A QUADRATURE FORMULA OF RATIONAL TYPE FOR INTEGRANDS WITH ONE ENDPOINT SINGULARITY

Electronic Transactions on Numerical Analysis. Volume 16, pp. 143-164, 2003.Copyright©2003, Kent State University. ISSN 1068-9613.ETNAKent State University etna@A QUADRATURE FORMULA OF RATIONAL TYPE FOR INTEGRANDS WITHONE ENDPOINT SINGULARITYJ.ILL´ANAbstract.The paper deals with the construction of an efficient quadrature formula of rational type to evaluate the integral of functions which are analytic in the interval of integration,except at the endpoints.Basically our approach consists in introducing a change of variable into the integralwhere and,,.We evaluate the new form by a quadrature approximant which is based on Hermite interpolation by means of rational functions.The nodes of are derived from a fundamental result proved by Ganelius[Anal.Math.,5(1979),pp.19-33]in connection with the problem of approximating the function,,by means of rational functions.Wefind such that as,for all.For functions in,,which satisfy an integral Lipschitz condition of order,the following estimate is deducedIf then the upper bound for is that which is exact for the optimal quadrature error in,.We report some numerical examples to illustrate the behavior of the method for several values of the parameters.Key words.interpolatory quadrature formulas,rational approximation,order of convergence,boundary singu-larities.AMS subject classifications.41A25,41A55,65D30,65D32.1.Introduction.In applications the solution of a problem often involves the numeri-cal integration of functions with singularities.The traditional approach in this subject has been based on the use of Gauss quadrature formulas of polynomial type,though more re-cently some rational versions have also been considered.The latter concerns classes of Gauss quadrature formulas which require the largest degree of exactness for simple rational func-tions.These rational rules are connected with multipoint Pad´e approximation of Stieltjes functions[12,13,14,15,16,17,28],and with the evaluation of integrals whose integrand has poles close to the integration interval[7,8,9,10].Another relevant area is that concerned with the use of functions which transform the integration interval,and increase the efficiency of the numerical procedures.These trans-formations modify the distribution of the quadrature nodes in such a way that the new ones,namely,exhibit a higher concentration near those endpoints where the sin-gularities should be located.As far as we know the strategy offitting a change of variable into an integration rule to increase the efficiency of a numerical procedure,have only been investigated in the polynomial context[1,20,21,22,23,24,25].When compared to polynomials,rational functions are now considered a nicer class to approximate functions with a variety of singularities.This conclusion has a starting point in1964when Newman[26]proved that the function,,can be uniformly approximated by rational functions much faster than by polynomials.At present,there are additional reasons to assert that the following formulation(1.1)144A quadrature formula of rational type for integrands with one endpoint singularitywhere is a convenient rational function,is expedient when is an analytic function with a finite number of singularities on,or with some poles close to.A basic argument in favour of the latter is that the term in the error,can be better approximated than by polynomials,if annihilates the poles of closer to.The class of functions to be integrated are those defined on the interval,,which admit analytic continuation to the space.Rational approximation of functions in a Hardy space was earlier investigated by Gonchar in[11]as a continuation to Newman’s research[26].In this paper we deal with a method of numerical integration which works,theoretically speaking,on the Hardy space and is based on interpolation by means of rational functions. For each this formula has the form given below(1.2)where,,,,,, ;is the-th derivative of,and is the quadrature error.Formula(1.2)is a special case of those studied by Bojanov[4],Newman[27],Andersson [2],Andersson&Bojanov[3].To understand why rational functions play a role in this theory it is sufficient to consider the general formulation given below(1.3)where,is afinite and positive measure on,is some ra-tional function of order with poles,,,and.It is well-known that is a continuous linear functional on,and that Cauchy’s integral formula can be applied to yield thefinite sum on the right term of(1.3).The exact bound,for functions in which satisfy an integral Lipschitz condition. Besides,we expect that the numerical behavior of such a procedure compares favorably with some other remarkable quadrature rules(cf.