CSci 5304, F12
Quiz 3
11/28/2012
Name:
The rules: You can and are encouraged to discuss answers to the questions. You need to
write your own anwser at the end of the discussion phase.
1. Let A be a ma
CSci 5304, F15
First Midterm Test
Oct 13th, 2015
Only lecture notes from class web-site allowed - no books. Duration: 75 mn. No calculators.
There are 4 questions to answer. Weights indicated in brack
Householder QR
Householder reflectors are matrices of the form
P = I 2wwT ,
where w is a unit vector (a vector of 2-norm unity)
w
Px
x
w
9-1
Geometrically, P x represents a
mirror image of x with res
THE SINGULAR VALUE DECOMPOSITION
The SVD existence - properties.
Pseudo-inverses and the SVD
Use of SVD for least-squares problems
Applications of the SVD
The Singular Value Decomposition (SVD)
Th
Least-Squares Systems and The QR factorization
Orthogonality
Least-squares systems.
The Gram-Schmidt and Modified Gram-Schmidt processes.
The Householder QR and the Givens QR.
Orthogonality The Gr
CSci 5304
Practice Exercises
Set #6
10 17 2016
1 Read the notes (set 8) and text
on the Gram-Schmidt algorithm (Algorithms 7.1 and 8.1 in text). Then
use the classical Gram-schmidt algorithm to orthog
CSci 5304
Practice Exercises
Set #7
10 19 2016
1 Given the vector x = [1, 0, 2, 2]T find all unit
vectors w such that (I 2wwT )x = e1 where is some
scalar. [You need to find 2 solutions]
2 [Use Matlab
CSci 5304
Practice Exercises
1 Obtain the LU factorization (A = LU ) of the
matrix on the right. Is the
Set #5
10 11 2016
4 2 2
2 2 1
2 1 5
matrix SPD?
If so what is its Cholesky factorization?
2 (a)
CSci 5304
Practice Exercises
Set #8
10 31 2016
1 Let Q = [q1, q2, , qn] an m n matrix with orthonormal columns (so m n). Show that any vector
x in Rm can be written as x = Qy + w where y Rn
and w span
CSci 5304
Practice Exercises
1 Obtain the LU factorization (A = LU ) of the
matrix on the right. Is the
Set #5
10 18 2016
4 2 2
2 2 1
2 1 5
matrix SPD?
If so what is its Cholesky factorization?
2 (a)
CSci 5304, F17
Homework # 4
Due Date: 11/16/2017
1. Let A be a matrix with singular values 1 2 r > 0, and E a perturbation matrix
such that kEk2 < r . Show that rank(A + E) rank(A).
2. Consider the pr
CSci 5304, F17
Homework # 4
Due Date: 11/16/2017
1.
Answer:
Obviously, rank(A) = r. Argument is by contradiction. Suppose rank(A + E) = k < r, then
kEk2 = kA (A + E)k2
min
rank(B)=k
kA Bk2 = k+1 r ,
CSci 5304, F17
Homework # 4
Due Date: 11/07/2017
1. You will find in the class web-site [see matlab page], a sample set of seasonal farm employment
data (ti , yi ) over about an 18 month period, where
CSci 5304, F15
Homework # 2
Due Date: 1006-2015
1. Consider
following two algorithms to compute the function f (x) = (ex
Pthe
i
1)/x =
i=0 x /(i + 1)! which arises in many applications:
Algorithm 1
SPECIAL LINEAR SYSTEMS OF EQUATIONS
Symmetric positive definite matrices.
