lecture18 - Lecture 18 Iterative solution of linear...

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Unformatted text preview: Lecture 18 Iterative solution of linear systems* Newton refinement Conjugate gradient method 62 Review of Part II Methods and Formulas Basic Matrix Theory: Identity matrix: AI = A , IA = A , and I v = v Inverse matrix: AA 1 = I and A 1 A = I Norm of a matrix: | A | max | v | =1 | A v | A matrix may be singular or nonsingular. See Lecture 10. Solving Process: Gaussian Elimination produces LU decomposition Row Pivoting Back Substitution Condition number: cond( A ) max | x | / | x | | A | | A | + | b | | b | = max parenleftbigg Relative error of output Relative error of inputs parenrightbigg . A big condition number is bad; in engineering it usually results from poor design. LU factorization: PA = LU. Solving steps: Multiply by P: d = P b Forwardsolve: L y = d Backsolve: U x = y Eigenvalues and eigenvectors: A nonzero vector v is an eigenvector ( ev ) and a number is its eigenvalue ( ew ) if A v = v . Characteristic equation: det( A I ) = 0 Equation of the eigenvector: ( A I ) v = Complex ews: Occur in conjugate pairs: 1 , 2 = i and ev s must also come in conjugate pairs: w = u i v . Vibrational modes: Eigenvalues are frequencies squared. Eigenvectors are modes. 63 64 REVIEW OF PART II Power Method:- Repeatedly multiply x by A and divide by the element with the largest absolute value....
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This note was uploaded on 02/09/2012 for the course MATH 344 taught by Professor Young,t during the Fall '08 term at Ohio University- Athens.

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lecture18 - Lecture 18 Iterative solution of linear...

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