Nonsymmetric Eigenvalue Problems: LAPACK Computational Routines

Developer Reference for Intel® oneAPI Math Kernel Library for C

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Date 3/31/2023

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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

z^HA = λz^H (left eigenvectors z).

Nonsymmetric eigenvalue problems have the following properties:

The number of eigenvectors may be less than the matrix order (but is not less than the number of distinct eigenvalues of A).
Eigenvalues may be complex even for a real matrix A.
If a real nonsymmetric matrix has a complex eigenvalue a+bi corresponding to an eigenvector z, then a-bi is also an eigenvalue. The eigenvalue a-bi corresponds to the eigenvector whose elements are complex conjugate to the elements of z.

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 = QHQ^H 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 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".

Computational Routines for Solving Nonsymmetric Eigenvalue Problems
Operation performed	Routines for real matrices	Routines for complex matrices
Reduce to Hessenberg form `A = QHQ^H`	?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

Decision Tree: Real Nonsymmetric Eigenvalue Problems

Decision Tree: Complex Non-Hermitian Eigenvalue Problems

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Developer Reference for Intel® oneAPI Math Kernel Library for C

Nonsymmetric Eigenvalue Problems: LAPACK Computational Routines