Numerical Linear Algebra, CME 302
E. Darve
Autumn 2015
CME 302: Homework 1
Due Wednesday, October 7th, 2015
Problem 1: (Givens Rotation on Sparse Matrices)
20 points. Let a matrix A Rnn be of the form
1 2 3 . . . n
2 2
3
A = 3
.
.
.
.
.
n
n
Please desi

CME 302: Homework 3
Problem 1
Solution: Let y = Xx, then
xT X T AXx = y t Ay 0.
So we just need to show that equality only holds if x = 0. Since X is rank r we know that
the kernel of X is just the zero vector. Since y T Ay = 0 if and only if y = 0 and we

CME 302, Fall 2015
Homework 1
Problem 1.
Solution:
To simplify notation, we denote Gij as the Givens rotation between the i-th and the j-th row of A. We get
the following algorithm.
.
. .
.
.
.
.
. .
.
G
.
n1,n .
0
+ +
+
+ +
.
. .
.
G

Numerical Linear Algebra, CME 302
E. Darve
Autumn 2014
CME 302: Homework 2
Due Wednesday, October 8th, 2014
Problem 1: (Givens Rotation on Sparse Matrices)
20 points. Let a matrix A Rnn be of the form
1 2 3 . . . n
2 2
3
A = 3
.
.
.
.
.
n
n
Please desi

Numerical Linear Algebra, CME 302
E. Darve
Autumn 2014
CME 302: Homework 1
Due Wednesday, October 1st, 2014
Problem 1: (Norm Equivalencies)
In a nite dimensional space, all norms are equivalent. In this problem, you will be asked to verify
this theorem fo

CME 302: Homework 4 Solutions
Problem 1.
1. The code simply builds a diagonal matrix with the given eigenvalues and then performs a
similatrity transform.
Matlab code:
n=8;
A=diag ( 3 . ( 0 : ( n 1 ) ) ) ;
B=randn ( n ) ;
A=B\ (AB ) ;
Python code:
import

CME 302: NUMERICAL LINEAR ALGEBRA
FALL 2005/06
ASSIGNMENT 3 SOLUTIONS
GENE H. GOLUB
1. (a) Let a and b be vectors such that a 2 = b 2 = 1. Construct a Householder matrix
H = I 2uu , u 2 = 1 such that H a = b.
Solution: We want
(I 2uu )a = b.
In other word

Numerical Linear Algebra, CME 302
E. Darve
Autumn 2015
CME 302 Schedule
GVL: Golub, Van Loan
TB: Trefethen, Bau
D: Davis
H: homework
Date
Topic
Reading
Week 1 Sep 2125
Introduction; range and null space, rank; vector and matrix norms; singular
value decom

Numerical Linear Algebra, CME 302
E. Darve
Autumn 2014
CME 302: Homework 3
Due Wednesday, October 15, 2014
Problem 1: (Floating Point)
For this problem, make sure to review the denition of oating point numbers that we saw in
class. You will be asked to de

CME302 Homework 7 (1) Problem 33.2 from T.B. (2) Problem 36.1 from T.B. (3) We have seen the use of shifts in inverse iterations in Lecture 27. The section `Connection with Shifted Inverse Iteration' on pg.220 of T.B. claims that the QR algorithm which se

CME302 Homework 6 (1) Problem 28.2 (a) (b) (c) (2) This problem is based on Problem 29.1 from Trefethen. We would like you to put together a MATLAB program that finds all the eigenvalues of a real symmetric matrix, using only elementary building blocks. U

Numerical Linear Algebra, CME 302
E. Darve
Autumn 2016
CME 302: Homework 3
Due Wednesday, October 26th, 2016
Total number of points = 100.
Problem 1: (Givens rotation on sparse matrices)
10 points. Let a matrix A Rnn be of the form
1 2 3 . . . n
2 2
3
A

Chapter 10
Arnoldi and Lanczos algorithms
10.1
An orthonormal basis for the Krylov space Kj (x)
The natural basis of the Krylov subspace Kj (x) = Kj (x, A) is evidently cfw_x, Ax, . . . , Aj1 x.
Remember that the vectors Ak x converge to the direction of

