This document describes implementing the inverse power method to find the least eigenvalue of a matrix. It provides: 1) An objective to create an M-file to implement the inverse power method and check for convergence. 2) A brief theory section explaining that the inverse power method is similar to the power method but uses A^-1 instead of A to find the largest eigenvalue of A^-1, which corresponds to the smallest eigenvalue of A. 3) An M-file function that performs the inverse power method iterations, calculates the error at each step, and returns the least eigenvalue and corresponding eigenvector once the error converges below a tolerance.

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
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2. OBJECTIVE
1. Create an M-file to implement
Inverse Power Method to find least
eigen value.
1. Write the program to find the
convergence.
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3. Theory
The inverse power method is an iterative method to find
the smallest eigen value and corresponding eigen vector
of a matrix A. It is very similar to the power method, but
involves iterations that involve multiplication of the iterates
by A-1 (provided inverse exists) rather than A. We then
apply the standard power method to which will return the
largest eigenvalue of (say, λ’ ) and the corresponding
eigenvector (say,v’ ). The inverse of this eigenvalue, that
is, 1/λ’ will give the smallest eigenvalue of the matrix .
Please note that the eigenvector corresponding to λ in that
is, remains same as the eigenvector for 1/λ’ in A.
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4. Function
File
function [vec,value]=invpow(start,A,toler)
start=start/norm(start); %initialisation
err=toler+1;
i=0;
A=inv(A); %condition for iteration
while err>toler
i=i+1 %iteration count
vec=A*start;
value=0;
for j=1:length(start)
if start(j)~=0
if 0~=vec(j)/start(j)
value=vec(j)/start(j);
end
end
end
value
vec
err=norm((vec/value)-start);
start=vec/value;
end %display eigen vector and value
vec
value=1/value
end
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5. Output
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6. Plot of iteration count and
falling errors
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Codes:
x=1:i-1
err=err(2:i)
plot(x,err)

7. Conclusion
• Hence we can verify the least
Eigen value by the built in function
• Trace gives the sum of eigenvalues to
calculate the remaining by subtracting
the dominant and least eigen value in
a 3*3 matrix
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