Lanczos method

[hoshang]

Hi all
I wonder how one could get eigenvalue and eigenvector in Lanczos method.

 

[JStructsteel]

I would google it.

 

[SWComposites]

per ChatGPT:

The Lanczos method is an iterative algorithm used to find a few eigenvalues and eigenvectors of a large symmetric matrix. Here are the general steps to obtain the eigenvalues and eigenvectors using the Lanczos method:

Choose a starting vector x_0 and normalize it so that ||x_0|| = 1.

Compute the first Lanczos vector v_1 = Ax_0 - beta_0 x_0, where beta_0 = 0.

Compute the Lanczos vectors v_i and the Lanczos coefficients alpha_i and beta_i recursively using the following formula:

alpha_i = v_i^T A v_i
v_{i+1} = (A - alpha_i I) v_i - beta_i v_{i-1}
beta_{i+1} = ||v_{i+1}||

Use the Lanczos vectors to construct a small tridiagonal matrix T of dimension m x m, where m is the number of Lanczos vectors computed.

Compute the eigenvalues and eigenvectors of T using an eigensolver such as the QR algorithm or the Jacobi algorithm.

Use the Lanczos vectors and the eigenvectors of T to construct the eigenvectors of A.

The Lanczos method is an efficient algorithm for computing a few eigenvalues and eigenvectors of a large symmetric matrix, and it is often used in numerical linear algebra and quantum chemistry. However, it is important to note that the Lanczos method may not be accurate for computing all the eigenvalues and eigenvectors of a matrix, especially if the matrix has a large number of eigenvalues or if the eigenvalues are clustered. In such cases, other methods such as the Arnoldi method or the Davidson method may be more appropriate.

 

[JoshPlumSE]

https://en.wikipedia.org/wiki/Lanczos_algorithm

The Lanczos method is merely a more "efficient" way to computationally get the eigen vectors. Not sure what you're asking.

 

[hoshang]

Compute the eigenvalues and eigenvectors of T using an eigensolver such as the QR algorithm or the Jacobi algorithm.

Thanks. Sorry for my mis-spelling my query. I meant to "Compute the eigenvalues and eigenvectors of T using an eigensolver such as the QR algorithm or the Jacobi algorithm".
After tridiagonal matrix construction step, how one could use QR algorithm to get eigenvalues and eigenvectors?

 

[JoshPlumSE]

Probably not the best question for the structural engineering forum. You might want to delete this thread and create a new one in the Finite Element Engineering Analysis forum... which tends to be more technical related to numerical methods.