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Unformatted text preview: Louisiana State University CE 2720 (Fall 2009) MidTerm Exam 1 (Practice) Formulas are provided in the last page. Problem 1 (25 points) Determine the root of the following function using the bisection method: 1.04 1.26 0.0307 Use initial guesses of xl=1.4 and xu=2.5 (conduct five iterations and estimate the error for each iteration). 1 Louisiana State University CE 2720 (Fall 2009) MidTerm Exam 1 (Practice) Problem 2 (20 points) Use the Newton‐Raphson method to find the root of: 2 Use an initial guess of x=1.5 (conduct four iterations with error estimation for each iteration). 2 Louisiana State University CE 2720 (Fall 2009) MidTerm Exam 1 (Practice) Problem 3 (25 points) Consider the following function: 2 8 5 Use the Golden‐section search method to find the minimum of the function in the range between ‐2 and 1 (conduct five iterations). Estimate the error for each iteration. 3 Louisiana State University CE 2720 (Fall 2009) MidTerm Exam 1 (Practice) Problem 4 (20 points) Use Gauss Elimination to solve the systems of equations: 4x + 8y +4z = 80 2x + y ‐4z = 7 3x –y +2z = 22 4 Louisiana State University CE 2720 (Fall 2009) MidTerm Exam 1 (Practice) Problem 5 (10 Points) Indicate whether these statements are true or false: 1‐ An optimization problem can be solved using the Newton‐Raphson method. 2‐ Minimizing the cost of a highway project is an example of a problem that can be solved using optimization. 3‐ A bracketing method such as the bisection method requires two values of x that bound the root but a change in sign is not necessarily needed between these two values. 4‐ The bisection method always converges to the root if setup correctly but it converges slowly. 5‐ The benefit of the secant method compared to Newton‐Raphson is that it does not require estimating the derivative. 6‐ If no divergence problems are encountered, Newton‐Raphson converges faster than the bisection method. 7‐ The modified secant method is considered a bracketing method. 8‐ The modified secant method converges faster than the secant method. 9‐ The constant φ in the golden search method varies from one iteration to the next. 10‐ For a given function, a local maximum is smaller than a global maximum. 5 Louisiana State University CE 2720 (Fall 2009) MidTerm Exam 1 (Practice) Formula Sheet: εa = x new − x old r r x100% x new r xr = xl + xu 2 φ = 1.618034 d = (φ − 1)( x u − x l ) x1 = x l + d x2 = xu − d – – If f(x1)>f(x2): – x1 becomes the new xu – x2 becomes the new x1 – xL remains the same If f(x1)<f(x2): – x1 becomes the new x2 – x2 becomes the new xL – xu remains the same ε a = (2 − φ) xu − xl 100% x opt 6 ...
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 Fall '08

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