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review_5 - CS205 Review Session#4 Notes Secant Formula...

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Unformatted text preview: CS205 Review Session #4 Notes Secant Formula Consider the following alternate form of the secant method: x k +1 = x k- 1 f ( x k )- x k f ( x k- 1 ) f ( x k )- f ( x k- 1 ) This formula leads to an indefinite form ( ) as the current iterate asymptotically approaches the true solution. If x * is an analytic root of f ( x ) s.t. f ( x * ) = 0, we can see that x k ≈ x k- 1 ≈ x * and f ( x k ) ≈ f ( x k- 1 ) ≈ 0 as x k- 1 , x k → x * . This observation is significant, since the aforementioned update formula relies on the division of two quantities that are very close to zero, which may lead to numerical instability. Similar behavior may be seen in the more usual form of the secant update formula: x k +1 = x k- f ( x k ) x k- x k- 1 f ( x k )- f ( x k- 1 ) Mean Value Theorem Consider a real-valued function f ( x ) that is differentiable and continuous on the interval [ x k , x k- 1 ]. Then, the Mean Value Theorem states that: ∃ ˆ x ∈ [ x k , x k- 1 ] s . t . f ( x k )- f ( x k- 1 ) = f (ˆ x )( x k- x k- 1 ) Furthermore, it is clearly the case that f (ˆ x ) ≈ f ( x * ) as x k , x k- 1 → x * ....
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This note was uploaded on 01/05/2012 for the course CS 205A taught by Professor Fedkiw during the Fall '07 term at Stanford.

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review_5 - CS205 Review Session#4 Notes Secant Formula...

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