A quadrature formula based on Chebyshev rational functions

Transcription

1 A quadrature formula based on Chebyshev ratonal functons J. Van Deun and A. Bultheel Department of Computer Scence, K.U.Leuven, Belgum E-mal: {jors.vandeun Abstract Several generalsatons to the classcal Gauss quadrature formulas have been made over the last few years. When the ntegrand has sngulartes near the nterval of ntegraton, formulas based on ratonal functons gve more accurate results than the classcal quadrature rules based on polynomals. In ths paper we present one such generalsaton whch uses results from the theory of orthogonal ratonal functons. Compared to smlar exstng formulas, t has the advantage of mproved stablty and smaller quadrature weghts. Introducton When faced wth the problem of numercally ntegratng a gven functon over the nterval [, ], the choce of a sutable quadrature formula wll usually depend on at least two dfferent consderatons. Frst there s the queston of accuracy: Whch quadrature formula gves a more accurate result, for the same number of functon evaluatons of the ntegrand? Maxmsng the number of functons that s ntegrated exactly, we may hope to reduce the quadrature error for other functons as well. Gauss quadrature formulas then seem the most obvous choce; for a gven number of nodes, they have a maxmal doman of exactness. Classcal Gauss rules le Gauss-Legendre or Gauss-Chebyshev choose nodes as the zeros of orthogonal polynomals (respectvely Legendre and Chebyshev) and are often used when the ntegrand s smooth (polynomal-le), snce they ntegrate exactly n polynomal spaces of largest possble dmenson. When there are poles present n the ntegrand, Gauss formulas that ntegrate n ratonal spaces wll gve better results. Several such generalsatons are dscussed n [Gau99, VAV93, WL00]. But then we arrve at the second ssue, whch s equally mportant,.e. that of the actual constructon of the quadrature formula. If the effort nvolved n computng the nodes and weghts s too large, the effcency ganed by superor accuracy may be totally lost. The nodes and weghts n the (polynomal) Gauss- Chebyshev formula are explctly nown and have very smple expressons, but when there s no Chebyshev weght functon n the ntegral, ths formula behaves very poorly. For an ntegral wthout weght functon, the Gauss-Legendre rule

2 s more adequate, but ts nodes and weghts are not nown explctly and have to be computed, ether by solvng a trdagonal egenvalue problem [Gau99] or, more effcently, usng Newton teratons [Swa02]. In ths case the nodes are also often precomputed (for several degrees) and hard-coded nto the program. However, for more general weghts, or for the formulas based on ratonal functons, the computatons for the Gauss rules are even more nvolved. Frst one has to compute the recurrence coeffcents for the orthogonal functons. For an n-pont rule, ths comes down to computng approxmately 2n nner products (ntegrals!), as explaned n [Gau68, Gau99], or performng a two-dmensonal recurrence f modfed moments are used [Gau70]. Then these coeffcents are gathered n a trdagonal matrx whose egenvalues are the quadrature nodes (the weghts are obtaned from the correspondng egenvectors). To compute the ntegral of a smooth functon wthout a weght functon, a good alternatve s Fejér s frst rule [DR84, p ]. The computatonal effort s mnmal both nodes and weghts are nown explctly and, usng n nodes, t ntegrates polynomals of degree n exactly. We recall that ths formula uses the same nodes as the Gauss-Chebyshev rule (whch are the zeros of the Chebyshev polynomals), wh le the weghts are determned to gve maxmal accuracy for polynomals. Snce t does not use the nodes of the Legendre rule (correspondng to an ntegral wth out weght functon), ts doman of exactness s the space of polynomals of degree n, nstead of 2n for the true Gauss- Legendre rule. In the presence of sngulartes, quadrature formulas whch ntegrate ratonal functons wth fxed poles are more accurate. In ths paper we present a new quadrature rule, a ratonal varant of Fejér s rule, whch requres less computatonal effort than the formulas from [Gau99] and n some cases of nterest s more stable and gves more accurate results than the rule presented n [WL00]. In [VDBGV06] we constructed ratonal generalzatons of the well-nown classcal Gauss-Chebyshev quadrature formula. These ratonal quadrature rules ntegrate functons wth arbtrary real poles outsde the nterval [, ], wth respect to the dfferent Chebyshev weght functons ( x) α ( + x) β, wth α and β belongng to {±/2}. The poles are fxed n advance and there can be poles at nfnty, n whch case the formulas also ntegrate polynomals of a certan degree. If there are only m dstnct poles (possbly repeated), then these formulas can be constructed n order O(mn) operatons, for arbtrarly hgh degree n. The man reason for the effcent computaton of these rules s that we have explct representatons for the so-called Chebyshev orthogonal ratonal functons. These were also ntroduced n the same artcle. Puttng all poles at nfnty just gves the Chebyshev polynomals and correspondng Gauss- Chebyshev quadrature formula, whch shows that t s ndeed a true ratonal generalsaton. Unle the formulas from [VDBGV06], the quadrature rule constructed n the present paper s not a Gaussan rule, but nstead a ratonal generalzaton of Fejér s rule. We use the nodes of the ratonal Gauss-Chebysh ev formula from [VDBGV06], but there s no weght functon n the ntegral. The quadrature weghts are determned to gve maxmal accuracy n certan r atonal functon spaces. The reason for studyng ths rule nstead of a ratonal Gauss-Legendre 2

