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1、SIAMJ.NUMER.ANAL.()1996SocietyforIndustrialandAppliedMathematicsVol.33,No.3,pp.971-996,June1996007GENERALIZEDGAUSSIANQUADRATURERULESFORSYSTEMSOFARBITRARYFUNCTIONS*J.MAt,V.ROKHLINt,ANDS.WANDZURA*Abstract.AnumericalalgorithmispresentedfortheconstructionofgeneralizedGaussianquadratur
2、erules,originallyintroducedbyS.KarlinandW.Studdenoverthreedecadesago.ThequadraturerulestobediscussedpossessmostofthedesirablepropertiesoftheclassicalGaussianintegrationformulae,suchaspositivityoftheweights,rapidconvergence,mathematicalelegance,etc.Thealgorithmisapplicabletoawidecl
3、assoffunctions,includingsmoothfunctions(notnecessarilypolynomials),aswellasfunctionswithend-pointsingularities,suchasthoseencounteredinthesolutionofintegralequations,complexanalysis,potentialtheory,andseveralotherareas.Theperformanceofthealgorithmisillustratedwithseveralnumericale
4、xamples.Keywords,numericalintegration,quadraturerule,GaussianquadratureAMSsubjectclassifications.65D30,65D321.Introduction.ClassicalGaussianquadraturerulesareextremelyefficientwhenthefunctionstobeintegratedarewellapproximatedbypolynomials.Whenthefunctionstobeintegratedareverydiffe
5、rentfrompolynomials,Gaussianquadraturesdonotperformwell;manyparticularlydifficultandimportantproblemsinvolvetheintegrationoffunctionsoftheformn(1)f(x)EiOi,i=1whereeachofthefunctions(/9hasitsownsingularityatoneoftheendsoftheinterval,andthefunctionfcanonlybeevaluatedintoto,thecoeffi
6、cientsO/ibeingunavailable.Thisproblemisencounteredinthesolutionofintegralequationswithsingularkernels,inthenumericalcomplexanalysis,inthenumericalsolutionofellipticpartialdifferentialequationsonregionswithcomers,andinmanyothersituations.Whilesuchproblemsarenormallydealtwithbymeans
7、ofvariousadhocprocedures(see,forexample,1],[8]),theseschemeslacktherapidconvergence,stability,andeleganceoftheGaussianrules.Infact,in[6],afar-reachinggeneralizationoftheclassicalGaussianquadraturesisintro-duced,replacingthepolynomialswithfunctionsfromanextremelywideclass.Thequadra
8、-turerulesof[6]possessmostofthede
