Generalized Nonsymmetric Eigenvalue Problems: LAPACK Computational Routines

This topic describes LAPACK routines for solving generalized nonsymmetric eigenvalue problems, reordering the generalized Schur factorization of a pair of matrices, as well as performing a number of related computational tasks.

A generalized nonsymmetric eigenvalue problem is as follows: given a pair of nonsymmetric (or non-Hermitian) n-by-n matrices A and B, find the generalized eigenvaluesλ and the corresponding generalized eigenvectorsx and y that satisfy the equations

Ax = λBx (right generalized eigenvectors x)

and

y^HA = λy^HB (left generalized eigenvectors y).

Table "Computational Routines for Solving Generalized Nonsymmetric Eigenvalue Problems" lists LAPACK routines used to solve the generalized nonsymmetric eigenvalue problems and the generalized Sylvester equation.

Computational Routines for Solving Generalized Nonsymmetric Eigenvalue Problems
Routine name	Operation performed
gghrd	Reduces a pair of matrices to generalized upper Hessenberg form using orthogonal/unitary transformations.
ggbal	Balances a pair of general real or complex matrices.
ggbak	Forms the right or left eigenvectors of a generalized eigenvalue problem.
gghd3	Reduces a pair of matrices to generalized upper Hessenberg form.
hgeqz	Implements the QZ method for finding the generalized eigenvalues of the matrix pair (H,T).
tgevc	Computes some or all of the right and/or left generalized eigenvectors of a pair of upper triangular matrices
tgexc	Reorders the generalized Schur decomposition of a pair of matrices (A,B) so that one diagonal block of (A,B) moves to another row index.
tgsen	Reorders the generalized Schur decomposition of a pair of matrices (A,B) so that a selected cluster of eigenvalues appears in the leading diagonal blocks of (A,B).
tgsyl	Solves the generalized Sylvester equation.
tgsyl	Estimates reciprocal condition numbers for specified eigenvalues and/or eigenvectors of a pair of matrices in generalized real Schur canonical form.