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

###

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 is an orthogonal (or unitary)
matrix whose columns
are the eigenvectors. This is the classical **spectral factorization**
of .

There are four types of driver routines for symmetric and Hermitian
eigenproblems, and these are listed below.
Originally LAPACK had just the first two (the simple and expert drivers), and
the last 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 in
the LAPACK Users' Guide[1]).

- 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 in the LAPACK
Users' Guide[1]).

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

** Next:** Nonsymmetric Eigenproblems (NEP)
** Up:** Standard Eigenvalue and Singular
** Previous:** Standard Eigenvalue and Singular
** Contents**
** Index**
Susan Blackford
2001-08-19