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首页 1-2 Gaussian Elimination Algorithm.pdf

1-2 Gaussian Elimination Algorithm.pdf

1-2 Gaussian Elimination Algori…

上传者: 小柯 2012-09-15 评分 0 0 0 0 0 0 暂无简介 简介 举报

简介:本文档为《1-2 Gaussian Elimination Algorithmpdf》,可适用于高等教育领域,主题内容包含GaussianEliminationAlgorithmGaussianEliminationAlgorithmtkzgjnueducntkzgjn符等。

GaussianEliminationAlgorithmGaussianEliminationAlgorithmtkzgjnueducntkzgjnueducnGaussianEliminationAlgorithmContentsElementaryOperationsonaLinearSystemMatrixElementaryRowOperationsonaMatrixGaussianEliminationAlgorithmApplicationstkzgjnueducnGaussianEliminationAlgorithmContentsElementaryOperationsonaLinearSystemMatrixElementaryRowOperationsonaMatrixGaussianEliminationAlgorithmApplicationstkzgjnueducnGaussianEliminationAlgorithmContentsElementaryOperationsonaLinearSystemMatrixElementaryRowOperationsonaMatrixGaussianEliminationAlgorithmApplicationstkzgjnueducnGaussianEliminationAlgorithmContentsElementaryOperationsonaLinearSystemMatrixElementaryRowOperationsonaMatrixGaussianEliminationAlgorithmApplicationstkzgjnueducnGaussianEliminationAlgorithmContentsElementaryOperationsonaLinearSystemMatrixElementaryRowOperationsonaMatrixGaussianEliminationAlgorithmApplicationstkzgjnueducnGaussianEliminationAlgorithmContentsElementaryOperationsonaLinearSystemMatrixElementaryRowOperationsonaMatrixGaussianEliminationAlgorithmApplicationstkzgjnueducnGaussianEliminationAlgorithmElementaryOperationsonaLinearSystemExampleSolvexxx=xx=xxx=Solution:SeeLay,,ExampletkzgjnueducnGaussianEliminationAlgorithmElementaryOperationsonaLinearSystemExampleSolvexxx=xx=xxx=Solution:SeeLay,,ExampletkzgjnueducnGaussianEliminationAlgorithmElementaryOperationsonaLinearSystemSolution:Keepxin()anduseittoeliminatexfrom():SysIxxx=xx=xxx=()()=()′=SysIIxxx=xx=xx=ArethesetwolinearsystemsequivalenttkzgjnueducnGaussianEliminationAlgorithmElementaryOperationsonaLinearSystemSolution:Keepxin()anduseittoeliminatexfrom():SysIxxx=xx=xxx=()()=()′=SysIIxxx=xx=xx=ArethesetwolinearsystemsequivalenttkzgjnueducnGaussianEliminationAlgorithmElementaryOperationsonaLinearSystemSolution:Keepxin()anduseittoeliminatexfrom():SysIxxx=xx=xxx=()()=()′=SysIIxxx=xx=xx=ArethesetwolinearsystemsequivalenttkzgjnueducnGaussianEliminationAlgorithmElementaryOperationsonaLinearSystemSolution:Multiply()byinordertoobtainasthecoefficientofx:SysIIxxx=xx=xx=()=()′=SysIIIxxx=xx=xx=ArethesetwolinearsystemsequivalenttkzgjnueducnGaussianEliminationAlgorithmElementaryOperationsonaLinearSystemSolution:Multiply()byinordertoobtainasthecoefficientofx:SysIIxxx=xx=xx=()=()′=SysIIIxxx=xx=xx=ArethesetwolinearsystemsequivalenttkzgjnueducnGaussianEliminationAlgorithmElementaryOperationsonaLinearSystemSolution:Multiply()byinordertoobtainasthecoefficientofx:SysIIxxx=xx=xx=()=()′=SysIIIxxx=xx=xx=ArethesetwolinearsystemsequivalenttkzgjnueducnGaussianEliminationAlgorithmElementaryOperationsonaLinearSystemSolution:Usethexin()toeliminatethexin():SysIIIxxx=xx=xx=()()=()′=SysIVxxx=xx=x=ThesetwolinearsystemsareequivalenttkzgjnueducnGaussianEliminationAlgorithmElementaryOperationsonaLinearSystemSolution:Usethexin()toeliminatethexin():SysIIIxxx=xx=xx=()()=()′=SysIVxxx=xx=x=ThesetwolinearsystemsareequivalenttkzgjnueducnGaussianEliminationAlgorithmElementaryOperationsonaLinearSystemSolution:SinceSysISysIISysIIISysIV,theoriginalrectanglesystemxxx=xx=xxx=isequivalenttothefinaltrianglesystemxxx=xx=x=whichiseasiertosolvetkzgjnueducnGaussianEliminationAlgorithmElementaryOperationsonaLinearSystemSolution:SinceSysISysIISysIIISysIV,theoriginalrectanglesystemxxx=xx=xxx=isequivalenttothefinaltrianglesystemxxx=xx=x=whichiseasiertosolvetkzgjnueducnGaussianEliminationAlgorithmElementaryOperationsonaLinearSystemSolution:Forthetrianglesystemxxx=xx=x=substitutingx=into()toobtainx=Thensubstitutingx=andx=into()toobtainx=Consequently,theoriginallinearsystemhastheuniquesolutionx=,x=,x=tkzgjnueducnGaussianEliminationAlgorithmElementaryOperationsonaLinearSystemSolution:Forthetrianglesystemxxx=xx=x=substitutingx=into()toobtainx=Thensubstitutingx=andx=into()toobtainx=Consequently,theoriginallinearsystemhastheuniquesolutionx=,x=,x=tkzgjnueducnGaussianEliminationAlgorithmElementaryOperationsonaLinearSystemSolution:Forthetrianglesystemxxx=xx=x=substitutingx=into()toobtainx=Thensubstitutingx=andx=into()toobtainx=Consequently,theoriginallinearsystemhastheuniquesolutionx=,x=,x=tkzgjnueducnGaussianEliminationAlgorithmElementaryOperationsonaLinearSystemRoughlyspeaking,analgorithm(Ž{,asystematicprocedure)forsolvinglinearsystemsis:usethexterminthestequationtoeliminatethextermsintheotherequationsthenusethexterminthendequationtoeliminatethextermsintheotherequationsbelowandsoon,untilyoufinallyobtainasimpleequivalentsystem,possiblyinthetriangleformtkzgjnueducnGaussianEliminationAlgorithmElementaryOperationsonaLinearSystemRoughlyspeaking,analgorithm(Ž{,asystematicprocedure)forsolvinglinearsystemsis:usethexterminthestequationtoeliminatethextermsintheotherequationsthenusethexterminthendequationtoeliminatethextermsintheotherequationsbelowandsoon,untilyoufinallyobtainasimpleequivalentsystem,possiblyinthetriangleformtkzgjnueducnGaussianEliminationAlgorithmElementaryOperationsonaLinearSystemRoughlyspeaking,analgorithm(Ž{,asystematicprocedure)forsolvinglinearsystemsis:usethexterminthestequationtoeliminatethextermsintheotherequationsthenusethexterminthendequationtoeliminatethextermsintheotherequationsbelowandsoon,untilyoufinallyobtainasimpleequivalentsystem,possiblyinthetriangleformtkzgjnueducnGaussianEliminationAlgorithmElementaryOperationsonaLinearSystemRoughlyspeaking,analgorithm(Ž{,asystematicprocedure)forsolvinglinearsystemsis:usethexterminthestequationtoeliminatethextermsintheotherequationsthenusethexterminthendequationtoeliminatethextermsintheotherequationsbelowandsoon,untilyoufinallyobtainasimpleequivalentsystem,possiblyinthetriangleformtkzgjnueducnGaussianEliminationAlgorithmElementaryOperationsonaLinearSystemThreeelementaryoperationsareusedtosimplifyalinearsystem:InterchangetwoequationsMultiplyalltermsinanequationbyanonzeroconstantReplaceoneequationbythesumofitselfandamultipleofanotherequationItiseasytoseethatthethreeelementaryoperationslistedabovecantransformalinearsystemintoanequivalentsystemtkzgjnueducnGaussianEliminationAlgorithmElementaryOperationsonaLinearSystemThreeelementaryoperationsareusedtosimplifyalinearsystem:InterchangetwoequationsMultiplyalltermsinanequationbyanonzeroconstantReplaceoneequationbythesumofitselfandamultipleofanotherequationItiseasytoseethatthethreeelementaryoperationslistedabovecantransformalinearsystemintoanequivalentsystemtkzgjnueducnGaussianEliminationAlgorithmElementaryOperationsonaLinearSystemThreeelementaryoperationsareusedtosimplifyalinearsystem:InterchangetwoequationsMultiplyalltermsinanequationbyanonzeroconstantReplaceoneequationbythesumofitselfandamultipleofanotherequationItiseasytoseethatthethreeelementaryoperationslistedabovecantransformalinearsystemintoanequivalentsystemtkzgjnueducnGaussianEliminationAlgorithmElementaryOperationsonaLinearSystemThreeelementaryoperationsareusedtosimplifyalinearsystem:InterchangetwoequationsMultiplyalltermsinanequationbyanonzeroconstantReplaceoneequationbythesumofitselfandamultipleofanotherequationItiseasytoseethatthethreeelementaryoperationslistedabovecantransformalinearsystemintoanequivalentsystemtkzgjnueducnGaussianEliminationAlgorithmElementaryOperationsonaLinearSystemThreeelementaryoperationsareusedtosimplifyalinearsystem:InterchangetwoequationsMultiplyalltermsinanequationbyanonzeroconstantReplaceoneequationbythesumofitselfandamultipleofanotherequationItiseasytoseethatthethreeelementaryoperationslistedabovecantransformalinearsystemintoanequivalentsystemtkzgjnueducnGaussianEliminationAlgorithmMatrixNoticethatinsolvingsystemsofequationsbyelimination,thecoefficientsandtheconstanttermsplayacentralroleTheessentialinformationofalinearsystemcanberecordedcompactlyinarectangulararraycalledmatrixtkzgjnueducnGaussianEliminationAlgorithmMatrixNoticethatinsolvingsystemsofequationsbyelimination,thecoefficientsandtheconstanttermsplayacentralroleTheessentialinformationofalinearsystemcanberecordedcompactlyinarectangulararraycalledmatrixtkzgjnueducnGaussianEliminationAlgorithmMatrixtkzgjnueducnGaussianEliminationAlgorithmMatrixDefinition(Matrix)Amatrixisarectangulararrayofnumberswrittenwithinbrackets,suchasA=,B=tkzgjnueducnGaussianEliminationAlgorithmMatrixDefinition(Matrix)Eachnumberinamatrixiscalledanentry(þ)ofthematrixThematrixhasmrows()andncolumns()size(Œ):amn(“mbyn”)matrixtkzgjnueducnGaussianEliminationAlgorithmMatrixDefinition(Matrix)Eachnumberinamatrixiscalledanentry(þ)ofthematrixThematrixhasmrows()andncolumns()size(Œ):amn(“mbyn”)matrixtkzgjnueducnGaussianEliminationAlgorithmMatrixDefinition(Matrix)Eachnumberinamatrixiscalledanentry(þ)ofthematrixThematrixhasmrows()andncolumns()size(Œ):amn(“mbyn”)matrixtkzgjnueducnGaussianEliminationAlgorithmMatrixDefinition(Matrix)Eachnumberinamatrixiscalledanentry(þ)ofthematrixThematrixhasmrows()andncolumns()size(Œ):amn(“mbyn”)matrixtkzgjnueducnGaussianEliminationAlgorithmMatrixRelatedtothesystemxxx=xx=xxx=are,,coefficientconstantaugmented(O)matrixmatrixmatrixtkzgjnueducnGaussianEliminationAlgorithmMatrixRelatedtothesystemxxx=xx=xxx=are,,coefficientconstantaugmented(O)matrixmatrixmatrixtkzgjnueducnGaussianEliminationAlgorithmMatrixRelatedtothesystemxxx=xx=xxx=are,,coefficientconstantaugmented(O)matrixmatrixmatrixtkzgjnueducnGaussianEliminationAlgorithmMatrixRelatedtothesystemxxx=xx=xxx=are,,coefficientconstantaugmented(O)matrixmatrixmatrixtkzgjnueducnGaussianEliminationAlgorithmMatrixRelatedtothesystemxxx=xx=xxx=are,,coefficientconstantaugmented(O)matrixmatrixmatrixtkzgjnueducnGaussianEliminationAlgorithmElementaryRowOperationsonaMatrixThethreeelementaryoperationsonlinearsystemslistedearliercorrespondtothefollowingthreeelementaryrowoperations(ÐC†)onaugmentedmatrix:Interchangetworows(RiRj)Multiplyallentriesinarowbyanonzeroconstantk(kRiRi)Replaceonerowbythesumofitselfandamultipleofanotherrow(kRiRjRj)tkzgjnueducnGaussianEliminationAlgorithmElementaryRowOperationsonaMatrixThethreeelementaryoperationsonlinearsystemslistedearliercorrespondtothefollowingthreeelementaryrowoperations(ÐC†)onaugmentedmatrix:Interchangetworows(RiRj)Multiplyallentriesinarowbyanonzeroconstantk(kRiRi)Replaceonerowbythesumofitselfandamultipleofanotherrow(kRiRjRj)tkzgjnueducnGaussianEliminationAlgorithmElementaryRowOperationsonaMatrixThethreeelementaryoperationsonlinearsystemslistedearliercorrespondtothefollowingthreeelementaryrowoperations(ÐC†)onaugmentedmatrix:Interchangetworows(RiRj)Multiplyallentriesinarowbyanonzeroconstantk(kRiRi)Replaceonerowbythesumofitselfandamultipleofanotherrow(kRiRjRj)tkzgjnueducnGaussianEliminationAlgorithmElementaryRowOperationsonaMatrixThethreeelementaryoperationsonlinearsystemslistedearliercorrespondtothefollowingthreeelementaryrowoperations(ÐC†)onaugmentedmatrix:Interchangetworows(RiRj)Multiplyallentriesinarowbyanonzeroconstantk(kRiRi)Replaceonerowbythesumofitselfandamultipleofanotherrow(kRiRjRj)tkzgjnueducnGaussianEliminationAlgorithmElementaryRowOperationsonaMatrixDefinition(RowEquivalent)TwomatricesAandBaresaidtoberowequivalent,denotedbyAB,ifthereisasequenceofelementaryrowoperationsthattransformonematrixintotheothertkzgjnueducnGaussianEliminationAlgorithmElementaryRowOperationsonaMatrixNote:Iftheaugmentedmatricesoftwolinearsystemsarerowequivalent,thenthetwosystemshavethesamesolutionsetElementaryrowoperationsarereversibletkzgjnueducnGaussianEliminationAlgorithmElementaryRowOperationsonaMatrixNote:Iftheaugmentedmatricesoftwolinearsystemsarerowequivalent,thenthetwosystemshavethesamesolutionsetElementaryrowoperationsarereversibletkzgjnueducnGaussianEliminationAlgorithmElementaryRowOperationsonaMatrixNote:Iftheaugmentedmatricesoftwolinearsystemsarerowequivalent,thenthetwosystemshavethesamesolutionsetElementaryrowoperationsarereversibletkzgjnueducnGaussianEliminationAlgorithmElementaryRowOperationsonaMatrixThefollowingprocedureoutlineshowtofindallsolutionsofalinearsystem:WritetheaugmentedmatrixofthesystemUseelementaryrowoperationstoobtainanequivalentaugmentedmatrixinasimplerformRewriteeachnonzeroequationtkzgjnueducnGaussianEliminationAlgorithmElementaryRowOperationsonaMatrixThefollowingprocedureoutlineshowtofindallsolutionsofalinearsystem:WritetheaugmentedmatrixofthesystemUseelementaryrowoperationstoobtainanequivalentaugmentedmatrixinasimplerformRewriteeachnonzeroequationtkzgjnueducnGaussianEliminationAlgorithmElementaryRowOperationsonaMatrixThefollowingprocedureoutlineshowtofindallsolutionsofalinearsystem:WritetheaugmentedmatrixofthesystemUseelementaryrowoperationstoobtainanequivalentaugmentedmatrixinasimplerformRewriteeachnonzeroequationtkzgjnueducnGaussianEliminationAlgorithmElementaryRowOperationsonaMatrixThefollowingprocedureoutlineshowtofindallsolutionsofalinearsystem:WritetheaugmentedmatrixofthesystemUseelementaryrowoperationstoobtainanequivalentaugmentedmatrixinasimplerformRewriteeachnonzeroequationtkzgjnueducnGaussianEliminationAlgorithmElementaryRowOperationsonaMatrixExampleSolvethefollowinglinearsystemxxx=xxx=xxx=tkzgjnueducnGaussianEliminationAlgorithmElementaryRowOperationsonaMatrixSolution:TheaugmentedmatrixofthelinearsystemisRR()RRR()RRRtkzgjnueducnGaussianEliminationAlgorithmElementaryRowOperationsonaMatrixSolution:()RR()RRR()RRtkzgjnueducnGaussianEliminationAlgorithmElementaryRowOperationsonaMatrixSolution:FortheaugmentedmatrixThecorrespondinglinearsystemisxxx=xx=x=Usingsubstitution,itiseasytoseethatthelinearsystemhastheuniquesolutionx=,x=,x=tkzgjnueducnGaussianEliminationAlgorithmElementaryRowOperationsonaMatrixExampleSolvethefollowinglinearsystemxxx=xxx=xxx=tkzgjnueducnGaussianEliminationAlgorithmElementaryRowOperationsonaMatrixSolution:Theaugmentedmatri

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