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CS3911 Intro. to Numerical Methods with Fortran Final { Fall 2007 1 CS3911 Introduction to Numerical Methods with Fortran Final Exam Fall 2007 200 points { 9 pages Name: • Write to the point. If I don't understand what you are saying, I believe, in most cases, you don't understand the subject. • Justify your answer with a convincing argument. If there is no justification when it is needed, you will receive ZERO point for that question even though you have provided a correct answer. I consider a good and correct justification more important than a right answer. Thus, if you provide a very vague answer without a convincing argument to show your answer being correct, you will likely receive a very low grade. • For computation-based problems, always show the com- putation steps. Only providing an answer receives ZERO point unless the problem only requires an answer. • Your calculations should have at least 5 significant digits. • Do those problems you know how to do first. Otherwise, you may not be able to complete this exam on time. None of the following problems require extensive calculations. CS3911 Intro. to Numerical Methods with Fortran Final { Fall 2007 2 1. Accuracy and Reliability: (a) [15 points] Euler proved the following result centuries ago: n 1 lim 1+ = e = 2.71828 . . . n→∞ n A programmer wrote a program to verify this result using single precision (i.e., about 7 signif- icant digits) and a Fortran 90 statement E = (1.0 + 1.0/n)**n in a DO loop, where n is an INTEGER and E is a REAL, and obtained the following table with n = 10, 100, . . ., 10000000 and 100000000. n Computed (1 + 1/n)n Absolute Error 10 2.593743 0.12453866 100 2.7048113 0.013470411 1000 2.7170507 1.2309551E-3 10000 2.718597 3.1518936E-4 100000 2.7219622 3.6804676E-3 1000000 2.5952267 0.12305498 10000000 3.2939677 0.575686 100000000 1.0 1.7182817 The problem is that the computed value (1 + 1/n)n is about 2.718597 when n = 10000; however, accuracy of further results gets worse if n > 10000, and eventually the computed result becomes 1, which is obviously incorrect. This programmer is sure that his/her program logic is correct. So, what is the numerical problem (or problems) in this computation? You should not make any correction to the Fortran 90 statement. What you have to do is to point out the numerical problem (or problems) with a clear and to-the-point explanation. Vague answers such as \because of cancelation" and meaningless answers such as \should use double precision" will receive zero point. CS3911 Intro. to Numerical Methods with Fortran Final { Fall 2007 3 2. Linear Algebra: (a) [15 points] Matrix A and its LU-decomposition L and U are shown below: 1 2 1 0 1 2 A= A=L·U = · 2 7 2 1 0 3 Find A−1 using the given LU-decomposition. Show all calculation steps clearly. Otherwise, you will receive zero point. (b) [15 points] Use Jacobi method to find all eigenvalues and their corresponding eigenvectors of the following symmetric matrix A. You should clearly provide all computation details, and clearly match each eigenvalue with its corresponding eigenvector. Otherwise, you will risk low grade. Additionally, you will receive zero point if you do not use Jacobi method. 1 0 1 A= 0 3 0 1 0 1 CS3911 Intro. to Numerical Methods with Fortran Final { Fall 2007 4 3. Interpolation: Suppose there are five data points (x0, y0 ) = (−2, 0), (x1, y1) = (−1, 3), (x2, y2) = (0, 4), (x3 , y3) = (1, 3) and (x4, y4 ) = (2, 0). Do the following problems: (a) [10 points] Given a polynomial of degree n as follows, Pn (x) = a0 + a1 x + a2x2 + a3x3 + · · · + an xn suggest an efficient way to evaluate Pn (x) so that the complexity is O(n). You should provide an algorithm and its complexity analysis. A method that does not achieve O(n) receives zero point. (b) [15 points] Find the Lagrange interpolating polynomial for the given data points. You should show all computation steps. Only providing an answer and/or using a wrong method receives zero point. CS3911 Intro. to Numerical Methods with Fortran Final { Fall 2007 5 (c) [15 points] Find the Newton interpolating polynomial for the given data points using the divided difference method. You should show all computation steps. Only providing an answer and/or using a wrong method receives zero point. 