On the convergence of interpolatory-type quadrature rules for evaluating Cauchy integrals

Transcription

1 Journal of Computational and Applied Mathematics 49 () On the convergence of interpolatory-type quadrature rules for evaluating Cauchy integrals Philsu Kim a;, Beong In Yun b a Major in Mathematics, Dong-A University, 8 Hadan-Dong, Saha-Gu, Busan 64-74, South Korea b Department of Informatics and Statistics, Kunsan ational University, Kunsan, South Korea Received 4 August ; received in revised form 6 March Abstract The aim of this work is to analyse the stability and the convergence for the quadrature rule of interpolatorytype, based on the trigonometric approximation, for the discretization of the Cauchy principal value integrals f()=( t)d. We prove that the quadrature rule has almost optimal stability property behaving in the form O((log +)=sin x), x = cos t. Using this result, we show that the rule has an exponential convergence rate when the function f is dierentiable enough. When f possesses continuous derivatives up to order p and the derivative f (p) (t) satises Holder continuity of order, we can also prove that the rule has the convergence rate of the form O((A + B log + )= p+p ), where is as small as we like, A and B are constants depending only on x. c Elsevier Science B.V. All rights reserved. MSC: primary 65D3; secondary 65D3; 4A Keywords: Cauchy principal value integral; Quadrature rule; Trigonometric interpolation. Introduction Recently, Kim and Choi [5] have considered interpolatory-type quadrature rules for one-dimensional Cauchy principal value integrals ( f() t ) f() Q(f; t)= d = lim + d; t (.) t t+ t This work was supported by Korea Research Foundation Grant (KRF--5-DP4). Corresponding author. Department of Mathematics, 373- Kusong-Dong, Yousong-Gu, Taejon 35-7, South Korea. addresses: kimps@donga.ac.kr (P. Kim), biyun@ks.kunsan.ac.kr (B.I. Yun) //$ - see front matter c Elsevier Science B.V. All rights reserved. PII: S ()48-8

2 38 P. Kim, B.I. Yun / Journal of Computational and Applied Mathematics 49 () based on the trigonometric interpolations. By making use of the change of variable = cos y and t = cos x, the Cauchy integral Q(f; t) can be written as follows: f(cos y) sin y Q(f(cos ); cos x) = cos y cos x dy h(y) sin y h(x) sin x = dy cos y cos x := H(h; x); say; (.) where h(y) = f(cos y). The second equation of (.) follows from the fact dy =; x (;) (.3) cos y cos x which can be proved by using the denition of the Cauchy principal value integral and the following indenite integral: cos y cos x dy = sin x log sin((x + y)=) sin((x y)=) : The rule of [5] for H(h; x) is based on the interpolation polynomial p h (x)= ak cos(kx); k= (.4) where the coecients ak are determined to satisfy the interpolation polynomial conditions h(xk )=p h (xk ); xk = k ; 6 k 6 (.5) and given as follows: a k = h(xj ) cos(kxj ); 6 k 6 : (.6) j= A summation symbol with double primes denotes a sum with rst and last terms halved. Combining (.4) and (.6), we rewrite the interpolation polynomial p h (x) as follows: p h (x)= h(xk )lk (x); k= (.7) where l k (x)= cos(jxk ) cos(jx): (.8) j=

3 P. Kim, B.I. Yun / Journal of Computational and Applied Mathematics 49 () Using (.) and (.7), the rule of [5] for H(h; x) is dened by approximating h(y) and h(x) by p h (y) and ph (x), respectively. The formula is given by where H(h; x) =H (h; x)+r (h; x) H (h; x)=h(p h ; x)=! k (x)= = h(xk )! k (x); (.9) k= lk (y) sin y l k (x) sin x dy cos y cos x cos(jxk )(I j+ (x) I j (x)); (.) j= where I j (x) are dened by sin(jy) sin(jx) I j (x)= dy; cos y cos x x (;) (.) and satisfy the following recurrence relations [5]: I j+ (x)=cosxi j+ (x) I j (x)+ ( )j + ; j + j (.) with initial values I (x)= I (x); I (x)=log tan and I (x) = (.3) or equivalently, I j+ (x)= sin((j +)x) sin x I (x)+ j k= ( ) k + k + sin((j k +)x) ; j (.4) sin x which can be derived by using relation (.) and mathematical induction (see Lemma.). We note that rule (.9) is exact when the function h is a trigonometric polynomial of degree 6. That is, rule (.9) is of interpolatory-type. Dene (x) by (x)=! k (x) : j= Then (x) is very important numerically. Indeed, if h and h are two functions such that sup h (x) h (x) 6 ; x [;] (.5)