3 rule, s that we do not have explct representatons for the Legendre orthogonal ratonal functons, whch maes the computatonal cost of the correspondng Gauss rule much hgher, as explaned above. Before we conclude ths secton, let us explan the man dfference between our rule and the ratonal Fejér rule from [WL00], whch has the same doman of exactness. In [WL00], the authors use the zeros of Chebyshev polynomals as quadrature nodes, and then determne the weghts to ntegrate ratonal functons exactly. The computatonal effort s moderate, requrng a twodmensonal recurrence. For poles close to the nterval, ths gves weghts whch are very large and of mxed sgn. We, however, use the zeros of Chebyshev ratonal functons, as explaned n the next secton. For poles close to the nterval ths maes a bg dfference, because these zeros tend to cluster near the poles and the quadrature weghts turn out to be much smaller and not of mxed sgn. Numercal examples n secton 5 wll llustrate ths. We further menton that the authors n [WL00] also dscuss a ratonal Gauss-Legendre rule. The computatonal effort of ths rule (beng a Gaussan rule) s very hgh, as explaned above. The outlne of ths paper s as follows. In the next secton we present the necessary theoretcal prelmnares. The man results are gven n secton 3, but snce the computatons are rather techncal, all the proofs have been moved to the end of the paper, n secton 7. The numercal ssues of constructng ths quadrature formula are dscussed n secton 4. In secton 5, fnally, we gve several numercal examples and compare our formulas to some of the abovementoned alternatves. 2 Prelmnares The man concern n ths paper s the computaton of the ntegral I(f) = f(x)dx I n (f) = n A n f(x n ) (2.) for a functon f(x) whch has (real) sngulartes close to the nterval [, ]. We wll approxmate I(f) by a quadrature formula I n (f) whch s exact for a certan class of ratonal functons wth prescrbed poles. Therefore, let us fx n advance a sequence of real numbers (poles) A = {α, α 2,...} outsde the nterval [, ]. Some or all of these poles may be equal to nfnty. The actual choce of the poles of course depends on the locaton of the sngulartes of f(x), as llustrated by the examples n secton 5. It s mportant to eep n mnd throughout the entre exposton, that we can always return to the polynomal case f we fx all poles at nfnty, as wll be clear from the defntons that follow. Wth ths sequence of poles we now assocate spaces of ratonal functons L n. Frst defne the polynomal π n (x) = = n ) ( xα = 3

