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Unformatted text preview: Homework 3 Solution 1. Determine the approximated root of the following equation: f ( x ) = 5 x 3 3 x 2 + 6 x 2 (a) [1pt] Using three iterations of the NewtonRaphson method with the initial guess, x = 1. (b) [1pt] Using three iterations of the secant method with the initial guesses, x 1 = 0 and x = 1. Solution: (a) NewtonRaphson iteration First order derivative: f ( x ) = 15 x 2 6 x + 6 x 1 = x f ( x ) f ( x ) = 1 6 15 = 0 . 6 x 2 = x 1 f ( x 1 ) f ( x 1 ) = 0 . 6 1 . 6 7 . 8 = 0 . 3949 x 3 = x 2 f ( x 2 ) f ( x 2 ) = 0 . 3949 . 2095 5 . 9698 = 0 . 3598 , f (0 . 3598) = 0 . 0033 (b) secant method iteration x 1 = x f ( x )( x 1 x ) f ( x 1 ) f ( x ) = 1 6 (0 1) 2 6 = 0 . 25 x 2 = x 1 f ( x 1 )( x x 1 ) f ( x ) f ( x 1 ) = 0 . 25 . 6094 (1 . 25) 6 + 0 . 6094 = 0 . 3192 x 3 = x 2 f ( x 2 )( x 1 x 2 ) f ( x 1 ) f ( x 2 ) = 0 . 3192 . 2281 (0 . 25 . 3192) . 6094 + 0 . 2281 = 0 . 3606 2. For a given equation, A~x = ~ b where A is an nbyn matrix and ~x and ~ b are nby1 vectors, wellconditioned systems have det A 6 = 0 and illconditioned systems have det A 0, i.e., the determinant is not zero but close to zero. Given the equations . 5 x 1 x 2 = 9 . 5 , 1 . 02 x 1...
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This note was uploaded on 11/14/2011 for the course EAME 250 taught by Professor Lee during the Spring '11 term at Case Western.
 Spring '11
 LEE

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