4 384 P. Kim, B.I. Yun / Journal of Computational and Applied Mathematics 49 () then from (.9), we see that H (h h ; x) 6 (x): (.6) This inequality shows that (x) gives a stability of the quadrature rule (.9). That is, when increases, the magnitude of the right-hand side of (.6) depends only on that of (x). Hence, we shall call (x) the stability factor of the quadrature rule (.9). Mathematically, the stability analysis for a quadrature rule enables us to analyse the convergence for the rule with a dierentiable function. Indeed, if g is any function for which H(g; x) and H (g; x) exist, then from (.9), we have R (h; x)=h(h; x) H (h; x) = H(h g; x)+h(g; x) H (g; x)+h (g h; x): For a given positive integer, if we choose g to be any trigonometric polynomial of degree 6, then since p g (x)=g(x) and H(g; x) H (g; x)=h(g p g ; x) =, we have that R (h; x)=h(r ; x) H (r ; x); (.7) where r (x)=h(x) p g (x). Two equations (.6) and (.7) show that the growth of the remainder R (h; x) depends on behaviours of r (x) and (x). Our purpose in this paper is to analyse the stability and convergence for rule (.9). In Section, we will show that the stability factor (X ) has the following behaviour (see Theorem.5): ( ) log + (x) 6 C sin ; (.8) x where C is a constant independent of and x. Monegato [7] considered interpolatory-type quadrature rules for (.) based on a Lagrange interpolation polynomial at given node points. When the chosen node points are the zeros of th-degree Legendre polynomial, Elliott and Paget [] showed that the stability factor has the behaviour of the form O(K + L log ) for some constants K and L depending on the pole value t. Using this result and the knowledge of the Lebesgue constant of the form (see [6]) max 6t6 l k (t) = O(log ); (.9) k= where l k (t) is the Lagrange interpolation polynomial associated with the set of nodes {cos(k= )}, Monegato [8] showed that the stability factor has the behaviour of the form O((log ) ) provided that the node points are chosen as cos(k= ). Hence, we can see that the present quadrature rule has better stability properties than that of [8]. From the result (.8) and some preliminary results for r (x) =h(x) p h (x) given in (3.4) (3.6), we prove that the error R (h; x) has a uniform bound ( ) log R (h; x) =O (.)

5 P. Kim, B.I. Yun / Journal of Computational and Applied Mathematics 49 () provided that f(t) is( + )-times continuously dierentiable. This result enables us to use the polynomial p h common to the set of approximations {H (h; x)} for a set of values of x in (;) and to construct an automatic algorithm by using the same procedure as in [4]. A similar exponential convergence for rule (.9) is proved in [5] for an analytic function f. It is known that the integral Q(f; t) of (.) exists for Holder continuous functions of any order greater than zero (see [, Lemma ]). By the aid of the bound (.8) for (x), we will prove that the error R (h; x) has the following convergence rate (see Theorem 3.3): ( ) A + B log + R (h; x) =O ; (.) p+ where is as small as we like, A and B are constants depending only on x, when the function f() possesses continuous derivatives up to order p and the derivative f (p) () satises Holder continuity of order. For simplicity of our analysis, throughout this paper, we assume that the number is of the form = m ; m=; 3; 4 ::: :. Estimation of the stability factor (x) In this section we shall examine the behaviour of the stability factor (x) of (.9). To do this, we state some formulae for trigonometric polynomials of the form: for not multiple of, n sin(n) cos(=) cos(j)= (.) sin(=) j= and n sin(j)= j= cos(=) cos((n +=)) sin(=) (.) which are used several times in this paper. According to the recurrence relation (.), we have the following expression for I k (x). Lemma.. The functions I k (x) dened in (.) have the explicit forms I k (x)= sin(kx) sin x I k (x)+ j= ( ) j + j + sin((k j )x) ; k ; (.3) sin x where I (x) is given in (.3). Proof. We rst note Eq. (.3) is correct for k =. This is easily veried by using I (x) = and I (x)=cosxi (x)+4; which follows from the recurrence relation (.). ext; we use induction

