INTEGRAL FORMULAS FOR CHEBYSHEV POLYNOMIALS AND THE ERROR TERM OF INTERPOLATORY QUADRATURE FORMULAE FOR ANALYTIC FUNCTIONS

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1 MATHEMATICS OF COMPUTATION Volume 75 Number 255 July 2006 Pages S (06)0859-X Article electronically published on May 2006 INTEGRAL FORMULAS FOR CHEBYSHEV POLYNOMIALS AND THE ERROR TERM OF INTERPOLATORY QUADRATURE FORMULAE FOR ANALYTIC FUNCTIONS SOTIRIOS E NOTARIS Abstract We evaluate explicitly the integrals π n(t)/(r t)dt r with the π n being any one of the four Chebyshev polynomials of degree n These integrals are subsequently used in order to obtain error bounds for interpolatory quadrature formulae with Chebyshev abscissae when the function to be integrated is analytic in a domain containing [ ] in its interior Introduction The usefulness of Chebyshev polynomials in numerical analysis is undisputed This is largely due to their nice properties many of which follow from the trigonometric representations these polynomials satisfy on the interval [ ] An immediate consequence of these representations is that the zeros of Chebyshev polynomials can be given by explicit formulas and this makes them attractive choices for nodes of interpolatory quadrature formulae Notable examples of the latter are the wellknown Fejér and Clenshaw Curtis rules In the following we enlarge the list of properties of Chebyshev polynomials by showing that the integrals () dt r r t with the π n being any one of the four Chebyshev polynomials of degree n canbe computed explicitly This finds immediate application in estimating the error of certain interpolatory formulae with Chebyshev abscissae by means of Hilbert space or contour integration methods when the function to be integrated is analytic in a domain containing [ ] in its interior The quality of these estimates and a comparison with already existing ones is illustrated with a numerical example Received by the editor May and in revised form October Mathematics Subject Classification Primary 33C45 65D32 Key words and phrases Integral formulas Chebyshev polynomials interpolatory quadrature formulae error bounds This work was supported in part by a grant from the Research Committee of the University of Athens Greece and in part by a Pythagoras OP Education grant to the University of Athens from the Ministry of National Education Greece and the European Union 27 c 2006 American Mathematical Society Reverts to public domain 28 years from publication

2 28 S E NOTARIS 2 Integral formulas for Chebyshev polynomials The nth degree Chebyshev polynomials T n U n V n and W n of the first second third and fourth kind respectively are characterized by the well-known representations sin (n +)θ (2) T n (cos θ) =cosnθ U n (cos θ) = sin θ cos (n +/2)θ sin (n +/2)θ (22) V n (cos θ) = W n (cos θ) = cos (θ/2) sin (θ/2) We first give some simple relations among these polynomials which will be useful in the subsequent development Lemma 2 Let U (t) =0Then (23) (24) (25) T n (t) = 2 U n(t) U n2 (t)} n = 2 V n (t) =U n (t) U n (t) n =0 2 W n (t) =U n (t)+u n (t) n =0 2 Proof Formulas (23) and (24) follow from the corresponding formulas for the difference of sines sin (n +)θ sin (n )θ =2cosnθ sin θ sin (n +)θ sin nθ =2cos(n +/2)θ sin (θ/2) and (2) (22) Also replacing θ by π + θ in the second equation in (2) and (22) gives (26) U n (t) =() n U n (t) n =0 2 (27) W n (t) =() n V n (t) n =0 2 which together with (24) implies (25) We can now present the announced explicit forms for the integrals () The pattern was discovered by computing analytically enough of these integrals for the first few values of n We examine separately the cases r > and r < Proposition 22 Let r R with r > (i) If π n = T n then (28) r t dt = π n(r)ln while if π n = U n π n = V n or π n = W n then (29) (ii) If π n = T n then (20) r t dt = π n(r)ln r + t dt = π n(r)ln ( ) [(n+)/2] r + 4 r ( ) [(n+)/2] r + 4 r ( ) [(n+)/2] r + +4 r π n2k+ (r) π n2k+ (r) π n2k+ (r)

