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Symmetric Eigenproblems (SEP)

The symmetric eigenvalue problem is to find the eigenvalues, $\lambda$, and corresponding eigenvectors, $z \ne 0$, such that

\begin{displaymath}
Az = \lambda z, \quad A = A^T, \mbox{ where } A \mbox{ is real}.
\end{displaymath}

For the Hermitian eigenvalue problem we have

\begin{displaymath}
Az = \lambda z, \quad A = A^H.
\end{displaymath}

For both problems the eigenvalues $\lambda$ are real.

When all eigenvalues and eigenvectors have been computed, we write:

\begin{displaymath}
A = Z \Lambda Z^T
\end{displaymath}

where $\Lambda$ is a diagonal matrix whose diagonal elements are the eigenvalues, and Z is an orthogonal (or unitary) matrix whose columns are the eigenvectors. This is the classical spectral factorization of A.

There are four types of driver routines for symmetric and Hermitian eigenproblems. Originally LAPACK had just the simple and expert drivers described below, and the other two were added after improved algorithms were discovered. Ultimately we expect the algorithm in the most recent driver (called RRR below) to supersede all the others, but in LAPACK 3.0 the other drivers may still be faster on some problems, so we retain them.

Different driver routines are provided to take advantage of special structure or storage of the matrix A, as shown in Table 2.5.


next up previous contents index
Next: Nonsymmetric Eigenproblems (NEP) Up: Standard Eigenvalue and Singular Previous: Standard Eigenvalue and Singular   Contents   Index
Susan Blackford
1999-10-01