PLUExample

PLUExample - Pivoted LU Factorization(Crout version Ref(S&H...

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Page 1 of 6 D:\classes\cs316\sp03\noteslectures\PLUExample.doc 5/14/2003 Pivoted LU Factorization (Crout version) Ref: (S&H) Applied Numerical Methods for Engineers ,R . Schilling and S. Harris, Brooks/Cole, 2000. The purpose of these notes is to show how to compute the factors of a square matrix, A , of order n so that: (1) PA = LU (Crout version) where P is a (general) permutation matrix of order n; L is lower triangular (i.e. L =[ l ij ] and l ij = 0 for j>i) and U is unit upper triangular; i.e. U u ij ] with u ii =1.0 and u ij =0.0 for i>j. It is a useful property of P that it is nonsingular and (2) P -1 = P T See separate notes for more details on permutation matrices. The Crout version of LU decomposition assigns 1.0 to the values on the diagonal of U . In the Doolittle version, the L matrix has 1.0 values on the diagonals: (3) PA = L U (Doolittle version) Equations (1) or (3) are useful for solving sets of systems of equations: (4) Ax k = b k for k=1,2,. ..K I.e. solving Ax = b when solutions for a set of various b vectors is sought. In LU decomposition we do not repeat the steps of Gaussian elimination for a set of augmented matrices with last column ( b vector) being replaced for each new problem, but rather we reuse the LU factorization. Here are the steps in the Crout method for solving one set of the equations in (4). 1. Obtain the factors L , U , and P so that PA = LU . Note that Ax = b can then be rewritten as PAx = LUx = Pb c .

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PLUExample - Pivoted LU Factorization(Crout version Ref(S&H...

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