Nonsymmetric Eigenvalue Problems: LAPACK Computational Routines
topicdescribes LAPACK routines for solving nonsymmetric eigenvalue problems, computing the Schur factorization of general matrices, as well as performing a number of related computational tasks.
or the equation
Nonsymmetric eigenvalue problems have the following properties:
- Eigenvalues may be complex even for a real matrixA.
- If a real nonsymmetric matrix has a complex eigenvaluecorresponding to an eigenvectora+biz, thenis also an eigenvalue. The eigenvaluea-bicorresponds to the eigenvector whose elements are complex conjugate to the elements ofa-biz.
To solve a nonsymmetric eigenvalue problem with LAPACK, you usually need to reduce the matrix to the upper Hessenberg form and then solve the eigenvalue problem with the Hessenberg matrix obtained. Table
"Computational Routines for Solving Nonsymmetric Eigenvalue Problems"lists LAPACK routines to reduce the matrix to the upper Hessenberg form by an orthogonal (or unitary) similarity transformation
as well as routines to solve eigenvalue problems with Hessenberg matrices, forming the Schur factorization of such matrices and computing the corresponding condition numbers.
The corresponding routine names in the Fortran 95 interface are without the first symbol.
The decision tree in Figure
"Decision Tree: Real Nonsymmetric Eigenvalue Problems"helps you choose the right routine or sequence of routines for an eigenvalue problem with a real nonsymmetric matrix. If you need to solve an eigenvalue problem with a complex non-Hermitian matrix, use the decision tree shown in Figure
"Decision Tree: Complex Non-Hermitian Eigenvalue Problems".
Routines for real matrices
Routines for complex matrices