naginterfaces.library.lapackeig.dggevx

naginterfaces.library.lapackeig.dggevx(balanc, jobvl, jobvr, sense, a, b)

dggevx 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.

Optionally it also computes a balancing transformation to improve the conditioning of the eigenvalues and eigenvectors, reciprocal condition numbers for the eigenvalues, and reciprocal condition numbers for the right eigenvectors.

For full information please refer to the NAG Library document for f08wb

https://support.nag.com/numeric/nl/nagdoc_30.1/flhtml/f08/f08wbf.html

Parameters

balancstr, length 1

Specifies the balance option to be performed.

$balanc=\text{'N'}$

Do not diagonally scale or permute.

$balanc=\text{'P'}$

Permute only.

$balanc=\text{'S'}$

Scale only.

$balanc=\text{'B'}$

Both permute and scale.

Computed reciprocal condition numbers will be for the matrices after permuting and/or balancing.

Permuting does not change condition numbers (in exact arithmetic), but balancing does.

In the absence of other information, $balanc=\text{'B'}$ is recommended.

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.

sensestr, length 1

Determines which reciprocal condition numbers are computed.

$sense=\text{'N'}$

None are computed.

$sense=\text{'E'}$

Computed for eigenvalues only.

$sense=\text{'V'}$

Computed for eigenvectors only.

$sense=\text{'B'}$

Computed for eigenvalues and 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. If $jobvl=\text{'V'}$ or $jobvr=\text{'V'}$ or both, then $A$ contains the first part of the real Schur form of the ‘balanced’ versions of the input $A$ and $B$.

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 generalized eigenvectors $u_j$ are stored one after another in the columns of $vl$, in the same order as the corresponding eigenvalues. Each eigenvector will be scaled so the largest component will have $\left|\text{real part}\right|+\left|\text{imag. part}\right|=1$.

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

vrfloat, ndarray, shape $(:,:)$

If $jobvr=\text{'V'}$, the right generalized eigenvectors $v_j$ are stored one after another in the columns of $vr$, in the same order as the corresponding eigenvalues. Each eigenvector will be scaled so the largest component will have $\left|\text{real part}\right|+\left|\text{imag. part}\right|=1$.

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

iloint

$ilo$ and $ihi$ are integer values such that $a[i-1,j-1]=0$ and $b[i-1,j-1]=0$ if $i>j$ and $j=1,2,\dots ,ilo-1$ or $i=ihi+1,\dots ,n$.

If $balanc=\text{'N'}$ or $\text{'S'}$, $ilo=1$ and $ihi=n$.

ihiint

$ilo$ and $ihi$ are integer values such that $a[i-1,j-1]=0$ and $b[i-1,j-1]=0$ if $i>j$ and $j=1,2,\dots ,ilo-1$ or $i=ihi+1,\dots ,n$.

If $balanc=\text{'N'}$ or $\text{'S'}$, $ilo=1$ and $ihi=n$.

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

Details of the permutations and scaling factors applied to the left side of $A$ and $B$.

If ${\text{pl}}_j$ is the index of the row interchanged with row $j$, and ${\text{dl}}_j$ is the scaling factor applied to row $j$, then:

$lscale[\text{j}-1]={\text{pl}}_{\text{j}}$, for $\text{j}=1,2,\dots ,ilo-1$;

$lscale={\text{dl}}_{\text{j}}$, for $\text{j}=ilo,\dots ,ihi$;

$lscale={\text{pl}}_{\text{j}}$, for $\text{j}=ihi+1,\dots ,n$.

The order in which the interchanges are made is $\text{n}$ to $ihi+1$, then $1$ to $ilo-1$.

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

Details of the permutations and scaling factors applied to the right side of $A$ and $B$.

If ${\text{pr}}_j$ is the index of the column interchanged with column $j$, and ${\text{dr}}_j$ is the scaling factor applied to column $j$, then:

$rscale[\text{j}-1]={\text{pr}}_{\text{j}}$, for $\text{j}=1,2,\dots ,ilo-1$;

if $rscale={\text{dr}}_{\text{j}}$, for $\text{j}=ilo,\dots ,ihi$;

if $rscale={\text{pr}}_{\text{j}}$, for $\text{j}=ihi+1,\dots ,n$.

The order in which the interchanges are made is $\text{n}$ to $ihi+1$, then $1$ to $ilo-1$.

abnrmfloat

The $1$-norm of the balanced matrix $A$.

bbnrmfloat

The $1$-norm of the balanced matrix $B$.

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

If $sense=\text{'E'}$ or $\text{'B'}$, the reciprocal condition numbers of the eigenvalues, stored in consecutive elements of the array. For a complex conjugate pair of eigenvalues two consecutive elements of $rconde$ are set to the same value. Thus $rconde[j-1]$, $rcondv[j-1]$, and the $j$th columns of $vl$ and $vr$ all correspond to the $j$th eigenpair.

If $sense=\text{'V'}$, $rconde$ is not referenced.

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

If $sense=\text{'V'}$ or $\text{'B'}$, the estimated reciprocal condition numbers of the eigenvectors, stored in consecutive elements of the array. For a complex eigenvector two consecutive elements of $rcondv$ are set to the same value.

If $sense=\text{'E'}$, $rcondv$ is not referenced.

Raises

NagValueError

(errno $-1$)

On entry, error in parameter $balanc$.

Constraint: $balanc=\text{'N'}$, $\text{'P'}$, $\text{'S'}$ or $\text{'B'}$.

(errno $-2$)

On entry, error in parameter $jobvl$.

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

(errno $-3$)

On entry, error in parameter $jobvr$.

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

(errno $-4$)

On entry, error in parameter $sense$.

Constraint: $sense=\text{'N'}$, $\text{'E'}$, $\text{'V'}$ or $\text{'B'}$.

(errno $-5$)

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.

For details of the balancing option, see Notes for dggbal.

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