lec07_09202006 - 10.34, Numerical Methods Applied to...

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10.34, Numerical Methods Applied to Chemical Engineering Professor William H. Green Lecture #7: Introduction to Eigenvalues and Eigenvectors. Newton’s Method (Multi-dimensional) F (x true ) = 0 Newton: Taylor expansion around x guess If Δ x is small. Works well when x guess is close F (x guess ) + J (x guess ) Δ x ~ 0.0 x true x guess + Δ x Select x guess ; usually difficult to get a good guess compute F (x guess ), J (x guess ) guess x n m mn x F J = factorize J Æ L U solve L U Δ x = -F backsub: L V = -F ; U Δ x = V x new = x guess + Δ x if ||x new – x guess || < tolx CONVERGENCE if ||F (x new )|| < atolf rtol doesn’t work for F (x ) = 0 x guess Å x new Iterate from compute F (x guess ) If J is singular or poorly conditioned, will not be able to solve. If Δ x is big, method will not work. In general, radius of convergence is small - can bound Δ x size - can stop iteration after a certain number, for example, 20 iterations to see Assumption of Newton’s Method is x guess is VERY GOOD How close does x guess have to be to guarantee convergence? radius of convergence Backtrack Line Search If you think x new is too big, you can backtrack by looking at: ||F (x guess )||- ||F (x new )|| x
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lec07_09202006 - 10.34, Numerical Methods Applied to...

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