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MPI下矩阵相乘的实现与分析
Introduction
Matrixmultiplicationisafundamentaloperationinlinearalgebraandhasnumerousapplicationsinvariousfieldssuchascomputergraphics,scientificcomputing,andmachinelearning.TheMPI(MessagePassingInterface)isapopularparallelprogrammingmodelusedtodevelopparallelprogramsfordistributedmemorysystems.Inthispaper,wewilldiscusstheimplementationandanalysisofmatrixmultiplicationusingMPI.
MPIforParallelComputing
MPIisastandardprotocolthatallowsmultipleprocessesrunningondifferentnodesofadistributedmemorysystemtocommunicateandsynchronizewitheachother.Itprovidesasetoflibraryfunctionsthatcanbeusedtodistributecomputationanddataacrossmultipleprocesses.MPIprogramsgenerallyfollowamaster-workermodel,whereoneprocess(themaster)dividestheworkamongmultipleworkerprocessesandthencollectstheresults.
MatrixMultiplicationAlgorithm
Thestandardalgorithmformatrixmultiplicationisthenestedloopalgorithm,whichcomputeseachelementoftheresultingmatrixbysummingtheproductsofcorrespondingelementsoftheinputmatrices.LetA,B,andCbethreematricesofdimensionsm×k,k×n,andm×n,respectively.Thealgorithmcanbeimplementedasfollows:
1.Dividetheworkamongtheprocesses:
-MasterprocessdividestheinputmatricesAandBintoblocksanddistributesthemtoworkerprocesses.
-EachworkerprocessreceivesasubsetofrowsofAandasubsetofcolumnsofB.
2.MultiplytheassignedrowsofAwiththeassignedcolumnsofBandcalculatethecorrespondingelementsoftheresultingmatrixC:
-EachworkerprocesscomputesasubsetoftheelementsofCbyperformingtheinnerproductofitsassignedrowofAandcolumnofB.
-Theproductofeachelementisaccumulatedinalocalvariable.
3.GathertheresultsandcalculatethefinalmatrixC:
-ThemasterprocesscollectsthecomputedblocksofCfromworkerprocessesandcombinesthemtocreatethefinalmatrix.
MPIImplementation
TheMPIlibraryprovidesvariousfunctionsforcommunicationandsynchronizationbetweenprocesses.SomeimportantfunctionsusedintheMPIimplementationofmatrixmultiplicationinclude:
1.MPI_Init():InitializestheMPIenvironment.
2.MPI_Comm_size():Returnsthetotalnumberofprocesse