naginterfaces.library.lapackeig.zgeev

naginterfaces.library.lapackeig.zgeev(jobvl, jobvr, a)

zgeev computes the eigenvalues and, optionally, the left and/or right eigenvectors for an $n\times n$ complex nonsymmetric matrix $A$.

For full information please refer to the NAG Library document for f08nn

https://support.nag.com/numeric/nl/nagdoc_30/flhtml/f08/f08nnf.html

Parameters

jobvlstr, length 1

If $jobvl=\text{'N'}$, the left eigenvectors of $A$ are not computed.

If $jobvl=\text{'V'}$, the left eigenvectors of $A$ are computed.

jobvrstr, length 1

If $jobvr=\text{'N'}$, the right eigenvectors of $A$ are not computed.

If $jobvr=\text{'V'}$, the right eigenvectors of $A$ are computed.

acomplex, array-like, shape $(n,n)$

The $n\times n$ matrix $A$.

Returns

acomplex, ndarray, shape $(n,n)$

$a$ has been overwritten.

wcomplex, ndarray, shape $\left(n\right)$

Contains the computed eigenvalues.

vlcomplex, ndarray, shape $(:,:)$

If $jobvl=\text{'V'}$, the left eigenvectors $u_j$ are stored one after another in the columns of $vl$, in the same order as their corresponding eigenvalues; that is $u_j=vl[\text{i}-1,j-1]$, for $\text{i}=1,2,\dots ,n$.

If $jobvl=\text{'N'}$, $vl$ is not referenced.

vrcomplex, ndarray, shape $(:,:)$

If $jobvr=\text{'V'}$, the right eigenvectors $v_j$ are stored one after another in the columns of $vr$, in the same order as their corresponding eigenvalues; that is $v_j=vr[\text{i}-1,j-1]$, for $\text{i}=1,2,\dots ,n$.

If $jobvr=\text{'N'}$, $vr$ is not referenced.

Raises

NagValueError

(errno $-1$)

On entry, error in parameter $jobvl$.

Constraint: $jobvl=\text{'N'}$ or $\text{'V'}$.

(errno $-2$)

On entry, error in parameter $jobvr$.

Constraint: $jobvr=\text{'N'}$ or $\text{'V'}$.

(errno $-3$)

On entry, error in parameter $\text{n}$.

Constraint: $n\ge 0$.

Warns

NagAlgorithmicWarning

(errno $i>0$)

The $QR$ algorithm failed to compute all the eigenvalues, and no eigenvectors have been computed; elements $⟨value⟩$ to $\text{n}$ of $w$ contain eigenvalues which have converged.

Notes

The right eigenvector $v_j$ of $A$ satisfies

$$A{v}_j={\lambda }_j{v}_j$$

where ${\lambda }_j$ is the $j$th eigenvalue of $A$. The left eigenvector $u_j$ of $A$ satisfies

$$u_j^{H}A={\lambda }_j{u}_j^H$$

where $u_j^H$ denotes the conjugate transpose of $u_j$.

The matrix $A$ is first reduced to upper Hessenberg form by means of unitary similarity transformations, and the $QR$ algorithm is then used to further reduce the matrix to upper triangular Schur form, $T$, from which the eigenvalues are computed. Optionally, the eigenvectors of $T$ are also computed and backtransformed to those of $A$.

References

Anderson, E, Bai, Z, Bischof, C, Blackford, S, Demmel, J, Dongarra, J J, Du Croz, J J, Greenbaum, A, Hammarling, S, McKenney, A and Sorensen, D, 1999, LAPACK Users’ Guide, (3rd Edition), SIAM, Philadelphia, https://www.netlib.org/lapack/lug

Golub, G H and Van Loan, C F, 1996, Matrix Computations, (3rd Edition), Johns Hopkins University Press, Baltimore