NAG Library Routine Document

F08YGF (DTGSEN)

1  Purpose

2  Specification

3  Description

4  References

5  Parameters

1:     IJOB – INTEGERInput

On entry: specifies whether condition numbers are required for the cluster of eigenvalues ($p$ and $q$) or the deflating subspaces (${\mathrm{Dif}}_u$ and ${\mathrm{Dif}}_l$).

$\mathbf{IJOB}=0$

Only reorder with respect to SELECT. No extras.

$\mathbf{IJOB}=1$

Reciprocal of norms of ‘projections’ onto left and right eigenspaces with respect to the selected cluster ($p$ and $q$).

$\mathbf{IJOB}=2$

The upper bounds on ${\mathrm{Dif}}_u$ and ${\mathrm{Dif}}_l$. $F$-norm-based estimate ($\mathbf{DIF}\left(1:2\right)$).

$\mathbf{IJOB}=3$

Estimate of ${\mathrm{Dif}}_u$ and ${\mathrm{Dif}}_l$. $1$-norm-based estimate ($\mathbf{DIF}\left(1:2\right)$). About five times as expensive as $\mathbf{IJOB}=2$.

$\mathbf{IJOB}=4$

Compute PL, PR and DIF as in $\mathbf{IJOB}=0$, $1$ and $2$. Economic version to get it all.

$\mathbf{IJOB}=5$

Compute PL, PR and DIF as in $\mathbf{IJOB}=0$, $1$ and $3$.

Constraint: $0\le \mathbf{IJOB}\le 5$.

2:     WANTQ – LOGICALInput

On entry: if $\mathbf{WANTQ}=\mathrm{.TRUE.}$, update the left transformation matrix $Q$.

If $\mathbf{WANTQ}=\mathrm{.FALSE.}$, do not update $Q$.

3:     WANTZ – LOGICALInput

On entry: if $\mathbf{WANTZ}=\mathrm{.TRUE.}$, update the right transformation matrix $Z$.

If $\mathbf{WANTZ}=\mathrm{.FALSE.}$, do not update $Z$.

4:     SELECT(N) – LOGICAL arrayInput

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

To select a complex conjugate pair of eigenvalues ${\lambda }_j$ and ${\lambda }_{j+1}$, corresponding to a $2$ by $2$ diagonal block, either $\mathbf{SELECT}\left(j\right)$ or $\mathbf{SELECT}\left(j+1\right)$ or both must be set to .TRUE.; a complex conjugate pair of eigenvalues must be either both included in the cluster or both excluded.

5:     N – INTEGERInput

On entry: $n$, the order of the matrices $S$ and $T$.

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

6:     A(LDA,$*$) – 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 matrix $S$ in the pair $\left(S,T\right)$.

On exit: the updated matrix $\hat{S}$.

7:     LDA – INTEGERInput

On entry: the first dimension of the array A as declared in the (sub)program from which F08YGF (DTGSEN) is called.

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

8:     B(LDB,$*$) – 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 matrix $T$, in the pair $\left(S,T\right)$.

On exit: the updated matrix $\hat{T}$ 

9:     LDB – INTEGERInput

On entry: the first dimension of the array B as declared in the (sub)program from which F08YGF (DTGSEN) is called.

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

10:   ALPHAR(N) – REAL (KIND=nag_wp) arrayOutput

On exit: see the description of BETA.

11:   ALPHAI(N) – REAL (KIND=nag_wp) arrayOutput

On exit: see the description of BETA.

12:   BETA(N) – REAL (KIND=nag_wp) arrayOutput

On exit: $\mathbf{ALPHAR}\left(\mathit{j}\right)/\mathbf{BETA}\left(\mathit{j}\right)$ and $\mathbf{ALPHAI}\left(\mathit{j}\right)/\mathbf{BETA}\left(\mathit{j}\right)$ are the real and imaginary parts respectively of the $\mathit{j}$th eigenvalue, for $\mathit{j}=1,2,\dots ,\mathbf{N}$.

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

Conjugate pairs of eigenvalues correspond to the $2$ by $2$ diagonal blocks of $\hat{S}$. These $2$ by $2$ blocks can be reduced by applying complex unitary transformations to $\left(\hat{S},\hat{T}\right)$ to obtain the complex Schur form $\left(\tilde{S},\tilde{T}\right)$, where $\tilde{S}$ is triangular (and complex). In this form $\mathbf{ALPHAR}+i\mathbf{ALPHAI}$ and BETA are the diagonals of $\tilde{S}$ and $\tilde{T}$ respectively.

13:   Q(LDQ,$*$) – REAL (KIND=nag_wp) arrayInput/Output

Note: the second dimension of the array Q must be at least $\max \left(1,\mathbf{N}\right)$ if $\mathbf{WANTQ}=\mathrm{.TRUE.}$, and at least $1$ otherwise.

On entry: if $\mathbf{WANTQ}=\mathrm{.TRUE.}$, the $n$ by $n$ matrix $Q$.

On exit: if $\mathbf{WANTQ}=\mathrm{.TRUE.}$, the updated matrix $Q\hat{Q}$.

If $\mathbf{WANTQ}=\mathrm{.FALSE.}$, Q is not referenced.

14:   LDQ – INTEGERInput

On entry: the first dimension of the array Q as declared in the (sub)program from which F08YGF (DTGSEN) is called.

Constraints:

• if $\mathbf{WANTQ}=\mathrm{.TRUE.}$, $\mathbf{LDQ}\ge \max \left(1,\mathbf{N}\right)$;
• otherwise $\mathbf{LDQ}\ge 1$.

15:   Z(LDZ,$*$) – REAL (KIND=nag_wp) arrayInput/Output

Note: the second dimension of the array Z must be at least $\max \left(1,\mathbf{N}\right)$ if $\mathbf{WANTZ}=\mathrm{.TRUE.}$, and at least $1$ otherwise.

