The Schur form depends on the order of the eigenvalues
on the diagonal of ** T** and this may optionally be chosen by the
user. Suppose the user chooses that
,
, appear in the upper left corner of

The following routines perform this re-ordering and also compute condition numbers for eigenvalues, eigenvectors, and invariant subspaces:

- 1.
- xTREXC will move an eigenvalue (or
**2**-by-**2**block) on the diagonal of the Schur form from its original position to any other position. It may be used to choose the order in which eigenvalues appear in the Schur form. - 2.
- xTRSYL solves the Sylvester matrix equation
for
, given matrices*X*,*A*and*B*, with*C*and*A*(quasi) triangular. It is used in the routines xTRSNA and xTRSEN, but it is also of independent interest.*B* - 3.
- xTRSNA computes the condition numbers of the eigenvalues and/or
right eigenvectors of a matrix
in Schur form. These are the same as the condition numbers of the eigenvalues and right eigenvectors of the original matrix*T*from which*A*is derived. The user may compute these condition numbers for all eigenvalue/eigenvector pairs, or for any selected subset. For more details, see section 4.8 and [12].*T* - 4.
- xTRSEN moves a selected subset of the eigenvalues
of a matrix
in Schur form to the upper left corner of*T*, and optionally computes the condition numbers of their average value and of their right invariant subspace. These are the same as the condition numbers of the average eigenvalue and right invariant subspace of the original matrix*T*from which*A*is derived. For more details, see section 4.8 and [12]*T*

See Table 2.11 for a complete list of the routines.