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Unformatted text preview: problem? Problem 3 [Heath 5.11, p.249] Suppose you are using the secant method to ﬁnd a root x * of a nonlinear equation f ( x ) = 0. Show that if at any iteration it happens to be the case that either x k = x * or x k-1 = x * (but not both), then it will also be true that x k +1 = x * . 1 Problem 4 [Heath 6.8, p.302] Consider the function f : R 2 → R deﬁned by f ( x ) = 1 2 ( x 2 1-x 2 ) 2 + 1 2 (1-x 1 ) 2 1. At what point does f attain a minimum? 2. Perform one iteration of Newton’s method for minimizing f using as starting point x = ± 2 2 ² T 3. In what sense is this a good step? 4. In what sense is this a bad step? [Note: The Newton method for optimization of a function of multiple variables f : R n → R gives the update step s k = x k +1-x k as the solution to the system H ( x k ) s k =-∇ f ( x k ), where H ( x k ) is the Hessian matrix of f evaluated at x k . See also Heath, section 6.5.3.] 2...
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- Fall '07
- Numerical Analysis, Continuous function, Secant method, Root-finding algorithm