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Unformatted text preview: Lecture 14: Cholesky Factorization Daniel Frances c 2012 Contents 1 Cholesky Factorization for SPD (sym +ve definite) matrices 1 1.1 The Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2 Only need to compute lower diagonal elements . . . . . . . . . . . . . . . . . 2 1.3 An example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.4 O n analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.5 Computing the Cholesky Factorization . . . . . . . . . . . . . . . . . . . . . 4 1.6 Preponderance of SPD matrices . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.7 Why does Cholesky require SPD? . . . . . . . . . . . . . . . . . . . . . . . . 5 1 Cholesky Factorization for SPD (sym +ve definite) matrices Now that we have seen the ease with which triangular matrices can be inverted, there is a real incentive for reducing matrices for triangular form. In Gaussian elimination we reduced the matrix to an upper triangular form in n 3 flops and then • took another n 3 flops to invert the matrix for a total of 2 n 3 flops • or, took another n 2 flops to solve Ay = b with one back substitution. With Cholesky Decomposition we will factorize an SPD matrix so that A = LU where L is a lower triangular matrix, and U is an upper triangular matrix in 1 3 n 3 flops, a third of the 1 time it takes to produce a triangular matrix with Gaussian elimination!. To solve a set of equations Ay = b or equivalently LUy = b we need two steps: 1. Solve Lx i = b with back substitution in n 2 flops 2. Solve Uy = x with forward substitution in n 2 flops In this way the total cost for solving a set of equations is O n = 1 3 n 3 , basically the cost of factorization which is 3 times as fast a Gaussian elimination. For matrix inversion it’s a different story. After we factorize the matrix for n 3 / 3 flops, we have to solve n sets of equations Ay i = e i to calculate each of the columns of the inverse....
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 Spring '12
 Frances
 SPD, Cholesky factorization, SPD matrices, SPD matrix

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