CSE/MATH 6643: Numerical Linear Algebra
Haesun Park
[email protected]
School of Computational Science and Engineering
College of Computing
Georgia Institute of Technology
Atlanta, GA 30332, USA
Lecture 18
Eigenvalue Problem
Symmetric Eigenvalue Decompos
CSE6643/MATH6643, Numerical Linear Algebra, Spring 2014
Mon. Wed. 3:05-4:25 pm
Instr Center 105, Prof. Haesun Park
Prerequisites: Introductory Numerical Analysis and Linear Algebra background, or Permission of the
instructor.
Instructors Oce Hours: Mon. 2
HW 2 Solutions
March 3, 2014
1
The information in the question implies that
A + ei eT
j
(1)
is non-singular for all = 0. From the Sherman-Morrison-Woodbury formula we have the condition
1 + eT A1 ei = 0 R, = 0
j
which, in turn implies that (A
1
(2)
)ij =
HW 1 Solutions
February 24, 2014
1
A + iB = (CE DF ) + i (DE + CF ) .
(1)
W = CE + DE CF DF.
(2)
Also,
Now, note that A = W + CF DE , and B = DE + CF . Thus the three multiplications required for computing A and B are CF , DF , and W = (C + D) (E F ).
2
U
CSE/MATH 6643: Numerical Linear Algebra
Haesun Park
[email protected]
School of Computational Science and Engineering
College of Computing
Georgia Institute of Technology
Atlanta, GA 30332, USA
Lecture 11
Least Squares Problem Full Rank
Least Square
CSE/MATH 6643: Numerical Linear Algebra
Haesun Park
[email protected]
School of Computational Science and Engineering
College of Computing
Georgia Institute of Technology
Atlanta, GA 30332, USA
Lecture 7
Cholesky Decomposition and Banded Systems
Definit
CSE/MATH 6643: Numerical Linear Algebra
Haesun Park
[email protected]
School of Computational Science and Engineering
College of Computing
Georgia Institute of Technology
Atlanta, GA 30332, USA
Lecture 1
What is Numerical Analysis?
Three great branc
CSE/MATH 6643: Numerical Linear Algebra
Haesun Park
[email protected]
School of Computational Science and Engineering
College of Computing
Georgia Institute of Technology
Atlanta, GA 30332, USA
Lecture 8
QR Factorization
QR Factorization
For solving
CSE/MATH 6643: Numerical Linear Algebra
Haesun Park
[email protected]
School of Computational Science and Engineering
College of Computing
Georgia Institute of Technology
Atlanta, GA 30332, USA
Lecture 9
QR Factorization
In general, we need to nd Ho
CSE/MATH 6643: Numerical Linear Algebra
Haesun Park
[email protected]
School of Computational Science and Engineering
College of Computing
Georgia Institute of Technology
Atlanta, GA 30332, USA
Lecture 3
LU Factorization
LU Factorization
How to find
CSE/MATH 6643: Numerical Linear Algebra
Haesun Park
[email protected]
School of Computational Science and Engineering
College of Computing
Georgia Institute of Technology
Atlanta, GA 30332, USA
Lecture 2
Introduction
Introduction
Topics
1. Linear sy
CSE/MATH 6643: Numerical Linear Algebra
Haesun Park
[email protected]
School of Computational Science and Engineering
College of Computing
Georgia Institute of Technology
Atlanta, GA 30332, USA
Lecture 6
Symmetric Positive Denite Matrix & Cholesky
Decom
CSE/MATH 6643: Numerical Linear Algebra
Haesun Park
[email protected]
School of Computational Science and Engineering
College of Computing
Georgia Institute of Technology
Atlanta, GA 30332, USA
Lecture 5
LU Factorization
Applications of P A = LU
Lin
CSE/MATH 6643, Spring 2016
HW1 Solutions
1
P1.1.1, page 13
v = A(:, 1)
v(1) = v(1) xr
for i = r 1 : 1 : 1 do
v = Av xi v
end for
After the algorithm v would be the rst column of M . The complexity of this algorithm is 1 + (r 1)(2n2
n + 2n) = O(rn2 ) ops.