3 INTEGRAL FORMULAS FOR CHEBYSHEV POLYNOMIALS 29 while if π n = U n π n = V n or π n = W n then ( r + (2) r + t dt = π n(r)ln r ) [(n+)/2] +4 π n2k+ (r) By [ ] we denote the integer part of a real number while the notation means that the last term in the sum must be halved when n is odd Proof (i)wefirstprove(29)withπ n = U n To this end we apply induction on n Forn = an easy computation using the second equation in (2) gives ( ) U (t) r + r t dt = U (r)ln 4 r Assume now that the formula is true for the indices n andn and we want to prove it for the index n + It is well known that the Chebyshev polynomials of the second kind satisfy the three-term recurrence relation U m+ (t) =2tU m (t) U m (t) m =0 2 (22) U (t) =0 U 0 (t) = Applying (22) with m = n dividing by r t and taking the integral on both sides we have U n+ (t) tu n (t) (23) dt =2 r t r t dt U n (t) dt r t Furthermore adding and subtracting the term ru n (t) in the numerator of the first integral on the right side of (23) we get (24) U n+ (t) dt = 2 r t U n (t)dt +2r U n (t) r t dt U n (t) dt r t We now consider two cases for n Whenn is even (24) by virtue of the induction hypothesis and 2 if m is even U m (t)dt = m + 0 if m is odd yields U n+ (t) dt = 2rU n (r) U n (r)} ln r t which in view of (22) gives (25) ( ) r + r 4 2rU n (r) U n2 (r)} + 3 2rU n3(r) U n4 (r)} U n+ (t) dt = U n+ (r)ln r t + + n 2rU (r) U 0 (r)} + } n + ( ) r + r 4 U n (r)+ 3 U n2(r)+ + n U 2(r)+ n + }

4 220 S E NOTARIS Also for n odd we obtain in a like manner ( ) U n+ (t) r + dt = U n+ (r)ln r t r 4 U n (r)+ 3 U n2(r)+ + n 2 U 3(r)+ } n U (r) which together with (25) combines to U n+ (t) dt = U n+ (r)ln r t ( ) [((n+)+)/2] r + 4 r proving the induction claim and concluding the induction Also from (23) T n (t) r t dt = U n (t) 2 r t dt } U n2 (t) dt r t U n+2k+ (r) and inserting (29) with π n = U n and π n2 = U n2 we find taking into account that U (t) =0 T n (t) r t dt = 2 U n(r) U n2 (r)} ln ( ) r + r 2 U n(r) U n3 (r)} U n3(r) U n5 (r)} n3 n2 2 U 3(r) U (r)} + 2 U 2(r) U 0 (r)} + 2 n n 2 U (r) U (r)} if n is even if n is odd which by (23) shows (28) The proof of (29) with π n = V n or π n = W n is similar to that of (28) if we use (24) and (25) in place of (23) (ii) Replacing θ by π + θ in (2) and (22) gives T n (t) =() n T n (t) n =0 2 V n (t) =() n W n (t) n =0 2 which inserted together with (26) and (27) into (28) and (29) and changing variables from t to t in the integrals involved there yield (20) and (2) We now turn to the integrals () with r < which in view of the singularity at r or r will be computed in the Cauchy principal value sense We recall that if p is an interior point of an interval [a b] inr and f is a function defined at every point of [a b] except perhaps p then the Cauchy principal value of the integral of f over [a b] denoted by b f(t)dt isgivenby a (26) a b f(t)dt = lim ɛ 0 + pɛ a b f(t)dt + f(t)dt p+ɛ } }

5 INTEGRAL FORMULAS FOR CHEBYSHEV POLYNOMIALS 22 (cf [2 Section 32]) An interesting case of a Cauchy principal value integral is the so-called Hilbert transform of a function f b f(t) dt x (a b) a x t (cf [4 Section 6]) The integrals () with r < are precisely of this type Proposition 23 Let r R with r < (i) If π n = T n then (27) r t dt = π n(r)ln while if π n = U n π n = V n or π n = W n then (28) (ii) If π n = T n then (29) r t dt = π n(r)ln r + t dt = π n(r)ln while if π n = U n π n = V n or π n = W n then (220) r + t dt = π n(r)ln ( ) [(n+)/2] +r 4 r ( ) [(n+)/2] +r 4 r ( ) [(n+)/2] +r +4 r ( ) [(n+)/2] +r +4 r π n2k+ (r) π n2k+ (r) π n2k+ (r) π n2k+ (r) By [ ] we denote the integer part of a real number while the notation that the last term in the sum must be halved when n is odd means Proof The proof follows the steps of the proof in Proposition 22 (i) We first prove (28) with π n = U n by applying induction on n For n = utilizing (26) we find ( ) U (t) +r r t dt = U (r)ln 4 r Then assuming that (28) is true for the indices n andn and proceeding precisely as in the case r > we show that it is also true for the index n + concluding that way the induction The proof of (27) and (28) with π n = V n or π n = W n is almost identical to the proof of the corresponding cases in Proposition 22 (ii) Formulas (29) and (220) can be derived most easily if we note that r + t dt = (r) t dt and then apply formulas (27) and (28) respectively with r replaced by r

