Nonsymmetric Eigenvalue Problems: LAPACK Computational Routines

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

A nonsymmetric eigenvalue problem is as follows: given a nonsymmetric (or non-Hermitian) matrix A, find the eigenvaluesλ and the corresponding eigenvectorsz that satisfy the equation

Az = λz (right eigenvectors z)

or the equation

zHA = λzH (left eigenvectors z).

Nonsymmetric eigenvalue problems have the following properties:

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 A = QHQH as well as routines to solve eigenvalue problems with Hessenberg matrices, forming the Schur factorization of such matrices and computing the corresponding condition numbers.

Computational Routines for Solving Nonsymmetric Eigenvalue Problems

Operation performed

Routines for real matrices

Routines for complex matrices

Reduce to Hessenberg form A = QHQH

?gehrd,

?gehrd

Generate the matrix Q

?orghr

?unghr

Apply the matrix Q

?ormhr

?unmhr

Balance matrix

?gebal

?gebal

Transform eigenvectors of balanced matrix to those of the original matrix

?gebak

?gebak

Find eigenvalues and Schur factorization (QR algorithm)

?hseqr

?hseqr

Find eigenvectors from Hessenberg form (inverse iteration)

?hsein

?hsein

Find eigenvectors from Schur factorization

?trevc

?trevc

Estimate sensitivities of eigenvalues and eigenvectors

?trsna

?trsna

Reorder Schur factorization

?trexc

?trexc

Reorder Schur factorization, find the invariant subspace and estimate sensitivities

?trsen

?trsen

Solves Sylvester's equation.

?trsyl

?trsyl