[20,23]).The paper is organized as follows.Section2is devoted to define nodes and coefficients for integration formulas of the form(1.2).Section3deals with the construction of a change of variable which wefit into the integral,to reach an order of convergence which is optimal in according to the theoretical results obtained by Andersson[2].Finally, Section4contains some remarks and numerical examples to show the power of the method.2.Construction of the quadrature formula.The next result can be derived from[6] and is basic for our approach.J.Ill´a n145 L EMMA2.1.For,positive integer there exists a constant which only depends on,such that(2.1)where,;and for.Lemma2.1is based on the strong connection between rational approximation and some equilibrium problems.This result was used by Ganelius to obtain the exact order of conver-gence for the best uniform approximation of,,by means of rational functions of orden.Next we describe some of the most relevant aspects of Ganelius’technique to obtain estimate(2.1).Given the Green function of the right half plane,namelymust be assumed,and for allThe following step is to take as that given in Lemma2.1,tofind(2.2)Notice that implies.At this point the nature of the number and the role played by it in this theory should be clear enough to the reader.Of course,much more details can be seen in the paper of Ganelius.After obtaining(2.2)the proof of lemma2.1is practically concluded.To derive the inequality(2.1)from(2.2),it only remains to replace the distribution by a discrete measure .Despite all the technical difficulties of this proof,the suitable location of the mass-points ,where the discrete distribution is supported,doesn’t seem very hard to obtain.The importance of the points for our work,comes from the fact that the nodes in(1.2) will be expressed in terms of the former.Andersson[2]applied Lemma2.1tofind the exact rate of convergence of the optimal quadrature error in Hardy spaces.We also applied this result in Section3to obtain an upper146A quadrature formula of rational type for integrands with one endpoint singularitybound for the quadrature error generated by formula(1.2)after introducing a change of variable(see also[18]).The notation in Lemma2.1will be kept in the rest of the article.The proof of the following result is straightforward.L EMMA2.2.Let,,,and be real numbers such thatLet and be the circle with center at and radius .In addition,let consider the following rational functions(2.3)where is given bywith,,.Then,and satisfy the following properties1..2..3.is an increasing function on.4.,.5.If then.L EMMA2.3.Let and be the rational functions defined in Lemma2.2.Let be the rational function given by the following equation(2.4)where is an analytic function in the unit disk.If is a non-negative and continuous function on then the rational function interpolates to at the zeros of,and the quadrature formula(2.5)is of the form(1.3)and it integrates exactly any rational function,J.Ill´a n147 where is the following polynomial of degree at mostLikewise,the zeros of are the poles of.They appear in the statement of the exactness condition for formula(2.5),and their formulation can be deduced from the following representation of(2.8)whose corresponding integrals can be evaluated by means of residues to obtain that all of them are zero.The inequality makes possible that for some values of and.It is the reason for which formula(2.8)can be considered as a suitable manner of representing the poles of.In case of it means that quadrature formula(2.5)is exact for some polynomials as well.We observe that quadrature formula(2.5)depends on the parameters,,,and. Moreover,if we put for some suitable,then we obtain a quadrature approximant of the same form as that in(1.2)with.3.An upper bound for the quadrature error.In this section we construct a special analytic change of variable,,to be introduced into the integral148A quadrature formula of rational type for integrands with one endpoint singularity Next we evaluate,with,, and,using the corresponding quadrature approximant (2.5)to approximate with an error of orderFor the function is a conformal mapping from the unit disk onto itself which satisfies and.Here we will only consider the case.To construct we need the following sequences.D EFINITION3.1.For every and,,,we define the sequence(3.1)J.Ill´a n1493.,,4.5., where and.Proof.It is trivial that property1is true for.For we use thatis a non decreasing