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LUfactorization - Silvana Ilie MTH510 Lecture Notes 1 LU...

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Silvana Ilie - MTH510 Lecture Notes 1 LU Factorization PROBLEM: Find the solution of the following system of linear equations: Ax = b where A is an n × n matrix, x and b are n × 1 (column vectors). NOTE: Gaussian elimination used in Chapter 9 to find solutions of systems of linear equations becomes inefficient when solving linear equations with the same A and different b . Say A is a 3 × 3 matrix and we can decompose A = L · U where L = 1 0 0 l 21 1 0 l 31 l 32 1 , U = u 11 u 12 u 13 0 u 22 u 23 0 0 u 33 Say d is such that Ld = b . If we solve Ux = d , then x satisfies Ax = b . Indeed if Ux - d = 0 apply L ( L · U ) x - Ld = 0 Ax - b = 0 A · x = b since A = L · U . Therefore, instead of solving Ax = b , we can, equivalently, do: 1. Decompose A : find L (Lower triangular) and U (Upper triangular) such that A = L · U 2. Solve Ld = b : determine d by forward substitution 3. Solve Ux = d : determine x by backward substitution 1.1 Gaussian elimination as LU Factorization R 1 R 2 R 3 a 11 a 12 a 13 a 21 a 22 a 23 a 31 a 32 a 33 | {z } = A
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