NAG Library Routine Document

f08quf  (ztrsen)

1

Purpose

2

Specification

3

Description

4

References

5

Arguments

1:     $\mathbf{job}$ – 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:     $\mathbf{compq}$ – 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:     $\mathbf{select}\left(*\right)$ – 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:     $\mathbf{n}$ – IntegerInput

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

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

5:     $\mathbf{t}\left(\mathbf{ldt},*\right)$ – 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:     $\mathbf{ldt}$ – 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:     $\mathbf{q}\left(\mathbf{ldq},*\right)$ – 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:     $\mathbf{ldq}$ – 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:     $\mathbf{w}\left(*\right)$ – Complex (Kind=nag_wp) arrayOutput

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

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

10:   $\mathbf{m}$ – 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:   $\mathbf{s}$ – 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:   $\mathbf{sep}$ – 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:   $\mathbf{work}\left(\max \left(1,\mathbf{lwork}\right)\right)$ – 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.

14:   $\mathbf{lwork}$ – 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 $\mathbf{job}=\text{'E'}$, $\mathbf{lwork}\ge \max \left(1,\mathit{m}\times \left(\mathbf{n}-\mathit{m}\right)\right)$ or $\mathbf{lwork}=-1$;
• if $\mathbf{job}=\text{'V'}$ or $\text{'B'}$, $\mathbf{lwork}\ge \max \left(1,2\mathit{m}\times \left(\mathbf{n}-\mathit{m}\right)\right)$ 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:   $\mathbf{info}$ – 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.

7

Accuracy

8

Parallelism and Performance

9

Further Comments

10

Example

10.1

Program Text

Program Text (f08qufe.f90)

10.2

Program Data

Program Data (f08qufe.d)

10.3

Program Results

Program Results (f08qufe.r)