6 222 S E NOTARIS 3 Error bounds for interpolatory quadrature formulae An interpolatory quadrature formula for approximating the integral f(t)dt based on the n distinct points τ τ 2 τ n ordered decreasingly in the interval ( ) is that constructed by integrating the inteprolating polynomial p n (f; τ τ 2 τ n ; t) which leads to n (3) f(t)dt = w ν f(τ ν )+R n (f) ν= By definition (3) has degree of exactness at least n ; ie R n (f) = 0 for all f P n One way to obtain an estimate for the error term of (3) is by using a Hilbert space method proposed by Hämmerlin in [6] If f is a single-valued holomorphic function in the disc C r = z C : z <r} r> then it can be written as f(z) = a k z k z C r k=0 Define (32) f r =sup a k r k : k N 0 and R n (t k ) 0} which is a seminorm in the space X r = f : f holomorphic in C r and f r < } Then the error term R n in (3) is a continuous linear functional in (X r r ) and itserrornormisgivenby R n (t k ) R n = r k k=d+ where d is precise degree of exactness of (3) (cf [ Section ]) If in addition (33 i ) R n (t k ) 0 k =0 2 or (33 ii ) () k R n (t k ) 0 k =0 2 then letting n = (t τ ν ) ν= we can derive the representations (34 i ) R n = r π n (r) or (34 ii ) R n = r t dt r π n (r) r + t dt respectively (see [ Section 2]) Consequently if f X R (35) R n (f) R n f r <r R

7 INTEGRAL FORMULAS FOR CHEBYSHEV POLYNOMIALS 223 and optimizing the bound on the right of (35) as a function of r we get (36) R n (f) inf <r R ( R n f r ) An alternate way to obtain an estimate for the error term of (3) is by using a contour integration method If f is a single-valued holomorphic function in a domain D containing [ ] in its interior and C r = z C : z = r} r> is a contour in D surrounding [ ] then the R n ( ) can be represented as (37) R n (f) = K n (z)f(z)dz 2πi C r where the kernel K n is given by ( ) (38) K n (z) =R n z with π n (z) = From (37) there immediately follows = π n (z) z t dt n (z τ ν ) ν= (39) R n (f) r max z C r K n (z) max z C r f(z) Now (30) max z C r K n (z) = Kn (r) if R n satisfies (33 i ) K n (r) if R n satisfies (33 ii ) (see [5 Sections 2 3 and 4]) hence in this case (39) gives in view of (38) (30) and (34 i ) (34 ii ) (3) R n (f) R n max z =r f(z) with the R n given by (34 i )or(34 ii ) accordingly as R n satisfies (33 i )or(33 ii ) respectively Incidentally (3) can also be derived from (35) if f r is estimated by max z =r f(z) whichforf X R exists at least for r<r Consequently the bound on the right of (3) can be optimized as a function of r giving ( ) (32) R n (f) inf R n max f(z) <r<r z =r Among all interpolatory formulae of particular interest are those based on the zeros of Chebyshev polynomials of any one of the four kinds primarily because their nodes and weights can be expressed in explicit form Formulae of this kind were introduced by Fejér in 933; however their practical importance was shown by Clenshaw and Curtis in 960 who used them in automatic computing A detailed description of all interpolatory formulae with Chebyshev abscissae is given in [7] Here we consider formulae of type (3) with the τ ν being zeros of the nth degree Chebyshev polynomial of the second third or fourth kind The nodes and weights