CME 302, Fall 2016
Assignment 2
Problem 1.
Solution. Let us write the matrix A in block form as
A11 A12
L
0
U11
= 11
A21 A22
L21 L22
0
U12
U22
Therefore,
A22 = L21 U12 + L22 U22 .
Let X = A(r + 1 : n, r + 1 : n) be the matrix obtained after r steps of t

Numerical Linear Algebra, CME 302
E. Darve
Autumn 2014
CME 302: Homework 6
Due Wednesday, November 12, 2014
Problem 1: (Convergence rate of shifted QR algorithm)
In this problem we consider the convergence rate of the QR algorithm with a single-shift stra

Numerical Linear Algebra, CME 302
E. Darve
Autumn 2016
CME 302: Homework 6
Due Wednesday, November 30, 2016
Total number of points = 100.
Problem 1: (Chebyshev acceleration)
Let us assume that we are using a stationary iterative method to solve Ax = b wit

Numerical Linear Algebra, CME 302
E. Darve
Autumn 2014
CME 302: Homework 5
Due Wednesday, November 5, 2014
Problem 1: (Convergence of the orthogonal iteration algorithm)
1. 5 points. Write some Python or Matlab code to create a matrix in Rnn of size n = 8

Numerical Linear Algebra, CME 302
E. Darve
Autumn 2014
CME 302: Homework 4
Due Wednesday, October 22, 2014
Problem 1: (Positive denite matrices)
10 points. Show that for any matrix A Rnn and X Rnr , if A is positive denite and X is
rank r, then X T AX is

CME 302: Assignment 6
Problem 1.
1. First, we can see that e(k) = Gk e(0) from the iteration x(k) = Gx(k1) + M 1 b. With pk (1) =
Pk
j=1 vj (k) = 1, we have that
y (k) x =
k
X
vj (k)x(k) x
j=0
=
k
X
vj (k)x(k)
j=0
=
k
X
k
X
vj (k)x
j=0
vj (k) x
(k)
x =
j

CME 302: Assignment 7
Due: December 9
Problem 1.
1. We have
C 1 M AC = C 1 (C 2 A)C = CAC,
which means M A is similar to CAC. To show that CAC is SPD, we notice that
x Rn ,
x 6= 0
Cx 6= 0
(since C is nonsingular)
=
T
(Cx) A(Cx) > 0
(since A is SPD)
xT (CA

Numerical Linear Algebra, CME 302
E. Darve
Autumn 2016
CME 302: Homework 2
Due Wednesday, October 19th, 2016
Total number of points = 100.
Problem 1: (Schur Complement)
20 points. This result is very useful to analyze and create numerical algorithms to so

CME 302, Fall 2016
Assignment 5
Problem 1.
Solution.
1. Define Gpq to be the Givens rotation matrix
the QR step, we have
T 0
A=
0 = G12 0
0 0
0 0
with respect to the p-th row and the q-th row. Then in
= = (G12 G23 G34 )T
0
0 0
0 0

Numerical Linear Algebra, CME 302
E. Darve
Autumn 2016
CME 302: Homework 4
Due Wednesday, November 9th, 2016
Total number of points = 100.
Problem 1: (Convergence of the orthogonal iteration algorithm)
1. 5 points. Write Julia code to create a matrix in R

CME 302, Fall 2016
Assignment 1
Problem 1.
Solution.
1. First, observe that we may write a floating point number as dt e where d is an integer such that
t d t+1 1, and L e U . Now, for any x G, we can write x = y k where
1 y < , L k U . Without loss of g

Numerical Linear Algebra, CME 302
E. Darve
Autumn 2016
CME 302: Homework 1
Due Wednesday, October 12th, 2016
Total number of points = 100.
Problem 1: (Floating Point)
For this problem, make sure to review the definition of floating point numbers that we s

CME 302, Fall 2016
Assignment 3
Problem 1
Solution: To simplify notation, we denote Gij as the Givens rotation between the
the j-th row of A. We get the following algorithm.
.
.
. . .
. . .
.
.
G
.
.
n1,n
0
+ +
+
+ +
.
.
. . .
.
.
G

CME 302: Homework 4 Solutions
Problem 1.
1. The code simply builds a diagonal matrix with the given eigenvalues and then performs a
similatrity transform.
Matlab code:
n=8;
A=diag ( 3 . ( 0 : ( n 1 ) ) ) ;
B=randn ( n ) ;
A=B\ (AB ) ;
Julia code:
n
D
X
A