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.

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.