8 224 S E NOTARIS of these formulae are given by (33) τ ν (2) =cosθ ν (2) θ ν (2) = ν n + π (34) τ ν (3) =cosθ ν (3) θ ν (3) = 2ν 2n + π (35) τ ν (4) =cosθ ν (4) θ ν (4) = 2ν 2n + π (see [7 Equations (24) (26)]) and w ν (i) = 2 [(n)/2] cos 2kθ ν (i) 2 n + α 4k 2 or alternatively w (i) ν = 4sinθ(i) ν n + α [(n+)/2] ν = 2n ν = 2n ν = 2n } cos 2[(n +)/2]θ(i) ν 2[(n +)/2] ν = 2n i=2 3 4 sin ()θ ν (i) ν = 2n i=2 3 4 where if i =2 α = /2 if i =3 4 and [ ] denotes the integer part of a real number Furthermore the w ν (i) ν = 2n i=2 3 4 are all positive (see [7 Section 2]) Formula (3) with τ ν = τ ν (2) knownasthefejér rule of the second kind or the Filippi rule has precise degree of exactness 2[(n + )/2] (cf [7 Section 22]) and it is positive definite satisfying (36 e ) R (2) n (f) = 2 n (n +) for n even and f C n [ ] and (36 o ) R (2) n (f) = n + 2 n n(n +2) f (n) (ξ e ) ξ e [ ] n! f (n+) (ξ o ) ξ o [ ] (n +)! for n odd and f C n+ [ ] (see [ Section 7]) From (36 e )and(36 o ) one can immediately deduce that (37 e ) R n (2) (t 2k ) 0 for all k [(n +)/2] while by symmetry (37 o ) R n (2) (t 2k+ ) = 0 for all k 0 On the other hand formula (3) with τ ν = τ ν (3) or τ ν = τ ν (4) has precise degree of exactness n and it is nondefinite (cf [7 Sections 22 and 24]) However as the next lemma shows the error term of this formula follows a pattern analogous to that of (37 e ) Lemma 3 Consider the interpolatory quadrature formula (3) (i) If τ ν = τ ν (3) then the error term R n = R n (3) satisfies (38) () k R n (3) (t k ) > 0 for all k K n

9 INTEGRAL FORMULAS FOR CHEBYSHEV POLYNOMIALS 225 (ii) If τ ν = τ ν (4) then the error term R n = R n (4) satisfies (39) R n (4) (t k ) > 0 for all k K n In both cases K n is a constant K n n Proof (i) Assume first that k =2l is even From (3) taking into account that w ν = w ν (3) > 0 ν= 2nweget n } (320) R n (3) (t 2l 2 )= n(2l +) w(3) ν (τ ν (3) ) 2l ν= Since 0 < τ ν (3) < ν = 2nwehave (3) lim (2l +)(τ ν ) 2l =0 ν = 2n l which implies 2 (32) n(2l +) w(3) ν (τ ν (3) ) 2l > 0 for all l Knν e ν = 2n Now taking Kn e =max ν n Knν} e we obtain from (320) in view of (32) (322) R n (3) (t 2l ) > 0 for all l Kn e If on the other hand k =2l + is odd then (3) gives (323) R (3) n (t 2l+ )= n ν= w (3) ν (τ (3) ν ) 2l+ We consider two cases for n Firstn =2m is even Then (323) takes the form m R n (3) (t 2l+ )= w ν (3) (τ ν (3) ) 2l+ + w (3) (3) nν+ (τ nν+ )2l+} (324) with the τ (3) ν (325) satisfying = (326) τ (3) (327) Hence implying lim l (328) w(3) nν+ w (3) ν ν= m ν= w (3) ν (τ (3) ν ) 2l+ w(3) nν+ w ν (3) τ (3) ν > 0 ν = 2m τ (3) nν+ < 0 ν > τ (3) nν+ ν = 2m ν = 2m ( ) (3) 2l+ τ nν+ =0 ν = 2m τ ν (3) ( ) (3) 2l+ τ nν+ > 0 for all l Knν o τ ν (3) ( ) (3) 2l+ τ nν+ τ ν (3) ν = 2m and taking Kn o =max ν m Knν} o we obtain from (324) in view of the positivity of w ν (3) (325) and (328) (329) R n (3) (t 2l+ ) < 0 for all l Kn o

