NAG Library Routine Document

f08xbf (dggesx)

1

Purpose

2

Specification

3

Description

4

References

5

Arguments

1:     $\mathbf{jobvsl}$ – Character(1)Input

On entry: if $\mathbf{jobvsl}=\text{'N'}$, do not compute the left Schur vectors.

If $\mathbf{jobvsl}=\text{'V'}$, compute the left Schur vectors.

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

2:     $\mathbf{jobvsr}$ – Character(1)Input

On entry: if $\mathbf{jobvsr}=\text{'N'}$, do not compute the right Schur vectors.

If $\mathbf{jobvsr}=\text{'V'}$, compute the right Schur vectors.

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

3:     $\mathbf{sort}$ – Character(1)Input

On entry: specifies whether or not to order the eigenvalues on the diagonal of the generalized Schur form.

$\mathbf{sort}=\text{'N'}$

Eigenvalues are not ordered.

$\mathbf{sort}=\text{'S'}$

Eigenvalues are ordered (see selctg).

Constraint: $\mathbf{sort}=\text{'N'}$ or $\text{'S'}$.

4:     $\mathbf{selctg}$ – Logical Function, supplied by the user.External Procedure

If $\mathbf{sort}=\text{'S'}$, selctg is used to select generalized eigenvalues to be moved to the top left of the generalized Schur form.

If $\mathbf{sort}=\text{'N'}$, selctg is not referenced by f08xbf (dggesx), and may be called with the dummy function f08xaz.

The specification of selctg is:

Fortran Interface

Function selctg ( ar, ai, b) Logical :: selctg Real (Kind=nag_wp), Intent (In) :: ar, ai, b

C Header Interface

#include <nagmk26.h> Nag_Boolean  selctg (const double *ar, const double *ai, const double *b)

1:     $\mathbf{ar}$ – Real (Kind=nag_wp)Input

2:     $\mathbf{ai}$ – Real (Kind=nag_wp)Input

3:     $\mathbf{b}$ – Real (Kind=nag_wp)Input

On entry: an eigenvalue $\left(\mathbf{ar}\left(j\right)+\sqrt{-1}\times \mathbf{ai}\left(j\right)\right)/\mathbf{b}\left(j\right)$ is selected if $\mathbf{selctg}\left(\mathbf{ar}\left(j\right),\mathbf{ai}\left(j\right),\mathbf{b}\left(j\right)\right)$ is .TRUE.. If either one of a complex conjugate pair is selected, then both complex generalized eigenvalues are selected.

Note that in the ill-conditioned case, a selected complex generalized eigenvalue may no longer satisfy $\mathbf{selctg}\left(\mathbf{ar}\left(j\right),\mathbf{ai}\left(j\right),\mathbf{b}\left(j\right)\right)=\mathrm{.TRUE.}$ after ordering. $\mathbf{info}=\mathbf{n}+\mathbf{2}$ in this case.

selctg must either be a module subprogram USEd by, or declared as EXTERNAL in, the (sub)program from which f08xbf (dggesx) is called. Arguments denoted as Input must not be changed by this procedure.

5:     $\mathbf{sense}$ – Character(1)Input

On entry: determines which reciprocal condition numbers are computed.

$\mathbf{sense}=\text{'N'}$

None are computed.

$\mathbf{sense}=\text{'E'}$

Computed for average of selected eigenvalues only.

$\mathbf{sense}=\text{'V'}$

Computed for selected deflating subspaces only.

$\mathbf{sense}=\text{'B'}$

Computed for both.

If $\mathbf{sense}=\text{'E'}$, $\text{'V'}$ or $\text{'B'}$, $\mathbf{sort}=\text{'S'}$.

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

6:     $\mathbf{n}$ – IntegerInput

On entry: $n$, the order of the matrices $A$ and $B$.

Constraint: $\mathbf{n}\ge 0$.

7:     $\mathbf{a}\left(\mathbf{lda},*\right)$ – Real (Kind=nag_wp) arrayInput/Output

Note: the second dimension of the array a must be at least $\max \left(1,\mathbf{n}\right)$.

On entry: the first of the pair of matrices, $A$.

On exit: a has been overwritten by its generalized Schur form $S$.

8:     $\mathbf{lda}$ – IntegerInput

On entry: the first dimension of the array a as declared in the (sub)program from which f08xbf (dggesx) is called.

Constraint: $\mathbf{lda}\ge \max \left(1,\mathbf{n}\right)$.

9:     $\mathbf{b}\left(\mathbf{ldb},*\right)$ – Real (Kind=nag_wp) arrayInput/Output

Note: the second dimension of the array b must be at least $\max \left(1,\mathbf{n}\right)$.

On entry: the second of the pair of matrices, $B$.

On exit: b has been overwritten by its generalized Schur form $T$.