function with respect to as well as the induction hypothesis on. The existence of the limit in property3is guaranteed by property2,and its value is obtained by taking limits on both sides of equation(3.1).After assuming that property5is true for we multiply both sides by the factor to obtain the relation for.Property6follows from the following equality(see(3.1))(3.4)(3.7)(3.9)150A quadrature formula of rational type for integrands with one endpoint singularityfrom which we obtain the following formula(3.10)Proof.Equation(3.12)is proved by induction on using Lemma3.2and the definition of both sequences and.Equality(3.12)can be seen as a consequence of applying times the change of variable (3.11)to the integral in the left hand side of(3.12).The approximation formula which is given in(3.13)is the result of coupling the approx-imant in(2.5)by putting,,and with the change of variable in(3.12).Thefinal version of our quadrature formula is(3.13)where(3.14)(3.15)J.Ill´a n151 where is the quadrature error whose expression is given in(2.9),and(3.17)Let be the rational function which produces the quadrature approxima-tion in(3.13).It means that is the rational function associated with according to the procedure given by(2.5).The weight function given by(3.17)corresponds to the statement of Lemma2.3,and will play a relevant role in applying Lemma2.1to our approach.Though we are mainly interested in evaluating the integral,,,we need the following result.L EMMA3.7.Let be a positive integer.Let,and.Letbe the quadrature approximant of formula(3.13)with nodes(2.7) for which the points are selected as those given by Lemma2.1.For sufficiently large, ,,,and the following estimate holds.(3.18)where is the circle with center at and radius.From property5of Lemma2.2,and(3.19)we obtain the following inequality.(3.20)152F IG.3.1.Error curves when,,,,.IGIn(3.20)we can consider the distributionJ.Ill´a n153 From Lemma(3.2)the distance from the circle to is. We apply to the principle of maximum to obtain(see[5],pp.29,36)(3.21)ing lemmas2.1and3.3,and equations(3.20),(3.21)and(3.22)we derive the following estimate.(3.23)where does not depend1on the parameters and.Let and where will be selected conveniently. From property4of Lemma2.2and Lemma3.4we obtainwhere.The terms in the right side of(3.24)have the following asymptotical behaviour as, respectively.1But it does depend on as can be deduced from[6].154A quadrature formula of rational type for integrands with one endpoint singularityF IG.3.3.Error surface when,,,,.The contribution of the factors,and leads us to select ,,in order to cancel all those terms which tend to infinity as,with the exception of which is grouped together with.J.Ill´a n155T ABLE3.1Absolute errors when,,,.,0.0000000e+00 2.0e–010.0000000000e+009.8e–031.0000000e–012.0e–01 2.0000000000e–018.4e–033.0000000e–01 1.8e–01 5.0000000000e–01 6.1e–035.0000000e–01 1.6e–017.0000000000e–01 4.2e–037.0000000e–01 1.2e–018.0000000000e–01 3.0e–037.5000000e–01 1.1e–019.0000000000e–01 1.5e–038.0000000e–018.2e–029.5000000000e–01 3.4e–048.9000000e–019.5e–039.5910000000e–01 2.5e–068.9700000e–01 4.4e–049.5915000000e–01 4.1e–078.9730000e–01 3.5e–059.5915900000e–01 3.8e–088.9732485e–01 1.7e–069.5915970654e–01 2.3e–100.000000000000000e+00 6.4e–072.000000000000000e–01 1.2e–019.999000000000000e–01 5.9e–079.900000000000000e–01 3.4e–039.999300000000000e–01 5.6e–079.997260000000000e–01 4.6e–079.999600000000000e–01 5.1e–079.997265635000000e–01 1.2e–109.999942747252746e–01 1.2e–119.997265635029354e–01 3.5e–12D EFINITION3.8.The integral modulus of continuity of is given by(3.25)The analysis of error in terms of the modulus(3.25)is the following(3.26)It is well-known that,for all,and(cf.[5,18]). Besides,the behaviour of as does not depend essentially on,, but on the nature of in a neighbourhood of the endpoint.T HEOREM3.9.Let,,,and, .Let be a sequence such that1.,2.,3.156A quadrature formula of rational type for integrands with one endpoint singularityT ABLE3.3Absolute errors when,,,,.0.000000000000000e+00 2.1e–092.500000000000000e–018.5e–103.000000000000000e–01 1.9e–097.500000000000000e–01 3.5e–107.000000000000000e–01 1.6e–099.500000000000000e–019.8e–119.900000000000000e–01 1.4e–099.960000000000000e–01 5.5e–129.999571142857140e–017.5e–129.989981684981686e–01 2.0e–15is the approximant given by(3.13),and the positive constant depends neither on nor.Convergence follows from estimate(3.27).J.Ill´a n157F IG.3.7.Error for,,,,,,.If then every sequence of the form,satisfies the conditions of Theorem3.9.Besides,the rate of convergence of the quadrature error with respect to depends