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MA371/MA307 - FINAL EXAM
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- Wilfrid Laurier University (WLU )
MA371/MA307 - FINAL EXAM Question and Answer
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WILFRID LAURIER UNIVERSITYWATERLOO, ONTARIO
Session: Winter, 2021 Name:
Course no: MA307 ID #:
Title: Numerical Analysis Section:
Instructors: Dr. Roderick Melnik (Section B, TuTh 11:30-12:50pm)
Number of pages: 2 plus cover page
Length of examination: 2.5 hours
Examination aids allowed: See next page.
Note that some of the items below are not applicable, refer to Addendum of March 13, 2020 and instructions provided at MyLS. The doors of the examination room will
be opened approximately 10 minutes before the start of the examination. Candidates will be permitted to enter the examination room quietly up to one half hour after
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the scheduled start of the exam. Candidates arriving late will not be allowed any extra time.
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Candidates must not begin the examination or attempt to read the examination questions until instructed to do so.
**THE UNIVERSITY IS NOT RESPONSIBLE FOR THE LOSS OF VALUABLES BROUGHT INTO THE EXAM LOCATIONS
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OR CLASSROOMS WHERE EXAMS ARE BEING WRITTEN.
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Candidates once having entered, may not leave the exam room before completing and submitting the exam unless accompanied by a Proctor. Candidates are not permitted
to submit their examination and leave the examination room until 1 hour after the examination has begun, and in no case before their attendance has been taken. In
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no case may a candidate leave the room temporarily, for any reason, until 30 minutes after the start of the examination. In order that remaining candidates are not
disrupted, candidates must remain seated and may not leave the examination room during the last 15 minutes of the examination session.
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At the close of the examination period, candidates must stop writing immediately. The Presiding Officer may seize the papers of candidates who fail to observe this
requirement, and a penalty may be imposed at the discretion of the instructor. Candidates must submit all their work, according to the instructions of the Presiding
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Officer, including all materials and a copy of the examination paper with their name and student ID number written on it. Unused examination booklets may not be
taken from the examination room.
A candidate who leaves before the examination is over must hand in all completed and attempted work, notes made during the exam, and a copy of the examination paper
with their name and student ID number on it.
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Talk or any form of communication between candidates is absolutely forbidden. No information of any kind is to be written on the question paper or on scrap paper for
the purpose of assisting other candidates. Responses to questions must not be done in an exaggerated way or in a manner that will involve transmission of information
to others.
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Candidates must remain seated during the examination period. A candidate needing to speak to the proctor (e.g. to ask for additional supplies or to request permission
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to leave the examination room for any reason) should so indicate by raising his or her hand.
Questions concerning possible errors, ambiguities or omissions in the examination paper must be directed to the proctor who will investigate them through the proper
channels. The proctor is not permitted to answer questions other than those concerning the examination paper.
Candidates must not use or attempt to use any improper source of information. No candidates for an examination may bring into the examination room any books, notes
or other material containing information pertaining to the examination unless the examiner has given instructions that such material will be allowed and this instruction
is specified on the examination paper. Any item brought into the examination room is subject to inspection.
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No briefcases, backpacks or other bags and carriers may be brought to the desk site where the candidate is writing the examination. These bags should be left outside the examination room.
If books, notes etc. cannot be left outside the examination room, they must be put at the front of the examination room in a place designated by the proctor before a candidate takes a seat.
Candidates are advised not to bring valuables to the examination room.
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No electronic or communication devices will be allowed in the examination room, including cell phones, smartphones, pagers, etc. Cell phones will be taken away if
found and an Irregularity notice will be filed with the Integrity Office. Calculators are not allowed unless specified by the instructor and indicated on the examination
paper. Only non programmable calculators without lids, authorized by the instructor, will be allowed. It is the candidate’s responsibility to ascertain whether the use of
calculators is permitted, and, if it is, whether any restrictions are imposed on the types of calculators that may be brought to the examination. No pencil cases are allowed
on the desks.
