mws_gen_sle_txt_ludecomp(1) - Chapter 04.07 LU...

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04.07.1 Chapter 04.07 LU Decomposition After reading this chapter, you should be able to: 1. identify when LU decomposition is numerically more efficient than Gaussian elimination, 2. decompose a nonsingular matrix into LU, and 3. show how LU decomposition is used to find the inverse of a matrix. I hear about LU decomposition used as a method to solve a set of simultaneous linear equations. What is it? We already studied two numerical methods of finding the solution to simultaneous linear equations – Naïve Gauss elimination and Gaussian elimination with partial pivoting. Then, why do we need to learn another method? To appreciate why LU decomposition could be a better choice than the Gauss elimination techniques in some cases, let us discuss first what LU decomposition is about. For a nonsingular matrix [ ] A on which one can successfully conduct the Naïve Gauss elimination forward elimination steps, one can always write it as [ ] [ ][ ] U L A = where [ ] L = Lower triangular matrix [ ] U = Upper triangular matrix Then if one is solving a set of equations ] [ ] ][ [ C X A = , then [ ][ ][ ] [ ] C X U L = as [ ][ ] ( ) U L A ] [ = Multiplying both sides by [ ] 1 L , [ ] [ ][ ][ ] [ ] [ ] C L X U L L 1 1 = [ ][ ][ ] X U I = [ ] [ ] C L 1 as [ ] [ ] ( ) ] [ 1 I L L = [ ][ ] [ ] [ ] C L X U 1 = as [ ][ ] ( ) ] [ U U I = Let [ ] [ ] [ ] Z C L = 1
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