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# HW3 - 5 x 1-10 x 2 =-95 10 2 x 1-20 x 2 =-188(a[0.5pt...

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EMAE 250.100: Homework 3 January 27, 2011 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. 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 , first rewrite the above equations in the form of A ~x = ~ b where ~x = ( x 1 , x 2 ) T and ~ b = ( b 1 , b 2 ) T . (a) [0.5pt] Compute the determinant of A . On the basis of (a) and (b), what would you expect regarding the system’s condition? (b) [0.5pt] Solve for x 1 and x 2 . (c) [1pt] Solve again, but with a 11 modified slightly to 0.52. Interpret your results. 3. Scale the above equations by multiplying 10, such that

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Unformatted text preview: 5 x 1-10 x 2 =-95; 10 . 2 x 1-20 x 2 =-188 . (a) [0.5pt] Compute the determinant of A and discuss the systems’s condition. (b) [0.5pt] Solve for x 1 and x 2 . (c) [1pt] Solve again, but with a 11 modiﬁed slightly to 5.02. Interpret your results. 1 4. For given the following three equations: x 1 + 2 x 3 = 3; 2 x 1 + x 2 + x 3 = 2; 4 x 1-2 x 2 + x 3 = 7 (a) [0.5pt] Rewrite the above equation in the form of A ~x = ~ b where A is a 3-by-3 matrix and ~x = ( x 1 ,x 2 ,x 3 ) T , and determine whether A is invertible or not. (b) [1.5pt] Find ~x using the Gauss Elimination method. (c) [1pt] If A is invertible (check by computing the determinant of A ), ﬁnd A-1 using the Gauss-Jordan Method. (d) [1pt] Find an LU decomposition of A . 2...
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