10:   $\mathbf{ldb}$ – IntegerInput

On entry: the first dimension of the array b as declared in the (sub)program from which f08xbf (dggesx) is called.

Constraint: $\mathbf{ldb}\ge \max \left(1,\mathbf{n}\right)$.

11:   $\mathbf{sdim}$ – IntegerOutput

On exit: if $\mathbf{sort}=\text{'N'}$, $\mathbf{sdim}=0$.

If $\mathbf{sort}=\text{'S'}$, $\mathbf{sdim}=$ number of eigenvalues (after sorting) for which selctg is .TRUE.. (Complex conjugate pairs for which selctg is .TRUE. for either eigenvalue count as $2$.)

12:   $\mathbf{alphar}\left(\mathbf{n}\right)$ – Real (Kind=nag_wp) arrayOutput

On exit: see the description of beta.

13:   $\mathbf{alphai}\left(\mathbf{n}\right)$ – Real (Kind=nag_wp) arrayOutput

On exit: see the description of beta.

14:   $\mathbf{beta}\left(\mathbf{n}\right)$ – Real (Kind=nag_wp) arrayOutput

On exit: $\left(\mathbf{alphar}\left(\mathit{j}\right)+\mathbf{alphai}\left(\mathit{j}\right)\times i\right)/\mathbf{beta}\left(\mathit{j}\right)$, for $\mathit{j}=1,2,\dots ,\mathbf{n}$, will be the generalized eigenvalues. $\mathbf{alphar}\left(\mathit{j}\right)+\mathbf{alphai}\left(\mathit{j}\right)\times i$, and $\mathbf{beta}\left(\mathit{j}\right)$, for $\mathit{j}=1,2,\dots ,\mathbf{n}$, are the diagonals of the complex Schur form $\left(S,T\right)$ that would result if the $2$ by $2$ diagonal blocks of the real Schur form of $\left(A,B\right)$ were further reduced to triangular form using $2$ by $2$ complex unitary transformations.

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

Note:  the quotients $\mathbf{alphar}\left(j\right)/\mathbf{beta}\left(j\right)$ and $\mathbf{alphai}\left(j\right)/\mathbf{beta}\left(j\right)$ may easily overflow or underflow, and $\mathbf{beta}\left(j\right)$ may even be zero. Thus, you should avoid naively computing the ratio $\alpha /\beta$. However, alphar and alphai will always be less than and usually comparable with ${‖A‖}_2$ in magnitude, and beta will always be less than and usually comparable with ${‖B‖}_2$.

15:   $\mathbf{vsl}\left(\mathbf{ldvsl},*\right)$ – Real (Kind=nag_wp) arrayOutput

Note: the second dimension of the array vsl must be at least $\max \left(1,\mathbf{n}\right)$ if $\mathbf{jobvsl}=\text{'V'}$.

On exit: if $\mathbf{jobvsl}=\text{'V'}$, vsl will contain the left Schur vectors, $Q$.

If $\mathbf{jobvsl}=\text{'N'}$, vsl is not referenced.

16:   $\mathbf{ldvsl}$ – IntegerInput

On entry: the first dimension of the array vsl as declared in the (sub)program from which f08xbf (dggesx) is called.

Constraints:

• if $\mathbf{jobvsl}=\text{'V'}$, $\mathbf{ldvsl}\ge \max \left(1,\mathbf{n}\right)$;
• otherwise $\mathbf{ldvsl}\ge 1$.

17:   $\mathbf{vsr}\left(\mathbf{ldvsr},*\right)$ – Real (Kind=nag_wp) arrayOutput

Note: the second dimension of the array vsr must be at least $\max \left(1,\mathbf{n}\right)$ if $\mathbf{jobvsr}=\text{'V'}$.

On exit: if $\mathbf{jobvsr}=\text{'V'}$, vsr will contain the right Schur vectors, $Z$.

If $\mathbf{jobvsr}=\text{'N'}$, vsr is not referenced.

18:   $\mathbf{ldvsr}$ – IntegerInput

On entry: the first dimension of the array vsr as declared in the (sub)program from which f08xbf (dggesx) is called.

Constraints:

• if $\mathbf{jobvsr}=\text{'V'}$, $\mathbf{ldvsr}\ge \max \left(1,\mathbf{n}\right)$;
• otherwise $\mathbf{ldvsr}\ge 1$.

19:   $\mathbf{rconde}\left(2\right)$ – Real (Kind=nag_wp) arrayOutput

On exit: if $\mathbf{sense}=\text{'E'}$ or $\text{'B'}$, $\mathbf{rconde}\left(1\right)$ and $\mathbf{rconde}\left(2\right)$ contain the reciprocal condition numbers for the average of the selected eigenvalues.