On entry: if $\mathbf{WANTZ}=\mathrm{.TRUE.}$, the $n$ by $n$ matrix $Z$.

On exit: if $\mathbf{WANTZ}=\mathrm{.TRUE.}$, the updated matrix $Z\hat{Z}$.

If $\mathbf{WANTZ}=\mathrm{.FALSE.}$, Z is not referenced.

16:   LDZ – INTEGERInput

On entry: the first dimension of the array Z as declared in the (sub)program from which F08YGF (DTGSEN) is called.

Constraints:

• if $\mathbf{WANTZ}=\mathrm{.TRUE.}$, $\mathbf{LDZ}\ge \max \left(1,\mathbf{N}\right)$;
• otherwise $\mathbf{LDZ}\ge 1$.

17:   M – INTEGEROutput

On exit: the dimension of the specified pair of left and right eigenspaces (deflating subspaces).

18:   PL – REAL (KIND=nag_wp)Output

19:   PR – REAL (KIND=nag_wp)Output

On exit: if $\mathbf{IJOB}=1$, $4$ or $5$, PL and PR are lower bounds on the reciprocal of the norm of ‘projections’ $p$ and $q$ onto left and right eigenspaces with respect to the selected cluster. $0<\mathbf{PL}$, $\mathbf{PR}\le 1$.

If $\mathbf{M}=0$ or $\mathbf{M}=\mathbf{N}$, $\mathbf{PL}=\mathbf{PR}=1$.

If $\mathbf{IJOB}=0$, $2$ or $3$, PL and PR are not referenced.

20:   DIF($*$) – REAL (KIND=nag_wp) arrayOutput

Note: the dimension of the array DIF must be at least $2$.

On exit: if $\mathbf{IJOB}\ge 2$, $\mathbf{DIF}\left(1:2\right)$ store the estimates of ${\mathrm{Dif}}_u$ and ${\mathrm{Dif}}_l$.

If $\mathbf{IJOB}=2$ or $4$, $\mathbf{DIF}\left(1:2\right)$ are $F$-norm-based upper bounds on ${\mathrm{Dif}}_u$ and ${\mathrm{Dif}}_l$.

If $\mathbf{IJOB}=3$ or $5$, $\mathbf{DIF}\left(1:2\right)$ are $1$-norm-based estimates of ${\mathrm{Dif}}_u$ and ${\mathrm{Dif}}_l$.

If $\mathbf{M}=0$ or $n$, $\mathbf{DIF}\left(1:2\right)$ $={‖\left(A,B\right)‖}_F$.

If $\mathbf{IJOB}=0$ or $1$, DIF is not referenced.

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

On exit: if $\mathbf{INFO}=\mathbf{0}$, $\mathbf{WORK}\left(1\right)$ returns the minimum LWORK.

If $\mathbf{IJOB}=0$, WORK is not referenced.

22:   LWORK – INTEGERInput

On entry: the dimension of the array WORK as declared in the (sub)program from which F08YGF (DTGSEN) is called.

If $\mathbf{LWORK}=-1$, a workspace query is assumed; the routine only calculates the minimum sizes of the WORK and IWORK arrays, 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 $\mathbf{LWORK}\ne -1$,

• if $\mathbf{N}=0$, $\mathbf{LWORK}\ge 1$;
• if $\mathbf{IJOB}=1$, $2$ or $4$, $\mathbf{LWORK}\ge \max \left(4\times \mathbf{N}+16,2\times \mathbf{M}\times \left(\mathbf{N}-\mathbf{M}\right)\right)$;
• if $\mathbf{IJOB}=3$ or $5$, $\mathbf{LWORK}\ge \max \left(4\times \mathbf{N}+16,4\times \mathbf{M}\times \left(\mathbf{N}-\mathbf{M}\right)\right)$;
• otherwise $\mathbf{LWORK}\ge 4\times \mathbf{N}+16$.

23:   IWORK($*$) – INTEGER arrayWorkspace

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

On exit: if $\mathbf{INFO}=\mathbf{0}$, $\mathbf{IWORK}\left(1\right)$ returns the minimum LIWORK.

If $\mathbf{IJOB}=0$, IWORK is not referenced.

24:   LIWORK – INTEGERInput

On entry: the dimension of the array IWORK as declared in the (sub)program from which F08YGF (DTGSEN) is called.

If $\mathbf{LIWORK}=-1$, a workspace query is assumed; the routine only calculates the minimum sizes of the WORK and IWORK arrays, 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 $\mathbf{LIWORK}\ne -1$,

• if $\mathbf{IJOB}=1$, $2$ or $4$, $\mathbf{LIWORK}\ge \mathbf{N}+6$;
• if $\mathbf{IJOB}=3$ or $5$, $\mathbf{LIWORK}\ge \max \left(2\times \mathbf{M}\times \left(\mathbf{N}-\mathbf{M}\right),\mathbf{N}+6\right)$;
• otherwise $\mathbf{LIWORK}\ge 1$.

25:   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.

$\mathbf{INFO}=1$

Reordering of $\left(S,T\right)$ failed because the transformed matrix pair $\left(\hat{S},\hat{T}\right)$ would be too far from generalized Schur form; the problem is very ill-conditioned. $\left(S,T\right)$ may have been partially reordered. If requested, $0$ is returned in $\mathbf{DIF}\left(1:2\right)$, PL and PR.

7  Accuracy

8  Further Comments

9  Example

9.1  Program Text

Program Text (f08ygfe.f90)

9.2  Program Data

Program Data (f08ygfe.d)

9.3  Program Results

Program Results (f08ygfe.r)