While Err > 10^-7 & TotalIterations 10^-7 & TotalIterations <= 100ĮigenVectorEstimate2= 1 / sqrt(abs(normV)). Keep performing last step until a positive eigenvalue is arrived atīelow is the updated MatLab code where i am shifting by twice the first eigenvalue. Write a Matlab function, called myeig, to implement the shifted inverse power method for the eigenvalue problem Ax x.If it is negative, perform a shift by the first eigenvalue and run the inverse iteration again Now, in the chapter 8 of the book A First Course in Numerical Methods (about Eigenvalues and Singular Values), at page 228 (at least in this version that I've), there's the following inverse iteration method: Main Questions.Vibrating mechanical systems and resonances. Awareness of other methods for approximating eigenvalues Engineering Computation ECL4-2 Motivation Many problems can be cast as eigenvalue - eigenvector problems. If the calculated eigenvalue is positive, accept the result Iterative Power method for approximating the dominant eigenvalue 2.Run Inverse Iteration Method to arrive at the lowest absolute eigenvalue.The statement lambda eig (A) produces a column vector containing the eigenvalues of A. The method to find the lowest positive eigenvalue is as follows: The solution to this equation is expressed in terms of the matrix exponential x(t) etAx(0). The same EigenProblem can be shifted by μ as follows: NewEigenValueEstimate2 = y2'* EigenVectorEstimate2*(normV / abs(normV)) Įrr = abs((NewEigenValueEstimate2 - EigenValueEstimate2) / EigenValueEstimate2) ĮigenValueEstimate2 = NewEigenValueEstimate2 įprintf('Lowest Positive Absolute EigenValue is %f\n\n',EigenValueEstimate2+Shift) While Err > 10^-5 & TotalIterations 10^-5 & TotalIterations <= 100ĮigenVectorEstimate2= 1 / sqrt(abs(normV)).* EigenVectorEstimate2 %Initial Run to Find the Smallest Absolute Value This is my code so far in MatLab used for testing: %Generalized EigenProblem: Ke.x + λ.Kg.x = 0įprintf('Matlab EigenValues: %f, %f, %f \n',d(1),d(2),d(3)) However, i do not know how to perform the shift on the matrices to find a positive eigenvalue. I am trying to write a Matlab function that takes a matrix and an iteration count and performs inverse power iteration to output the smallest eigenvalue. I have successfully implemented the inverse iteration to find the absolute lowest eigenvalue. Remember that eigenvectors are not unique and are accurate up to scale.I am trying to find the lowest positive eigenvalue of the generalized eigenproblem of buckling using the inverse iteration method. On a side note with regards to eig, MATLAB most likely scaled that eigenvector using another norm. Y_old=y_final %// Change - Save old eigenvector Tol=1e-10 %// Change - make tolerance more small to ensure convergence Here is how I modified your code to facilitate this: function =power_method(A,x) The power method estimates both the prominent eigenvector and eigenvalue, so it's probably a good idea to check to see if both converged. You're only checking the eigenvalue for convergence. You have the correct implementation, but you're not checking both the eigenvector and eigenvalue for convergence. Is this acceptable to see this inaccuracy, or am I making some mistake? This paper is meant to be a survey of existing algorithms for the eigenvalue computation problem. The eigenvalue coincides, but the eigenvector should be approaching. The methods to be examined are the power iteration method, the shifted inverse iteration method, the Rayleigh quotient method, the simultaneous iteration method, and the QR method. When comparing with the eigenvalues and eigenvectors of the above matrix in MATLAB, I did: =eig(A) Here's what I wrote so far: %function to implement power method to compute dominant I would like to implement the Power Method for determining the dominant eigenvalue and eigenvector of a matrix in MATLAB.
0 Comments
Leave a Reply. |
AuthorWrite something about yourself. No need to be fancy, just an overview. ArchivesCategories |