on the behavior of the sequenceThe following result states a class of functions for which we should expect a good behavior158A quadrature formula of rational type for integrands with one endpoint singularityT ABLE3.4Absolute errors when,,,,.0.000000000000e+00 3.2e–020.0000000e+00 5.8e–055.000000000000e–01 3.0e–02 2.0000000e+00 4.9e–057.000000000000e–01 1.4e–02 5.0000000e–01 2.1e–057.500000000000e–01 5.7e–03 6.0000000e–01 2.8e–067.710000000000e–01 1.4e–03 6.1000000e–01 5.1e–077.750000000000e–01 5.4e–04 6.1210000e–01 1.3e–087.760000000000e–01 3.1e–04 6.1213000e–01 6.3e–097.773000000000e–01 1.1e–05 6.1213307e–01 5.6e–097.773300000000e–01 4.0e–067.0000000e–01 2.7e–057.773385518591e–01 2.0e–069.0000000e–01 2.5e–04in the proof of Lemma3.7.We observe that we must consider for,and otherwise.Now the term in estimate (3.27)becomesLet be a function in such that it satisfies a Lipschitz condition of order.Ifand,,then we haveJ.Ill´a n159T ABLE3.5Absolute errors when,,,,,.0.000000000000e+00 6.6e–061.00000000e–012.2e–055.000000000000e–01 2.9e–064.00000000e–01 1.3e–059.000000000000e–01 4.1e–078.00000000e–01 2.9e–069.950000000000e–017.4e–099.15200000e–01 1.8e–089.970800000000e–01 2.9e–119.15823525e–01 2.8e–10T ABLE3.6Absolute errors when,,,,.0.000000000000000e+00 6.6e–062.00000000000000e–01 5.0e–069.500000000000000e–01 1.8e–079.00000000000000e–01 4.1e–079.995000000000000e–01 1.7e–099.99967920000000e–01 1.3e–109.999787058820000e–01 6.1e–129.99967921568627e–01 2.1e–130.000000000e+008.4e+001.000000000e–018.3e+003.000000000e–018.1e+005.000000000e–017.8e+007.000000000e–017.5e+009.000000000e–017.1e+009.999990000e–01 2.3e+009.999999500e–018.3e–019.999999900e–01 2.6e–029.999999905e–01 4.8e–04T ABLE3.8Absolute errors when,,,,,.0.00000000000000e+008.7e+001.00000000000e–018.6e+002.00000000000000e–018.5e+004.00000000000e–018.3e+006.00000000000000e–018.0e+008.00000000000e–017.5e+009.99000000000000e–01 4.6e+009.99999999000e–018.0e–019.99999999970000e–018.3e–019.99999999919e–01 6.9e–03160A quadrature formula of rational type for integrands with one endpoint singularityT ABLE3.9Absolute errors when,,,,.0.00000000000000e+008.7e+001.0000000000000e–018.4e+005.00000000000000e–018.1e+005.0000000000000e–018.1e+009.00000000000000e–017.0e+009.0000000000000e–017.0e+009.99999000000000e–01 2.5e+009.9999999000000e–01 1.6e+009.99999999958000e–019.2e–019.9999999994258e–01 1.6e–010.000000e+00 2.7e–011.000000e–012.5e–012.000000e–01 2.2e–013.000000e–01 2.0e–014.000000e–01 1.7e–016.000000e–019.6e–028.000000e–01 2.1e–038.039000e–01 4.1e–058.039760e–01 3.3e–078.039765e–01 6.6e–08T ABLE3.11Absolute errors when,,,,,.0.00000e+00 2.4e–011.0000000e–012.2e–012.00000e–01 2.0e–016.0000000e–01 1.2e–016.00000e–01 1.2e–019.9000000e–01 5.1e–039.90000e–01 5.2e–039.9998100e–01 2.4e–059.99990e–01 1.0e–059.9998158e–01 2.6e–07T ABLE3.12Absolute errors when,,,,,.0.000000000000e+00 6.4e–111.0e–01 1.4e–102.000000000000e–01 4.8e–113.0e–01 1.3e–106.000000000000e–01 2.7e–115.0e–01 1.2e–108.000000000000e–01 1.7e–117.0e–01 1.0e–109.780000000000e–01 1.1e–129.0e–019.2e–11J.Ill´a n161T ABLE3.13Absolute errors when,,,.,0.00000000e+00 1.4e–010.0000e+00 3.5e–021.00000000e–01 1.4e–01 1.0000e–012.9e–023.00000000e–01 1.3e–01 2.0000e–01 2.3e–024.00000000e–01 6.0e–02 3.0000e–01 1.8e–025.00000000e–01 1.3e–01 5.0000e–019.9e–039.00000000e–017.8e–027.0000e–01 3.8e–039.72000000e–01 1.8e–039.0000e–01 1.2e–039.72700000e–01 6.0e–069.3000e–01 5.0e–049.72702200e–01 2.3e–079.3950e–01 2.6e–059.72702285e–01 2.0e–099.3991e–01 5.2e–080.000000000e+00 1.9e–091.000000e–012.9e–022.000000000e–01 1.2e–095.000000e–01 1.0e–025.000000000e–01 3.8e–109.000000e–01 5.5e–047.000000000e–01 3.7e–119.952700e–017.6e–097.318000000e–01 4.8e–149.952725e–019.8e–14we only refer to the case.By means of the affine transformation,,we apply our method to estimate, for several values of and.If then it means that no change of variable is made.The values of the parameter which we have used to test our approach are. In spite of the theoretical results in the previous section we have obtained very good results for,though instability shows up for in case of.In this article we report Fig.3.4for,Fig.3.7to validate the role of in the procedure and Table3.13for the case.We use the exactness condition for the rational functions to implement a numerical procedure for the quadrature rule(3.13).It means that the coefficientsof the approximant are calculated as the solution of a lin-ear system of equations which we trasform in,where is an upper triangular matrix obtained via QR factorization.Table3.17shows that after scaling the con-dition number of should vary