EigenValue-Problems-and-QR-Algorithm_Apog-J..pptx

EIGENVALUE
PROBLEMS AND
QR ALGORITHM
Group 1
Presented by: Jasmine Apog

EIGENVALUE
PROBLEMS
CHARACTERISTICS - VALUE PROBLEMS ARE A SPECIAL
CLASS OF BOUNDARY-VALUE PROBLEMS THAT ARE
COMMON IN ENGINEERING PROBLEM CONTEXTS
INVOLVING VIBRATIONS, ELASTICITY AND OTHER
OSCILLATING SYSTEMS.

Matrix eigenvalue problem
The set of all eigenvalues of A, denoted by λ(A), is called the spectrum of A.

Estimation of the Dominant
Eigenvalue

Different cases of dominant
eigenvalue
Algorithm of the Power Method

Inverse Power Method: Estimation
of the Smallest Eigenvalue
1
2
3
Example

Inverse Power Method: Estimation
of the Smallest Eigenvalue

Shifted Inverse Power Method:
Estimation of the Eigenvalue Nearest a
Specified Value
1
2
4
3
5
6
7
8

Notes for Shifted Inverse Power Method
The initial vector
x_1and the
subsequent
vectors x_k will be
normalized to
have a length of 1
Equation 9.9 is solved as (A-αI)
x_(k+1)=x_k using Doolittle’s method,
which employs LU factorization of the
coefficient matrix A-αI. This proves
Useful, especially if α happens to be
very close to an eigenvalue of A,
causing
A-αI to be near singular.
Setting α=0 leads to
the estimation of
the smallest
magnitude
eigenvalue of A.

Example:
Solution:
>> A = [ 4 -3 3 -9;-3 6 -3 11;0 8 -5 8;3 -3 3 -8];
>> xl = [0;1;0;1]; % Initial vector
>> [e_val, e_vec] = PowerMethod (A,x1) % Default values for tol and kmax
e_val =
-5.0000 % Dominant Eigenvalue
e_vec =
0.5000
- 0.5000
- 0.5000
0.5000

Example: (Using ShiftInvPower)
Solution:
>> [e_val, e_vec] = ShiftInvPower (A, 0, x1)
e_val =
1.0001 % Smallest Magnitude Eigenvalue
e_vec =
- 0.8165
0.4084
0.0001
- 0.4082
Solution:
>> [e_val, e_vec] = ShiftInvPower (A, -1.5, x1) % alpha=-1.5
eigenval =
-2.0000 % Third eigenvalue
eigenvec =
0.5774
- 0.5774
0.0000
0.5774

Example: (Using Shifted Power Method)
Solution:
>> A1 = A + 2*eye(4,4);
>> [e_val, e_vec] = PowerMethod(A1, x1)
e_val =
5.0000 % Fourth eigenvalue = 5+(-2) =3
e_vec =
- 0.0000
0.7071
0.7071
0.0000
All eigenvalues and eigenvectors:

QR Algorithm
• Efficiently provides approximations for all eigenvalues/eigenvectors of a
matrix.
• Developed independently in the late 1950's by John G.F. Francis (England)
and Vera Kublanovskaya (USSR)

QR Factorization
Method
QR Factorization (or decomposition) Method - is a matrix decomposition technique
that expresses a matrix A as the product of an orthogonal matrix Q and an upper triangular
matrix R.
Any matrix can be decomposed into a product, M = QR ; Q-orthogonal and R - Upper triangular
Q-orthogonal - an adjective that describes two things in a right angles to each other.
R - Upper triangular - a special type of square matrix where all the elements below the main diagonal is
zero.
QR Algorithm - is an iterative method for computing the eigenvalues and eigenvectors of a
matrix. It relies on repeated QR Factorizations.

QR Factorization Process
Setting
01 03 04
02
Factorize
multiply in reverse
order to form a
new matrix
Apply QR
Factorization
multiply in reverse
order to form a new
matrix
05

Determination of and Matrices
1
2
3
4

Determination of and Matrices
6
5

Example for QR Factorization
Find the QR Decomposition of Matrix A=

THANK YOU
Have a blessed Sunday! :)
Group 1

EigenValue-Problems-and-QR-Algorithm_Apog-J..pptx

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