6 386 P. Kim, B.I. Yun / Journal of Computational and Applied Mathematics 49 () with respect to k. For the induction step of k tok; we use the recurrence relations (.) in I k (x)=cosxi k (x) I k (x)+ ( )k + k = sin(kx) sin x I k 3 (x)+4cosx j= ( ) j + j + sin((k j )x) ; sin x k 4 j= ( ) j + j + sin((k j 3)x) : sin x This proves the formula (.3). In the following lemma, we consider the alternative expression for l k (x). Lemma.. The functions lk (x) dened in (.8) have the following alternative expressions: l k (x)= ( )k for 6 k 6. Furthermore; we have sin(x) sin x cos x k cos x ; x x k (.4) l k (x k )= { ; k ; ; k =;:::;: (.5) Proof. Using the formula (.) and the denition of lk (x); we can easily check Eq. (.5). Since cos x cos y = cos(x + y) + cos(x y); the equation of lk (x) given in (.8) can be written as l k (x)= (cos(j(x + xk )) + cos(j(x xk ))): j= Hence; identity (.) gives lk (x)= cos x+x k sin x x k sin( (x + x k )) + sin x+x k cos x x k sin( (x x k )) (cos xk cos x) (.6) Since x k = k=; sin((x ± x k ))=( ) k sin(x): Thus; the desired Eq. (.4) follows from Eq. (.6). Using the above two Lemmas. and., and formula (.), we have the following expression for! k (x).

7 P. Kim, B.I. Yun / Journal of Computational and Applied Mathematics 49 () Lemma.3. The quadrature weights! k (x) dened in (.) have the explicit expressions! k (x)= ( ) k sin xi (x) sin xk I (xk ) cos xk ; x xk ; cos x ( ) k (( +)I +(x) ( )I (x)); x= x k ; (.7) where I (x) is given in (.3). Proof. Substituting (.4) into (.) yields! k (x)= ( )k sin(y) sin y (cos y cos x) (cos xk dy; (.8) cos y) where we used the property (.3). To show the rst equation of (.7); we rst assume that x x k. Then; since (cos y cos x) (cos xk cos y) ( ) = cos xk cos x cos y cos x cos y cos xk Eq. (.8) for! k (x) can be written as! k (x)= ( )k where sin(y) sin y A(x)= cos y cos x dy: A(x) A(x k ) cos x k cos x ; (.9) Then from property (.3); we can rewrite A(x) as follows: A(x) = sin x = sin x = sin x sin(y) cos y cos x dy + sin(y) cos y cos x dy sin(y) cos y cos x dy sin y sin x cos y cos x sin(y)dy (cos y + cos x) sin(y)dy = sin xi (x) : (.)

8 388 P. Kim, B.I. Yun / Journal of Computational and Applied Mathematics 49 () Substituting (.) into (.9); we can obtain the desired rst expression of (.7). We now consider the case x = xk. Then from (.8) and integration by parts; we nd that! k (x)= ( )k+ sin(y) sin y d ( ) dy dy cos y cos x =( ) k cos(y) sin y ( )k dy + cos y cos x sin(y) cos y cos y cos x dy: Since cos(y) sin y = (sin(( +)y) sin(( )y))= and sin(y) cos y = (sin(( +)y) + sin(( )y))=; by using the denition of I k (x); the above expression for! k (x) becomes ( +! k (x)=( ) k I +(x) ) I (x) : Hence; we complete the proof. The previous two Lemmas. and.3 allow us to obtain the following bound for! k (x). Lemma.4. Let! k (x) be the weights expressed by (.7) and let x k in (.5). Then we have that 9:5! k cos x cos xk (x) 6 ; x x k ; 4 sin x ; x= x k for k. be the node points dened (.) Proof. To showthe rst inequality of (.); if we dene a function t k (x) by k t k (x)= sin(( j)x) sin x; k j= and t (x)=; then from (.3); we have = sin xi (x) = sin(x) sin xi (x)+4 j + (t j(x) t j (x)) Since t k (x)= j= = 8t j (x) = sin(x) sin xi (x)+ (j +)(j +3) + t = (x) : (.) sin(( )x) j= k (sin((j +)x) sin((j )x)) j= + cos(( )x) k (cos((j +)x) cos((j )x)); j=

9 we can see that P. Kim, B.I. Yun / Journal of Computational and Applied Mathematics 49 () sin(( )x) t k (x)= (sin((k +)x) + sin x)+ = sin(( k )x) sin((k +)x): Thus; sin xi (x) 6 4+ sin x log tan x : If we let t = sin x; then sin x log tan x = tlog(( + t )=t) cos(( )x) (cos((k +)x) cos x) 6 t log t Hence; we see that 6 4 e : sin xi (x) 6 4(+=e) 5:5: (.3) On the other hand; from (.); we can easily see that sin x k I (x k ) 6 4: (.4) owcombining the rst equation of (.7); (.3) and (.4); we can get the desired rst inequality of (.). The second inequality of (.) easily follows from the second equation of (.7) and the inequality (.3). Lemmas. and.3 enable us to nd the following main theorem for the stability factor (x). Theorem.5. The stability factor (x) dened in (.5) satises ( ) log + (x) 6 C sin ; x (;) (.5) x for some constant C independent of and x. Proof. To show(.5); from Lemma.3; we can assume that x (xl ;x l+ ) for some xed l; l =; without loss of generality. We then split the stability factor (x) into four parts as follows: where (x)=a (x)+a (x)+a 3 (x)+a 4 (x); (.6) A (x)=(! (x) +! (x) )=; A (x)=! l (x) +! l+(x)

