NAG Library Routine Document

F08QUF (ZTRSEN)

1  Purpose

2  Specification

3  Description

4  References

5  Parameters

1:     JOB – CHARACTER(1)Input

On entry: indicates whether condition numbers are required for the cluster of eigenvalues and/or the invariant subspace.

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

No condition numbers are required.

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

Only the condition number for the cluster of eigenvalues is computed.

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

Only the condition number for the invariant subspace is computed.

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

Condition numbers for both the cluster of eigenvalues and the invariant subspace are computed.

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

2:     COMPQ – CHARACTER(1)Input

On entry: indicates whether the matrix $Q$ of Schur vectors is to be updated.

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

The matrix $Q$ of Schur vectors is updated.

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

No Schur vectors are updated.

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

3:     SELECT($*$) – LOGICAL arrayInput

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

On entry: specifies the eigenvalues in the selected cluster. To select a complex eigenvalue ${\lambda }_j$, $\mathbf{SELECT}\left(j\right)$ must be set .TRUE..

4:     N – INTEGERInput

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

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

5:     T(LDT,$*$) – COMPLEX (KIND=nag_wp) arrayInput/Output

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

On entry: the $n$ by $n$ upper triangular matrix $T$, as returned by F08PSF (ZHSEQR).

On exit: T is overwritten by the updated matrix $\tilde{T}$.

6:     LDT – INTEGERInput

On entry: the first dimension of the array T as declared in the (sub)program from which F08QUF (ZTRSEN) is called.

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

7:     Q(LDQ,$*$) – COMPLEX (KIND=nag_wp) arrayInput/Output

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

On entry: if $\mathbf{COMPQ}=\text{'V'}$, Q must contain the $n$ by $n$ unitary matrix $Q$ of Schur vectors, as returned by F08PSF (ZHSEQR).

On exit: if $\mathbf{COMPQ}=\text{'V'}$, Q contains the updated matrix of Schur vectors; the first $m$ columns of $Q$ form an orthonormal basis for the specified invariant subspace.

If $\mathbf{COMPQ}=\text{'N'}$, Q is not referenced.

8:     LDQ – INTEGERInput

On entry: the first dimension of the array Q as declared in the (sub)program from which F08QUF (ZTRSEN) is called.

Constraints:

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

9:     W($*$) – COMPLEX (KIND=nag_wp) arrayOutput

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

On exit: the reordered eigenvalues of $\tilde{T}$. The eigenvalues are stored in the same order as on the diagonal of $\tilde{T}$.

10:   M – INTEGEROutput

On exit: $m$, the dimension of the specified invariant subspace, which is the same as the number of selected eigenvalues (see SELECT); $0\le m\le n$.

11:   S – REAL (KIND=nag_wp)Output

On exit: if $\mathbf{JOB}=\text{'E'}$ or $\text{'B'}$, S is a lower bound on the reciprocal condition number of the average of the selected cluster of eigenvalues. If $\mathbf{M}=0\text{ or }\mathbf{N}$, $\mathbf{S}=1$.

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

12:   SEP – REAL (KIND=nag_wp)Output

On exit: if $\mathbf{JOB}=\text{'V'}$ or $\text{'B'}$, SEP is the estimated reciprocal condition number of the specified invariant subspace. If $\mathbf{M}=0\text{ or }\mathbf{N}$, $\mathbf{SEP}=‖T‖$.

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

13:   WORK($\max \left(1,\mathbf{LWORK}\right)$) – 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.

14:   LWORK – INTEGERInput

On entry: the dimension of the array WORK as declared in the (sub)program from which F08QUF (ZTRSEN) is called, unless $\mathbf{LWORK}=-1$, in which case a workspace query is assumed and the routine only calculates the minimum dimension of WORK.

Constraints:

• if $\mathbf{JOB}=\text{'N'}$, $\mathbf{LWORK}\ge 1$ or $\mathbf{LWORK}=-1$;
• if $\mathbf{JOB}=\text{'E'}$, $\mathbf{LWORK}\ge \max \left(1,m\times \left(\mathbf{N}-m\right)\right)$ or $\mathbf{LWORK}=-1$;
• if $\mathbf{JOB}=\text{'V'}$ or $\text{'B'}$, $\mathbf{LWORK}\ge \max \left(1,2m\times \left(\mathbf{N}-m\right)\right)$ or $\mathbf{LWORK}=-1$.

The actual amount of workspace required cannot exceed ${\mathbf{N}}^2/4$ if $\mathbf{JOB}=\text{'E'}$ or ${\mathbf{N}}^2/2$ if $\mathbf{JOB}=\text{'V'}$ or $\text{'B'}$.

15:   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.

7  Accuracy

8  Further Comments

9  Example

9.1  Program Text

Program Text (f08qufe.f90)

9.2  Program Data

Program Data (f08qufe.d)

9.3  Program Results

Program Results (f08qufe.r)