite) Newton Cotes rules for Hadamard finite-part integrals. The superconvergence of (composite) Newton Cotes rules for Hadamard finitepart integrals was first studied in [23,25], where the superconvergence rate of the trapezoidal rule and Simpson s rule was presented, respectively. This paper focuses on the superconvergence of arbitrary degree Newton Cotes rules for Hadamard finite-part integrals. We prove both theoretically and numerically that the (composite) Newton Cotes rules reach the superconvergence rate O(h k+ ) when the local coordinate of the singular point s is the zero of the function 23

3 The superconvergence of Newton Cotes rules 45 S k (τ) : ψ k (τ) + [ψ k (2i + τ)+ ψ k ( 2i + τ)], τ (, ), (.3) i where ψ k is a function of second kind associated with a polynomial of equallydistributed zeros. The rest of this paper is organized as follows. In Sect. 2, after introducing some basic formulas of the general (composite) Newton Cotes rules and some notations, we present our main result of superconvergence. The complete proof is given in Sect. 3. In Sect. 4, we prove the existence of superconvergence points and present some properties of these superconvergence points. In Sect. 5, we present several numerical examples to validate our analysis. Finally, we give some concluding remarks in the last section. 2 The superconvergence of Newton Cotes rules Let a x < x < < x n b be a partition of interval [a, b]. To construct a piecewise Lagrange interpolation polynomial of degree k, we introduce a further partition at each subinterval, and a linear transformation x i x i < x i < < x ik x i+ x ˆx i (τ) : (τ + )(x i+ x i )/2 + x i, τ [, ] from the reference element [, ] to the subinterval [x i, x i+ ].Hereweassume that both meshes are quasi-uniform. We define the piecewise Lagrange polynomial interpolation by F kn (x) k l ki (x) f (x ij ) (x x ij )l ki (x ij), x [x i, x i+ ], (2.) where l ki (x) k (x x ij ). Replacing f (x) in (.)byf kn (x) gives the general (composite) Newton Cotes rule Q kn ( f ) : F n kn(x) (x s) 2 dx a k i ω (k) ij f (x ij ) a f (x) (x s) 2 dx E kn( f ), (2.2) 23

4 46 J.Wu,W.Sun where E kn ( f ) denotes the error functional and ω (k) ij l ki (x ij) x i+ x i (x s) 2 k m,m j (x x im )dx. (2.3) The classical (composite) trapezoidal rule and Simpson s rule, two special cases of the quadrature formula (2.2), were studied by Linz [3] where explicit formulae of the Cotes coefficients ω (k) ij (k, 2) were presented. The error estimate obtained in [3] is where E kn ( f ) Cγ 2 (h, s)h k, k, 2, (2.4) γ(h, s) min i n s x i, h max h x i+ x i. (2.5) i n The above estimate shows that the accuracy depends upon a factor γ 2 (h, s), a quantity that tends to infinity when the singular point s approaches a mesh point. A more precise estimate was given in [9] where E kn ( f ) C min{γ (h, s), ln γ(h, s) + ln h }h k, k, 2. (2.6) Here we present the error estimate for the (composite) Newton Cotes rule with an arbitrary degree in the following theorem. The proof can be obtained along the line in [9,23]. Theorem 2. Assume that f (x) C k+α [a, b], <α, and s x i for any i,,...,n. Then, for the general (composite) Newton Cotes rule Q kn ( f ) defined in (2.2), there exists a positive constant C, independent of h and s, such that where γ(h, s) is defined in (2.5). E kn ( f ) C ln γ(h, s) h k+α, (2.7) Compared with Riemann integrals, the global convergence rate of the (composite) Newton Cotes rule for finite-part integrals is one order lower for odd k and two orders lower for even k. However, numerical results show that the error estimate in (2.7) is optimal. The main issue of this paper is the superconvergence of the general (composite) Newton Cotes rule. For simplicity, hereafter we always assume that both meshes {x i } and {x ij } are uniform. It is not difficult to extend our analysis to certain quasi-uniform meshes. 23