10 39 P. Kim, B.I. Yun / Journal of Computational and Applied Mathematics 49 () and A 3 (x)= = k= k l;l+! k (x) ; A 4 (x)= k==+! k (x) : From the rst inequality of (.); we can estimate A (x) as follows: A (x) 6 9:5 ( ) cos x + + cos x = 9:5 sin x : (.7) To estimate A (x); we make use of the generalized law of mean with the rst equation of (.7) and equation (.3). We then have that for some k and k between x and xk ;! k (x)= ( )k sin x sin(x) cos xk cos x I (x) = +4 = ( )k j= j + sin(( j )x) sin x sin(( j )xk ) sin x k cos xk cos x ( cos( k ) sin k + sin( k ) cos k I (x) sin k = +4 (( j ) cos(( j ) k ) sin k (j + ) sin k j= + sin(( j) k ) cos k ) which gives =! k (x) 6 I (x) +8 j I (x) +4+8 j= = 6 I (x) + 4 log(e ) for k = l; l +. Thus; we have that x + dx ; A (x) 6 4 I (x) + 8 log(e ): (.8)

11 P. Kim, B.I. Yun / Journal of Computational and Applied Mathematics 49 () To estimate A 3 (x); we observe that cos x cos xk = sin x + x k sin x x k x x x k x (l k); k =; ;:::;l ; x (k l ); k = l +;:::;=: Thus; inequality (.) gives [ l A 3 (x) 6 9:5 x k= [=] l k + k=l+ ] k l 6 9:5 x k= 6 9:5 x By using the inequality k log(e ): (.9) cos x cos x k x k x (k l ); x l x xl+ xk from inequality (.) we can also estimate A 4 (x) as follows: A 4 (x) 6 9:5 k=[=]+ k l 6 9:5 k= k = 9:5 log(e ): (.) Substituting (.7) (.) into (.6); we can get the asymptotic behaviour (.5). We can claim that the behaviour (.5) for (x) is almost optimal in considering the Lebesgue constant dened in (.5) associated with the Lagrange interpolation polynomials at the nodes k=. In order to test the estimation (.5), according to the formulas (.4) and (.7), we calculate the stability factor (x) varying the node number from to 8 for xed pole values i=, i =; ;:::;5. In Table, we list these values. For each xed pole value x, Table shows that

12 39 P. Kim, B.I. Yun / Journal of Computational and Applied Mathematics 49 () Table Evaluation of (x) according to (.4) and (.3) x Stability (x) 4(x) 8(x) 6(x) 3(x) 64(x) 8(x) = = = = = the stability factor (x) increases by about when the number of nodes varies as = k, k =; ;:::;7. Hence, we see that estimation (.5) substantiates the actual growth presented in Table. 3. Convergence analysis If we let p (x)= (cos x cos xj ); 6 x 6 (3.) j= then for the Lagrange interpolation formula (.4), we have h(x)=p h (x)+r (x); (3.) where p h p (x) (x)= (cos x cos x j= j ) (cos x k= j cos xk ) h(x j ) (3.3) k j and r (x)= p (x)f (+) ((x)) ; (x) : (3.4) ( + )! Here, the mean-value point (x) depends continuously on x and d dx f(+) ((x)) = f(+) ((x)) (3.5) + for some (x) (see [3,, (8)] and the references therein). If we let t = cos x and t j = cos xj, then the polynomial p (x) of (3.) becomes p (x)=p (cos t) = + (t )U (t);

