NAG Library Routine Document

F08PPF (ZGEESX)

1  Purpose

2  Specification

3  Description

4  References

5  Parameters

1:     $\mathrm{JOBVS}$ – CHARACTER(1)Input

On entry: if $\mathbf{JOBVS}=\text{'N'}$, Schur vectors are not computed.

If $\mathbf{JOBVS}=\text{'V'}$, Schur vectors are computed.

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

2:     $\mathrm{SORT}$ – CHARACTER(1)Input

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

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

Eigenvalues are not ordered.

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

Eigenvalues are ordered (see SELECT).

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

3:     $\mathrm{SELECT}$ – LOGICAL FUNCTION, supplied by the user.External Procedure

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

If $\mathbf{SORT}=\text{'N'}$, SELECT is not referenced and F08PPF (ZGEESX) may be called with the dummy function F08PNZ.

An eigenvalue $\mathbf{W}\left(j\right)$ is selected if $\mathbf{SELECT}\left(\mathbf{W}\left(j\right)\right)$ is .TRUE..

The specification of SELECT is:

FUNCTION SELECT ( W) LOGICAL SELECT COMPLEX (KIND=nag_wp)  W

1:     $\mathrm{W}$ – COMPLEX (KIND=nag_wp)Input

On entry: the real and imaginary parts of the eigenvalue.

SELECT must either be a module subprogram USEd by, or declared as EXTERNAL in, the (sub)program from which F08PPF (ZGEESX) is called. Parameters denoted as Input must not be changed by this procedure.

4:     $\mathrm{SENSE}$ – 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 right invariant subspace 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 $\text{'B'}$.

5:     $\mathrm{N}$ – INTEGERInput

On entry: $n$, the order of the matrix $A$.

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

6:     $\mathrm{A}\left(\mathbf{LDA},*\right)$ – COMPLEX (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 $n$ by $n$ matrix $A$.

On exit: A is overwritten by its Schur form $T$.

7:     $\mathrm{LDA}$ – INTEGERInput

On entry: the first dimension of the array A as declared in the (sub)program from which F08PPF (ZGEESX) is called.

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

8:     $\mathrm{SDIM}$ – INTEGEROutput

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

If $\mathbf{SORT}=\text{'S'}$, $\mathbf{SDIM}=$ number of eigenvalues for which SELECT is .TRUE..

9:     $\mathrm{W}\left(*\right)$ – COMPLEX (KIND=nag_wp) arrayOutput

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

On exit: contains the computed eigenvalues, in the same order that they appear on the diagonal of the output Schur form $T$.

10:   $\mathrm{VS}\left(\mathbf{LDVS},*\right)$ – COMPLEX (KIND=nag_wp) arrayOutput

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

On exit: if $\mathbf{JOBVS}=\text{'V'}$, VS contains the unitary matrix $Z$ of Schur vectors.

If $\mathbf{JOBVS}=\text{'N'}$, VS is not referenced.

11:   $\mathrm{LDVS}$ – INTEGERInput

On entry: the first dimension of the array VS as declared in the (sub)program from which F08PPF (ZGEESX) is called.

Constraints:

• if $\mathbf{JOBVS}=\text{'V'}$, $\mathbf{LDVS}\ge \max \left(1,\mathbf{N}\right)$;
• otherwise $\mathbf{LDVS}\ge 1$.

12:   $\mathrm{RCONDE}$ – REAL (KIND=nag_wp)Output

On exit: if $\mathbf{SENSE}=\text{'E'}$ or $\text{'B'}$, contains the reciprocal condition number for the average of the selected eigenvalues.

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

13:   $\mathrm{RCONDV}$ – REAL (KIND=nag_wp)Output

On exit: if $\mathbf{SENSE}=\text{'V'}$ or $\text{'B'}$, RCONDV contains the reciprocal condition number for the selected right invariant subspace.

If $\mathbf{SENSE}=\text{'N'}$ or $\text{'E'}$, RCONDV is not referenced.

14:   $\mathrm{WORK}\left(\max \left(1,\mathbf{LWORK}\right)\right)$ – COMPLEX (KIND=nag_wp) arrayWorkspace

On exit: if $\mathbf{INFO}=\mathbf{0}$, the real part of $\mathbf{WORK}\left(1\right)$ contains the minimum value of LWORK required for optimal performance.

15:   $\mathrm{LWORK}$ – INTEGERInput

On entry: the dimension of the array WORK as declared in the (sub)program from which F08PPF (ZGEESX) is called.

If $\mathbf{LWORK}=-1$, a workspace query is assumed; the routine only calculates an upper bound on 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.

If $\mathbf{SENSE}=\text{'E'}$, $\text{'V'}$ or $\text{'B'}$, $\mathbf{LWORK}\ge 2\times \mathbf{SDIM}\times \left(\mathbf{N}-\mathbf{SDIM}\right)$, where SDIM is the number of selected eigenvalues computed by this routine.

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 $\mathbf{LWORK}<\max \left(1,2\times \mathbf{N}\right)$, but if $\mathbf{SENSE}=\text{'E'}$, $\text{'V'}$ or $\text{'B'}$ this may not be large enough.

Suggested value: for optimal performance, LWORK must generally be larger than the minimum; increase the workspace by, say, $\mathit{nb}\times \mathbf{N}$, where $\mathit{nb}$ is the optimal block size for F08NSF (ZGEHRD).

Constraint: $\mathbf{LWORK}\ge \max \left(1,2\times \mathbf{N}\right)$.

16:   $\mathrm{RWORK}\left(*\right)$ – REAL (KIND=nag_wp) arrayWorkspace

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

17:   $\mathrm{BWORK}\left(*\right)$ – 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.

18:   $\mathrm{INFO}$ – 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.

$\mathbf{INFO}=1\hspace{0.17em}\text{to}\hspace{0.17em}\mathbf{N}$

If $\mathbf{INFO}=i$ and $i\le \mathbf{N}$, the $QR$ algorithm failed to compute all the eigenvalues.

$\mathbf{INFO}=\mathbf{N}+1$

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

$\mathbf{INFO}=\mathbf{N}+2$

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

7  Accuracy

8  Parallelism and Performance

9  Further Comments

10  Example

10.1  Program Text

Program Text (f08ppfe.f90)

10.2  Program Data

Program Data (f08ppfe.d)

10.3  Program Results

Program Results (f08ppfe.r)