5 The superconvergence of Newton Cotes rules 47 Let φ k (τ) k (τ τ j ) k ( τ 2 j k ) k and denote by ψ k the function of second kind associated with φ k, defined by φ k(τ) dτ, t <, 2 τ t ψ k (t) φ k (τ) dτ, t >. 2 τ t (2.8) (2.9) It is known that if φ k is the Legendre polynomial, ψ k defines the Legendre function of the second kind (see e.g., []). By (.2), we have φ k(τ) dτ, t <, 2 (τ t) 2 ψ k (t) φ k (τ) dτ, t >. 2 (τ t) 2 (2.) The superconvergence results of Newton Cotes rules are given in the following theorem. Theorem 2.2 Assume f (x) C k++α [a, b], <α and τ is a zero of S k (τ) defined by (.3) and (2.). Then, for the general (composite) Newton Cotes rule Q kn ( f ) defined in (2.2), there holds at s ˆx i (τ ) for even k, and for odd k, where E kn ( f ) C[ + η(s)h α ]h k+α, <α (2.) E kn ( f ) { C[ + η(s)h α ]h k+α, <α<, C[η(s) + ln h ]h k+, α (2.2) η(s) max{(b s),(s a) }. (2.3) By comparing Theorems 2. and 2.2, one can see that the superconvergence rate of the (composite) Newton Cotes rules at certain points is one order higher than their global convergence rate. We list the superconvergence points, the zeros of S k (τ), with 6 digits in Table for different k. The proof of Theorem 2.2 will be given in next section. 23

6 48 J.Wu,W.Sun Table Superconvergence points of Newton Cotes rules k superconvergence points (τk ) k ± k 2 k 3 ± , ± k 4, ± k 5 ± , ± , ± The proof of the main result We begin the analysis by investigating the properties of ψ k, defined in (2.9). In the following, C will denote a generic positive constant which is independent of h and s but which may depend on k,α and bounds of the derivatives of f (x). LetP l and Q l denote the Legendre polynomial of degree l and the associated Legendre function of the second kind, respectively. Lemma 3. Let ψ k (t) be defined in (2.9). Then and where and 23 k + ω 2i Q 2i (t), k 2k, ψ k (t) i k ω 2i Q 2i (t), k 2k i k a i Q 2i (t), k 2k, ψ k (t) i k b i Q 2i (t), k 2k, i ω i 2i + 2 a i (4i + ) b i (4i ) (3.) (3.2) φ k (τ)p i (τ)dτ (3.3) i ω 2 j, j i ω 2 j 2. j (3.4)

7 The superconvergence of Newton Cotes rules 49 Proof For k 2k, 2k φ k (τ) ( τ j k ) τ k ( ) k τ 2 j 2 and the polynomial φ k (τ) is an odd function. In terms of Legendre polynomials, φ k (τ) k + i j k 2 ω 2i P 2i (τ), (3.5) where ω 2i is defined in (3.3). The first part of (3.) follows immediately from the definition of ψ k (t). Since k + i ω 2i k + i we can rewrite the first part of (3.) by ψ k (t) k i ω 2i P 2i () φ k (), a i 4i + [Q 2i+(t) Q 2i (t)] with a i (4i + ) i j ω 2 j, which leads to the first part of (3.2) byusingthe recurrence relation (cf. []) Q l+ (τ) Q l (τ) (2l + )Q l(τ), l, 2, 3,... (3.6) The proof for the second parts of (3.) and (3.2) is similar. Lemma 3.2 Let ψ k (t) be defined by (2.9). Then for τ (, ) and m, we have [ ψ k (2i + τ) + ψ k ( 2i + τ) ] im+ C m +[+( )k ]/2 (3.7) and 2m i { C, α<, 2(m i) + τ α ψ k (2(m i) + τ) (3.8) C(ln m) [ ( )k]/2, α. Proof By the classical identity [] Q l (t) ( τ 2 ) l 2 l+ dτ, t >, (t τ) l+ l,, 2,..., (3.9) 23