10 226 S E NOTARIS Table 3 Values of K n n 20 n K n n K n If on the other hand n =2m + is odd then (3) gives m (t 2l+ )= R (3) n ν= w ν (3) (τ ν (3) ) 2l+ + w (3) (3) nν+ (τ nν+ )2l+} w (3) (3) m+ (τ m+ )2l+ with the τ ν (3) satisfying (325) (327) and in addition τ (3) m+ > 0 thus in a like manner as in the case of n even we get (329) Now taking K n =max2kn e 2Kn o +} (322) and (329) are merged to (38) (ii) Replacing ν by n ν + in (34) and using (35) gives τ ν (4) = τ (3) nν+ which inserted into (3) yields w (4) ν = w (3) nν+ ν = 2n ν = 2n R (4) n (f( )) = R n (3) (f( )) The latter together with (38) implies (39) The proof of Lemma 3 presents a method for computing the constants K n in (38) (39) Their values for n 20 are shown in Table 3 Also having examined (38) (39) for the remaining values of k we have found that for n 20 both of them hold true when n k K n So summarizing (330) () k R n (3) (tk ) > 0 for all k n n 20 (33) R n (4) (t k ) > 0 for all k n n 20 In addition our numerical results suggest the following Conjecture 32 Consider the interpolatory quadrature formula (3) (i) If τ ν = τ ν (3) then the error term R n = R n (3) satisfies () k R n (3) (t k ) > 0 for all k n (ii) If τ ν = τ ν (4) then the error term R n = R n (4) satisfies R n (4) (t k ) > 0 for all k n

11 INTEGRAL FORMULAS FOR CHEBYSHEV POLYNOMIALS 227 Now in view of (37 e ) (37 o ) and (330) (33) representation (34 i )with R n = R n (2) or R n = R n (4) by means of (29) with π n = U n or π n = W n and representation (34 ii )withr n = R n (3) by means of (2) with π n = V n leadto the following Proposition 33 Consider the interpolatory quadrature formula (3) (i) If τ ν = τ ν (2) then the norm of the error functional R n = R n (2) is given by ( ) r + (332) R n (2) = r ln 4r [(n+)/2] U n2k+ (r) r U n (r) (ii) If τ ν = τ ν (3) thenfor n 20 the norm of the error functional R n = R n (3) is given by (333) R n (3) = r ln ( ) r + + 4r [(n+)/2] r V n (r) V n2k+ (r) (iii) If τ ν = τ ν (4) thenfor n 20 the norm of the error functional R n = R n (4) is given by ( ) r + (334) R n (4) = r ln 4r [(n+)/2] W n2k+ (r) r W n (r) In [ Equations (75) (76)] Akrivis has shown that R n (2) 8(n +)rτ n+ τ 2n+2 2(n +) 2 + τ 2 n 2 +2n3 + τ 4 n 2 +2n 5 + τ 6 τ n+2 } 2n + τ 2 for n(even) 4 8(n +)rτ n+2 τ 2n+2 n(n +2) + τ 2 n 2 +2n8 + τ 4 n 2 +2n 24 + τ 6 τ n+ } 2n + τ 2 for n(odd) 5 R (2) 6rτ3 3( τ 4 ) ( R (2) 24rτ3 2 τ 6 R (2) 32rτ5 3 τ 8 where τ = r r 2 This bound for R n (2) 8 + τ 2 5 ( 5 + τ 2 7 is very close to its exact value given in (332) so in that sense our result for R n (2) is a refinement of Akrivis s result The explicit forms (332) (334) for the norm of the error functionals lead to bounds of type (36) and (32) for the error term of formula (3) with the τ ν given by (33) (35) The quality of these bounds is illustrated in the following section Remark Besides formula (3) with the τ ν given by (33) (35) there are no other interpolatory formulae with Chebyshev abscissae including the well-known Fejér rule of the first kind or Pólya rule and the Clenshaw Curtis rule whose error term satisfies either (33 i )or(33 ii ) which are essential in deriving representations (34 i )and(34 ii ) ) )

