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1、ActaNumerica(2005),pp.233297cCambridgeUniversityPress,2005DOI:10.1017/S0962492904000236PrintedintheUnitedKingdomRandommatrixtheoryAlanEdelmanDepartmentofMathematics,MassachusettsInstituteofTechnology,Cambridge,MA02139,USAE-mail:edelman@math.mit.eduN.RajRaoDepartmentofElectricalEngineeringandCompute
2、rScience,MassachusettsInstituteofTechnology,Cambridge,MA02139,USAE-mail:raj@mit.eduRandommatrixtheoryisnowabigsubjectwithapplicationsinmanydiscip-linesofscience,engineeringandfinance.Thisarticleisasurveyspecificallyorientedtowardstheneedsandinterestsofanumericalanalyst.Thissur-veyincludessomeoriginalm
3、aterialnotfoundanywhereelse.Weincludetheimportantmathematicswhichisaverymoderndevelopment,aswellasthecomputationalsoftwarethatistransformingthetheoryintousefulpractice.CONTENTS1Introduction2342Linearsystems2343Matrixcalculus2354Classicalrandommatrixensembles2435Numericalalgorithmsstochastically2546C
4、lassicalorthogonalpolynomials2577Multivariateorthogonalpolynomials2628Hypergeometricfunctionsofmatrixargument2649Painlev´eequations26510Eigenvaluesofabillionbybillionmatrix27511Stochasticoperators27812Freeprobabilityandinfiniterandommatrices28313Arandommatrixcalculator28514Non-Hermitianandstructuredr
5、andommatrices28815Asegue290References291234A.EdelmanandN.R.Rao1.IntroductionTextson‘numericalmethods’teachthecomputationofsolutionstonon-randomequations.Typicallyweseeintegration,differentialequations,andlinearalgebraamongthetopics.Wefind‘random’theretoo,butonlyinthecontextofrandomnumbergeneration.The
6、modernworldhastaughtustostudystochasticproblems.Alreadymanyarticlesexistonstochasticdifferentialequations.Thisarticlecov-erstopicsinstochasticlinearalgebra(andoperators).Here,theequationsthemselvesarerandom.Likethephysicsstudentwhohasmasteredthelecturesandnowmustfacethesourcesofrandomnessinthelaborat
7、ory,nu-mericalanalysisisheadinginthisdirectionaswell.Theironytonewcomersisthatoftenrandomnessimposesmorestructure,notless.2.LinearsystemsThelimitationsonsolvinglargesystemsofequationsarecomputermemory
