routine PDLAHQR in ScaLAPACK

[makarem]

Dear all,

I want to compute eigenvalues and eigenvectors of a matrix already in the upper Hessenberg form. I see that the PDLAHQR routine in scaLAPACK computes eigenvalues and Schur vectors of a hessenberg matrix.
So, i want to know if Schur vectors represents eigenvectors ? if it isn't the case is there other routine in scaLAPACK which computes eigenvalues and eigenvectors of a hessenberg matrix?
Finally, if schur vectors represents eigenvectors, should i before calling the PDLAHQR routine do a call to the PDHSEQR routine to compute the matrix Z from the Schur decomposition?

Thank you in advance for your help.
Best regards
Makarem

[Julien Langou]

Hello Makarem,

Schur vectors do not represent eigenvectors ( except for the first one when you have a 1-by-1 eigenvalue for the first diagonal block).

(The Schur form relation is A = V T V^T where T is quasi upper triangular and V is orthogonal so V^T V=I).
(In the real case, quasi upper triangular means that you may have 2-by-2 diagonal blocks which corresponds to complex conjugate eigenvalues.
real eigenvalues come as 1-by-1 diagonal blocks. In the complex case, the matrix T is upper triangular.)

You can read the eigenvalues off the diagonal of the Schur matrix, but you cannot read easily the eigenvectors of a Schur basis.

What we are missing is a ScaLAPACK subroutine such as LAPACK DTREVC.

Cheers,
Julien.