Midterm 1 (♣) MACM 316 - D100 Summer 2021 Instructor: Pengyu Liu June 11, 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 3 questions on 7 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 5 2 10 3 10 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. Circle the correct answer. [1] (a) Which real number is represented by the IEEE 64-bit floating point number? 1 10000001001 0100100100000000000000000000000000000000000000000000 • A. −292 • B. 292 • C. −1316 • D. 1316 • E. none of the above [1] (b) The following figure shows the graph of f(x) = x3 − 6x2 + 9x− 3. In solving the equation f(x) = x3 − 6x2 + 9x − 3 = 0 by the bisection method (middle point), which one of the following values would the sequence {pi} approximately converge to, given the initial interval [0, 4]? • A. 0.4679 • B. 1.0001 • C. 1.6527 • D. 2.9931 • E. 3.8794 MACM 316 - D100 [1] (c) Let an = 1 n(n+ 1) and N = 9999999. In IEEE 64-bit floating point arithmetic, which statement is true? • A. 0 <∑Nj=1 an =∑Nj=1 aN−j < 2. • B. 0 <∑Nj=1 an <∑Nj=1 aN−j < 1. • C. 0 <∑Nj=1 aN−j <∑Nj=1 an < 1. • D. 1 <∑Nj=1 aN−j <∑Nj=1 an < 2. • E. 1 <∑Nj=1 an <∑Nj=1 aN−j < 2. [1] (d) Which binary number represents the decimal number 0.7? • A. 0.10010011 • B. 0.11010011 • C. 0.01010011 • D. 0.10110011 • E. none of the above [1] (e) Compute √ 2021611 − √2021609 directly and without using subtraction in the floating point systems with base 10 and 4-digit mantissa using rounding. Which of the following are the correct answers? • A. directly: 0.0000, without subtraction: 0.7033× 10−3 • B. directly: 0.0001, without subtraction: 0.7033× 10−3 • C. directly: 0.0001, without subtraction: 0.3517× 10−3 • D. directly: 0.0000, without subtraction: 0.3517× 10−3 • E. none of the above MACM 316 - D100 2. Let f(x) = (x− 3)2. Answer the following questions (present detailed solutions). [2] (a) Write down the iteration formulas for solving the equation f(x) = (x− 3)2 = 0 by Newton’s method and by the secant method. [3] (b) Start with initial value p0 = 2, and compute the next three iterates p1, p2, p3 in solving the equation f(x) = (x − 3)2 = 0 by Newton’s method. (write your answers in fractions) MACM 316 - D100 [3] (c) Compute the absolute error and the relative error between p1, p2, p3 in (b) and the actual solution to the equation p = 3. (write your answers in fractions) [2] (d) Let e1 = |p1−p|, e2 = |p2−p|, e3 = |p3−p| be the absolute errors in (c), use these values to determine if the iteration sequence is linearly convergent or quadratically convergent. MACM 316 - D100 3. Let f(x) = x2 − 2x− 8. Answer the following questions (present detailed solutions). [2] (a) Show that solving the equation f(x) = x2− 2x− 8 = 0 is equivalent to finding the fixed point of the following functions. g1(x) = 0.5x 2 − 4 and g2(x) = √ 2x+ 8 [2] (b) x = −2 and x = 4 are two solutions to the equation f(x) = x2−2x−8 = 0. Show that g2(x) will give a convergent iteration sequence to p = 4, provided a sufficiently closed initial value p0. What is the speed of convergence? [2] (c) Given that p0 = 3, compute the next two iterates p1 and p2 in solving the equation f(x) = x2−2x−8 = 0 by g2(x). (round your answers to at least 4 decimal places) MACM 316 - D100 [2] (d) Show that g1(x) will not give a convergent iteration sequence to p = 4, for any initial value p0. Write down the iteration function G(x) using Aitken-Steffensen method to accelerate the iteration sequence given by g1(x). [2] (e) Given that p0 = 3, compute the next two iterates p1 and p2 in solving the equation f(x) = x2−2x−8 = 0 by G(x). (round your answers to at least 4 decimal places) MACM 316 - D100
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