1. The document describes an M-file that implements the power method to find the dominant eigenvector and eigenvalue of a matrix. 2. The power method is an iterative algorithm that starts with an initial vector and repeatedly multiplies it by the matrix to converge on the dominant eigenvector. 3. The M-file calculates the error between iterations to determine convergence within a specified tolerance, then displays the dominant eigenvector and eigenvalue.

1. PRACTICAL
Name- Saloni Singhal
M.Sc. (Statistics) II-Sem.
Roll No: 2046398
Course- MATH-409 L
Numerical Analysis Lab
Submitted To: Dr. S.C. Pandey

2. OBJECTIVE
1. Create an M-file to implement
Power method to find dominant
eigen value.
1. Write the program to find the
convergence.

3. Theory
The power method is a simple iteration method that can
be used to find λ and for a given matrix A, where λ is the
largest eigenvalue and is the corresponding
Eigenvector. we start with an initial approximation of the
dominant eigenvector of A, which must be non-zero.
Thus, we obtain a sequence of eigenvectors given by =
When k is large, we can obtain a good approximation of
the dominant eigenvector of A by properly scaling the
sequence.

4. Function
File
function [vec,value]=powerm(start,A,toler)
start=start/norm(start);
%initialisation
err=toler+1;
i=0
%iteration condition
while err>toler
i=i+1 %iteration count
vec=A*start
value=max(vec./start)
err=norm((vec/value)-start);
start=vec/value;
end
%display eigen vector and value
vec
value
end

5. Output

6. Plot of Iterations Vs Error
x=1:i-1
err=err(2:i)
plot(x,err)

7. Conclusion
• Hence, we can verify the dominant
Eigen value by the built in function
• Trace gives the sum of eigenvalues to
calculate the remaining eigenvalues

8. Applications
 It helps to determine system electric response, least and
maximum tolerance such as mechanical movement in
robotics
 Google Page rank algorithm: largest eigenvalue of graph
of internet is how pages are ranked for display to user
 Spectral clustering in Biology, business, image processing,
criminology, etc.
 Optimization: using largest eigenvalue of real symmetric
matrix is the maximize value of normalized form
 Principal component analysis (PCA) determines dominant
eigenvalue to capture most significant information from
redundant data
 To model and simulate real life phenomena using Markov
chain in AI and ML