4 whose zeros are the frst n poles, and the bass functons b 0, b (x) = x, =, 2,... π (x) The ratonal functon space L n s the lnear span of the frst n+ bass functons, L n = span{b 0,..., b n }. Note that, f all poles are at nfnty, the bass functons b (x) just become the monomals x and L n = P n, the space of polynomals of degree n. The Chebyshev polynomals T n (x) are well-nown and we only brefly recall some of ther propertes. They satsfy the orthogonalty condtons T n (x)x dx = 0, = 0,, 2,..., n x 2 and are gven explctly by T n (x) = cos(n arccos x). From ths equaton we readly obtan expressons for the zeros x n of T n (x), ( x n = cos π 2 ), =, 2,..., n. 2n Now consder the quadrature formula I n (f) whose nodes are exactly these zeros x n and whose weghts are determned such that the approxmaton (2.) s exact f f(x) s a polynomal of degree n or less. Ths s Fejér s frst quadrature rule and the weghts are nown explctly [DR84, p ]. A logcal ratonal generalsaton of Fejér s frst rule s obtaned n the followng way. Consder a ratonal functon ϕ n L n whch satsfes the orthogonalty condtons dx ϕ n (x)b (x) = 0, = 0,, 2,..., n. x 2 These functons are unquely defned, up to a normalsaton factor whch s rrelevant for our dscusson, and we shall call them Chebyshev ratonal functons. They were dscussed n great detal n [VDBGV06], where explct expressons were gven (whch we repeat below) and a method presented to compute ther zeros. Now let x n denote these zeros of ϕ n (x) and tae them as nodes n a quadrature formula whose weghts are determned such that (2.) s exact for any f L n. Ths quadrature formula s the ratonal generalsaton of Fejér s frst rule whch we wll study n the rest of ths paper. Expressons for the weghts A n are gven n the followng theorem, whch s gven n a more general settng n [VDB04]. We rephrase t accordng to our dscusson. Theorem 2.. Let {x n } n = denote the zeros of the Chebyshev ratonal functon ϕ n (x). Defne the functon n (x) as n (x) = n ν ϕ (x) =0 4

5 where ν = and defne the numbers λ n as ϕ (x)dx λ n = [ n ϕ 2 n)] (x, =, 2,..., n. =0 Now consder the quadrature formula n I n (f) = A n f(x n ) whch approxmates the ntegral I(f) = = f(x)dx such that ths approxmaton s exact for any functon f L n. weghts A n are gven by Then the A n = n (x n )λ n for =, 2,..., n. As explaned n [VDB04], the numbers ν are n fact modfed moments and the λ n are the quadrature weghts for the ratonal Gauss-Chebyshev formula studed n [VDBGV06]. For more nformaton we refer to these artcles. Before we present the explct expressons for the Chebyshev ratonal functons, we brefly dscuss the Jouows transformaton x = J(z), whch s of great mportance n the rest of the paper. Ths transformaton s defned as x = J(z) = ( z + ) 2 z and t maps any pont z nsde the complex unt dsc to a pont x n the complex plane outsde the nterval [, ]. The complex unt crcle s mapped to the nterval [, ]. The nverse mappng s denoted by z = J (x) and s chosen so that z < for any x outsde the nterval [, ]. Note that the pont x = corresponds to z = 0 and that the set {x : x R \ [, ]} corresponds to the nterval {z : z (, )}. The followng theorem can be found n [VDBGV06]. Theorem 2.2. The Chebyshev orthogonal ratonal functons ϕ n are gven by ( ) β 2 ϕ n (x) = n zbn (z) 2 β n z +, n =, 2,... (z β n )B n (z) where x = J(z) and α = J(β ), wth J the Jouows transform defned above. The functons B n (z) are defned by, B 0, B n (z) = z β β z z β 2 β 2 z z β n, n =, 2,... β n z 5

6 Furthermore, the Chebyshev ratonal functons ϕ n (x) are normalsed such that ϕ 2 dx n(x) = π. x 2 If all poles are at nfnty, then all β are equal to zero and we get ϕ n (x) = (z n + z n )/ 2. Apart from a constant factor whch s due to our dfferent normalsaton, ths s exactly the expresson for the Chebyshev polynomal T n (x) as gven n [Sze75, p. 296]. It was shown n [VDBGV06] that the Gaussan nodes and weghts {x n } and {λ n } can be computed very effcently for arbtrary poles A. In fact, the effort s perfectly comparable to that of computng the classcal (polynomal) Gauss-Legendre rule usng the method from [Sze75]. The man cost n both algorthms comes from n Newton teratons to determne the nodes, startng from accurate ntal values. Ths should be compared to the effort of constructng the ratonal Gauss-Legendre quadrature rule from [WL00] or [Gau99]. As we explaned n the ntroducton, ths requres performng a two-dmensonal recurrence startng from modfed moments or evaluatng 2n ntegrals, and then solvng an n-dmensonal trdagonal egenvalue problem. Just to gve the reader an dea of what we are talng about, computng the ratonal Gauss-Chebyshev nodes n Matlab 7 on a Pentum III wth a CPU speed of 733 MHz for n = 600 n the worst-case scenaro where all poles are dfferent, taes lttle more than 5 seconds. Solvng the egenvalue problem taes more than 4 mnutes. Ths sad, the problem of computng the ratonal Fejér rule thus reduces to the computaton of the modfed moments ν. The followng secton descrbes two dfferent approaches to compute these ν. Whch approach s preferable, depends on the locaton of the poles, as llustrated n sectons 4 and 5. 3 Computng the modfed moments As we mentoned n the ntroducton, ths secton only contans the results. All proofs can be found n secton 7. Both methods are based on decomposng ϕ nto functons whch can be ntegrated exactly. 3. Chebyshev polynomals Ths method s based on an explct representaton of the numerator of ϕ n terms of Chebyshev polynomals, as gven n the followng theorem. Theorem 3.. The Chebyshev ratonal functon ϕ (x) can be wrtten as [ ] ϕ (x) = β 2 2 = ( + β2 ) =0 c () + c () T (x) π (x) where a = a 0 /2 + a + a and the coeffcents c () recursvely from can be computed c (+) = c () (β + β + )c () + β β + c () + (3.) 6

