EE103Spring2012Instructor:ProfessorStephenJacobsen
EE 103
Applied Numerical Computing
Spring 2012
An Introduction to Numerical Computing and Analysis
for
engineering and computer science students
Lectures: 8:00-9:50, Tu, Th
(most often, Lectures 8:00-9:20
L. Vandenberghe EE103
9/30/10
Homework 1 solutions
1. Exercise 1.2. (a) Not linear or ane. Choose x = (1, 0), y = (0, 1), = = 1/2. Then f (x + y ) = 0 = f (x) + f (y ) = 1. (b) Linear. We have f (x) = aT x for a = (1, 0, . . . , 0, 1). (c) Ane. Working ou
EE 103, Winter 09, Prof SEJ: HW 2 Sol. Applied Numerical Computing Instructor: Prof. S. E. Jacobsen HW2 Solution Students: Distributed HW solutions are a component of the course and should be fully understood. SEJ Prob 1: The following are the diary
EE103, HW 4, Winter 2009, Prof. S. E. Jacobsen
EE103 Applied Numerical Computing, Winter 2009 HW 4 Due: 2/24/09 Your HW answers must contain your ID, Last Name, First Name, and the number of the Discussion Section in which you are enrolled. Class: Th
EE 103
Applied Numerical Computing
Instructor: Lara Dolecek
Homework 3 Solution
Monday, October 14, 2013
Due: Monday, October 21, 2013
Reading: Chapters 3 5
1. Yes. Since AT = A, A2 = AT A = I , and A4 = A2 A2 = I .
A1
, where A1 is n n and A2 is
A2
(m n)
EE 113 Digital Signal Processing
Solutions for Homework II
Instructor: Professor Ali H. Sayed
TA: Zaid Towfic, Xiaochuan Zhao
October 8, 2013
Problem 3.6. What is the period of the sequence x(n) = cos
6n
3
?
Solution.
By denition, the period N has to sati
EE103, HW5, Winter 2009, Prof. S. E. Jacobsen
EE103 Applied Numerical Computing, Winter 2009 HW 5 Due: 03/03/09 Your HW answers must contain your ID, Last Name, First Name, and the number of the Discussion Section in which you are enrolled. Class: Th
EE103, HW 1, Winter 2009, Prof. S. E. Jacobsen
EE103 Applied Numerical Computing, Winter 2009 HW 1 Due: 01/20/2009 Your HW answers must contain your ID, Last Name, First Name, and the number of the Discussion Section in which you are enrolled. Class:
EE 103
Applied Numerical Computing
Instructor: Lara Dolecek
Homework 7 Solution
Monday, November 18, 2013
Due: Monday, November 25, 2013
Reading: Chapter 12 14
1. For a tolerance = 1010 , results are
x0 = 0, does not converge due to zero gradient.
x0 =
EE 103
Applied Numerical Computing
Instructor: Lara Dolecek
Homework 8
Monday, November 25, 2013
Due: Monday, December 2, 2013
Reading: Chapters 13, 14,16-17
1. Dene
n1
g (x) = x a
2
(xk+1 xk )2 + b
+
k=1
where a is an n-vector and b is a positive scalar.
EE 103
Applied Numerical Computing
Instructor: Lara Dolecek
Homework 6 Solution
Monday, November 11, 2013
Due: Monday, November 18, 2013
Reading: Chapters 1012
1. Since the matrix A is right invertible for,
A=
1
1
17/3 16/3
1
1
2/3 1/3
The solution will
EE 103
Applied Numerical Computing
Instructor: Lara Dolecek
MATLAB project
Monday, November 11, 2013
Due: Friday, December 6, 2013
100 points total
In this project we will further analyze algorithms for solving linear and non-linear systems
of equations.
EE 113 Digital Signal Processing
Solutions for Homework V
Instructor: Professor Ali H. Sayed
TA: Zaid Towfic, Xiaochuan Zhao
October 30, 2013
Problem 10.3. Invert the transform
X (z ) =
1
z+
12
8
z+
Solution.