a little as ranges from to.Many numerical experiments have shown that the equation works good enough in the computer.In our opinion this procedure needs a parameter which ranges from to.If we make small it means that either the value of is very forced to be near the point,which yields a very high concentration of nodes and poles near the point,or.Instability is associated with small values of though it can be observed higher accuracy as well(see Fig.3.5-3.6).Theorem3.10shows that if is close enough to one then the quadrature error should be small.This effect is certainly produced by the change of variable,though one can detect that the decreasing behaviour of the162A quadrature formula of rational type for integrands with one endpoint singularityT ABLE3.15Absolute errors when,,,,.0.00000000000000e+00 1.6e–081.00e–01 3.8e–062.00000000000000e–01 1.1e–084.00e–01 1.8e–064.00000000000000e–01 6.1e–097.00e–01 4.8e–077.00000000000000e–01 1.6e–099.00e–01 6.1e–089.90000000000000e–01 2.2e–129.99e–01 1.3e–130.0000e+00 6.4e–031.0000000e–017.5e–022.0000e–01 5.5e–035.0000000e–01 5.5e–025.0000e–01 3.9e–038.0000000e–01 3.2e–028.0000e–01 2.0e–039.9000000e–01 2.0e–039.9700e–01 3.8e–059.9234833e–01 2.3e–06error with respect to is magnified a lot by the condition.When is too small the slope of the error curve with respect to is also small except for values of very close to one.Such a behaviour can be seen in Table3.1–3.16for several values of the parameters, ,and.The variable plays the role of counterpart of,in the sense that values of close to produce a concentration of nodes on the right side.We displayfigures3.1–3.6to illustrate the behavior of the error as function of the parameters and,particularly when they are close to one simultaneously.The error surfaces have been defined over the rectangular grid,,,for which,.Before making any conclusion on whether the selection is a reasonable decision for the numerical procedure one has to take into account that condition implies that some derivatives of the integrand are participating in the calculations.Despite the fact of having used simbolic tools to simulate all the derivatives,it is not surprising that a loss of accuracy is observed in Table3.16with respect to Table3.2.Naturally,this comparison clearly indicates that unless we were able to improve the algorithm,the case seems to be preferable.As for the use of an integral representation formula to calculate all the coefficients of the quadrature rule(3.13),for the moment we have not been able to reach precision enough in the experiments.For all cases a feature of the integration method(3.13) is that the error strongly depends on the behavior of near.A class of functions with singularities located at interior points to which can be applied this integration rule,is one as that defined in[19],namely,piecewise functions.J.Ill´a n163T ABLE3.17Condition number of matrices,.1.2e+00 4.1e+00 4.9e+012.6e+03 1.9e+067.7e+10 1.4e+130.11.6e+007.0e+008.4e+01 5.2e+03 4.3e+062.0e+117.7e+130.91.7e+019.1e+018.8e+02 1.9e+05 4.6e+08 5.0e+13 1.3e+170.011.2e+00 1.8e+00 4.5e+01 6.9e+03 4.2e+07 6.3e+12 6.6e+190.52.0e+00 4.2e+033.8e+049.0e+107.7e+12 1.7e+19 3.7e+230.99The reader should consult[23]where the-point Gauss-Legendre rule and several smooth-ing transformations are applied together to evaluate the integral of functions with one and two endpoint singularities.To illustrate the efficiency of the method in[23],the authors compare the corresponding numerical results with those generated by the trapezoidal rule when the latter has been modified by a change of variable.Our approach is based on interpolation of rational functions and it is different from that presented by Monegato and Scuderi in[23].The latter deals with Gaussian quadrature for-mulas of polynomial type and smoothing transformations for which some derivatives van-ish at the endpoints of the integration interval.Instead,we consider the change of variable which simply modifies the distribution of nodes to diminish the adverse effect of the endpoint singularity.However,for the purposes of comparison,we mainly refer to the numerical results in[23]because here we have practically tested the same functions as those reported in that paper.In spite of the ill conditioned matrices which arise in our implementation,a conclusion is that the accuracy which we can obtain with the rational quadrature rule(3.13)is competent, particularly for those functions for which estimate(3.28)holds.All the computations in this work have been performed on a PC using MatLab. 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