7 wth the ntal condtons c () 0 = β and c () =, and the conventon that c () = 0 for or > (from whch t also follows that c () = for all ). The functon T (x) s a Chebyshev polynomal of degree. Furthermore, f all poles are equal to α = J(β), then explct expressons for the coeffcents c () are gven by c () = ( β) ( 2 ). (3.2) Usng ths theorem, the computaton of the modfed moments ν comes down to computng the moments T (x)/π (x)dx. Ths can be done recursvely, as descrbed n [WL00] for the case where all poles are dfferent from each other. If there are repeated poles, some (techncal) modfcatons are needed, but we do not go nto detals. 3.2 Ratonal non-orthogonal bass functons Now we decompose ϕ (x) as follows, ϕ (x) = =0 a () f (x) (3.3) where f 0 and ( ) α x j f (x) =, x α =, 2,..., j =, 2,..., #α and #α denotes the multplcty of the pole α. Equaton (3.3) can be wrtten n matrx form for = 0,..., n as Φ = AF, where Φ = [ϕ 0 ϕ n ] T, the lower trangular matrx A contans the coeffcents a () and F = [f 0 f n ] T. The followng theorem s needed for the computaton of the matrx A. Theorem 3.2. Put by defnton B = [b () ] = A. The entres b () are gven by b () = dx f (x)ϕ (x) π. (3.4) x 2 It s clear that b (0) 0 =. Explct expressons for the other coeffcents are avalable n the followng two cases.. If all poles are equal to α, we have b () β 2 = 2 b () 0 = 2 /2 j=0 2 ( )/2 j=0 ( ) ( β) 2j ( j 2 /2 ( ) ( β) 2j,, (3.5) j ) ( mod 2), where x denotes the largest nteger less than or equal to x. Note that the last term n the second formula only appears when s even. 7

8 2. If all poles are dfferent from each other, we have b () = β 2 2 b () 0 = β. In all formulas α = J(β ) and α = J(β). β 2 β β B (β ),, For the general case where there are both repeated and dfferent poles, the explct formulas are too complcated to be of use. We come bac to ths n secton 4. To be able to compute the modfed moments ν, we also need the ntegrals f (x)dx. The next theorem shows how they can be computed. Theorem 3.3. For α >, defne the ntegrals I m (α) = ( ) αx m dx. x α They satsfy the (bacward) recurrence relaton and the seres expansons ( I (α) m = ) ( ) m α 2 2 m α I(α) m α 2 I(α) m+, I m (α) = 2 m + + 4m α 2 I (α) m = 4m α j=0 j=0 (2j + m + 3)(2j + m + ) α 2j, (2j + m + 2)(2j + m) α 2j, m odd. m even, As wll be explaned n secton 4, we use the bacward recurrence relaton for stablty reasons. The seres expansons are of course needed to start ths recurrence. If all poles are dfferent from each other, we only need ntegrals of type I (α), whch can be computed explctly, I (α) = (α 2 ) log α + α 2α. The prevous two theorems contan all the nformaton we need to solve the lower trangular system thus provdng the modfed moments ν. Fdx = A Φdx, 8

