naginterfaces.library.lapackeig.dtgsna

naginterfaces.library.lapackeig.dtgsna(job, howmny, a, b, mm, select=None, vl=None, vr=None)

dtgsna estimates condition numbers for specified eigenvalues and/or eigenvectors of a matrix pair in generalized real Schur form.

For full information please refer to the NAG Library document for f08yl

https://support.nag.com/numeric/nl/nagdoc_30.3/flhtml/f08/f08ylf.html

Parameters

jobstr, length 1

Indicates whether condition numbers are required for eigenvalues and/or eigenvectors.

$job=\text{'E'}$

Condition numbers for eigenvalues only are computed.

$job=\text{'V'}$

Condition numbers for eigenvectors only are computed.

$job=\text{'B'}$

Condition numbers for both eigenvalues and eigenvectors are computed.

howmnystr, length 1

Indicates how many condition numbers are to be computed.

$howmny=\text{'A'}$

Condition numbers for all eigenpairs are computed.

$howmny=\text{'S'}$

Condition numbers for selected eigenpairs (as specified by $select$) are computed.

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

The upper quasi-triangular matrix $S$.

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

The upper triangular matrix $T$.

mmint

The number of elements in the arrays $s$ and $dif$.

selectNone or bool, array-like, shape $(:)$, optional

Note: the required length for this argument is determined as follows: if $howmny=\text{'S'}$: $n$; otherwise: $1$.

Specifies the eigenpairs for which condition numbers are to be computed if $howmny=\text{'S'}$. To select condition numbers for the eigenpair corresponding to the real eigenvalue ${\lambda }_j$, $select[j-1]$ must be set $True$. To select condition numbers corresponding to a complex conjugate pair of eigenvalues ${\lambda }_j$ and ${\lambda }_{j+1}$, $select[j-1]$ and/or $select\left[j\right]$ must be set to $True$.

If $howmny=\text{'A'}$, $select$ is not referenced.

vlNone or float, array-like, shape $(:,:)$, optional

Note: the required extent for this argument in dimension 1 is determined as follows: if $job\text{in}(\text{'E'},\text{'B'})$: $n$; otherwise: $1$.

Note: the required extent for this argument in dimension 2 is determined as follows: if $job\text{in}(\text{'E'},\text{'B'})$: $mm$; otherwise: $0$.

If $job=\text{'E'}$ or $\text{'B'}$, $vl$ must contain left eigenvectors of $(S,T)$, corresponding to the eigenpairs specified by $howmny$ and $select$. The eigenvectors must be stored in consecutive columns of $vl$, as returned by dggev3() or dtgevc().

If $job=\text{'V'}$, $vl$ is not referenced.

vrNone or float, array-like, shape $(:,:)$, optional

Note: the required extent for this argument in dimension 1 is determined as follows: if $job\text{in}(\text{'E'},\text{'B'})$: $n$; otherwise: $1$.

Note: the required extent for this argument in dimension 2 is determined as follows: if $job\text{in}(\text{'E'},\text{'B'})$: $mm$; otherwise: $0$.

If $job=\text{'E'}$ or $\text{'B'}$, $vr$ must contain right eigenvectors of $(S,T)$, corresponding to the eigenpairs specified by $howmny$ and $select$. The eigenvectors must be stored in consecutive columns of $vr$, as returned by dggev3() or dtgevc().

If $job=\text{'V'}$, $vr$ is not referenced.

Returns

sfloat, ndarray, shape $(:)$

If $job=\text{'E'}$ or $\text{'B'}$, the reciprocal condition numbers of the selected eigenvalues, stored in consecutive elements of the array. For a complex conjugate pair of eigenvalues two consecutive elements of $s$ are set to the same value. Thus $s[j-1]$, $dif[j-1]$, and the $j$th columns of $VL$ and $VR$ all correspond to the same eigenpair (but not in general the $j$th eigenpair, unless all eigenpairs are selected).