We use partial fractions:
X (z ) =
1
3
,
1
1
<
EE113: Digital Signal Processing
Prof. Ali H. Sayed
Fall 2013
October 31, 2013
SOLUTIONS FOR MIDTERM EXAMINATION
1. A causal system is described by the following second-order dierence equation:
3
1
y (n) y (n 1) + y (n 2) = x(n),
4
8
where x(n) denotes ca
EE 103
Applied Numerical Computing
Discussion Set 1 Solution
Week of Sept. 30, 2013
Instructor: Lara Dolecek
Reading: Chapters 1 and 2
1. Consider x =
1
y
2
+
1
z,
2
where x =
0
0
.
.
.
, y =
1
0
.
.
.
, and z =
1
0
.
.
.
.
0
0
0
+
= 1, so f is neither
EE 103
Applied Numerical Computing
Instructor: Lara Dolecek
Homework 2
Monday, October 7, 2013
Due: Monday, October 14, 2013
Reading: Chapter 3
1. Prove that if the matrix A is non-singular then AT is also non-singular.
2. Consider all 2 2 matrices with b
EE 103
Applied Numerical Computing
Instructor: Lara Dolecek
Homework 1
Monday, Sept. 30, 2013
Due: Monday, October 7, 2013
Reading: Chapters 1 and 2
1. (a) Compute the inner product, correlation coecient and angle between the two
vectors for x = [1 2 3]T
EE 103
Applied Numerical Computing
Instructor: Lara Dolecek
Homework 1 Solution
Monday, September 30, 2013
Due: Monday, October 7, 2013
Reading: Chapters 1 and 2
1. (a) xT y = 20
xT
20
= x y = 1429 = 0.9926
y
= arccos() = 0.1219
(b) xT y = 28
xT
= x y
EE 103
Applied Numerical Computing
Discussion Set 2 Solution
Week of October 7, 2013
Instructor: Lara Dolecek
Reading: Chapter 3
3
, nullspace(A) = cfw_0.
u
(b) range(B ) = 2 , nullspace(B ) = u u
.
u
u + 2 v
u, v
u
(c) range(C ) =
, nullspace(C ) =
EE 103
Applied Numerical Computing
Discussion Set 2
Week of October 7, 2013
Instructor: Lara Dolecek
Reading: Chapter 3
Concepts covered this week:
Basics of solving linear equations
Range space and null space
Left and right inverse
Necessary and suci
EE 103, Fall 2013
Applied Numerical Computing
Instructor: Lara Dolecek
Final Study Guide
Reading: Chapters 1- 17 in the reader
You should be able to do the following:
Perform basic operations on vectors and matrices (addition, subtraction, multiplication
EE 103
Applied Numerical Computing
Discussion Set 3
Week of October 14, 2013
Instructor: Lara Dolecek
Reading: Chapters 3 5
Concepts covered this week:
Further properties of left/right inverse
Orthogonal matrices
Complexity of matrix algorithms; ops
u
EE 103
Applied Numerical Computing
Discussion Set 3 Solution
Week of October 14, 2013
Instructor: Lara Dolecek
1. By denition, an orthogonal matrix M is dened as M T M = I . Since A = AT ,
I = A2 = AA = AT A, so A is orthogonal.
2. (B 1 A 1 )(AB ) = B 1 (
EE 103
Applied Numerical Computing
Discussion Set 4 Solution
Week of October 21, 2013
Instructor: Lara Dolecek
Reading: Chapters 4-8
1. Consider a 2n 1 vector
x
T
A
A
A
cA
x=
y
T
y
, where y and z are both n 1 vectors.
z
z
T
A
A
A
cA
y
z
= (y
z )T A (y
z
EE 103
Applied Numerical Computing
Instructor: Lara Dolecek
Discussion Set 6 & 7
Week of November 4, 2013
Week of November 11, 2013
Reading: Chapters 10-12
Concepts covered this week:
Underdetermined linear systems
Solving non-linear equations with one
EE 103
Applied Numerical Computing
Instructor: Lara Dolecek
Discussion Set 8
Week of November 18, 2013
Week of November 25, 2013
Reading: Chapters 13, 14, 16 and 17
Concepts covered in these two weeks:
Unconstrained minimization
Computing gradient and H
L. Vandenberghe
EE103
12/09/10
Final Exam Solutions
Problem 1 (20 points). A matrix of the form
P =I
2
uuT ,
uT u
where u is a nonzero n-vector, is called a Householder matrix of order n.
1. (7 points) Show that P is orthogonal.
2. (7 points) Is P positiv