9 4 Numercal ssues Whle the prevous secton only presented the theoretcal formulas to compute the modfed moments, n ths secton we dscuss some numercal consderatons to determne whch method s more sutable, dependng on the locaton of the poles. Some examples are gven as llustraton. All computatons from ths and the followng secton were done n Matlab 7 on a Pentum III (Coppermne) wth a CPU speed of 733 MHz. 4. Chebyshev polynomals The applcablty of ths method depends on the sze of the coeffcents d () = c () + c () and the ntegrals I () = T (x)/π (x)dx. We frst consder two lmt cases. In the case where all poles are at nfnty (correspondng to the classcal Fejér rule), we obvously have ϕ (x) = 2 T (x), whch was already mentoned satsfy d () =, and the reman bounded. From a numercal pont of vew, computng the before. The coeffcents d () values of I () = 0 for < and d () modfed moments ν n ths case s a perfectly well-condtoned problem. It s safe to say that ths wll also be true for poles far from the nterval [, ] (e.g. poles tendng to nfnty), snce n that case the ratonal functon ϕ s stll very much polynomal-le. The other lmt case corresponds to all poles equal to α =. Ths s of course not allowed n practce, but t s nterestng to study the growth of the coeffcents d (). It follows from equaton (3.) that n ths case d (+) = d () 2d() + d () +. Together wth the fact that d () =, t can easly be proved by nducton that we have the followng explct formula, ( ) d () 2 = ( ), whch means that for large and small ths coeffcent becomes very large. The same wll be true f all poles are equal to α very close to or, as follows from equaton (3.2). In general, snce the coeffcents are contnuous functons of the poles, f all poles are close to the nterval (and at the same sde), these coeffcents can become very large. In that case also the ntegrals I () can be very large, but the modfed moments ν are typcally of order O(). It s well-nown that summng very large numbers to obtan a small number s an ll-condtoned problem. The prevous consderatons are llustrated n table, whch shows the maxmum absolute values of d (n), I (n) and the relatve error on ν n for dfferent locatons of the poles when n = 5. Note that the accuracy seems to be hgher for poles whch are more or less symmetrc wth respect to the orgn. Ths s due to the computaton of the ntegrals I () usng the algorthm from [WL00]. 9

10 Table : Sze of coeffcents and auxlary ntegrals n the method based on Chebyshev polynomals and relatve error of modfed moment ν n when n = 5. α max d (n) max I (n) rel. err. ν n 2.68e e 7.95e 09 ( ) 2.00e e e e e e e e e + 8 ( ) ( + 2 ) 3.58e e e 02 As mentoned by the authors, ther algorthm seems to be unstable for poles whch are all at the same sde of the nterval. However, some dgts are also lost n the case of α = ( ) 2. It s not mmedately clear what causes ths loss. Obvously, when there are many poles close to the nterval (at one or both sdes), all dgts are lost because the ntegrals I () become too large. 4.2 Ratonal non-orthogonal bass functons If there are both repeated and dfferent poles, t s very dffcult to derve explct formulas for the nner products (3.4). However, the functons f and ϕ all belong to L n, so we can compute the nner products exactly usng the n-pont ratonal Gauss-Chebyshev quadrature formula whch we already computed (see secton 2), snce t s exact n the space L n L n, as explaned n [VDBGV06]. In the case where all poles are equal, we can smplfy the computatons as follows. If, for fxed, we only consder the sum n (3.5) and denote t by 2 b = β 2 2 b () 2 then t s easly checed that we have the recurrence formula ( ) b = β 2 b + and b 0 =. For the other coeffcents t holds that b () 2 = βb() 2. Smlarly, f all poles are dfferent, there s no need to recompute the B (β ) for each. Obvously, B (β ) = B 2 (β ) β β β β. Tang nto account the above consderatons serously reduces the computatonal cost of ths approach. In fact, ths maes the dfference between a complexty of order O(n 2 ) and O(n 3 ). As for the ntegrals I m (α), we use the bacward recurrence relaton because t s stable. The homogeneous soluton of ths recurrence conssts of the two 0

