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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.

with row k. This process is referred to as partial (row) pivoting. Partial column pivoting and complete (row and column) pivoting are also possible, but not very popular. Example Consider again the matrix A = 1 1 1 2 2+ε 5 4 6 8 The largest element in the first column is the 4 in the (3,1) position. This is our first

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