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### Eigenvalues, Eigenvectors, and Schur Factorization

Let A be a square n-by-n matrix. A scalar is called an eigenvalue and a nonzero column vector v the corresponding right eigenvector if . A nonzero column vector u satisfying is called the left eigenvector . The first basic task of the routines described in this section is to compute, for a given matrix A, all n values of and, if desired, their associated right eigenvectors v and/or left eigenvectors u.

A second basic task is to compute the Schur factorization of a matrix A. If A is complex, then its Schur factorization is , where Z is unitary and T is upper triangular. If A is real, its Schur factorization is , where Z is orthogonal, and T is upper quasi-triangular (1-by-1 and 2-by-2 blocks on its diagonal). The columns of Z are called the Schur vectors of A. The eigenvalues of A appear on the diagonal of T; complex conjugate eigenvalues of a real A correspond to 2-by-2 blocks on the diagonal of T.

These two basic tasks can be performed in the following stages:

1. A general matrix A is reduced to upper Hessenberg form H which is zero below the first subdiagonal. The reduction may be written with Q orthogonal if A is real, or with Q unitary if A is complex. The reduction is performed by subroutine PxGEHRD, which represents Q in a factored form, as described in section 3.4. The routine PxORMHR (or in the complex case PxUNMHR) is provided to multiply another matrix by Q without forming Q explicitly.
2. The upper Hessenberg matrix H is reduced to Schur form T, giving the Schur factorization (for H real) or (for H complex). The matrix S (the Schur vectors of H) may optionally be computed as well. Alternatively S may be postmultiplied into the matrix Q determined in stage 1, to give the matrix Z = Q S, the Schur vectors of A. The eigenvalues are obtained from the diagonal of T. All this is done by subroutine PxLAHQR.

The algorithm used in PxLAHQR is similar to the LAPACK routine xLAHQR. Unlike xLAHQR, however, instead of sending one double shift through the largest unreduced submatrix, this algorithm sends multiple double shifts and spaces them apart so that there can be parallelism across several processor row/columns. Another critical difference is that this algorithm applies multiple double shifts in a block fashion, as opposed to xLAHQR which applies one double shift at a time, and xHSEQR from LAPACK which attempts to achieve a blocked code by combining the double shifts into one single large multi-shift. For complete details, please refer to [79].

See table 3.9 for a complete list of the routines.

Table 3.9: Computational routines for the nonsymmetric eigenproblem

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