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Name one advantage to using Cholesky factorization to solve Az = b when A is positive definite rather than using Gaussian Elimination to factor PA = LU.

Name one advantage to using Cholesky factorization to solve Az = b when A is positive definite rather than using Gaussian Elimination to factor PA = LU.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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**Question:** Name one advantage to using Cholesky factorization to solve \( Ax = b \) when \( A \) is positive definite rather than using Gaussian Elimination to factor \( PA = LU \). **Explanation:** This problem asks for a comparison between Cholesky factorization and Gaussian elimination in solving linear systems, specifically when the matrix \( A \) is positive definite. Cholesky factorization has the advantage of being more computationally efficient because it takes advantage of the symmetry and positive definiteness of \( A \). The process involves decomposing \( A \) into \( LL^T \), where \( L \) is a lower triangular matrix. This reduces both memory usage and computation time compared to the general LU decomposition with partial pivoting used in Gaussian elimination, which does not utilize the symmetry of \( A \).
Transcribed Image Text:**Question:** Name one advantage to using Cholesky factorization to solve \( Ax = b \) when \( A \) is positive definite rather than using Gaussian Elimination to factor \( PA = LU \). **Explanation:** This problem asks for a comparison between Cholesky factorization and Gaussian elimination in solving linear systems, specifically when the matrix \( A \) is positive definite. Cholesky factorization has the advantage of being more computationally efficient because it takes advantage of the symmetry and positive definiteness of \( A \). The process involves decomposing \( A \) into \( LL^T \), where \( L \) is a lower triangular matrix. This reduces both memory usage and computation time compared to the general LU decomposition with partial pivoting used in Gaussian elimination, which does not utilize the symmetry of \( A \).
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