課程名稱︰數值方法 課程性質︰必修 課程教師︰張恆華 開課學院：工學院 開課系所︰工科海洋系 考試日期（年月日）︰2011/6/22 (三) 考試時限（分鐘）：3小時 (9:10am ~ 12:10pm) 是否需發放獎勵金：是 （如未明確表示，則不予發放） Numerical Methods: 505 29120/ESOE 2024 This is an open book, open notes exam. Most electronic devices are forbidden on your person, including cell phones,iPods, iPads, headphones, PDAs, and computers. Turn your cell phone off and leave all electronics EXCEPT calculators in your backpack or bag, or risk getting a zero on the exam. 試題 : [ 1 2 3 ] 1. Determine the LU factorization of the matrix a = [ 4 5 6 ] using the Gauss [ 3 2 2 ] elimination method, i.e, show that a = LU. (10 pts) 2. Carry out the first three iterations of the solution of the following system of equations using the Gauss-Seidel iterative method. For the first guess of the solution, take the value of all the unknowns to be zero. 8 x1 + 2 x2 + 3 x3 = 51 2 x1 + 5 x2 + x3 = 23 -3 x1 + x2 + 6 x3 = 20 (10 pts) 3. Determine the solution of the simultaneous nonlinear equations 2 y = -x + x + 0.75 2 y + 5xy = x Use the Newton-Raphson method and employ initial guesses of x = y = 1.2 . Show the estimated solution and the corresponding approximate percent relative error of each variable for the first three iterations. (18 pts) 4. An investigator has reported the data tabulated below for an experiment to determine the growth rate of bacteria k (per d) as a function of oxygen concentration c (mg/L). It is known that such data can be modeled by the following equation: 2 k c max k = ————— 2 c + c s where c and k are parameters. (15 pts) s max ——————————————————— c 0.5 0.8 1.5 2.5 4 k 1.1 2.4 5.3 7.6 8.9 ——————————————————— (a) Use a transformation to linearize this equation. (5 pts) (b) Use linear regression to estimate c and k . (8 pts) s max (c) Predict the growth rate at c = 2 mg/L. (2 pts) 5. (a) Derive a general matrix form of an mth-order polynomial regression with [A]{p} = {b}, where [A] is a matrix of the calculated values of the basis functions, {b} is a column vector of the calculated values of independent and dependent variables, and {p} is a column vector containing the unknown coefficients. (12 pts) (b) Develop an M-file to implement the polynomial regression in (a). Pass the M-file two vectors holding the x and y values along with the desired order m. Return a vector p containing the coefficients. (13 pts) 6. The fuel economy of a car (miles per gallon) varies with its speed. In an experiment, the following five measurements are obtained: (12 pts) ————————————————————————— Speed (mph) 10 25 40 55 70 Fuel economy (mpg) 12 26 28 30 24 ————————————————————————— (a) Determine the fourth-order polynomial in the Lagrange form that passes through the points. (10 pts) (b) Use the polynomial to calculate the fuel economy at 65 mph. (2 pts) 7. Use linear splines interpolation with the data in Problem 6, to calculate the fuel economy at a speed of (a) 30 mph and (b) 65 mph. (10 pts) --

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