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using data from while loop and resetting initial conditions

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Michael Jacobson
Michael Jacobson 2021 年 4 月 27 日
回答済み: Chetan 2024 年 5 月 8 日 16:19
I am writing an algorithm for the power method, which is an iterative method for solving for all eigenvalues and eigenvectors of a matrix. Finding the first set of data, the dominant eigenvalue/vector is fairly straightforward.
What we do next is where i get confused. we are supposed to take the data and apply the following formula:. We then use Bv in the next iteration. I do not know how to take this value and input it into the new iteration.
The code:
clc;
%Assignement Conditions
FA = @(k_1, k_2, k_3,m_1, m_2) [-(k_2+k_1)/m_1 , k_2/m_1 ; k_2/m_2 -(k_3+k_2)/m_2];
A = FA(1,1,1,2,1);
%Stopping Criteria
e_s = .00005;
disp(['<strong> Stopping Criteria: </strong>' strcat(num2str(e_s),'%') ])
%Vector to display all Eigenvalues
Eigenvalue_summary = zeros(1,height(A));
%Dominant Eigenvalue
%Random Guess for Dominant Eigenvector
V_approximation = ones(height(A),1,'double');
%Iterative method for Dominant Eigenvalue/Eigenvector
Iteration = 0;
while (1)
disp('====================')
Iteration = Iteration + 1;
disp(['<strong>Iteration: </strong>' num2str(Iteration) ])
%Displaying Eigenvector to be used
disp(['<strong>Initial Guess: </strong>'])
for i = 1:length(V_approximation)
fprintf('%f\n',V_approximation(i))
end
%Sign of previous V (before iteration)
sign_old = sign(V_approximation);
%Approximations
Eigenvalue = norm(A*V_approximation);
Eigenvector = (A*V_approximation)/Eigenvalue;
%Sign of new V
sign_new = sign(Eigenvector);
%Reseting Eigenvalue based on sign of current and prev. V
if sign_new ~= sign_old
Eigenvalue = -Eigenvalue;
end
%Recording Approximations
for count = Iteration
C{count} = table(Iteration,Eigenvalue);
D{count} = table([Eigenvector]);
end
Eigen_Val = vertcat(C{:});
Eigen_Vec = vertcat(D{:});
if Iteration > 1
error = string(abs((table2array(Eigen_Val(Iteration,2)) - table2array(Eigen_Val(Iteration-1,2)))/table2array(Eigen_Val(Iteration,2)))*100);
else
error = string('----');
end
for count = Iteration
E{count} = table(Iteration,Eigenvalue,error);
end
Error = vertcat(E{:});
disp(['<strong>Eigenvalue: </strong>' num2str(Eigenvalue)])
disp(['<strong>Eigenvector: </strong>'])
for i = 1:length(Eigenvector)
fprintf('%f\n',Eigenvector(i))
end
disp(['<strong>Relative Error: </strong>' num2str(error) ])
%Resetting Eigenvector Approximation for next iteration
V_approximation = Eigenvector;
if str2num(error) < e_s,break,end
end
%Tables of Iteration Data
Eigen_Vec
Error
%Final Results
Dominant_Eigenvector = Eigenvector
Dominant_Eigenvalue = Eigenvalue
Eigenvalue_summary(1) = Dominant_Eigenvalue;
B1 = A - Eigenvalue_summary(1)*eye(height(A))
Note the final line of code, where B is determined
Thank you

回答 (1 件)

Chetan
Chetan 2024 年 5 月 8 日 16:19
I understand that you're implementing the power method in MATLAB to find the dominant eigenvalue and eigenvector of a matrix. And wish to modify the matrix for subsequent iterations to find more eigenvalues and eigenvectors.
You can modify the matrix by subtracting the outer product of the found eigenvector, scaled by the found eigenvalue, from the original matrix. This is known as deflation.
Here's how you can adjust your code to include this step:
clc;
% Assignment Conditions
FA = @(k_1, k_2, k_3, m_1, m_2) [-(k_2+k_1)/m_1 , k_2/m_1; k_2/m_2, -(k_3+k_2)/m_2];
A = FA(1, 1, 1, 2, 1);
% Stopping Criteria
e_s = 0.00005;
disp(['<strong> Stopping Criteria: </strong>', num2str(e_s), '%'])
% Vector for Eigenvalues
Eigenvalue_summary = zeros(1, height(A));
% Dominant Eigenvalue
V_approximation = ones(height(A), 1, 'double'); % Initial guess
Iteration = 0;
while true
Iteration = Iteration + 1;
disp(['<strong>Iteration: </strong>', num2str(Iteration)])
Eigenvalue = norm(A*V_approximation);
Eigenvector = (A*V_approximation) / Eigenvalue;
% Update for next iteration
V_approximation = Eigenvector;
% Check convergence
if Iteration > 1 && abs(Eigenvalue - Eigenvalue_summary(Iteration-1)) < e_s
break;
end
Eigenvalue_summary(Iteration) = Eigenvalue;
% Modify A for next eigenvalue (Deflation)
A = A - Eigenvalue * (Eigenvector * Eigenvector');
end
% Final Results
Dominant_Eigenvector = Eigenvector;
Dominant_Eigenvalue = Eigenvalue;
disp(['Dominant Eigenvalue: ', num2str(Dominant_Eigenvalue)]);
disp('Dominant Eigenvector: ');
disp(Dominant_Eigenvector);
This code iteratively finds the dominant eigenvalue and eigenvector, then deflates the matrix `A` for subsequent iterations.
Refer to the following MathWorks Documentation for more details:
Thanks
Chetan

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