ENGR-391-2201-AA FINAL EXAM TOPIC 4 COMPLETE
MATERIAL CONCORDIA UNIVERSITY
By checking the "yes" answer below I confirm
1. that I have neither given nor received unauthorized aid to answer the questions of this
assignment.
2. I agree to follow the rules in regard of online assignments as they are posted in the
announcement forum of this class and sent to me by email via moodle
Select one:
a. Yes I agree
b. No I do not agree
Your answer is correct.
The correct answer is: Yes I agree
A first order initial value problem is solved using Euler’s method.
If we reduce the step size h = xi+1 − xi to half of it’s previous value, how do you expect
the local and global truncation errors of the solution to change?
Select one:
a. local and global errors will be about a quarter of their previous values
b. local and global errors will be about a half of their previous values
c. local and global error will be a quarter and a half of their previous values,
respectively
d. local and global errors will be a half and a quarter of their previous values,
respectively
Your answer is correct.
The correct answer is: local and global error will be a quarter and a half of their previous
values, respectively
/
,We want to approximate the defined integral I = ∫ 35 f(x)dx.
When using the trapezoidal rule in its single version we get I ≃ 153.7.
Knowing that f(4) = 24.2, how much will be the approximation of I if we use the
composite trapezoidal rule with m = 2 sub-intervals ?
Select one:
a. Not enough information to answer the question
b. None of them
c. 101.05
d. 104.2
e. 94.9
f. 145.6
Your answer is incorrect.
f(3)+f(5)
We have that
Using the composite trapezoidal rule with m = 2 we have (h = 5−3 = 1):
I≃ 1 [f(3) + f(5) + 2 ⋅ f(4)] =
The correct answer is: 101.05
The numerical stability of a method for solving initial value problems depends on
Select one:
a. only the numerical method
b. only on the number of significant digits used
c. both the numerical method and the differential equation
d. only the differential equation
Your answer is correct.
It depends on the differential equation and as well on the chosen method (different methods
can lead to stable or unstable solutions for a same differential equation). It depends as well
on how the algorithm is used (for example which step size is chosen.
The correct answer is: both the numerical method and the differential equation
/
,The Lagrange polynomial that passes through the 3 data points is given by
yi | 9.2 | 9.9 | 2.3
How much is the value of L1 (x) in x = 7.6 ?
Give at least 4 significant figures
Answer: 1.067
L1 (x) is given by
Reference: lecture on Lagrange interpolation of topic 4 "Regression and interpolation"
The correct answer is: 3.977
Consider the linear equation ax = b with solution r = b
a.
How much is the maximal relative error magnification factor for this equation for the
approximation xr of the solution r?
Select one:
a.
b. |axr|
c. 1
d.
Your answer is correct.
Written in matrix form the system writes: [a]x = [b]. Here the coefficient matrix A is a 1x1
matrix.
The maximal error magnification factor is given by the conditioning number of the coefficient
matrix A.
In our case the coefficient matrix A is [a].
The correct answer is: 1
/
, Choose the correct statement
Select one:
a. The order of convergence for the bisection method is 1
2
b. For a given equation, every algorithm will converge eventually but with different
orders of convergence
c. The asymptotic error constant λ for the newton method is − 1
d. The fixed-point method converges linearly
Your answer is correct.
The correct answer is: The fixed-point method converges linearly
Consider the following data set:
x | 1.0| 2.0| 5.0| 7.0
y | 2.1| 2.9| 6.1| 8.3
The linear least square fitting will give the following model:
Select one:
Your answer is correct.
Reference: lectures on least square regression of topic 4
The correct answer is:
/
MATERIAL CONCORDIA UNIVERSITY
By checking the "yes" answer below I confirm
1. that I have neither given nor received unauthorized aid to answer the questions of this
assignment.
2. I agree to follow the rules in regard of online assignments as they are posted in the
announcement forum of this class and sent to me by email via moodle
Select one:
a. Yes I agree
b. No I do not agree
Your answer is correct.
The correct answer is: Yes I agree
A first order initial value problem is solved using Euler’s method.
If we reduce the step size h = xi+1 − xi to half of it’s previous value, how do you expect
the local and global truncation errors of the solution to change?
Select one:
a. local and global errors will be about a quarter of their previous values
b. local and global errors will be about a half of their previous values
c. local and global error will be a quarter and a half of their previous values,
respectively
d. local and global errors will be a half and a quarter of their previous values,
respectively
Your answer is correct.
The correct answer is: local and global error will be a quarter and a half of their previous
values, respectively
/
,We want to approximate the defined integral I = ∫ 35 f(x)dx.
When using the trapezoidal rule in its single version we get I ≃ 153.7.
Knowing that f(4) = 24.2, how much will be the approximation of I if we use the
composite trapezoidal rule with m = 2 sub-intervals ?
Select one:
a. Not enough information to answer the question
b. None of them
c. 101.05
d. 104.2
e. 94.9
f. 145.6
Your answer is incorrect.
f(3)+f(5)
We have that
Using the composite trapezoidal rule with m = 2 we have (h = 5−3 = 1):
I≃ 1 [f(3) + f(5) + 2 ⋅ f(4)] =
The correct answer is: 101.05
The numerical stability of a method for solving initial value problems depends on
Select one:
a. only the numerical method
b. only on the number of significant digits used
c. both the numerical method and the differential equation
d. only the differential equation
Your answer is correct.
It depends on the differential equation and as well on the chosen method (different methods
can lead to stable or unstable solutions for a same differential equation). It depends as well
on how the algorithm is used (for example which step size is chosen.
The correct answer is: both the numerical method and the differential equation
/
,The Lagrange polynomial that passes through the 3 data points is given by
yi | 9.2 | 9.9 | 2.3
How much is the value of L1 (x) in x = 7.6 ?
Give at least 4 significant figures
Answer: 1.067
L1 (x) is given by
Reference: lecture on Lagrange interpolation of topic 4 "Regression and interpolation"
The correct answer is: 3.977
Consider the linear equation ax = b with solution r = b
a.
How much is the maximal relative error magnification factor for this equation for the
approximation xr of the solution r?
Select one:
a.
b. |axr|
c. 1
d.
Your answer is correct.
Written in matrix form the system writes: [a]x = [b]. Here the coefficient matrix A is a 1x1
matrix.
The maximal error magnification factor is given by the conditioning number of the coefficient
matrix A.
In our case the coefficient matrix A is [a].
The correct answer is: 1
/
, Choose the correct statement
Select one:
a. The order of convergence for the bisection method is 1
2
b. For a given equation, every algorithm will converge eventually but with different
orders of convergence
c. The asymptotic error constant λ for the newton method is − 1
d. The fixed-point method converges linearly
Your answer is correct.
The correct answer is: The fixed-point method converges linearly
Consider the following data set:
x | 1.0| 2.0| 5.0| 7.0
y | 2.1| 2.9| 6.1| 8.3
The linear least square fitting will give the following model:
Select one:
Your answer is correct.
Reference: lectures on least square regression of topic 4
The correct answer is:
/
