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7 Gaussian Elimination and LU Factorization

7 Gaussian Elimination and LU FactorizationIn this final section on matrix Factorization methods for solvingAx=bwe want totake a closer look at Gaussian Elimination (probably the best known method for solvingsystems of linear equations).The basic idea is to use left-multiplication ofA Cm mby (elementary) lowertriangular matrices ,L1, L2, .. , Lm 1to convertAto upper triangular form, ,Lm 1Lm 2.. L2L1 =eLA= that the product of lower triangular matrices is a lower triangular matrix, andthe inverse of a lower triangular matrix is also lower triangular. Therefore, LA=U A=LU,whereL= L 1. This approach can be viewed astriangular Why Would We Want to Do This?Consider the systemAx=bwith LU factorizationA=LU. Then we haveL Ux =y= we can perform (a now familiar) 2-step solution the lower triangular systemLy=bforyby forward the upper triangular systemUx=yforxby back , consider the problemAX=B( , many different right-hand sides thatare associated with the same system matrix).

7 Gaussian Elimination and LU Factorization In this final section on matrix factorization methods for solving Ax = b we want to take a closer look at Gaussian elimination (probably the best known method for solving systems of linear equations). The basic idea is to use left-multiplication of A ∈Cm×m by (elementary) lower triangular matrices ...

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