Compute the eigenvectors in lapack using predetermined eigenvalues?

Question

I have a rather unusual challenge for lapack, and I have spent hours searching for a solution to it.

I have a generalized eigenvalue problem of the traditional form (A - x B = 0). Normally I would use for instance ?hegvx or ?hegvd to calculate the eigenvalues and the eigenvectors.

However the challenge I am facing now, is now that I already know the eigenvalues from the construction of the problem, and therefore I need an efficient lapack routine for calculating the eigenvectors only?

Anyone got a hack for this?

Answer 1

Given the generalised eigenvalue problem

(A - y B) x = 0

And an eigenvalue yn:

(A - y_n B) x_n = 0

We know A, B and yn, so we can form a new matrix Cn

C_n = A - y_n B

C_n x_n = 0

You can solve this with any linear algebra solver individually for each eigenvalue. According to the LAPACK docs on linear equations, for a general matrix, double precision, you might use DGETRS

Edit for degenerate eigenvalues:

The null space of the matrix Cn is what we are solving for here (as MvG commented). If

C_n j = 0 and
C_n k = 0

(i.e. degenerate e-vals) then given j^Tk = 0 (both are still eigenvectors of the AB system) we can say

Call a row of C_n r:

r.k = r.k - j^T.k = (r-j^T).k

Thus forming a matrix J whose rows are each j^T (there must be a name for this, but I don't know it):

(C_n - J) k = 0

Define

D_nj = C_n - J

And now solve for the new matrix D_nj. By construction this will be a new, orthogonal eigenvector of the original matrix with the same degenerate eigenvalue.