8 5 J.Wu,W.Sun we get and by (3.2), Q l (t) C, t > ( t ) l+ ψ k (t) C, t 2, (3.) ( t ) 2+[+( )k ]/2 which leads to (3.7) and (3.8). The proof is complete. Lemma 3.3 Assume s (x m, x m+ ) for some m and let c i 2(s x i )/h, i n. Then, we have 2k h k ψ k (c i) 2k h k x i+ x i x i+ Proof By the definition (.), we have x m+ x m x i (x s) 2 (x s) 2 k (x x ij )dx, i m, k (x x ij )dx, i m. k (x s) 2 (x x mj )dx s ε x m+ lim + k ε (x s) 2 (x x mj )dx 2 k (s x mj ) ε x m s+ε ( ) c h k m 2ε h lim 2 ε + φ k (τ) (τ c m ) 2 dτ h ε φ k(c m ) ( h 2 ) k c m + 2ε h φ k (τ) hk dτ (τ c m ) 2 2 k ψ k (c m), (3.) where the change of variable x ˆx m (τ) has been employed. The first identity in (3.) is then verified. The second identity can be obtained by applying the approach to the correspondent Riemann integral. Lemma 3.4 Assume f (x) C k++α [a, b], <α,n 2m + and s ˆx m (τk ) with τk (, ) being a zero of S k(τ). Then, for the general (composite) Newton Cotes rule Q kn ( f ) defined in (2.2), it holds that 23 E kn ( f ) Ch k+α (3.2)

9 The superconvergence of Newton Cotes rules 5 for even k, and E kn ( f ) { Ch k+α, <α<, C ln h h k+, α (3.3) for odd k. Proof Let ˆF k+,n (x) be a piecewise Lagrange interpolation polynomial of degree k+ which interpolates f (x) on the points {x i, x i,...,x ik, x i,k+ } at each subinterval [x i, x i+ ], where x i,k+ is an additional point in (x i, x i+ ), such as x i,k+ (x i + x i )/2. Then the error functional can be split into two parts, E kn ( f ) f (x) ˆF k+,n (x) (x s) 2 dx + ˆF k+,n (x) F kn (x) (x s) 2 dx. (3.4) a By Theorem 2., the first part can be bounded by O(h k+α ) since s ˆx m (τk ) and γ(h, s) ( + τk )/2or( τ k )/2, independent of h. Thus we only need to estimate the second part. Since both ˆF k+,n (x) and F kn (x) are the interpolation to f (x) on {x ij },wehave a ˆF k+,n (x) F kn (x) β ki k (x x ij ), x [x i, x i+ ], (3.5) where β ki is the leading coefficient of ˆF k+,n (x). It follows from Lemma 3.3 that ˆF k+,n (x) F kn (x) (x s) 2 dx hk a 2m 2 k i : I + I 2 + I 3, β ki ψ k (2(m i) + τ k ) (3.6) where I f (k+) (s)h k 2 k (k + )! h k I 2 2 k (k + )! I 3 hk 2 k 2m i 2m i 2m i ψ k (2(m i) + τ k ), [ ] f (k+) ( ˆx i ()) f (k+) (s) ψ k (2(m i) + τ k ), [ ] β ki f (k+) ( ˆx i ()) ψ k (k + )! (2(m i) + τ k ). 23

10 52 J.Wu,W.Sun Now we estimate these three terms one by one. First, by noting that S k (τ k ) and (.3), I f (k+) (s)h k 2 k (k + )! im+ [ψ k (2i + τ k ) + ψ k ( 2i + τ k )], which is bounded by O(h k+ ) for any positive integer k by (3.7) and by noting the fact h O(/m). Secondly, since for f (x) C k++α [a, b], <α, f (k+) ( ˆx i ()) f (k+) (s) C 2(m i) + τ k α h α, by Lemma 3.2, when k is odd, I 2 is bounded by O(h k+α ) for <α<and bounded by O( ln h h k+ ) for α, and when k is even, I 2 is always bounded by O(h k+α ). To estimate I 3, it suffices, by Lemma 3.2, to show that β ki f (k+) ( ˆx i ()) (k + )! Chα, (3.7) where β ki is defined in (3.5). From the standard Lagrange interpolation formula, which implies ˆF k+,n (x) k (x x i,k+ )l ki (x) f (x ij ) (x x ij )(x ij x i,k+ )l ki (x ij) + f ( x i,k+)l ki (x), x [x i, x i+ ], (3.8) l ki ( x i,k+ ) β ki k f (x ij ) (x ij x i,k+ )l ki (x ij) + f ( x i,k+) l ki ( x i,k+ ). (3.9) Taking f (x) ˆF k+,n (x) (x ˆx i ()) l in (3.8) for l k +, we have (x ˆx i ()) l k (x ij ˆx i ()) l (x x i,k+ )l ki (x) (x x ij )(x ij x i,k+ )l ki (x ij) + ( x i,k+ ˆx i ()) l l ki (x). l ki ( x i,k+ ) By comparing the leading coefficients on both sides, we get δ l,k+ k (x ij ˆx i ()) l (x ij x i,k+ )l ki (x ij) + ( x i,k+ ˆx i ()) l, l ki ( x i,k+ ) 23