The LDLT decomposition; The Cholesky factorization
Banded systems
Positive-Definite Matrices
A real matrix is said to be p
CSci 5304, F15
Homework # 1
Due Date: 09-22-2015
1. Let X be an m n matrix, with m n, that is of full rank. Show that
X T X is nonsingular. [Hint: By making judicious use of inner products,
show that
CSci 5304, F15
Homework # 5
Due Date: 11-24-2015
1. For this exercise, you can do all calculations by hand, and use matlab to verify or to
help. Consider the matrix:
1 0
1
0 1 1
A=
1 0
1
0 1
1
(a) Wha
CSci 5304, F15
Homework # 4
Due Date: 11-03-2015
1. This question is a modified version of Exercise 2.1.6 of the text. The Hald cement
data is used in several books and papers as an example of regress
SOLVING LINEAR SYSTEMS OF EQUATIONS
Background on linear systems
Gaussian elimination and the Gauss-Jordan algorithms
The LU factorization
Gaussian Elimination with pivoting
Background: Linear sys
CSCI 5304
Fall 2016
COMPUTATIONAL ASPECTS OF MATRIX THEORY
Class time : 9:45 11:00 TTh
Room
: Keller Hall 3-111
Instructor
: Yousef Saad
URL
: www-users.itlabs.umn.edu/classes/Fall-2016/csci5304/
Sep
Inner products and Norms
Inner product of 2 vectors
Inner product of 2 vectors x and y in Rn:
x1y1 + x2y2 + + xnyn in Rn
Notation: (x, y) or y T x
For complex vectors
(x, y) = x1y1 + x2y2 + + xnyn i
FLOATING POINT ARITHMETHIC - ERROR ANALYSIS
Brief review of floating point arithmetic
Model of floating point arithmetic
Notation, backward and forward errors
Roundoff errors and floating-point ari
CSci 5304, F13
2nd Midterm Test
Nov. 11, 2013
Lecture notes allowed - No books. Duration: 75 mn. No calculators. The base is: 100pts.
1. Answer the following questions - No justifications or proofs re
CSci 5304, F13
First Midterm Test
Oct 9th, 2013
Lecture notes allowed - No books. Duration: 75 mn. No calculators. The base is: 100pts.
1. Answer the following questions - No justifications or proofs
CSci 5304, F16
Second Midterm Test - Type A -
Return this sheet and blue book (s)
Oct 27, 2016
Your Name:
Class lecture notes (those posted on the class web site) are allowed. No books. Duration: 75
m
CSci 5304, F15
Second Midterm Test
Nov 12, 2015
Only lecture notes from class web-site allowed - no books. Duration: 75 mn. No calculators.
There are 4 questions to answer. Weights indicated in bracke
Estimating condition numbers.
Avoid the expense of computing A1 explicitly.
Choose a random or carefully chosen vector v.
Solve Au = v using factorization already computed.
Then kA1k kuk / kvk is
ERROR AND SENSITIVTY ANALYSIS FOR SYSTEMS
OF LINEAR EQUATIONS
Conditioning of linear systems.
Estimating errors for solutions of linear systems
Backward error analysis
Relative element-wise error
CSci 5304, F16
Final Exam
Dec. 22, 2016
Class lecture notes (those posted on the class web site) are allowed. No books. Duration: 90
mn. No calculators. Answer all 5 questions. Weights are indicated i
CSci 5304, F16
Final Exam
Dec. 22, 2016
Class lecture notes (those posted on the class web site) are allowed. No books. Duration: 90
mn. No calculators. Answer all 5 questions. Weights are indicated i
CSci 5304, F16
First Midterm Test - Type A -
Return this sheet and blue book (s)
Sep 29, 2016
Your Name:
Class lecture notes (those posted on the class web site) are allowed. No books. Duration: 75
mn
CSci 5304, F16
Third Midterm Test - Type A -
Return this sheet and blue book (s)
Nov. 29, 2016
Your Name:
Class lecture notes (those posted on the class web site) are allowed. No books. Duration: 75
m
CSci 5304, F17
Third Midterm Test
Nov. 28, 2017
Allowed: One formula sheet [back-to-back] (No lecture notes). Duration: 75 mn. No calculators.
Answer all 5 questions. Weights in brackets at the end of