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

20:   $\mathbf{rcondv}\left(2\right)$ – Real (Kind=nag_wp) arrayOutput

On exit: if $\mathbf{sense}=\text{'V'}$ or $\text{'B'}$, $\mathbf{rcondv}\left(1\right)$ and $\mathbf{rcondv}\left(2\right)$ contain the reciprocal condition numbers for the selected deflating subspaces.

if $\mathbf{sense}=\text{'N'}$ or $\text{'E'}$, rcondv is not referenced.

21:   $\mathbf{work}\left(\max \left(1,\mathbf{lwork}\right)\right)$ – Real (Kind=nag_wp) arrayWorkspace

On exit: if $\mathbf{info}=\mathbf{0}$, $\mathbf{work}\left(1\right)$ returns the optimal lwork.

22:   $\mathbf{lwork}$ – IntegerInput

On entry: the dimension of the array work as declared in the (sub)program from which f08xbf (dggesx) is called.

If $\mathbf{lwork}=-1$, a workspace query is assumed; the routine only calculates the bound on the optimal size of the work array and the minimum size of the iwork array, returns these values as the first entries of the work and iwork arrays, and no error message related to lwork or liwork is issued.

Constraints:

• if $\mathbf{sense}=\text{'E'}$, $\text{'V'}$ or $\text{'B'}$, $\mathbf{lwork}\ge \max \left(8\times \left(\mathbf{n}+1\right)+16,2\times \mathbf{n}+2\times \mathbf{sdim}\times \left(\mathbf{n}-\mathbf{sdim}\right)\right)$;
• otherwise $\mathbf{lwork}\ge \max \left(1,8\times \mathbf{n},6\times \mathbf{n}+16\right)$.

Note: that $2\times \mathbf{sdim}\times \left(\mathbf{n}-\mathbf{sdim}\right)\le \mathbf{n}\times \mathbf{n}/2$. Note also that an error is only returned if $\mathbf{lwork}<8\times \left(\mathbf{n}+1\right)+16$, but if $\mathbf{sense}=\text{'E'}$, $\text{'V'}$ or $\text{'B'}$ this may not be large enough. Consider increasing lwork by $\mathit{nb}$, where $\mathit{nb}$ is the optimal block size.

23:   $\mathbf{iwork}\left(\max \left(1,\mathbf{liwork}\right)\right)$ – Integer arrayWorkspace

On exit: if $\mathbf{info}=\mathbf{0}$, $\mathbf{iwork}\left(1\right)$ returns the minimum liwork.

24:   $\mathbf{liwork}$ – IntegerInput

On entry: the dimension of the array iwork as declared in the (sub)program from which f08xbf (dggesx) is called.

If $\mathbf{liwork}=-1$, a workspace query is assumed; the routine only calculates the bound on the optimal size of the work array and the minimum size of the iwork array, returns these values as the first entries of the work and iwork arrays, and no error message related to lwork or liwork is issued.

Constraints:

• if $\mathbf{sense}=\text{'N'}$ or $\mathbf{n}=0$, $\mathbf{liwork}\ge 1$;
• otherwise $\mathbf{liwork}\ge \mathbf{n}+6$.

25:   $\mathbf{bwork}\left(*\right)$ – Logical arrayWorkspace

Note: the dimension of the array bwork must be at least $1$ if $\mathbf{sort}=\text{'N'}$, and at least $\max \left(1,\mathbf{n}\right)$ otherwise.

If $\mathbf{sort}=\text{'N'}$, bwork is not referenced.

26:   $\mathbf{info}$ – IntegerOutput

On exit: $\mathbf{info}=0$ unless the routine detects an error (see Section 6).

6

Error Indicators and Warnings

$\mathbf{info}<0$

If $\mathbf{info}=-i$, argument $i$ had an illegal value. An explanatory message is output, and execution of the program is terminated.

$\mathbf{info}=1\hspace{0.17em}\text{to}\hspace{0.17em}\mathbf{n}$

The $QZ$ iteration failed. No eigenvectors have been calculated but $\mathbf{alphar}\left(j\right)$, $\mathbf{alphai}\left(j\right)$ and $\mathbf{beta}\left(j\right)$ should be correct from element $〈\mathit{\text{value}}〉$.

$\mathbf{info}=\mathbf{n}+1$

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

$\mathbf{info}=\mathbf{n}+2$

After reordering, roundoff changed values of some complex eigenvalues so that leading eigenvalues in the generalized Schur form no longer satisfy $\mathbf{selctg}=\mathrm{.TRUE.}$. This could also be caused by underflow due to scaling.

$\mathbf{info}=\mathbf{n}+3$

The eigenvalues could not be reordered because some eigenvalues were too close to separate (the problem is very ill-conditioned).

7

Accuracy

8

Parallelism and Performance

9

Further Comments

10

Example

10.1

Program Text

Program Text (f08xbfe.f90)

10.2

Program Data

Program Data (f08xbfe.d)

10.3

Program Results

Program Results (f08xbfe.r)