4. Numerical Differentiation: (a) [15 points] Use the 3-point backward difference method to compute the derivative of f (x) = ex sin(x) at x = 1 with ∆ = 0.01. You should show all computation steps. Only providing an answer and/or using a wrong method receives zero point. CS3911 Intro. to Numerical Methods with Fortran Final { Fall 2007 6 (b) [20 points] Use Richardson's extrapolation method to compute the derivative of f (x) = ex sin(x) at x = 1 with initial ∆ = 0.1. You should carry out the extrapolation steps un- til the computation for ∆ = 0.025 completes, and clearly indicate the desired answer. You should show all computation steps. Only providing an answer and/or using a wrong method receives zero point. 5. Numerical Integration: The following integral has an exact solution: 1 ex dx = e1 − e0 = e − 1 = 1.71828 . . . 0 (a) [20 points] Use Romberg's method to compute the above integration and use the trapezoid method for column 0. You should carry out all steps until the computation for ∆ = 0.25 completes, and clearly indicate the desired answer. You should show all computation steps. Only providing an answer and/or using a wrong method receives zero point. CS3911 Intro. to Numerical Methods with Fortran Final { Fall 2007 7 (b) [20 points] Use the iterative 3-point Simpson method to compute the above integration. You should start with ∆ = 0.5, update the result in each iteration, and stop after the result of ∆ = 0.125 is obtained. You should show all computation steps. Only providing an an- swer and/or using a wrong method receives zero point. 6. ODE Initial Value Problems: Solve y = xy with initial value y(0) = 1. (a) [10 points] Use Euler's method to solve the given equation on [0, 4] with ∆ = 1, and fill your results in the following table. Iteration x y Initial Value 0 1 1 1 2 2 3 3 4 4 CS3911 Intro. to Numerical Methods with Fortran Final { Fall 2007 8 (b) [15 points] Solve the given equation on [0, 4] using Ralston's second-order Runge-Kutta method with ∆ = 1, and fill your results in the following table. Iteration x y Initial Value 0 1 1 1 2 2 3 3 7. Random Numbers: (a) [15 points] Suppose a random vertical cut followed by a random horizontal cut are applied to a square of edge length 1. This yields four smaller rectangles as shown below. What is the probability of the area of the largest rectangle being larger than or equal to 0.7? The following shows a sequence of pre-generated random numbers. Each number has five digits, and a decimal point is assumed to appear to the left of the first digit. For example, random number 53479 should be interpreted as 0.53479. Therefore, all random numbers are in the range of [0,1). Use Monte Carlo method with 10 runs to find an approximation of this probability. You should show all computation steps. Only providing an answer and/or using a wrong method receives zero point. x1 to x8 53479 81115 98036 12217 59526 40238 40577 39351 x9 to x16 43211 69255 97344 70328 58116 91964 26240 44643 x17 to x24 83287 97391 92823 77578 66023 38277 74523 71118 CS3911 Intro. to Numerical Methods with Fortran Final { Fall 2007 9 Grade Report Problem Possible You Received 1 a 15 a 15 2 b 15 a 10 3 b 15 c 15 a 15 4 b 20 a 20 5 b 20 a 10 6 b 15 7 a 15 Total 200 Some Important and Complex Formulas The following has some important and complex formulas discussed in class. They are provided for your convenience. If you have anything in doubt, use the version discussed in class or ask questions. n n ai (x − xj ) i=0 j=0,j=i 1 (3fi − 4fi−1 + fi−2 ) 2∆ n i−1 f [x0 ] + f [x0 , x1, . . . , xi] (x − xj ) i=1 j=0 f0 + fn n−1 ∆ + fi 2 i=1 2(n − i) + 1 xi = cos π where i = 0, 1, . . ., n 2n + 2 1 (−fi+2 + 4fi+1 − 3fi ) 2∆ di,j − di−1,j di,j + 4j+1 − 1 m m ∆ (f0 + f2m ) + 4 f2i−1 + 2 f2i 3 i=1 i=1 n (x − x0 )(x − x1 ) · · · (x − xi−1 )(x − xi+1 ) · · · (x − xn ) yi i=0 (xi − x0)(xi − x1 ) · · · (xi − xi−1 )(xi − xi+1 ) · · · (xi − xn ) 1 c1 = 0 c2 = 1 p2 = a2,1 = 2 1 c1 = c2 = p2 = a2,1 = 1 2 1 2 3 c1 = c2 = p2 = a2,1 = 3 3 4 1

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