13 P. Kim, B.I. Yun / Journal of Computational and Applied Mathematics 49 () where U (t) is the Chebyshev polynomial of the second kind. That is, sin x sin x p (x)= : (3.6) Using two expressions (3.4) and (3.6), we have the error estimation for the quadrature rule (.9). Theorem 3.. Let f(t) be a ( + )-times continuously dierentiable function and h(x)= f(cos x). Then the quadrature rule (.9) has the following uniform error bound: R (h; x) 6 CM log ; (3.7) where C is independent of h and and { maxt [ ;] f (+) (t) M = max ( + )! ; max t [ ;] f (+) (t) ( + )! Proof. If we let r (x)=h(x) p h (x); then it follows from (.9) and (.7) that R (h; x) = H(r ; x) H (r ; x) 6 H(r ; x) + max r (x) x! k (x) : k= Hence; Eq. (.5); (3.4) and (3.6) give R (h; x) 6 H(r ; x) + max t [ ;] f (+) (t) log : (3.8) ( + )! To bound H(r ; x); we make use of the fact (3.) and r () = r () =. Then integration by parts yields r (y) sin y H(r ; x)= cos y cos x dy = r (y) log cos y cos x dy: Hence; from (3.4); (3.5) and (3.6); we see that H(r ; x) 6 max t [ ;] f ( +) (t) ( + )! } : log cos y cos x dy and the desired result follows at once from this and the inequality (3.8). In the rest of the section, we shall derive a bound for the error R (h; x) =H(h; x) H (h; x) when the function f(cos x) is Holder continuous. To this end, we quote the following known results (see Ref. [9, p. 79] and references therein).

14 394 P. Kim, B.I. Yun / Journal of Computational and Applied Mathematics 49 () Lemma 3.. Suppose the function f(t) possesses continuous derivatives up to order p and the derivative f (p) (t) satises Holder continuity of order. Then there exists a polynomial p (t) of order such that for r (k) (t)=f(k) (t) p (k) (t); r (k) (t) =O( p +k ); k =; ;:::;p (3.9) and r (k) () r(k) (t) =O( p+k + ); t k =; ;:::;p; (3.) where is as small as we like. Using this Lemma 3. and the previous Theorem.5, we have the following convergence theorem for the quadrature rule (.9). Theorem 3.3. Let us consider the quadrature rule (.9). Suppose the function f(t) possesses continuous derivatives up to order p and the derivative f (p) (t) satises Holder continuity of order. Then the remainder term R (h; x)=h(h; x) H (h; x) satises ( ) A + B log + R (h; x) =O ; (3.) p+ where is as small as one like; Aand B are constants depending only on x. Proof. Let p be any trigonometric polynomial of degree 6. Then by (.7); we nd that R (h; x)=h(r ; x) H (r ; x); (3.) where r (x)=h(x) p (x). The quadrature rule (.9) shows that H (r ; x) 6 max f(t) p (cos t) (x): t [ ;] oweqs. (.5) and (3.9) give H (r ; x) = O((K + L log ) p ); (3.3) where K and L are constants depending only on x. For estimating H(r ; x); we make use of the change of variable y = cos ; x = cos t and the fact (.3). If we let r (t)=f(t) p (cos t); then we nd that H(r ; x) 6 r (t) log t +t + r () r (t) d: t t Since d ; t the bound (3.) shows that H(r ; x) = O((K + L ) p ); (3.4) where K and L are constants depending on x and is as small as we like. Finally substituting (3.3) and (3.4) into (3.); we can complete the proof.

15 Acknowledgements P. Kim, B.I. Yun / Journal of Computational and Applied Mathematics 49 () The authors would like to express our sincere gratitude to reviewers for many helpful comments and valuable suggestions on the rst draft of this paper. In particular crucial improvements of Section and several other places are due to anonymous referee s kind encouragements and advices. References [] D. Elliott, D.F. Paget, On the convergence of a quadrature rule for evaluating certain Cauchy principal value integrals, umer. Math. 3 (975) [] D. Elliott, D.F. Paget, On the convergence of a quadrature rule for evaluating certain Cauchy principal value integrals: an addendum, umer. Math. 5 (976) [3] W. Fraser, M.W. Wilson, Remarks on the Clenshaw Curtis quadratures scheme, SIAM Rev. 8 (966) [4] T. Hasegawa, T. Torii, An automatic quadrature for Cauchy principal value integrals, Math. Comput. 56 (99) [5] P. Kim, U.J. Choi, A quadrature rule of interpolatory-type for Cauchy integrals, J. Comput. Appl. Math. 6 () 7. [6] J.H. McCabe, G.M. Phillips, On a certain class of Lebesgue constants, BIT 3 (973) [7] G. Monegato, The numerical evaluation of one-dimensional Cauchy principal value integrals, Computing 9 (98) [8] G. Monegato, Convergence of product formulas for the numerical evaluation of certain two-dimensional Cauchy principal value integrals, umer. Math. 43 (984) [9] G. Monegato, On the weights of certain quadratures for the numerical evaluation of Cauchy principal value integrals and their derivatives, umer. Math. 5 (987) [] R.D. Riess, L.W. Johnson, Error estimates for Clenshaw Curtis quadrature, umer. Math. 8 (97)