11 The superconvergence of Newton Cotes rules 53 where δ l,k+ is the Kronecker delta. Moreover, by Taylor s expansion, k f (l) ( ˆx i ()) f (x ij ) l! l k f ( x i,k+ ) l f (l) ( ˆx i ()) l! (x ij ˆx i ()) l + f (k+) (ξ ij ) (x ij ˆx i ()) k+, (k + )! (3.2) ( x i,k+ ˆx i ()) l + f (k+) (ˆξ i ) ( x i,k+ ˆx i ()) k+, (k + )! where ξ ij, ˆξ i (x i, x i+ ). Substituting (3.2) into(3.9), we obtain β ki f (k+) ( ˆx i ()) (k + )! (k + )! k (x ij ˆx i ()) k+ ( f (k+) (ξ ij ) f (k+) ( ˆx i ()) (x ij x i,k+ )l ki (x ij) + ( x i,k+ ˆx i ()) k+ ( f (k+) (ˆξ i ) f (k+) ( ˆx i ())). (3.2) (k + )!l ki ( x i,k+ ) Thus (3.7) follows immediately by noting f (k+) (x) C α [a, b] and the proof is complete. The Proof of Theorem 2.2 We assume s ˆx m (τk ) with its local coordinate τ k satisfying S k (τk ). If m orm n, the estimates in Theorem 2.2 can be directly obtained from Theorem 2. by noting η(s) O(h ). Here we only consider the case m < n/2 since the proof for the case n/2 m < n is similar. From (2.2), E kn ( f ) x 2m+ f (x) F kn(x) (x s) 2 dx + a x 2m+ f (x) F kn (x) (x s) 2 dx. (3.22) The first part can be estimated by Lemma 3.4. By the standard interpolation theory, The second part of (3.22) is bounded by f (x) F kn (x) Ch k+. (3.23) x 2m+ f (x) F kn (x) (x s) 2 dx Ch k+ ( Ch k+ x 2m+ (x s) 2 dx x 2m+ s b s (3.24) ) Cη(s)h k+. We obtain the desired estimates and the proof is complete. 23

12 54 J.Wu,W.Sun 4 The existence of superconvergence points In the above sections we have proved that the general (composite) Newton Cotes rule achieves its superconvergence at zeros of the function S k (τ), which is related to the derivative of the function of second kind associated with φ k (x), a polynomial of equally-distributed zeros. Those superconvergence points listed in Table are obtained by solving the equation S k (τ) and can be used for practical computation. Here we prove the existence of the zeros of S k (τ) for any positive integer k. Let J : (, ) (, ) (, + ) and the operator W : C(J) C(, ) be defined by W f (τ) f (τ) + [ f (2i + τ)+ f ( 2i + τ)], τ (, ). (4.) i Obviously, W is a linear operator. By Lemma 3., ψ k is a linear combination of Q l with l k and therefore belongs to C(J). By(.3), we can write S k (τ) Wψ k (τ). (4.2) Some properties of the operator W are summarized below. Lemma 4. Let the operator W be defined in (4.) and τ (, ). Then (i) W Q (τ) ; (ii) For j > and l, the differential operator D j d j /dτ j and W are communicable with respect to function Q l, i.e., (iii) For j >, D j (W Q l )(τ) W(Q ( j) l )(τ); (4.3) (iv) For j >, Proof Since 23 (2 j) W(P Q )(τ) > ; (4.4) lim W Q τ 2 j(τ) lim W Q τ + 2 j(τ). (4.5) Q (t) 2 ln + t t, t,

