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`.

Three types of driver routines are provided for symmetric or Hermitian eigenproblems:

- a
**simple**driver (name ending -EV) , which computes all the eigenvalues and (optionally) the eigenvectors of a symmetric or Hermitian matrix`A`; - a
**divide and conquer**driver (name ending -EVD) , which computes all the eigenvalues and (optionally) the eigenvectors of a symmetric or Hermitian matrix`A`using an algorithm based on divide and conquer ; the -EVD routines can be much faster than their -EV counterparts, but use more workspace; - an
**expert**driver (name ending -EVX) , which can compute either all or a selected subset of the eigenvalues, and (optionally) the corresponding eigenvectors.

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

In the future LAPACK will include routines based on the Jacobi algorithm [76][69][24], which are slower than the above routines but can be significantly more accurate.

Tue Nov 29 14:03:33 EST 1994