11 Table 2: Condton number for the matrx B n the method based on ratonal non-orthogonal bass functons when n = 5. α cond B e + 24 ( ) 2 4.e e e + 05 ( ) ( + 2 ).06e + 03 modes ( /α) and ( /α) whch both reman bounded for because α >. For the forward recurrence the homogeneous soluton conssts of ( α) and ( α) whch both become unbounded. Summng the seres expanson s a well-condtoned problem, snce all terms are of the same sgn. The convergence rate of course depends on both m and α. As an example, for m = 30 and α = 2, we needed 23 terms to reach full precson. However, for poles very close to the nterval, or for small values of m, t s preferable to use the forward recurrence snce t does not need the seres expansons (whch converge very slowly f α ). Fnally, the applcablty of ths method depends on the condton number of the matrx B, whch corresponds roughly to the number of dgts lost n solvng the lower trangular system. In table 2 we show the 2-norm condton number for dfferent locatons of the poles when n = 5. Contrary to the method based on Chebyshev polynomals, ths approach fals completely when the poles are far from the nterval, and seems to wor rather well for poles extremely close to the nterval. Ths s confrmed by fgure, whch shows the error growth (the relatve error dvded by the machne precson) for the modfed moments ν and the same locatons of the poles as n table 2. For poles tendng to nfnty, all dgts are lost very qucly. However, when the poles converge to the nterval exponentally, only a few dgts are lost. Also, the ntegrals I m (α) reman bounded as s obvous from ther defnton. Ths nd of behavour s exactly what we want and shows that ths method s perfectly complementary to the method based on Chebyshev polynomals. It should be menton ed, however, that the condton number grows rather qucly wth ncreasng n, so ths method wll only wor for moderate values of n, say n < 50 or so. But then agan, ntegrals wth a hgh number of poles extremely close to the nterval do not seem lely to arse very often n practce. 5 Examples In ths secton we apply our quadrature formula based on orthogonal ratonal functons (ORF) to some test ntegrals and compare the accuracy and stablty wth the results obtaned usng the formulas from [WL00]. In all examples, the exact result was obtaned from a multprecson computaton n Maple.

12 Fgure : Error growth of modfed moment ν for dfferent locatons of the poles n the method based on ratonal non-orthogonal bass functons. Error growth of ν ( ) ( ) ( + 2 ) Table 3: Relatve error for I wth ω =.. n ORF WL 2 4.5e 0.42e e e e e e e 3 6 < ɛ mach 4.44e 6 For the frst example we follow [WL00] and [Gau99] and compute I = πx/ω dx, ω >. sn(πx/ω) The ntegrand has poles at the nteger multples of ω, so we choose the sequence of poles A = {ω, ω, 2ω, 2ω,...}. Table 3 shows the relatve error n the quadrature rule usng our approach (ORF) based on Chebyshev polynomals and the method from Wedeman and Laure (WL) for the case ω =.. It seems that wth ncreasng n, our quadrature formula s a lttle more accurate, but the dfference s very small. For n = 6 Matlab returned a relatve error of 0 for our method, whch means less than machne precson, whch s approxmately 2.22e 6. In all cases the weghts were postve and less than. In table 4 we repeat ths experment but now wth ω =.00. The results are comparable to the prevous case, but t seems that we cannot get more than 2

13 Table 4: Relatve error for I wth ω =.00. n ORF WL e e e e e e e 3.56e e 4.75e 4 Table 5: Relatve error for I 2 wth α = 2.5. n ORF WL e e e 06.86e e e e e e 6 3.e approxmately 4 correct dgts due to roundoff errors. Agan all weghts were postve and less than. The next example s I 2 = (x + 3)(x + 2) dx. As noted by Van Assche n [VAV93], the ntegrand s n fact a Steltjes functon and can be well approxmated by a ratonal functon wth poles on the branch cut [ 3, 2]. Frst we tae all poles equal to α = 2.5. Table 5 gves the results. In our approach, we used the method based on ratonal non-orthogonal bass functons, but the other method (Chebyshev polynomals) gave smlar results. It s worth mentonng that for n = 6, even though the condton number of the matrx B was of order O(0 5 ), the quadrature sum s accurate up to machne precson. It s not uncommon n numercal ntegraton to obtan an accurate approxmaton to the ntegral even wth quadrature weghts whch are not very accurate. Also note that n ths case the WL formula seems to stagnate at more or less 2 correct dgts. We also notced several negatve weghts n ths formula, whle the weghts n our formula were all postve. Next we tae all poles dfferent from each other as n the second example of [VAV93]. They are the zeros of successve Chebyshev polynomals T 3 m(x), transformed to [ 3, 2] and ordered n such a way that they are dense on ths nterval. For more nformaton we refer to the artcle. The results are shown n table 6. Ths tme we used the method based on Chebyshev polynomals for the ORF rule. Note that wth ncreasng n, all dgts are lost n the WL rule, whle ths does not occur n our method. Ths s very remarable snce both methods use the ntegrals I (). The loss of accuracy must therefore occur not n the computaton 3

