HOCcholesky - Solving Systems of Linear Equations by...

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Page 1 of 5 5/14/2003 cholesky.doc Last Revised: November 5, 2002, by R. F. Taylor. Ref: Numerical Analysis: Mathematics of Scientific computing , Third Edition, D. Kincaid and W. Cheney, Brooks/Cole, 2002, ISBN 0-534- 38905-8, page 158. 1.0 When to Apply the Cholesky Method The Cholesky method is an effective alternative to LU decomposition when the n by n A matrix is real, symmetric, and positive definite. It is easy to determine when a matrix is symmetric since A T = A in this case. A matrix is positive definite if x T Ax > 0 for all choices of real n- vectors x . Note that x T Ax is just a single positive number which depends on x and A . This is not a very useful definition to test for positive definiteness since a positive number must result for all x vectors. A theorem in matrix algebra states that a matrix is positive definite if the determinants of the upper left principal submatrices are all positive. For example, consider the following matrix (Matlab notation): A=[4– 2– 4 ; -2 10 14; -4 14 56] To use the above theorem, we need to show that the following submatrices all have positive determinants: A 1=[4] A 2=[4– 2 ; -2 10] A 3=A=[4– 4 ; -2 10 14; -4 14 56] In this case we have det(A1)=4>0 , det(A2) = 16 > 0, and det(A3) = det(A) = 1296 and hence the matrix is positive definite.
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This note was uploaded on 04/26/2010 for the course CEG 616 taught by Professor Taylor during the Winter '10 term at Wright State.

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HOCcholesky - Solving Systems of Linear Equations by...

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