NAG Library Routine Document

F08YKF (DTGEVC)

1  Purpose

2  Specification

3  Description

4  References

5  Parameters

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

On entry: specifies the required sets of generalized eigenvectors.

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

Only right eigenvectors are computed.

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

Only left eigenvectors are computed.

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

Both left and right eigenvectors are computed.

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

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

On entry: specifies further details of the required generalized eigenvectors.

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

All right and/or left eigenvectors are computed.

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

All right and/or left eigenvectors are computed; they are backtransformed using the input matrices supplied in arrays VR and/or VL.

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

Selected right and/or left eigenvectors, defined by the array SELECT, are computed.

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

3:     $\mathrm{SELECT}\left(*\right)$ – LOGICAL arrayInput

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 the eigenvectors to be computed if $\mathbf{HOWMNY}=\text{'S'}$. To select the generalized eigenvector corresponding to the $j$th generalized eigenvalue, the $j$th element of SELECT should be set to .TRUE.; if the eigenvalue corresponds to a complex conjugate pair, then real and imaginary parts of eigenvectors corresponding to the complex conjugate eigenvalue pair will be computed.

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

Constraint: if $\mathbf{HOWMNY}=\text{'S'}$, $\mathbf{SELECT}\left(\mathit{j}\right)=\mathrm{.TRUE.}$ or $\mathrm{.FALSE.}$, for $\mathit{j}=1,2,\dots ,n$.

4:     $\mathrm{N}$ – INTEGERInput

On entry: $n$, the order of the matrices $A$ and $B$.

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

5:     $\mathrm{A}\left(\mathbf{LDA},*\right)$ – REAL (KIND=nag_wp) arrayInput

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

On entry: the matrix pair $\left(A,B\right)$ must be in the generalized Schur form. Usually, this is the matrix $A$ returned by F08XEF (DHGEQZ).

6:     $\mathrm{LDA}$ – INTEGERInput

On entry: the first dimension of the array A as declared in the (sub)program from which F08YKF (DTGEVC) is called.

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

7:     $\mathrm{B}\left(\mathbf{LDB},*\right)$ – REAL (KIND=nag_wp) arrayInput

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

On entry: the matrix pair $\left(A,B\right)$ must be in the generalized Schur form. If $A$ has a $2$ by $2$ diagonal block then the corresponding $2$ by $2$ block of $B$ must be diagonal with positive elements. Usually, this is the matrix $B$ returned by F08XEF (DHGEQZ).

8:     $\mathrm{LDB}$ – INTEGERInput

On entry: the first dimension of the array B as declared in the (sub)program from which F08YKF (DTGEVC) is called.

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

9:     $\mathrm{VL}\left(\mathbf{LDVL},*\right)$ – 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 $\mathbf{SIDE}=\text{'L'}$ or $\text{'B'}$ and at least $1$ if $\mathbf{SIDE}=\text{'R'}$.

On entry: if $\mathbf{HOWMNY}=\text{'B'}$ and $\mathbf{SIDE}=\text{'L'}$ or $\text{'B'}$, VL must be initialized to an $n$ by $n$ matrix $Q$. Usually, this is the orthogonal matrix $Q$ of left Schur vectors returned by F08XEF (DHGEQZ).

On exit: if $\mathbf{SIDE}=\text{'L'}$ or $\text{'B'}$, VL contains:

• if $\mathbf{HOWMNY}=\text{'A'}$, the matrix $Y$ of left eigenvectors of $\left(A,B\right)$;
• if $\mathbf{HOWMNY}=\text{'B'}$, the matrix $QY$;
• if $\mathbf{HOWMNY}=\text{'S'}$, the left eigenvectors of $\left(A,B\right)$ specified by SELECT, stored consecutively in the columns of the array VL, in the same order as their corresponding eigenvalues.

A complex eigenvector corresponding to a complex eigenvalue is stored in two consecutive columns, the first holding the real part, and the second the imaginary part.

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

10:   $\mathrm{LDVL}$ – INTEGERInput

On entry: the first dimension of the array VL as declared in the (sub)program from which F08YKF (DTGEVC) is called.

Constraints:

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

11:   $\mathrm{VR}\left(\mathbf{LDVR},*\right)$ – 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 $\mathbf{SIDE}=\text{'R'}$ or $\text{'B'}$ and at least $1$ if $\mathbf{SIDE}=\text{'L'}$.

On entry: if $\mathbf{HOWMNY}=\text{'B'}$ and $\mathbf{SIDE}=\text{'R'}$ or $\text{'B'}$, VR must be initialized to an $n$ by $n$ matrix $Z$. Usually, this is the orthogonal matrix $Z$ of right Schur vectors returned by F08XEF (DHGEQZ).

On exit: if $\mathbf{SIDE}=\text{'R'}$ or $\text{'B'}$, VR contains:

• if $\mathbf{HOWMNY}=\text{'A'}$, the matrix $X$ of right eigenvectors of $\left(A,B\right)$;
• if $\mathbf{HOWMNY}=\text{'B'}$, the matrix $ZX$;
• if $\mathbf{HOWMNY}=\text{'S'}$, the right eigenvectors of $\left(A,B\right)$ specified by SELECT, stored consecutively in the columns of the array VR, in the same order as their corresponding eigenvalues.

A complex eigenvector corresponding to a complex eigenvalue is stored in two consecutive columns, the first holding the real part, and the second the imaginary part.

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

12:   $\mathrm{LDVR}$ – INTEGERInput

On entry: the first dimension of the array VR as declared in the (sub)program from which F08YKF (DTGEVC) is called.

Constraints:

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

13:   $\mathrm{MM}$ – INTEGERInput

On entry: the number of columns in the arrays VL and/or VR.

Constraints:

• if $\mathbf{HOWMNY}=\text{'A'}$ or $\text{'B'}$, $\mathbf{MM}\ge \mathbf{N}$;
• if $\mathbf{HOWMNY}=\text{'S'}$, MM must not be less than the number of requested eigenvectors.

14:   $\mathrm{M}$ – INTEGEROutput

On exit: the number of columns in the arrays VL and/or VR actually used to store the eigenvectors. If $\mathbf{HOWMNY}=\text{'A'}$ or $\text{'B'}$, M is set to N. Each selected real eigenvector occupies one column and each selected complex eigenvector occupies two columns.

15:   $\mathrm{WORK}\left(6\times \mathbf{N}\right)$ – REAL (KIND=nag_wp) arrayWorkspace

16:   $\mathrm{INFO}$ – 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.

$\mathbf{INFO}>0$

If $\mathbf{INFO}=i$, the $2$ by $2$ block $\left(\mathbf{INFO}:\mathbf{INFO}+1\right)$ does not have complex eigenvalues.

7  Accuracy

8  Parallelism and Performance

9  Further Comments

10  Example

10.1  Program Text

Program Text (f08ykfe.f90)

10.2  Program Data

Program Data (f08ykfe.d)

10.3  Program Results

Program Results (f08ykfe.r)