NAG Library Routine Document

f08xaf  (dgges)

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 f08xaf (dgges), 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 $\mathbf{selctg}\left(\mathbf{ar}\left(j\right),\mathbf{ai}\left(j\right),\mathbf{b}\left(j\right)\right)=\mathrm{.TRUE.}$. 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 f08xaf (dgges) is called. Arguments denoted as Input must not be changed by this procedure.

5:     $\mathbf{n}$ – IntegerInput

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

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

6:     $\mathbf{a}\left(\mathbf{lda},*\right)$ – 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 $S$.

7:     $\mathbf{lda}$ – IntegerInput

On entry: the first dimension of the array a as declared in the (sub)program from which f08xaf (dgges) is called.

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

8:     $\mathbf{b}\left(\mathbf{ldb},*\right)$ – 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 $T$.

9:     $\mathbf{ldb}$ – IntegerInput

On entry: the first dimension of the array b as declared in the (sub)program from which f08xaf (dgges) is called.

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

10:   $\mathbf{sdim}$ – 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 $2$.)

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

On exit: see the description of beta.

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

On exit: see the description of beta.

13:   $\mathbf{beta}\left(\mathbf{n}\right)$ – 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 ${‖B‖}_2$.

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

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

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

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

15:   $\mathbf{ldvsl}$ – IntegerInput

On entry: the first dimension of the array vsl as declared in the (sub)program from which f08xaf (dgges) is called.

Constraints:

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

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

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

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

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

17:   $\mathbf{ldvsr}$ – IntegerInput

On entry: the first dimension of the array vsr as declared in the (sub)program from which f08xaf (dgges) is called.

Constraints:

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

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

On exit: if $\mathbf{info}=\mathbf{0}$, $\mathbf{work}\left(1\right)$ contains the minimum value of lwork required for optimal performance.

19:   $\mathbf{lwork}$ – IntegerInput

On entry: the dimension of the array work as declared in the (sub)program from which f08xaf (dgges) is called.

If $\mathbf{lwork}=-1$, a workspace query is assumed; the routine only calculates the optimal size of the work array, returns this value as the first entry of the work array, and no error message related to lwork is issued.

Suggested value: for optimal performance, lwork must generally be larger than the mimimum; add, say $\mathit{nb}\times \mathbf{n}$, where $\mathit{nb}$ is the optimal block size.

Constraints:

• if $\mathbf{n}=0$, $\mathbf{lwork}\ge 1$;
• otherwise $\mathbf{lwork}\ge \max \left(8\times \mathbf{n},6\times \mathbf{n}+16\right)$.

20:   $\mathbf{bwork}\left(*\right)$ – 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.

21:   $\mathbf{info}$ – 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. $\left(A,B\right)$ are not in Schur form, but $\mathbf{alphar}\left(j\right)$, $\mathbf{alphai}\left(j\right)$, and $\mathbf{beta}\left(j\right)$ should be correct for $j=\mathbf{info}+1,\dots ,\mathbf{n}$.

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

Unexpected error returned from f08xef (dhgeqz).

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

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 (f08xafe.f90)

10.2

Program Data

Program Data (f08xafe.d)

10.3

Program Results

Program Results (f08xafe.r)