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MTH510-Assignment2

# MTH510-Assignment2 - x 1-2 x 3 x 4 = 4 Use fractions...

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MTH 501/510 Assignment # 2 Fall 2011 DUE (at the beginning of your Lab): the week of October 17 1. Determine the root of f ( x ) = 2 e - x 2 - 1 (a) using Newton’s method (by hand) with x 0 = 1 until you have at least 3 significant figures. As part of your answer, include the value of | a | at each iteration. (b) using the secant method (by hand) with x 0 = 0 and x 1 = 1. Iterate until you have at least 2 significant figures. As part of your answer, include the value of | a | at each iteration. (c) using Newton’s method (with Matlab) with x 0 = 1 . 5 until you have at least 10 signif- icant figures. Include a printout or write-up of the commands and/or m-functions used in MATLAB, and of the output generated. Indicate clearly the approximate value of the root, as well as the approximate percent relative error associated with your estimate of the root. 2. Use Gaussian elimination with partial pivoting and backward substitution (by hand) to compute the solution of 2 x 1 + x 2 - x 3 - x 4 = 11 x 2 - x 3 + 2 x 4 = 7 x 1 - x 2 + 6 x 3 + x 4
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Unformatted text preview: x 1-2 x 3 + x 4 = 4 Use fractions throughout your calculations. 3. Let A = 1 0 3-1 1 2 2 1 . (a) Find (by hand) the LU factorization of A (without pivoting). (b) Use the LU factorization from (a) to ﬁnd the solution to Ax = b where b = [10 1 20] T . (c) Use the LU factorization from (a) to ﬁnd the determinant of A . 4. Let A = -60 10 2-. 1 . 4 60 10-2 . (a) Find || A || 1 , || A || ∞ and || A || f (by hand). (b) Use Matlab to ﬁnd || A || 2 , and also to check your answers in part (a). Provide a print-out or write-up of the commands entered in Matlab, as well as of the output obtained. (c) Using Matlab, ﬁnd the condition number of A based on the 2-norm. Do you expect the system Ax = b to be ill-conditioned? Explain. (Note: Material from Question 4 will not be covered on Test 1, but will be covered on Test 2)...
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