If $job=\text{'V'}$, $s$ is not referenced.

diffloat, ndarray, shape $(:)$

If $job=\text{'V'}$ or $\text{'B'}$, the estimated reciprocal condition numbers of the selected eigenvectors, stored in consecutive elements of the array. For a complex eigenvector two consecutive elements of $dif$ are set to the same value. If the eigenvalues cannot be reordered to compute $dif[j-1]$, $dif[j-1]$ is set to $0$; this can only occur when the true value would be very small anyway.

If $job=\text{'E'}$, $dif$ is not referenced.

mint

M, the number of elements of the arrays $s$ and $dif$ used to store the specified condition numbers; for each selected real eigenvalue one element is used, and for each selected complex conjugate pair of eigenvalues, two elements are used. If $howmny=\text{'A'}$, $m$ is set to $\text{n}$.

Raises

NagValueError

(errno $-1$)

On entry, error in parameter $job$.

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

(errno $-2$)

On entry, error in parameter $howmny$.

Constraint: $howmny=\text{'A'}$ or $\text{'S'}$.

(errno $-4$)

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

Constraint: $n\ge 0$.

(errno $-15$)

On entry, error in parameter $mm$.

Constraint: $mm\ge n$.

(errno $-15$)

On entry, error in parameter $mm$.

Constraint: $mm\ge \text{m}$.

Notes

dtgsna estimates condition numbers for specified eigenvalues and/or right eigenvectors of an $n\times n$ matrix pair $(S,T)$ in real generalized Schur form. The function actually returns estimates of the reciprocals of the condition numbers in order to avoid possible overflow.

The pair $(S,T)$ are in real generalized Schur form if $S$ is block upper triangular with $1\times 1$ and $2\times 2$ diagonal blocks and $T$ is upper triangular as returned, for example, by dggesx() or dgges3(), or dhgeqz() with $\text{job}=\text{'S'}$. The diagonal elements, or blocks, define the generalized eigenvalues $({\alpha }_{\text{i}},{\beta }_{\text{i}})$, for $\text{i}=1,2,\dots ,n$, of the pair $(S,T)$ and the eigenvalues are given by

$${\lambda }_i={\alpha }_i/{\beta }_i\text{,}$$

so that

$${\beta }_{i}S{x}_i={\alpha }_{i}T{x}_i\text{or}S{x}_i={\lambda }_{i}T{x}_i\text{,}$$

where $x_i$ is the corresponding (right) eigenvector.

If $S$ and $T$ are the result of a generalized Schur factorization of a matrix pair $(A,B)$

$$A=QS{Z}^T\text{,}B=QT{Z}^T$$

then the eigenvalues and condition numbers of the pair $(S,T)$ are the same as those of the pair $(A,B)$.

Let $(\alpha ,\beta )\ne (0,0)$ be a simple generalized eigenvalue of $(A,B)$. Then the reciprocal of the condition number of the eigenvalue $\lambda =\alpha /\beta$ is defined as

$$s\left(\lambda \right)=\frac{{({\left|y^{T}Ax\right|}^2+{\left|y^{T}Bx\right|}^2)}^{1/2}}{\left({\parallel x\parallel }_2{\parallel y\parallel }_2\right)}\text{,}$$

where $x$ and $y$ are the right and left eigenvectors of $(A,B)$ corresponding to $\lambda$. If both $\alpha$ and $\beta$ are zero, then $(A,B)$ is singular and $s\left(\lambda \right)=-1$ is returned.

The definition of the reciprocal of the estimated condition number of the right eigenvector $x$ and the left eigenvector $y$ corresponding to the simple eigenvalue $\lambda$ depends upon whether $\lambda$ is a real eigenvalue, or one of a complex conjugate pair.

