UNIVERSITY
DEPARTMENT RYERSON OF MATHEMATICS
MTH 510/501 - Numerical Analysis
MIDTERM - FALL 2009
Last Name (print):
First Name (print):
Student ID Number:
Signature:
Course
MTH 501
MTH 510
MTH 510
MTH 510
MTH 510
MTH 510
MTH 510
MTH 510
MTH 510
Instructor:
Course:
Section:
Section
1
1
2
3
4
5
6
7
8
Lab day/time
Fri 8-9am
Thurs 3-4pm
Wed 2-3pm
Thurs 9-10am
Wed 9-10am
Thurs 9-10am
Thurs 1-2pm
Wed 3-4pm
Wed 4-5pm
Room
LIB 393
LIB 393
LIB 393
LIB 393
KHW 379
KHW 377
KHW 377
LIB 393
LIB 393
Date: October 26, 2009, 3:30 pm
Instructor
Dr. Ilie
Dr. Ilie
Dr. Ilie
Dr. Ilie
Dr. Rohlf
Dr. Rohlf
Dr. Rohlf
Dr. Rohlf
Dr. Rohlf
Time Allowed: 90 minutes
INSTRUCTIONS:
1.
Verify that the test contains all 9 pages, including this cover page.
2.
Use a pen or pencil and write legibly for full marks.
3.
The examination has two parts: Part A consists of fullsolution questions with the mark for each full solution
as indicated. Answer all questions in the space provided.
Clearly explain your methods, and show all relevant steps
in the solution. An answer consisting only of the nal result
will be given little or no credit.
Part B contains Multiple Choice questions. Clearly
write your answer in the space provided. No part marks
will be given and no marks will be deducted for incorrect
answers. If more than one answer is given, a mark of zero
will be assigned to that question.
4.
This is a closed-book test.
5.
When not specied, six signicant digits of accuracy is
sucient.
6.
The last page is for rough work.
DO NOT SEPARATE THE PAGES
7.
Permitted Aids:
i) One handwritten 8.5 x 11 inch Formula sheet (both sides),
ii) Non-programmable scientic calculators.
For instructors use only.
Question(s)
Value
1
10
2
8
3
8
4
12
5
4
6
10
7-10
8
Total
60
Mark
MTH 510 - MIDTERM
- Page 2 -
FALL 2009
1. (10 marks) (a) Find the rst 5 terms of the MacLaurin series for
1 + x.
f (x) =
(b) Using part (a), add terms one at a time to estimate 1.02. After each new term is
added, compute the approximate percent relative error. Add terms ONLY until the absolute
value of the approximate error falls below an error criterion conforming to two signicant
gures.
MTH 510 - MIDTERM
- Page 3 -
FALL 2009
2. (8 marks) Use the Newton-Raphson method with x0 = 1 to approximate root the of
f (x) = x ex on the interval [1, 2] until the absolute value of the approximate percent
relative error is less than 2 %.
MTH 510 - MIDTERM
- Page 4 -
FALL 2009
3. (8 marks) (a) Using 3 iterations of the Bisection algorithm, estimate the root of
f (x) = x3 6x2 + 10
in the interval [1, 2]. What is the maximum error made in the estimate for the root of f (x)?
(b) How many iterations would be required for the true error to be less than 105 ?
MTH 510 - MIDTERM
- Page 5 -
FALL 2009
4. (12 marks) Consider the system Ax = b where
12
A = 1 7
22
3
2
4 and b = 3
38
4
9
(a) Find the LU decomposition of A = LU (without pivoting). Write L and U in the space
below
L=
and U =
(b) Use the LU decomposition of A to solve the given system.
(c) Find the determinant of A.
MTH 510 - MIDTERM
- Page 6 -
5. (4 marks) Express the decimal number 231.6875 as a binary number.
FALL 2009
MTH 510 - MIDTERM
- Page 7 -
FALL 2009
6. (10 marks) Use Gaussian elimination with partial pivoting and backward substitution
to nd the solution to
x1 + 3 x2 x3 = 4
2x1 + 4x2 + 6x3 = 2
x1 + 5x3 = 5
MTH 510 - MIDTERM
- Page 8 -
FALL 2009
Part B - Multiple Choice Questions
7. (2 marks) A matrix A has the eigenvalues 7, 0.5 and 1. Which of the following
statements is true:
(A) The system Ax = b cannot be solved for x, where x = [x1 x2 x3 ]T and b = [0 1 2]T .
(B) The condition number based on the 2-norm || ||2 is 7.
(C) ||A||f = 7
(D) None of the above.
ANSWER:
7.
8. (2 marks) The following commands have been entered in MATLAB:
A=[1 2 ; 2 3 ]
B=[4 -1 ; 1 -4 ]
The result of the command
A.*B
is equivalent to the matrix
(A) [4 2 ; 2 12]
(B) [4 2 ; 2 12]
(C) [2 7 ; 5 10]
(D) [6 9 ; 11 14]
ANSWER:
8.
9. (2 marks) Which of the following commands can be used in MATLAB to solve the
system Ax = b for a given matrix A and a given vector b:
(A)
(B)
(C)
(D)
x=A^{-1}*b
x=A.^{-1}*b
x=inverse(A)*b
None of the above
ANSWER:
9.
ANSWER:
10.
10. (2 marks) Typing the commands
a=[1 -3 2];
roots(a)
into MATLAB gives the values of the roots as
(A) 1 and 2
(B) 1.0000 and 0.5000
(C) 3 and 0
(D) None of the above
MTH 510 - MIDTERM
- Page 9 -
THIS PAGE IS FOR ROUGH WORK
Do not separate the pages
FALL 2009