naginterfaces.library.lapackeig.dgeev

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

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

For full information please refer to the NAG Library document for f08na

https://support.nag.com/numeric/nl/nagdoc_30.2/flhtml/f08/f08naf.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.

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

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

Returns

afloat, ndarray, shape $(n,n)$

$a$ has been overwritten.

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

$wr$ and $wi$ contain the real and imaginary parts, respectively, of the computed eigenvalues. Complex conjugate pairs of eigenvalues appear consecutively with the eigenvalue having the positive imaginary part first.

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

$wr$ and $wi$ contain the real and imaginary parts, respectively, of the computed eigenvalues. Complex conjugate pairs of eigenvalues appear consecutively with the eigenvalue having the positive imaginary part first.

vlfloat, 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. If the $j$th eigenvalue is real, then $u_j=vl[\text{i}-1,j-1]$, for $\text{i}=1,2,\dots ,n$. If the $j$th and $(j+1)$st eigenvalues form a complex conjugate pair, then $u_j=vl[\text{i}-1,j-1]+\text{i}\times vl[\text{i}-1,j]$ and $u_{j+1}=vl[\text{i}-1,j-1]-\text{i}\times vl[\text{i}-1,j]$, for $\text{i}=1,2,\dots ,n$.

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

vrfloat, 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. If the $j$th eigenvalue is real, then $v_j=vr[\text{i}-1,j-1]$, for $\text{i}=1,2,\dots ,n$. If the $j$th and $(j+1)$st eigenvalues form a complex conjugate pair, then $v_j=vr[\text{i}-1,j-1]+\text{i}\times vr[\text{i}-1,j]$ and $v_{j+1}=vr[\text{i}-1,j-1]-\text{i}\times vr[\text{i}-1,j]$, 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 $wr$ and $wi$ 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 orthogonal similarity transformations, and the $QR$ algorithm is then used to further reduce the matrix to upper quasi-triangular Schur form, $T$, with $1\times 1$ and $2\times 2$ blocks on the main diagonal. The eigenvalues are computed from $T$, the $2\times 2$ blocks corresponding to complex conjugate pairs and, optionally, the eigenvectors of $T$ are computed and backtransformed to the eigenvectors 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