Direct Methods

In this section, we briefly discuss methods for computing eigenvalues and eigenvectors of dense matrices. With the factorization (5.4) of , the GHEP (5.1) is reduced to the standard Hermitian eigenproblem (5.5). Then one may use the direct methods discussed in §4.2.

Specifically, in LAPACK [12], the following driver routines are provided for solving the GHEP (5.1) with positive definite:

- a simple driver
`xSYGV`computes all the eigenvalues and (optionally) eigenvectors. The underlying algorithm is the QR algorithm; see §4.2. - an expert driver
`xSYGVX`computes all or a selected subset of the eigenvalues and (optionally) eigenvectors. If few enough eigenvalues or eigenvectors are desired, the expert driver is faster than the simple driver. This driver routine uses the QR algorithm or bisection method and inverse iteration, whichever is more efficient. - a divide-and-conquer driver
`xSYGVD`solves the same problem as the simple driver. It is much faster than the simple driver for large matrices, but uses more workspace. The name divide-and-conquer refers to the underlying divide-and-conquer algorithm; see §4.2.