关闭

关闭

封号提示

内容

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 5 0 152 21 692 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

类似资料

该用户的其他资料

2-1 Vectors in R^n.pdf

1-1 集合与函数.pdf

1-2 函数的极限.pdf

1-1 集合与函数答案.pdf

1-3 三明治原理.pdf

职业精品

精彩专题

上传我的资料

精选资料

热门资料排行换一换

  • 进步与贫困 [美]亨利·乔治.p…

  • 社会研究方法+(美)艾尔.巴比(…

  • 陈嘉映:《语言哲学》.pdf

  • 鲍罗廷与武汉政权(蒋永敬).pdf

  • 淘宝天猫品牌运营计划自创.ppt

  • 魔法英语语法.pdf

  • 中国哲学史大纲(蔡仁厚着).pdf

  • ABAQUS有限元分析实例详解.…

  • 毛泽东农村调查文集.pdf

  • 资料评价:

    / 109
    所需积分:0 立即下载

    意见
    反馈

    返回
    顶部