13 The superconvergence of Newton Cotes rules 55 we have W Q (τ) 2 ln + τ τ + 2 i 2i + + τ lim ln i 2 2i + τ, ( ln 2i + + τ 2i + τ which proves the part (i). By the classical identity [] we get ) 2i τ + ln 2i + τ Q l (t) ( τ 2 ) l 2 l+ dτ, x >, l,, 2,..., (t τ) l+ Q ( j) C l (t), t >, j ( t ) l++ j and the series in W Q l (τ) and W(Q ( j) l )(τ) are convergent uniformly in any closed subset of (, ), which implies the part (ii) with l. By direct calculation, { W(Q ( j) )(τ) ( ) j+ ( j )! 2 (τ + ) j (τ ) j + (2i + τ ) j + ( 2i + τ + ) j ( ) j+ [ ( j )! lim 2 i (2i + + τ) j [ i (2i + τ + ) ]} j ( 2i + τ ) j ( 2i + τ) j ], which together with the part (i) proves the part (ii) with l. For the part (iii), since [ (2 j) (2 j )! P (t)q (t) 2 (t + ) 2 j + ] (t ) 2 j (2 j ) +( 2 j)q (t), j, 2,..., applying the operator W to both sides of the above identity and using (i) and (ii), we find { (2 j) (2 j )! W(P Q )(τ) 2 (τ + ) 2 j + + (2i + τ ) 2 j + (τ ) 2 j + [ (2i + τ + ) 2 j i ] } >. ( 2i + τ + ) 2 j + ( 2i + τ ) 2 j 23

14 56 J.Wu,W.Sun Finally, we prove the part (iv). Since P l (t) and Q l (t) are the Legendre polynomial and associated function of second kind, we have the identity Q l (t) P l (t)q (t) + f l (t) 2 ln + t t P l(t) + f l (t), t, l, where f l (t) is a polynomial of degree not higher than l. Moreover, { lim W Q [ τ 2 j(τ) lim Q 2 j (τ) + Q2 j (2i + τ)+ Q 2 j ( 2i + τ) ]} τ i [ lim Q2 j (τ) Q 2 j (2 τ) ] τ [ lim τ 2 ln + τ τ P 2 j(τ) + f 2 j (τ) 2 ln 3 τ ] τ P 2 j(2 τ) f 2 j (2 τ) 2 lim τ [ P2 j (2 τ) P 2 j (τ) ] ln( τ) lim τ P 2 j (ξ τ )( τ)ln( τ), where ξ τ (τ, 2 τ). By a similar argument, we reach lim τ + W Q 2 j(τ), which concludes the proof. Lemma 4.2 For j i >, D 2 j (W Q 2i )(τ) > (4.6) and D 2 j+ (W Q 2i )(τ) >. (4.7) Proof Since by Lemma 4. we have P (t) t, Q (t) P (t)q (t), D 2 j (2 j ) (2 j) (W Q ) W(2 jq + P Q ) 2 jd 2 j (W Q ) + W(P Q (2 j) W(P Q )> 23 (2 j) )

15 The superconvergence of Newton Cotes rules 57 and D 2 j+ (2 j+) (2 j+) (2 j+) (W Q 2 ) W(Q 2 Q + Q ) W(3Q 3D 2 j (W Q )>, which verifies (4.6) and (4.7) with i. In the general case, we have (2 j) ) and therefore, i 2i (t) [Q (2 j) Q (2 j+) Q 2i (t) k i (2 j) 2k+ j) j) (t) Q(2 (t)]+q(2 (t) 2k (2 j ) (2 j) (4k + )Q 2k (t) + Q (t), k i [Q k (2 j+) 2k i (4k )Q k i (t) Q (2 j) 2k (2 j+) 2k 2 j+) (t)]+q(2 (t) j+) (t) + Q(2 (t) D 2 j (W Q 2i ) (4k + )D 2 j (W Q 2k ) + D 2 j (W Q ), D 2 j+ (W Q 2i ) k i (4k )D 2 j (W Q 2k ). k By the mathematical induction (4.6) and (4.7) hold for all positive integers j, i with j i. Now we show the existence of superconvergence points in the following theorem. Theorem 4.3 For any positive integer k, the function S k (τ), defined in (.3), has at least one zero in (, ). Proof Clearly we see from the classical orthogonal function theory that and moreover, by Lemma 3. Q l ( t) ( ) l+ Q l (t), t, l,, 2,... ψ k ( t) ( )k+ ψ k (t). 23

