Midterm 2 (♣) MACM 316 - D100 Summer 2021 Instructor: Pengyu Liu July 16, 2020, 10:30 – 11:20 am Name: (please print) family name given name SFU ID: @sfu.ca student number SFU-email Signature: Instructions: 1. Do not open this booklet until 10:30am. 2. Write your name above in block letters. Write your SFU student number and email ID on the line provided for it. 3. Write your answer in the space provided below the ques- tion. If additional space is needed then use the back of the previous page. Your final answer should be simpli- fied as far as is reasonable. 4. To receive full credit for a particular question your so- lution must be complete and well presented. 5. This exam has 4 questions on 8 pages (not including this cover page). Once the exam begins please check to make sure your exam is complete. 6. You may used a calculator. No books, papers, or other electronic devices shall be used for and during the ex- amination. 7. During the examination, copying from, communi- cating with, or deliberately exposing written pa- pers to the view of, other examinees is forbidden. Question Maximum Score 1 6 2 6 3 6 4 7 Total 25 Academic Honesty Agreement Please read this academic honesty agreement before answering the ques- tions. By uploading your answers to the questions in this exam to Crowdmark, you acknowledge that no aid of any kind other than a calculator is used for and during the exam. You pledge that you do not rely on other sources than your knowledge and that you do not give help to or receive help from any other person in writing this exam. You also acknowledge that you may be asked to participate in a post-exam interview to demonstrate knowledge of the subject matter that is consistent with your performance on this exam. MACM 316 - D100 1. Consider the following matrix A and its inverse A−1. Answer the following questions with detailed solutions. A = 1 0 10 1 0 1 1 2 A−1 = 2 1 −10 1 0 −1 −1 1 [4] (a) Compute the 1-norm and the ∞-norm of A and condition numbers corresponding to the 1-norm and the ∞-norm. [2] (b) Given the following LU factorization of A and the vector b, solve the system of equations Ax = b using the matrices L and U . A = LU = 1 0 0 0 1 0 1 1 1 1 0 1 0 1 0 0 0 1 b = 4 2 7 blank page for additional work MACM 316 - D100 2. Answer the following questions regarding approximating a function f(x) on the interval [−1, 1] with detailed solutions. [2] (a) If we use equidistant interpolation points, and assume x0 = −1 and x6 = 1, what are x1, x2, x3, x4 and x5? Write your answers as integers or fractions. [2] (b) If we use Chebyshev interpolation points, and assume x0 = −1 and x6 = 1, what are x1, x2, x3, x4 and x5? Write your answers as integers or fractions. [2] (c) Write down the barycentric formula. Write down the definition of wi in the barycen- tric formula. Do not simplify or evaluate the formulas. blank page for additional work MACM 316 - D100 3. Answer the following questions regarding the matrix A and the vector b with detailed solutions. A = 3 64 8 0 1 b = 21 3 [3] (a) QR factorize the matrix A by the Gram-Schmidt process. [3] (b) Solve the least square problem with A and b using the matrices Q and R. blank page for additional work MACM 316 - D100 4. Consider the following data points (x0, f0) = (0, 1), (x1, f1) = (1, 0), (x2, f2) = (2, 1). Answer the following questions with detailed solutions. [2] (a) Write down the Vandermonde matrix associated with the three data points given above. Compute the polynomial interpolating the three data points by the Van- dermonde matrix. [2] (b) Compute all divided differences for the three data points, and use the divided dif- ferences to compute the Newton form of the the interpolation polynomial. [3] (c) In finding the natural cubic spline that interpolates the three data points, let p0(x) = a3x 3 + a2x 2 + a1x + a0 be the cubic polynomial on the interval [x0, x1] and p1(x) = b3(x − 2)3 + b2(x − 2)2 + b1(x − 2) + b0 be the cubic polynomial on the interval [x1, x2]. Our friend Harry the magician has solved a3 = 1/2 for us. Find the natural cubic spline by solving the rest of the unknowns. blank page for additional work
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