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lecture15(3)

# lecture15(3) - Lecture 15 LU Factorization Daniel Frances c...

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Lecture 15: LU Factorization Daniel Frances c 2012 Contents 1 LU Factorization 1 1.1 Permutations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2 Can any non-singular matrix be LU factorized? . . . . . . . . . . . . . . . . 3 1.3 Recursive LU Factorization Process . . . . . . . . . . . . . . . . . . . . . . . 4 2 An example 4 2.1 first 3x3 step in the recursion . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.2 next 2x2 step in the recursion . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.3 Putting it altogether . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.4 Checking it out . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 3 O n Analysis 5 4 Comparing the various factorizations 6 5 Supplementary Section: The divide by zero problem 7 1 LU Factorization This is the last method we are covering for matrix inversion. It works for any non-singular matrix, and competes directly with Gaussian elimination. It turns out it is not as efficient as Gaussian elimination, but it has a feature which makes LU preferable, not for matrix inversion ‘per se’, but for solving linear equations - a close cousin to matrix inversion - which we will deal with briefly a bit later. In the LU factorization approach, as with Cholesky, we first factorize a matrix as a product of a lower and upper triangular matrices and then use the result to invert the matrix in two steps: 1

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1. Solve Lx i = e i and then 2. Solve Uy i = x i . In this way the total cost for inversion would be the cost of factorization plus 2 n 3 . Unfor- tunately the factorization of a non-SPD matrix (with LU factorization) takes more effort than that for a SPD matrix (with Cholesky factorization) which renders it inferior to matrix inversion with Gaussian elimination. But more on the importance of LU factorization in coming sections. In a way LU factorization attempts to follow in the steps of Cholesky, but recognizing that it cannot take advantage of the properties of SPD matrices, i.e. the matrix is not symmetric diagonal elements can be zero or negative For these reasons it needs some new concepts formalized. 1.1 Permutations Because the upper left corner of the matrix may be zero it becomes necessary to change the order of the rows. For example if we wish to start inverting 0 1 2 3 4 5 6 7 9 we will likely want to permute rows 1 and 2. We will denote such a permutation by pre-multiplying A by a permutation matrix P , as follows PA = 0 1 0 1 0 0 0 0 1 0 1 2 3 4 5 6 7 9 = 3 4 5 0 1 2 6 7 8 Note that in order to express a row permutation as a permutation matrix, we have to adopt a specific mindset. Start with an identity matrix in mind, and then permute its rows to accomplish the same permutation in the matrix it is multiplying. Thus we exchanged the order of rows 1 and 2 of the identity matrix, because we wish to exchange the order of the first two rows of A.
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lecture15(3) - Lecture 15 LU Factorization Daniel Frances c...

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