16 58 J.Wu,W.Sun It follows from (.3) that S k ( τ) ( ) k+ S k (τ), τ (, ). (4.8) When k is even, τ is a zero of the function S k (τ). Now we need to prove the case that k is odd. Let k 2k and C k (τ) be the function of τ, defined by C k (τ) Wψ k (τ). (4.9) On the one hand, by an argument similar to that of (4.8), we have C k ( τ) ( ) k C k (τ), (4.) which implies that C k (τ) vanishes at τ when k is odd. By the second formula in (3.) and the part (i) of Lemma 4., we obtain k C k (τ) ω 2i W Q 2i (τ) ω 2i W Q 2i (τ), (4.) i k i which together with the part (iv) of Lemma 4. yields lim C k(τ). (4.2) τ By Rolle s theorem, the first derivative of C k (τ) has at least one zero in (, ).Onthe other hand, by the part (ii) of Lemma 4. and by (4.2), C k (τ) S k(τ). (4.3) As a result, S k (τ) has at least one zero in (, ) when k is odd. The proof is complete. Theorem 4.4 Let {a i } and {b i } be defined in (3.4). Ifa i, b i >, then S k (τ) has at most k ( ) k distinct zeros in (, ). Proof For k 2k, by Lemma 3., wehave k S k (τ) Wψ k (τ) a i W Q 2i (τ). (4.4) It follows from Lemma 4., Lemma 4.2 and the assumption a i > that 23 k i D k+ S k (τ) a i D 2k+ (W Q 2i )(τ) >. (4.5) i

17 The superconvergence of Newton Cotes rules 59 Similarly, for k 2k, we have k D k+ S k (τ) b i D 2k (W Q 2i )(τ) >. (4.6) i Hence, D k+ S k (τ) > for any positive integer k, which implies that S k (τ) has at most k + distinct zeros in [, ]. Otherwise,ifS k (τ) has k + 2 or more distinct zeros in [, ], by Rolle s Theorem, D k+ S k (τ) has at least one zero in (, ), which contradicts with D k+ S k (τ) >. In the case of k being even, by (4.4) and the part (iv) of Lemma 4., we see that lim S k(τ) lim S k(τ), τ τ + which shows that S k (τ) has two zeros at τ ±. Thus, in this case, S k (τ) has at most k k ( ) k zeros in (, ). The proof is then complete. In Theorem 4.4, we have presented an upper bound for the number of the zeros of S k (τ) when ψ k (t) is a positive linear combination of Q i(t)( i k). Our numerical test shows that the condition a i, b i > always holds for any positive k, although we cannot provide a theoretical analysis. We list in Table 2 the numbers of zeros of S k (τ), denoted by N k, until k 5. We see that for k 5, the upper bound given by Theorem 4.4 is reached except for the three cases where k, 3, 5. As an example, we present the graph of the function S (τ) in Fig. where S (τ) has been truncated when the absolute value is larger than 5.E 3. One can see from Fig. that S (τ) has only eight distinct zeros in (, ). The graph of the function S 5 (τ) is shown in Fig. 2 where we can see that S 5 (τ) has only four zeros. It has been proved theoretically in [23,25] that the numbers of the superconvergence points in the trapezoidal rule (k ) and Simpson s rule (k 2) reach the upper bounds. Theoretical analysis for the familiar Simpson s 3/8 rule(k 3) is given below. Theorem 4.5 S 3 (τ) has only four distinct zeros in (, ). Proof A straightforward calculation gives ψ 4 (t) 8 5 Q 3(t) Q (t). By Theorem 4.4, S 3 (τ) has at most four distinct zeros in (, ). Note that φ 3 (τ) (τ 2 /9)(τ 2 ) 6 8, τ (, ). 23

18 6 J.Wu,W.Sun Table 2 The number of zeros of S k (τ) k N k k ( ) k Upper bound reached or not 2 2 Y 2 Y Y Y Y Y Y Y 9 Y 9 9 Y 8 2 N 2 Y N Y N Fig. The function S (τ) in (, ) By an argument similar to that for (3.7), we obtain [ ψ 3 (2i + τ) + ψ 3 ( 2i + τ) ] 6 8( τ 2, τ (, ) ) i 23