14 Table 6: Relatve error for I 2 wth α dstrbuted over [ 3, 2]. n ORF WL e e e e e 4.20e e e 03 6.e e + 0 Table 7: Relatve error for I 3 wth ω =.. n ORF WL PF e e 0.86e 0 0.8e e e e e e e e e 06 of these ntegrals, but when combnng these ntegrals to obtan the quadrature weghts. We menton that some of the weghts n the WL rule were negatve and of order O(0 2 ). All weghts n the ORF rule were postve and less than. Computng ths ntegral wth a classcal (polynomal) Gauss-Legendre rule gves approxmately 9 correct dgts for n = 8 and 4 for n = 2. For the fnal example we loo at the ntegral I 3 = sn ω x dx, ω > whose ntegrand has an essental sngularty n ω. We tae all poles equal to α = ω and use the method based on ratonal non-orthogonal bass functons. In table 7 we compare our results wth the WL rule and the classcal (polynomalbased) Fejér rule (PF). Ths tme the WL rule fals completely, not only because of loss of accuracy n the constructon of the rule, but also because the weghts are very large and of mxed sgn. For n = 20 the largest weght was of order O(0 20 ). Note that our method performs very well (even though for n = 30 the condton number of the matrx B was equal to.59e + ). All the computed weghts were postve and less than. 6 Concluson For the cases where the WL quadrature rule from [WL00] wors well, there s not really a need to use our formulas. Although the results are a lttle better, ths s at the expense of consderably more computatons. However, for poles close to the nterval, or at the same sde of the nterval, the WL rule fals, ether because of roundng errors or because the weghts become very large and of mxed sgn. In ths case the examples ndcate that our rule performs 4

15 much better, especally when the computatons are done usng the ratonal non-orthogonal bass functons. The weghts reman small and the presence of poles attract the nodes to the endponts (we refer to [VDBGV06] for a detaled explanaton). 7 Proofs In all proofs n ths secton, we use x = J(z) and α = J(β ) where J s the Jouows transform. Proof of theorem 3.. From theorem 2.2 and the fact that we get that π n (x) = n = ( β z)(z β ) z n n = ( + β2 ) ϕ n (x) = β 2 n 2 If we put z (n ) (z β n ) n = (z β ) 2 + z n ( β n z) n = ( β z) 2 π n (x) n = ( + β2 ). then the numerator equals n z (n ) (z β n ) (z β ) 2 = = = n = (n ) c (n) z n n z (n ) (z β n ) (z β ) 2 + z n ( β n z) ( β z) 2 = n = (n ) n (z + z ) = 2 c (n) (c (n) = = + c (n) )T (x) + 2(T n (x) + c (n) 0 ). To fnd the coeffcents c (n) wrte z n (z β n+ ) n (z β ) 2 = n = z (z β n+ )(z β n )z (n ) (z β n ) (z β ) 2 = z (z β n+ )(z β n ) = n+ = n [c (n) n = (n ) (β n + β n+ )c (n) c (n) z = + β n β n+ c (n) + ]z 5

16 f we tae the conventon that c (n) = 0 for n or > n. Ths yelds the recurrence relaton (3.). If all poles are equal, formula (3.2) follows (by nducton) from ths recurrence relaton usng the well-nown recurrence for bnomal numbers. Proof of theorem 3.2. Formula (3.4) follows from the orthogonalty and the normalsaton of the ϕ. We only prove the explct expressons correspondng to all poles equal. If all poles are dfferent, the reasonng s analogous and the computatons are easer. If all poles are equal to α, the functons f have the form f (x) = ( ) αx, =, 2,... x α From the defnton of the Jouows transformaton t follows that f (x) = 2 ( βz z β + z β βz so the ntegral becomes { ( b () β 2 βz = 2R 2π 2 2 z β + z β ) } (z β) βz ( βz) dz where R denotes the real part and the ntegral s over the complex unt crcle. Usng the resdue theorem then gves { b () β 2 ( βz = Res 2 2 z β + z β ) } (z β) βz ( βz), β (snce β s real, the resdue at β s real as well, so the real part can be omtted from the formula). Usng the bnomal theorem, the ntegrand can be wrtten as ( βz z β + z β ) (z β) βz ( βz) = and the numerator n the summand equals ( βz) 2j = 2j m=0 ( 2j m j=0 ( j ) ) ( βz z β ) 2j z β ) ( β 2 ) 2j m ( β) m (z β) m. Snce the resdue s a lnear functon of ts argument, we fnally get b () β 2 = 2 2 ( )/2 j=0 ( ) ( β) 2j. j 6

