naginterfaces.library.lapackeig.dggev3

naginterfaces.library.lapackeig.dggev3(jobvl, jobvr, a, b)

dggev3 computes for a pair of $n\times n$ real nonsymmetric matrices $(A,B)$ the generalized eigenvalues and, optionally, the left and/or right generalized eigenvectors using the $QZ$ algorithm.

For full information please refer to the NAG Library document for f08wc

https://support.nag.com/numeric/nl/nagdoc_30.3/flhtml/f08/f08wcf.html

Parameters

jobvlstr, length 1

If $jobvl=\text{'N'}$, do not compute the left generalized eigenvectors.

If $jobvl=\text{'V'}$, compute the left generalized eigenvectors.

jobvrstr, length 1

If $jobvr=\text{'N'}$, do not compute the right generalized eigenvectors.

If $jobvr=\text{'V'}$, compute the right generalized eigenvectors.

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

The matrix $A$ in the pair $(A,B)$.

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

The matrix $B$ in the pair $(A,B)$.

Returns

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

$a$ has been overwritten.

bfloat, ndarray, shape $(n,n)$

$b$ has been overwritten.

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

The element $alphar[j-1]$ contains the real part of ${\alpha }_j$.

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

The element $alphai[j-1]$ contains the imaginary part of ${\alpha }_j$.

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

$\left(alphar\right[\text{j}-1]+alphai[\text{j}-1]\times i)/beta[\text{j}-1]$, for $\text{j}=1,2,\dots ,n$, will be the generalized eigenvalues.

If $alphai[j-1]$ is zero, then the $j$th eigenvalue is real; if positive, then the $j$th and $(j+1)$st eigenvalues are a complex conjugate pair, with $alphai\left[j\right]$ negative.

Note: the quotients $alphar[j-1]/beta[j-1]$ and $alphai[j-1]/beta[j-1]$ may easily overflow or underflow, and $beta[j-1]$ may even be zero.

Thus, you should avoid naively computing the ratio ${\alpha }_j/{\beta }_j$.

However, $max\left(\left|{\alpha }_j\right|\right)$ will always be less than and usually comparable with ${\parallel A\parallel }_2$ in magnitude, and $max\left(\left|{\beta }_j\right|\right)$ will always be less than and usually comparable with ${\parallel B\parallel }_2$.

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 the corresponding eigenvalues.

If the $j$th eigenvalue is real, then $u_j=vl[:,j]$, the $j$th column of $vl$.

If the $j$th and $(j+1)$th eigenvalues form a complex conjugate pair, then $u_j=vl[:,j]+i\times vl[:,j+1]$ and $u(j+1)=vl[:,j]-i\times vl[:,j+1]$.

Each eigenvector will be scaled so the largest component has $\left|\text{real part}\right|+\left|\text{imag. part}\right|=1$.

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 the corresponding eigenvalues.

If the $j$th eigenvalue is real, then $v_j=vr[:,j]$, the $j$th column of $VR$.

If the $j$th and $(j+1)$th eigenvalues form a complex conjugate pair, then $v_j=vr[:,j]+i\times vr[:,j+1]$ and $v_{j+1}=vr[:,j]-i\times vr[:,j+1]$.

Each eigenvector will be scaled so the largest component has $\left|\text{real part}\right|+\left|\text{imag. part}\right|=1$.

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: $n\ge 0$.

(errno $n+1$)

The $QZ$ iteration failed with an unexpected error, please contact NAG.

(errno $n+2$)

A failure occurred in dtgevc() while computing generalized eigenvectors.

Warns

NagAlgorithmicWarning

(errno $(i>0)\text{and}(i\le n)$)

The $QZ$ iteration failed. No eigenvectors have been calculated but $alphar\left[j\right]$, $alphai\left[j\right]$ and $beta\left[j\right]$ should be correct from element $⟨value⟩$.

Notes

A generalized eigenvalue for a pair of matrices $(A,B)$ is a scalar $\lambda$ or a ratio $\alpha /\beta =\lambda$, such that $A-\lambda B$ is singular. It is usually represented as the pair $(\alpha ,\beta )$, as there is a reasonable interpretation for $\beta =0$, and even for both being zero.

The right eigenvector $v_j$ corresponding to the eigenvalue ${\lambda }_j$ of $(A,B)$ satisfies

$$A{v}_j={\lambda }_{j}B{v}_j\text{.}$$

The left eigenvector $u_j$ corresponding to the eigenvalue ${\lambda }_j$ of $(A,B)$ satisfies

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

where $u_j^H$ is the conjugate-transpose of $u_j$.

All the eigenvalues and, if required, all the eigenvectors of the generalized eigenproblem $Ax=\lambda Bx$, where $A$ and $B$ are real, square matrices, are determined using the $QZ$ algorithm. The $QZ$ algorithm consists of four stages:

1. $A$ is reduced to upper Hessenberg form and at the same time $B$ is reduced to upper triangular form.

2. $A$ is further reduced to quasi-triangular form while the triangular form of $B$ is maintained. This is the real generalized Schur form of the pair $(A,B)$.

3. The quasi-triangular form of $A$ is reduced to triangular form and the eigenvalues extracted. This function does not actually produce the eigenvalues ${\lambda }_j$, but instead returns ${\alpha }_j$ and ${\beta }_j$ such that

   $${\lambda }_j={\alpha }_j/{\beta }_j\text{,}j=1,2,\dots ,n\text{.}$$

   The division by ${\beta }_j$ becomes your responsibility, since ${\beta }_j$ may be zero, indicating an infinite eigenvalue. Pairs of complex eigenvalues occur with ${\alpha }_j/{\beta }_j$ and ${\alpha }_{j+1}/{\beta }_{j+1}$ complex conjugates, even though ${\alpha }_j$ and ${\alpha }_{j+1}$ are not conjugate.

4. If the eigenvectors are required they are obtained from the triangular matrices and then transformed back into the original coordinate system.

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, 2012, Matrix Computations, (4th Edition), Johns Hopkins University Press, Baltimore

Wilkinson, J H, 1979, Kronecker’s canonical form and the $QZ$ algorithm, Linear Algebra Appl. (28), 285–303