The generalized Schur
form depends on the order of the eigenvalues on the
diagonal of **( S,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 reordering and also compute condition numbers for eigenvalues, eigenvectors and deflating subspaces:

- 1.
- xTGEXC
will move an eigenvalue pair (or a pair of 2-by-2 blocks)
on the diagonal of the generalized Schur form
**(**from its original position to any other position. It may be used to choose the order in which eigenvalues appear in the generalized Schur form. The reordering is performed with orthogonal (unitary) transformation matrices. For more details see [70,73].*S*,*T*) - 2.
- xTGSYL
solves the generalized Sylvester equations
and*AR*-*LB*=*sC*for*DR*-*LE*=*sF*and*L*, given*R*and*A*upper (quasi-)triangular and*B*and*D*upper triangular. It is also possible to solve a transposed system (conjugate transposed system in the complex case)*E*and*A*^{T}*X*+*D*^{T}*Y*=*sC***-**for*X B*^{T}-*Y E*^{T}=*sF*and*X*. The scaling factor*Y*is set during the computations to avoid overflow. Optionally, xTGSYL computes a Frobenius norm-based estimate of the ``separation'' between the two matrix pairs*s***(**and*A*,*B*)**(**. xTGSYL is used by the routines xTGSNA and xTGSEN, but it is also of independent interest. For more details see [71,74,75].*D*,*E*) - 3.
- xTGSNA
computes condition numbers of the
eigenvalues and/or left and right eigenvectors of a matrix pair
**(**in generalized Schur form. These are the same as the condition numbers of the eigenvalues and eigenvectors of the original matrix pair*S*,*T*)**(**, from which*A*,*B*)**(**is derived. The user may compute these condition numbers for all eigenvalues and associated eigenvectors, or for any selected subset. For more details see section 4.11 and [73].*S*,*T*) - 4.
- xTGSEN
moves a selected subset of the eigenvalues of a matrix pair
**(**in generalized Schur form to the upper left corner of*S*,*T*)**(**, and optionally computes condition numbers of their average value and their associated pair of (left and right) deflating subspaces. These are the same as the condition numbers of the average eigenvalue and the deflating subspace pair of the original matrix pair*S*,*T*)**(**, from which*A*,*B*)**(**is derived. For more details see section 4.11 and [73].*S*,*T*)

See Table 2.15 for a complete list of the routines, where, to save space, the word ``generalized'' is omitted.