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HW5 - minimum 3 Consider the following function f x,y,z =...

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EMAE 250.100: Homework 5 February 10, 2011 1. Consider the following function: f ( x ) = 1 + 4 x - 2 x 2 + cos x (a) [2pt] Find an maximum using three iterations of the Golden Section search with initial guesses x l = - 5 and x u = 5. (b) [2pt] Find an maximum using three iterations of the quadratic interpolation with the initial guesses of x 0 = - 5, x 1 = - 2, and x 2 = 3. (c) [2pt] Find an maximum using three iterations of the Newton’s method with an initial value x 0 = 5. 2. Consider the following function: f ( x, y ) = ( x - 3) 2 + ( y - 2) 2 (a) [1pt] Find the gradient vector and the Hessian matrix. (b) [0.5pt] Find an optimum and determine whether it is the maximum or the
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Unformatted text preview: minimum. 3. Consider the following function: f ( x,y,z ) = ln( x 2 + 3 xy + 2 y 2 + 4 z 2 ) Note: ‘ ln ’ is the natural logarithm and the derivative is computed by d dx (ln f ( x )) = d dx ( f ( x )) f ( x ) . (a) [0.5pt]Find the gradient vector. (b) [2pt]Perform one iteration of the gradient method with the initial point ~x = ( x ,y ,z ) T = (1 , 1 , 1) T ; check the function values and the gradient vector at ~x and ~x 1 and discuss the results. 1...
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