Reorderings and graphs
Let = cfw_i1, , in a permutation
A, =
a(i),j i,j =1,.,n = matrix A with its i-th row
replaced by row number (i).
A, = matrix A with its j -th column replaced by column
(j ).
Dene P = I,
= Permutation matrix Then:
(1) Each row (
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 regression analysis. The
right-hand side is the heat evolved i
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 systems
The Problem:
x such that:
A is an n n matrix, and
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/
September 3, 2016
About this class
Me:
Yousef Saad
TA:
J
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 in Cn
Note: (x, y) = y H x
2-1
Text 3; GvL 2.2-2.3; AB:
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 arithmetic
The basic problem: The set A of all possible r
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 brackets at the end of the question. Base
= 100pts.
1. Answe
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 required. [5 4 pts]
(a) Given a full rank m n matrix with
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 required. [5 4 pts]
(a) For any vectors u, v we have Tr
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
mn. No calculators. Answer all 5 questions. Weights are
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 brackets at the end of the question. Base
= 100pts.
1. Answer
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) What is the rank of A?
(b) Find orthonormal bases for the
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 X T Xy = 0 implies that Xy = 0 which in turn implies th
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
if x = 0
f = 1
else
f = (exp(x)-1)/x;
end
Algorithm 2
y
EIGENVALUE PROBLEMS
Background on eigenvalues/ eigenvectors/ Jordan form
The Schur form
Perturbation analysis, condition numbers.
Power method
The QR algorithm
Practical QR algorithms: use of Hessenberg form and
shifts
The symmetric eigenvalue prob
A few applications of the SVD
Many methods require to approximate the original data
(matrix) by a low rank matrix before attempting to solve
the original problem
Regularization methods require the solution of a least-
squares linear system Ax = b approxi
function T = tridiag(a, b, c, n)
% function T = tridiag(a, b, c, n)
% makes the matrix tridiag(a,b,c) of size n x n
%
sub = diag(ones(n-1,1),-1);
% T = a .* sub + b*eye(n) + c .* sub';
T = a .* sub + b*eye(n) + c .* sub';
%
function A = convdiff(nx,ny,alpha)
% function A = convdiff(nx,ny,alpha)
% generates a 2-D conv-diffusion matrix
% to get a laplacian enter alpha=0
tx=sptridiag(-1+alpha, 2, -1+alpha, nx) ;
ty=sptridiag(-1, 2, -1, ny);
A =kron(speye(ny,ny),tx)+kron(ty
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 a guess-timate of kA1k.
Estimated condition number 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 analysis
Perturbation analysis for linear systems (Ax =
Inner products and Norms
Inner product of 2 vectors (Read sec. 2.2 )
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 in Cn
Note: (x, y) = y H x
2-1
Text (1.
FLOATING POINT ARITHMETHIC - ERROR ANALYSIS
Brief review of oating point arithmetic
Model of oating point arithmetic
Notation, backward and forward errors
Roundo errors and oating-point arithmetic
The basic problem: The set A of all possible represent
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 systems
The Problem:
x such that:
A is an n n matrix, and
SPECIAL LINEAR SYSTEMS OF EQUATIONS
Symmetric positive denite matrices.
The LDLT decomposition; The Cholesky factorization
Banded systems
Positive-Denite Matrices
A real matrix is said to be positive denite if
(Au, u) > 0 for all u = 0 u Rn
Let A be
function [Q,R] = mgsa (A)
% [Q,R] = cgsa (A)
% modified Gram Schmidt QR factorization of A
[m,n] = size(A);
for j=1:n
q = A(:,j);
for i=1:j-1
r = q'*Q(:,i);
q = q - r*Q(:,i);
R(i,j) = r;
end
%- error exit for case rjj = 0
r = norm(q) ;
if (r=0.0), error('
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) Show that if A is Symmetric Positive Definite
(SPD) th
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 spancfw_Q. Show a geometric illustration of this
decomposit
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) Show that if A is Symmetric Positive Definite
(SPD) th