Translation dictionaries (e.g. English-French) or other dictionaries, (thesaurus, definitions, technical) are not allowed unless specified by the instructor and indicated on
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the examination paper. Electronic dictionaries are never allowed.
Except for bottled water (with label removed), no food or drink is allowed in the examination room. Candidates with health problems that warrant relaxation of this
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regulation should provide medical documentation to the presiding officer prior to the beginning of the examination. Such students should restrict themselves to those
items and packaging that will least distract other examinees.
Candidates are expected to write their examinations in an honest and straightforward manner. Where there are reasonable grounds for believing a violation of exam
protocol has occurred, the candidate will be subject to the disciplinary procedures and sanctions according to the University Calendar.
Only currently registered students will be permitted to write the final exam.
Examinations conducted at Wilfrid Laurier University will be bound by WLU regulations, regardless of where the candidate is registered.
Approved by Senate ( Oct 27/2003) Updated July 2016
Total Value: Numeric Value 67 marks
Show all your work. Insufficient justification will result in a loss of marks.
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, MA307–Final Examination April 15, 2021 Page 1 of 2
Numerical Analysis: MA307
(Final Exam)
Semester: Winter 2021
April 15, 2021
This is an open-book examination and any auxiliary materials, in particular your own lecture notes,
are allowed. Your careful explanations of all steps are absolutely essential in this test. Please write
clearly. All other details can be found at the MyLS of the course. The deadline for uploading your
assignment as a single pdf file is 21:40pm on April 15. No assignment files will be accepted after that1 .
Marks for each question are given in brackets. Good luck!
Question 1 (25 marks in total) From observations, we obtained the following data, representing a
dependency of y on x (i = 1, ..., 5):
Table 1:
xi 1 2 3 4 5
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yi 1.3 3.5 4.2 5.0 7.0
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✓
(a) (10 marks) Write down the system of algebraic equations to be solved in order to construct the
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least-squares polynomial of degree two for the data in Table 1. Writing this system in the form of
normal equations can help you to complete this task quicker. The matrix and the right-hand side of
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this system should be written with all numerical values substituted.
✓
(b) (13 marks) Solve the resulting system by using the LU decomposition. Take all diagonal elements
of matrix L to be equal to 1. All steps should be shown.
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✓
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(c) (2 marks) Based on your result from Question 1 (b), write explicitly the least-squares polynomial of
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degree two for the data given in Table 1, with all numerical coefficients in place.
Question 2 (30 marks in total) Consider function u(x, t) of two variables, x and t, in the following
domain: QT = {(x, t) : x 2 [0, 1], t 2 [0, 0.1]}.
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(a) (8 marks) Construct the grid in this domain with x = 1/3 and t = 0.05, circling all grid points.
Approximate @u/@t at grid point x = 2/3, t = 0.05 with (i) forward di↵erence derivative, (ii) backward
di↵erence derivative, (iii) central di↵erence derivative (sometimes it is called the three-point midpoint
formula). In doing so, denote the grid function approximating u(x, t) as yij , where i and j indicate
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discretization points in x and t, respectively. For example, the value of this function at x = 0, t = 0 will
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be denoted as y00 , and its value at the grid point where the approximation is sought will be denoted
as y21 . State the accuracy of the approximation in each of these three cases. Then, approximate
@ 2 u/@x2 at the same grid point (x = 2/3, t = 0.05) with the second-order central di↵erence derivative
(sometimes it is called the second derivative midpoint formula) and draw a schematic diagram to show
the points involved in this last approximation for the constructed grid, as well as the values of the grid
function at these points.
(b) (2 marks) Consider now the following equation
@u @ 2u
= c2 2 + f (x, t), 0 < x < 1, 0 < t 0.1, (1)
@t @x
1
If you are an Accessible Learning student, please see the postscript in the Notes on the Final Examination found at
the MyLS of the course.
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