If the eigenvalue $\lambda$ is real and $U$ and $V$ are orthogonal transformations such that

$$\begin{array}{c}{U}^T(A,B)V=(S,T)=\left(\begin{array}{cc}\alpha & \ast \\ 0& S_{22}\end{array}\right)\left(\begin{array}{cc}\beta & \ast \\ 0& T_{22}\end{array}\right)\text{,}\end{array}$$

where $S_{22}$ and $T_{22}$ are $(n-1)\times (n-1)$ matrices, then the reciprocal condition number is given by

$$Dif\left(x\right)\equiv Dif\left(y\right)=Dif((\alpha ,\beta ),(S_{22},T_{22}))={\sigma }_{min}\left(Z\right)\text{,}$$

where ${\sigma }_{min}\left(Z\right)$ denotes the smallest singular value of the $2(n-1)\times 2(n-1)$ matrix

$$\begin{array}{c}Z=\left(\begin{array}{cc}\alpha \otimes I& -1\otimes S_{22}\\ \beta \otimes I& -1\otimes T_{22}\end{array}\right)\end{array}$$

and $\otimes$ is the Kronecker product.

If $\lambda$ is part of a complex conjugate pair and $U$ and $V$ are orthogonal transformations such that

$$\begin{array}{c}{U}^T(A,B)V=(S,T)=\left(\begin{array}{cc}{S}_{11}& \ast \\ 0& S_{22}\end{array}\right)\left(\begin{array}{cc}{T}_{11}& \ast \\ 0& T_{22}\end{array}\right)\text{,}\end{array}$$

where $S_{11}$ and $T_{11}$ are two by two matrices, $S_{22}$ and $T_{22}$ are $(n-2)\times (n-2)$ matrices, and $(S_{11},T_{11})$ corresponds to the complex conjugate eigenvalue pair $\lambda$, $\overline{\lambda }$, then there exist unitary matrices $U_1$ and $V_1$ such that

$$\begin{array}{c}{U}_1^H{S}_{11}{V}_1=\left(\begin{array}{cc}{s}_{11}& s_{12}\\ 0& s_{22}\end{array}\right)\text{and}{U}_1^H{T}_{11}{V}_1=\left(\begin{array}{cc}{t}_{11}& t_{12}\\ 0& t_{22}\end{array}\right)\text{.}\end{array}$$

The eigenvalues are given by $\lambda =s_{11}/t_{11}$ and $\overline{\lambda }=s_{22}/t_{22}$. Then the Frobenius norm-based, estimated reciprocal condition number is bounded by

$$Dif\left(x\right)\equiv Dif\left(y\right)\le min(d_1,max(1,\left|Re\left(s_{11}\right)/Re\left(s_{22}\right)\right|),d_2)$$

where $Re\left(z\right)$ denotes the real part of $z$, $d_1=Dif((s_{11},t_{11}),(s_{22},t_{22}))={\sigma }_{min}\left(Z_1\right)$, $Z_1$ is the complex two by two matrix

$$\begin{array}{c}{Z}_1=\left(\begin{array}{cc}{s}_{11}& -s_{22}\\ t_{11}& -t_{22}\end{array}\right)\text{,}\end{array}$$

and $d_2$ is an upper bound on $Dif((S_{11},T_{11}),(S_{22},T_{22}))$; i.e., an upper bound on ${\sigma }_{min}\left(Z_2\right)$, where $Z_2$ is the $(2n-2)\times (2n-2)$ matrix

$$\begin{array}{c}{Z}_2=\left(\begin{array}{cc}{S}_{11}^T\otimes I& -I\otimes S_{22}\\ T_{11}^T\otimes I& -I\otimes T_{22}\end{array}\right)\text{.}\end{array}$$

See Sections 2.4.8 and 4.11 of Anderson et al. (1999) and Kågström and Poromaa (1996) for further details and information.

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

Kågström, B and Poromaa, P, 1996, LAPACK-style algorithms and software for solving the generalized Sylvester equation and estimating the separation between regular matrix pairs, ACM Trans. Math. Software (22), 78–103