12 228 S E NOTARIS 4 A numerical example We want to approximate the integral (4) e ωt dt = eω e ω ω > 0 ω by using formula (3) with τ ν = τ ν (2) or τ ν = τ ν (3) The function f(z) =e ωz = k=0 ωk z k k! is entire and in view of (32) and (37 o ) (42) ω 2[(n+)/2] r 2[(n+)/2] (2[(n+)/2]+)(2[(n+)/2]+2) <r (2[(n +)/2])! ω f (2) ω 2([(n+)/2]+k) r 2([(n+)/2]+k) (2[(n+)/2]+2k)(2[(n+)/2]+2k) r = <r (2([(n +)/2] + k))! ω (2[(n+)/2]+2k+)(2[(n+)/2]+2k+2) ω k = 2 with [ ] denoting the integer part of a real number The above formula holds as it stands if (2[(n +)/2] + )(2[(n +)/2] + 2) >ω; in case that (2[(n +)/2] + )(2[(n +)/2] + 2) ω the formula for f (2) r starts at the branch of (42) for which (2[(n +)/2] + 2k + )(2[(n +)/2] + 2k +2)>ω Similarly ω n r n <r n + (43) f (3) n! ω r = ω n+k r n+k n + k <r n + k + k = 2 (n + k)! ω ω assuming that n +>ω; otherwise the formula for f (3) r starts at the branch of (43) for which n + k +>ω Therefore in both cases f X In addition max f(z) = z =r eωr Hence the error functionals R n (2) (f) andr n (3) (f) can be estimated by means of (36) and (32) ie (44) (45) and (46) (47) R (2) n (f) inf <r< R (3) n (f) inf <r< R (2) n (f) inf <r< ( R n (3) (f) inf <r< R n (2) f (2) r ( R n (3) f (3) r ( R (2) ) ) n e ωr) ( R n (3) eωr) with the R (2) n and R (3) n given by (332) and (333)

13 INTEGRAL FORMULAS FOR CHEBYSHEV POLYNOMIALS 229 Also it is interesting to see how estimates (44) and (46) compare with already existing ones We chose two such estimates The first is a traditional one obtained from (36 e )and(36 o ) (48) R n (2) (f) ω n e ω 2 n (n +)! (n +)ω n+ e ω 2 n n(n +2)! if n is even if n is odd The second estimate given by Basu in [3] was obtained by the contour integration method of the previous section applied on the ellipse E ρ = z C : z = 2 (u + u ) u = ρe iθ 0 θ 2π} with foci at z = ± and sum of semiaxes equal to ρ ρ > R n (2) σnn+3 (f) ρ 2 + } 4(n +) (ρ + ρ ) 2 ρ n+ max f(z) 2n +3 ρ n+ ρ(n+) z E ρ where Since in our case we get R n (2) (f) n+ σ nn+3 =2 2k + max z E ρ f(z) = e 2 ω(ρ+ρ) n odd σnn+3 ρ 2 + } 4(n +) (ρ + ρ ) 2 ρ n+ 2n +3 ρ n+ ρ e (n+) 2 ω(ρ+ρ ) n odd which can be optimized as a function of ρ giving (49) R (2) n (f) inf <ρ< ( σnn+3 ρ 2 + } ) 4(n +) (ρ + ρ ) 2 ρ n+ 2n +3 ρ n+ ρ e (n+) 2 ω(ρ+ρ ) n odd Our results are summarized in Tables 4 and 42 for the case τ ν = τ ν (2) and in Table 43 for the case τ ν = τ ν (3) (Numbers in parentheses indicate decimal exponents) All computations were performed on a SUN Ultra 5 computer in quad precision (machine precision ) The value of r or ρ at which the infimum in each of bounds (44) (47) and (49) was attained is given in the column headed r opt and ρ opt respectively which is placed immediately before the column of the corresponding bound As n and r increase R n (2) and R n (3) decrease and close to machine precision they can even take a negative value This actually happens for τ ν = τ ν (2) when ω =05 n 5 and ω =0 n 9 and for τ ν = τ ν (3) when ω =05 n 6 and ω =0 n 9 The reason is that in all these cases the infimums in bounds (44) (47) are attained at values of r>20 Bounds (44) (45) provide an excellent estimate of the actual error and are always somewhat better than bounds (46) (47) respectively This is to be expected since (46) (47) can be derived from (44) (45) if f (2) r and f (3) r are estimated by max z =r f(z) In addition bounds (44) and (46) are consistently better than bounds (48) and (49) Bound (48) for formula (3) with τ ν = τ ν (2) isthebest among the error bounds that use higher order derivatives of the function f Bound

