{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

lecture14 - Lecture 14 Cholesky Factorization Daniel...

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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....
View Full Document

{[ snackBarMessage ]}

Page1 / 6

lecture14 - Lecture 14 Cholesky Factorization Daniel...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon bookmark
Ask a homework question - tutors are online