17 The coeffcent correspondng to ϕ 0 can be found by smlar reasonng, startng from { ( b () 0 = βz 2 Res z β + z β ) } βz z, β + 2 Res { ( βz z β + z β ) } βz z, 0. Some computatons yeld that the frst resdue equals { ( βz Res z β + z β ) } /2 βz z, β ( ) ( = ( ) β 2j ) j β 2j whle the second resdue obvousely equals { ( βz Res z β + z β ) } ( βz z, 0 = ( ) β + ). β Addng both terms and dong some algebra fnshes the proof. Proof of theorem 3.3. To fnd the recurrence relaton, note that we have ( ) [ d αx m ( = m ) αx m+ dx x α α 2 + x α 2α Integratng both sdes from to gves ( ) m = m α 2 j=0 ( αx x α ( I (α) m+ + 2αI(α) m + α 2 I (α) m ) m ( ) ] αx m + α 2. x α Rearrangng terms then yelds the recurrence relaton. The seres expansons are much more dffcult to derve. If we defne the generatng functon I (α) (z) as the (formal) seres I (α) (z) = m=0 I (α) m z m, then usng the recurrence relaton and followng the procedure outlned n [GKP89, Chap. 7], we fnd the explct expresson ). I (α) (z) = (α2 )z ( + z)(α + ) log ( + αz) 2 ( z)(α ) αz. A Taylor seres expanson for ths functon now gves expressons for the ntegrals I m (α). We omt the computatons, whch are cumbersome but straghtforward, and only gve the result, ( I m (α) = ( α) m (α 2 )m log α + ) α 2α + 2(α 2 )c (α) m, m = 0,,... 7 (7.)

18 where the constants c (α) m are gven by m/2 c (α) m = =0 m 2 ( α) m and an empty sum s equal to zero. These expressons, however, are not very useful (especally for large m and α), because of heavy cancellaton when summng both terms n (7.). In fact, f we compute a Laurent seres expanson for as a functon of α, all terms wth postve powers of α cancel aganst each other (there s only a fnte number of such terms) and we are left wth only powers of /α. Ths Laurent seres expanson corresponds to the last formulas n theorem 3.3. Agan we omt the computatons. I (α) m References [DR84] [Gau68] [Gau70] P.J. Davs and P. Rabnowtz. Methods of numercal ntegraton. Academc Press, 2nd edton, 984. W. Gautsch. Constructon of Gauss Chrstoffel quadrature formulas. Math. Comp., 22:25 270, 968. W. Gautsch. On the constructon of Gaussan quadrature rules from modfed moments. Math. Comp., 24: , 970. [Gau99] W. Gautsch. Algorthm 793: GQRAT - Gauss quadrature for ratonal functons. ACM Trans. Math. Software, 25:23 39, 999. [GKP89] R.L. Graham, D. Knuth, and O. Patashnc. Concrete mathematcs. Addson Wesley, 989. [Swa02] P.N. Swarztrauber. On computng the ponts and weghts for Gauss Legendre quadrature. SIAM J. Sc. Comput., 24(2): , [Sze75] [VAV93] [VDB04] G. Szegő. Orthogonal polynomals, volume 33 of Amer. Math. Soc. Colloq. Publ. Amer. Math. Soc., Provdence, Rhode Island, 4th edton, 975. Frst edton 939. W. Van Assche and I. Vanherwegen. Quadrature formulas based on ratonal nterpolaton. Math. Comp., 6: , 993. J. Van Deun and A. Bultheel. Modfed moments and orthogonal ratonal functons. Appl. Numer. Anal. Comput. Math., (3): , [VDBGV06] J. Van Deun, A. Bultheel, and P. González-Vera. On computng ratonal Gauss-Chebyshev quadrature formulas. Math. Comp., 75: , [WL00] J.A.C. Wedeman and D.P. Laure. Quadrature rules based on partal fracton expansons. Numer. Algorthms, 24:59 78,