14 230 S E NOTARIS Table 4 Error bounds and actual error in approximating the integral (4) using formula (3) with τ ν = τ (2) ν ω n r opt Bound (44) r opt Bound (46) Error (-7) (-6) 2347(-7) (-4) (-3) 4808(-4) (-5) (-5) 544(-5) (-) (-0) 505(-) (-9) (-8) 858(-9) (-3) (-3) 03(-3) (-8) (-7) 5529(-8) (-4) (-3) 272(-4) (-20) (-9) 44(-20) (-) (-) 084(-) (-5) (-4) 7568(-5) (-9) (-8) 9936(-0) (-4) (-3) 5005(-4) (+) 329 4(+2) 3266(+) (-) (0) 2309(-) (-4) (-3) 292(-4) (-7) (-6) 9407(-8) Table 42 Error bounds and actual error in approximating the integral (4) using formula (3) with τ ν = τ (2) ν ω n Bound (48) ρ opt Bound (49) Error (-7) (-6) 2347(-7) (-4) (-4) (-5) (-4) 544(-5) 0 330(-0) (-) (-9) (-7) 858(-9) (-3) (-2) 03(-3) (-7) (-8) (-4) (-2) 272(-4) (-9) (-20) (0) (0) 084(-) 0 280(-3) (-5) (-8) (-7) 9936(-0) (-2) (-4) (+4) (+3) 3266(+) 0 566(+2) (-) 5 536(-) (-2) 292(-4) (-4) (-8) (49) was obtained by the same method as that used in order to derive bound (46) except that in the case of (49) the method was applied on an elliptic instead of a circular contour Ellipses have the advantage of shrinking around the interval [ ] as the sum of the semiaxes ρ and that way give better results than

15 INTEGRAL FORMULAS FOR CHEBYSHEV POLYNOMIALS 23 Table 43 Error bounds and actual error in approximating the integral (4) using formula (3) with τ ν = τ (3) ν ω n r opt Bound (45) r opt Bound (47) Error (-6) (-5) 395(-6) (-4) (-3) 4672(-4) (-23) (-22) 9402(-23) (-4) (-4) 026(-4) (-) (-0) 4726(-) (-8) (-7) 3074(-8) (-3) (-2) 3539(-3) (-8) (-7) 4873(-8) (-3) (-2) 026(-3) (-20) (-9) 3889(-20) (-) (0) 79(-) (-5) (-4) 573(-5) (-9) (-8) 3827(-9) (-4) (-3) 4440(-4) (+) (+2) 3056(+) (-) (0) 92(-) (-4) (-3) 239(-4) (-7) (-6) 728(-8) circles However bound (49) uses an estimate for the kernel K n (cf (37) (38)) which is poor for ρ close to and reasonably good for large ρ (when the ellipse looks more and more like a circle) and this apparently accounts for (49) giving results substantially less sharp than bound (46) References [] G Akrivis Fehlerabschätzungen bei der numerischen Integration in einer und mehreren Dimensionen Doctoral Dissertation Ludwig-Maximilians-Universität München 982 [2] RG Bartle The Elements of Real Analysis 2nd ed John Wiley & Sons New York 976 MR (52:479) [3] NK Basu Error estimates for a Chebyshev quadrature method Math Comp v pp MR0277 (43:2848) [4] PJ Davis and P Rabinowitz Methods of Numerical Integration 2nd ed Academic Press San Diego 984 MR (86d:65004) [5] W Gautschi Remainder estimates for analytic functions in (TO Espelid and A Genz eds) Numerical Integration Kluwer Academic Publishers Dordrecht/Boston/London 992 pp MR98903 (94e:4049) [6] G Hämmerlin Fehlerabschätzung bei numerischer Integration nach Gauss in (B Brosowski and E Martensen eds) Methoden und Verfahren der mathematischen Physik vol 6 Bibliographisches Institut Mannheim-Wien-Zürich 972 pp MR (50:732) [7] SE Notaris Interpolatory quadrature formulae with Chebyshev abscissae J Comput Appl Math v pp MR (2002g:65027) Department of Mathematics University of Athens Panepistemiopolis 5784 Zografou Greece address: notaris@mathuoagr