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###

Symmetric Eigenproblems (SEP)

The **symmetric eigenvalue problem** is to find the **eigenvalues**,
, and corresponding **eigenvectors**,
, such that

For the **Hermitian eigenvalue problem** we have

For both problems the eigenvalues
are real.
When all eigenvalues and eigenvectors have been computed, we write:

where
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.

- A
**simple** driver (name ending -EV) computes all the eigenvalues
and (optionally) eigenvectors.

- An
**expert** driver (name ending -EVX) 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.

- A
**divide-and-conquer** driver (name ending -EVD) 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 algorithm (see sections 2.4.4 and 3.4.3).

- A
**relatively robust representation** (RRR) driver (name ending
-EVR) computes all or (in a later release) a subset of the eigenvalues,
and (optionally) eigenvectors. It is the fastest algorithm of all (except
for a few cases), and uses the least workspace. The name RRR refers
to the underlying algorithm (see sections 2.4.4 and 3.4.3).

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

** Next:** Nonsymmetric Eigenproblems
(NEP) ** Up:** Standard
Eigenvalue and Singular ** Previous:** Standard Eigenvalue and Singular ** Contents** ** Index**
*Susan Blackford*

*1999-10-01*