19 The superconvergence of Newton Cotes rules 6 Fig. 2 The function S 5 (τ) in (, ) and therefore, and S 3 () ψ 3 () + [ψ 3 (2i) + ψ 3 ( 2i)] 6 ψ 3 () > ( ) S 3 ψ 3 2 ( ) + 2 i [ψ 3 (2i + 2 ) + ψ 3 ( 2i + 2 )] ψ 3 i ( ) <. Also note that lim τ S 3 (τ) +. Thus, S 3 (τ) has two distinct zeros in (, ) and by (4.8), S 3 (τ) has another two zeros in (, ), which completes the proof. 5 Numerical examples In this section, we present some numerical examples to confirm our theoretical analysis given in the above sections. Example 5. First we consider the finite-part integral x 6 dx, s (, ). (5.) (x s) 2 By (.), the exact solution is s + 2s2 + 3s 3 + 6s 4 + s + 6s5 ln s. s 23

20 62 J.Wu,W.Sun Table 3 The error of Q 3n ( f ) and Q 4n ( f ) for evaluating (5.)ats x [n/2] + (τ + )h/2 Q 3n ( f ) Q 4n ( f ) n τ τ τ3 τ τ32 τ /3 τ τ4 τ τ E E E E E E E E E E E E E E E E E E E E E E E E E E E-8.43E E-.8458E- h α Table 4 The error of Q 3n ( f ) and Q 4n ( f ) for evaluating (5.)ats x n + (τ + )h/2 Q 3n ( f ) Q 4n ( f ) n τ τ τ3 τ τ32 τ /3 τ τ4 τ τ E E E E E-6.685E E E E E E E E E E E E E E E E E E E E E E E E E- h α We use the quadrature rules Q 3n ( f ) and Q 4n ( f ) defined by (2.2) to compute the approximate value of (5.), respectively. The error E 3n ( f ) at s x [n/2] +(τ +)h/2 with τ,τ3,τ 32 is presented in the left half of Table 3. The error E 4n( f ) at s x [n/2] + (τ + )h/2 with τ /3,τ4,τ 42 is presented in the right half of Table 3. Hereτ is not a superconvergence point for E 3n ( f ) and τ /3 is not a superconvergence point for E 4n ( f ), while τ τ3,τ 32 and τ τ 4,τ 42 are superconvergence points in (, ] for the quadrature rules E 3n ( f ) and E 4n ( f ), respectively, as given in Table. Numerical estimates of the convergence order are given in the last row, which are calculated from the last two meshes. Clearly the convergence orders at superconvergence points are O(h 4 ) and O(h 5 ), respectively, one order higher than those at non-superconvergence points, which confirms our theoretical analysis in Theorem 2.2. Numerical results at s x n + (τ + )h/2 aregivenin Table 4. One can see that at all three points, the convergence order of E 3n ( f ) is about O(h 3 ) and the convergence order of E 4n ( f ) is O(h 4 ), which coincides with our theoretical analysis since η(s) O(h ) in this case. Example 5.2 Secondly we consider an example with less regularity. Let f (x) x 4 + x 4+α,<α, a, b and s. In this case, f (x) C 4+α [, ] and the exact value of the finite-part integral (.) is(2 + 2α)/(9 + 3α). We still 23

21 The superconvergence of Newton Cotes rules 63 Table 5 The error of Q 3n ( f ) for approximating (x 4 + x 4+α )x 2 dx Mesh I Mesh II n α /3 α /2 α /3 α / E E E E E E E E E E E-6.539E E E E E E E E E-8 h α Table 6 The error of Q 3n ( f ) for approximating (x 4 + x 3+α )x 2 dx Mesh I Mesh II n α /3 α /2 α /3 α / E E-2.776E E E E E E E E E E E E E E E E E E-6 h α use quadrature rule Q 3n ( f ). Here two meshes strategies, denotes by Mesh I and Mesh II, respectively, are adopted. In the first, s is always placed at the midpoint of some subinterval, non-superconvergence point, and in the second, s is placed at the superconvergence point τ3 same as used in Example 5.. Both meshes are uniform except two subintervals near the ending points. Numerical results are presented in Table 5. One can see that the convergence orders in Mesh I and II are O(h 3 ) and O(h 3+α ), respectively, which is in good agreement with our theoretical analysis. Example 5.3 Finally, we consider an example in which f (x) x 4 + x 3+α (<α ), a and b. In this case, f (x) C 3+α [, ] and the exact value of the finite-part integral (.) is( + 2α)/(6 + 3α). Here we use the same meshes and singular point s as in Example 5.2. Numerical results are given in Table 6. We find that the superconvergence phenomenon disappears since the convergence rates are about O(h 2+α ) in all four cases, which implies that the assumption on the regularity of f (x) in Theorem 2.2 cannot be weakened. 6 Concluding remarks We have shown both theoretically and numerically the superconvergence of the general (composite) Newton Cotes rules for the evaluation of Hadamard finite-part integrals. The convergence rate at the superconvergence points is one order higher than the global convergence rate. In this paper the (composite) Newton Cotes rules are obtained 23