CSE/MATH 6643: Numerical Linear Algebra
Haesun Park
[email protected]
School of Computational Science and Engineering
College of Computing
Georgia Institute of Technology
Atlanta, GA 30332, USA
Lecture 4
LU Factorization
G.E. with Partial Pivoting
1
CSE6643/MATH6643: Numerical Linear Algebra
Haesun Park
[email protected]
School of Computational Science and Engineering
College of Computing
Georgia Institute of Technology
Atlanta, GA 30332, USA
Lecture 1
What is Numerical Analysis?
Three great branch
CSE/MATH 6643: Numerical Linear Algebra
Haesun Park
[email protected]
School of Computational Science and Engineering
College of Computing
Georgia Institute of Technology
Atlanta, GA 30332, USA
Lecture 17
Eigenvalue Problem
Schur vector = eigen vector w
CSE6643/MATH6643:Numerical Linear Algebra
Haesun Park
[email protected]
School of Computational Science and Engineering
College of Computing
Georgia Institute of Technology
Atlanta, GA 30332, USA
Lecture 2
Introduction
1 Introduction
1.1 Topics
1. Linea
CSE/MATH 6643: Numerical Linear Algebra
Haesun Park
[email protected]
School of Computational Science and Engineering
College of Computing
Georgia Institute of Technology
Atlanta, GA 30332, USA
Lecture 4
LU Factorization
3.6
G.E. with Partial Pivoti
CSE/MATH 6643: Numerical Linear Algebra
Haesun Park
[email protected]
School of Computational Science and Engineering
College of Computing
Georgia Institute of Technology
Atlanta, GA 30332, USA
Lecture 5
LU Factorization
3.9
Applications of P A
= LU
CSE/MATH 6643: Numerical Linear Algebra
Haesun Park
[email protected]
School of Computational Science and Engineering
College of Computing
Georgia Institute of Technology
Atlanta, GA 30332, USA
Lecture 8
QR Factorization
6
QR Factorization
For solving L
CSE/MATH 6643: Numerical Linear Algebra
Haesun Park
[email protected]
School of Computational Science and Engineering
College of Computing
Georgia Institute of Technology
Atlanta, GA 30332, USA
Lecture 6
Symmetric Positive Denite & Cholesky Decomp.
CSE/MATH 6643: Numerical Linear Algebra
Haesun Park
[email protected]
School of Computational Science and Engineering
College of Computing
Georgia Institute of Technology
Atlanta, GA 30332, USA
Lecture 9
QR Factorization
In general, we need to nd Househ
CSE/MATH 6643: Numerical Linear Algebra
Haesun Park
[email protected]
School of Computational Science and Engineering
College of Computing
Georgia Institute of Technology
Atlanta, GA 30332, USA
Lecture 11
Least Squares Problem Full Rank
7
7.1
Least Squa
CSE/MATH 6643: Numerical Linear Algebra
Haesun Park
[email protected]
School of Computational Science and Engineering
College of Computing
Georgia Institute of Technology
Atlanta, GA 30332, USA
Lecture 10
QR Factorization
Problem: Given a vector x, nd a
CSE/MATH 6643: Numerical Linear Algebra
Haesun Park
[email protected]
School of Computational Science and Engineering
College of Computing
Georgia Institute of Technology
Atlanta, GA 30332, USA
Lecture 12
Classical Gram-Schmidt (CGS) for Reduced QRD
Giv
CSE/MATH 6643: Numerical Linear Algebra
Haesun Park
[email protected]
School of Computational Science and Engineering
College of Computing
Georgia Institute of Technology
Atlanta, GA 30332, USA
Lecture 13
Least Squares Problem
7.4
Solved by Singular Val
CSE/MATH 6643: Numerical Linear Algebra
Haesun Park
[email protected]
School of Computational Science and Engineering
College of Computing
Georgia Institute of Technology
Atlanta, GA 30332, USA
Lecture 15
Least Square Problem Rank Decient
Computing QRD
CSE/MATH 6643: Numerical Linear Algebra
Haesun Park
[email protected]
School of Computational Science and Engineering
College of Computing
Georgia Institute of Technology
Atlanta, GA 30332, USA
Lecture 16
SVD and Lower Rank Approximation
For any matrix
CSE/MATH 6643: Numerical Linear Algebra
Haesun Park
[email protected]
School of Computational Science and Engineering
College of Computing
Georgia Institute of Technology
Atlanta, GA 30332, USA
Lecture 14
Least Square Problem SVD
7.4.4
Solving LS
min Ax