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hw3solution

# hw3solution - Homework 3 Solution 1 Determine the...

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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 Newton-Raphson method with the initial guess, x 0 = 1. (b) [1pt] Using three iterations of the secant method with the initial guesses, x - 1 = 0 and x 0 = 1. Solution: (a) Newton-Raphson iteration First order derivative: f 0 ( x ) = 15 x 2 - 6 x + 6 x 1 = x 0 - f ( x 0 ) f 0 ( x 0 ) = 1 - 6 15 = 0 . 6 x 2 = x 1 - f ( x 1 ) f 0 ( x 1 ) = 0 . 6 - 1 . 6 7 . 8 = 0 . 3949 x 3 = x 2 - f ( x 2 ) f 0 ( x 2 ) = 0 . 3949 - 0 . 2095 5 . 9698 = 0 . 3598 , f (0 . 3598) = 0 . 0033 (b) secant method iteration x 1 = x 0 - f ( x 0 )( x - 1 - x 0 ) f ( x - 1 ) - f ( x 0 ) = 1 - 6 × (0 - 1) - 2 - 6 = 0 . 25 x 2 = x 1 - f ( x 1 )( x 0 - x 1 ) f ( x 0 ) - f ( x 1 ) = 0 . 25 - - 0 . 6094 × (1 - 0 . 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 - - 0 . 2281 × (0 . 25 - 0 . 3192) - 0 . 6094 + 0 . 2281 = 0 . 3606 2. For a given equation, A~x = ~ b where A is an n-by-n matrix and ~x and ~ b are n-by-1 vectors, well-conditioned systems have ‘det A 6 = 0’ and ill-conditioned systems have ‘det A 0’, i.e., the determinant is not zero but close to zero. Given the equations 0 . 5 x 1 - x 2 = - 9 . 5 , 1 . 02 x 1 - 2 x 2 = - 18 . 8 , (a) [0 . 5pt]Compute the determinant of A. On the basis of (a) and (b), what would you expect regarding the system’s condition?

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hw3solution - Homework 3 Solution 1 Determine the...

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