NAG Library Routine Document

F08QKF (DTREVC)

1  Purpose

2  Specification

3  Description

4  References

5  Parameters

1:     JOB – CHARACTER(1)Input

On entry: indicates whether left and/or right eigenvectors are to be computed.

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

Only right eigenvectors are computed.

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

Only left eigenvectors are computed.

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

Both left and right eigenvectors are computed.

Constraint: $\mathbf{JOB}=\text{'R'}$, $\text{'L'}$ or $\text{'B'}$.

2:     HOWMNY – CHARACTER(1)Input

On entry: indicates how many eigenvectors are to be computed.

$\mathbf{HOWMNY}=\text{'A'}$

All eigenvectors (as specified by JOB) are computed.

$\mathbf{HOWMNY}=\text{'B'}$ or $\text{'O'}$

All eigenvectors (as specified by JOB) are computed and then pre-multiplied by the matrix $Q$ (which is overwritten).

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

Selected eigenvectors (as specified by JOB and SELECT) are computed.

Constraint: $\mathbf{HOWMNY}=\text{'A'}$, $\text{'B'}$, $\text{'O'}$ or $\text{'S'}$.

3:     SELECT($*$) – LOGICAL arrayInput/Output

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

On entry: specifies which eigenvectors are to be computed if $\mathbf{HOWMNY}=\text{'S'}$. To obtain the real eigenvector corresponding to the real eigenvalue ${\lambda }_j$, $\mathbf{SELECT}\left(j\right)$ must be set .TRUE.. To select the complex eigenvector corresponding to a complex conjugate pair of eigenvalues ${\lambda }_j$ and ${\lambda }_{j+1}$, $\mathbf{SELECT}\left(j\right)$ and/or $\mathbf{SELECT}\left(j+1\right)$ must be set .TRUE.; the eigenvector corresponding to the first eigenvalue in the pair is computed.

On exit: if a complex eigenvector was selected as specified above, then $\mathbf{SELECT}\left(j\right)$ is set to .TRUE. and $\mathbf{SELECT}\left(j+1\right)$ to .FALSE..

If $\mathbf{HOWMNY}=\text{'A'}$, $\text{'O'}$ or $\text{'B'}$, SELECT is not referenced.

4:     N – INTEGERInput

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

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

5:     T(LDT,$*$) – REAL (KIND=nag_wp) arrayInput

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 quasi-triangular matrix $T$ in canonical Schur form, as returned by F08PEF (DHSEQR).

6:     LDT – INTEGERInput

On entry: the first dimension of the array T as declared in the (sub)program from which F08QKF (DTREVC) is called.

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

7:     VL(LDVL,$*$) – REAL (KIND=nag_wp) arrayInput/Output

Note: the second dimension of the array VL must be at least $\max \left(1,\mathbf{MM}\right)$ if $\mathbf{JOB}=\text{'L'}$ or $\text{'B'}$ and at least $1$ if $\mathbf{JOB}=\text{'R'}$.

On entry: if $\mathbf{HOWMNY}=\text{'O'}$ or $\text{'B'}$ and $\mathbf{JOB}=\text{'L'}$ or $\text{'B'}$, VL must contain an $n$ by $n$ matrix $Q$ (usually the matrix of Schur vectors returned by F08PEF (DHSEQR)).

If $\mathbf{HOWMNY}=\text{'A'}$ or $\text{'S'}$, VL need not be set.

On exit: if $\mathbf{JOB}=\text{'L'}$ or $\text{'B'}$, VL contains the computed left eigenvectors (as specified by HOWMNY and SELECT). The eigenvectors are stored consecutively in the columns of the array, in the same order as their eigenvalues. Corresponding to each real eigenvalue is a real eigenvector, occupying one column. Corresponding to each complex conjugate pair of eigenvalues, is a complex eigenvector occupying two columns; the first column holds the real part and the second column holds the imaginary part.

If $\mathbf{JOB}=\text{'R'}$, VL is not referenced.

8:     LDVL – INTEGERInput

On entry: the first dimension of the array VL as declared in the (sub)program from which F08QKF (DTREVC) is called.

Constraints:

• if $\mathbf{JOB}=\text{'L'}$ or $\text{'B'}$, $\mathbf{LDVL}\ge \max \left(1,\mathbf{N}\right)$;
• if $\mathbf{JOB}=\text{'R'}$, $\mathbf{LDVL}\ge 1$.

9:     VR(LDVR,$*$) – REAL (KIND=nag_wp) arrayInput/Output

Note: the second dimension of the array VR must be at least $\max \left(1,\mathbf{MM}\right)$ if $\mathbf{JOB}=\text{'R'}$ or $\text{'B'}$ and at least $1$ if $\mathbf{JOB}=\text{'L'}$.

On entry: if $\mathbf{HOWMNY}=\text{'O'}$ or $\text{'B'}$ and $\mathbf{JOB}=\text{'R'}$ or $\text{'B'}$, VR must contain an $n$ by $n$ matrix $Q$ (usually the matrix of Schur vectors returned by F08PEF (DHSEQR)).

If $\mathbf{HOWMNY}=\text{'A'}$ or $\text{'S'}$, VR need not be set.

On exit: if $\mathbf{JOB}=\text{'R'}$ or $\text{'B'}$, VR contains the computed right eigenvectors (as specified by HOWMNY and SELECT). The eigenvectors are stored consecutively in the columns of the array, in the same order as their eigenvalues. Corresponding to each real eigenvalue is a real eigenvector, occupying one column. Corresponding to each complex conjugate pair of eigenvalues, is a complex eigenvector occupying two columns; the first column holds the real part and the second column holds the imaginary part.

If $\mathbf{JOB}=\text{'L'}$, VR is not referenced.

10:   LDVR – INTEGERInput

On entry: the first dimension of the array VR as declared in the (sub)program from which F08QKF (DTREVC) is called.

Constraints:

• if $\mathbf{JOB}=\text{'R'}$ or $\text{'B'}$, $\mathbf{LDVR}\ge \max \left(1,\mathbf{N}\right)$;
• if $\mathbf{JOB}=\text{'L'}$, $\mathbf{LDVR}\ge 1$.

11:   MM – INTEGERInput

On entry: the number of columns in the arrays VL and/or VR. The precise number of columns required, $\mathit{m}$, is $n$ if $\mathbf{HOWMNY}=\text{'A'}$, $\text{'O'}$ or $\text{'B'}$; if $\mathbf{HOWMNY}=\text{'S'}$, $\mathit{m}$ is obtained by counting $1$ for each selected real eigenvector and $2$ for each selected complex eigenvector (see SELECT), in which case $0\le \mathit{m}\le n$.

Constraint: $\mathbf{MM}\ge \mathit{m}$.

12:   M – INTEGEROutput

On exit: $\mathit{m}$, the number of columns of VL and/or VR actually used to store the computed eigenvectors. If $\mathbf{HOWMNY}=\text{'A'}$, $\text{'O'}$ or $\text{'B'}$, M is set to $n$.

13:   WORK($3\times \mathbf{N}$) – REAL (KIND=nag_wp) arrayWorkspace

14:   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