22 64 J.Wu,W.Sun by replacing the integrand function f (x) with its piecewise Lagrange interpolation. According to (.2), these Newton Cotes rules can also be obtained by differentiating with respect to s the corresponding (composite) Newton Cotes rules for Cauchy principle value integrals. Moreover, it is possible to extend the approach in this paper to the Cauchy principal value integral to obtain certain superconvergence result. The superconvergence phenomenon has been extensively studied for solving partial differential equations and singular integral equations by finite element method and collocation method, see e.g., [4,8,4,8]. The former gives a solution of a higher-order accuracy at certain superconvergence points and the latter produces a solution with a higher-order accuracy when some special points are used as collocation points. A popular approach is the spectral method with Gaussian type collocation points, which has been used for both partial different equations and singular integral equations. The results in this paper show a possible way to improve the accuracy of the collocation method for Hadamard finite-part integral equations by choosing the superconvergence points to be the collocation points. A collocation method based on the Simpson s rule and its superconvergence points was used in [24] to solve an integral equation of Hadamard kernel. Numerical results show that the method is of higher-order accuracy. However, no theoretical analysis has been done. In some practical applications, the integrand function in (.) is given by f (x) w(x)g(x) where w(x) is a weight function which may have certain kind of singularities at the endpoints a and b. In this case, Gaussian quadrature rules may have advantages due to the nature of the weight function of orthogonal polynomials. Acknowledgements The authors would like to thank the referees for their valuable suggestions. References. Andrews, L.C.: Special Functions of Mathematics for Engineers, 2nd edn. McGraw-Hill, Inc., New York (992) 2. Ainsworth, M., Guo, B.: An additive Schwarz preconditioner for p-version boundary element approximation of the hypersingular operator in three dimensions. Numer. Math. 85, (2) 3. Bao, G., Sun, W.: A fast algorithm for the electromagnetic scattering from a large cavity. SIAM J. Sci. Comput. 27, (25) 4. Brunner, H.: Collocation Methods for Volterra Integral and Related Functional Equations. Cambridge University Press, UK (24) 5. Choi, U.J., Kim, S.W., Yun, B.I.: Improvement of the asymptotic behaviour of the Euler Maclaurin formula for Cauchy principal value and Hadamard finite-part integrals. Int. J. Numer. Methods Eng. 6, (24) 6. Du, Q.K.: Evaluations of certain hypersingular integrals on interval. Int. J. Numer. Methods Eng. 5, 95 2 (2) 7. Elliott, D., Venturino, E.: Sigmoidal transformations and the Euler Maclaurin expansion for evaluating certain Hadamard finite-part integrals. Numer. Math. 77, (997) 8. Fairweather G., Ma, H., Sun, W.: Orthogonal spline collocation methods for the Navier Stokes equations in stream function and vorticity formulation. Numer. Methods PDEs (in press) 9. Hasegawa, T.: Uniform approximations to finite Hilbert transform and its derivative. J. Comput. Appl. Math. 63, (24). Hui, C.Y., Shia, D.: Evaluations of hypersingular integrals using Gaussian quadrature. Int. J. Numer. Methods Eng. 44, (999). Ioakimidis, N.I.: On the uniform convergence of Gaussian quadrature rules for Cauchy principal value integrals and their derivatives. Math. Comp. 44, 9 98 (985) 23

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