Innaturalscienceandengineeringtechnology,theresolutionsofmanyproblemsarereducedtosolvinglinearalgebraicequations.Withthedevelopmentofscienceandtechnology,thecomplexityofaproblemtobesolvedisincreasing.Therefore,highlyefficientsolutionoflargescalesparsematricesisofuniversalmeaninginscientificcomputation.Generalnumericalsolutionsoflinearequationsaredirectmethodsanditerativemethods.ThisarticlemainlyanalyzessuchdirectmethodsasGaussianEliminationandLU-Factorization,thenproprosesasquare-rootingmethodtofindthesymmetricalpositivedefinitematrices.Theiterativemethods,Jacobimethoditerativemethod,Gauss-SeidelMethod,andSuccessiveOverRelaxation(SOR)method,arealsoanalyzed.WefinallydiscusstheConjugateGradientmethodbasedontheanalysisofSORmethod.Wealsoprovidedetailedderivationofthesemethodsusingmathematicalanalysis.Matlabprogramsaredevelopedandtestedforsolvinglinearequations.
Keywords:linearequations,directmethods,square-rooting,iterativemethods,conjugategradientmethod.