NAG Library Routine Document

f08ykf  (dtgevc)

1

Purpose

2

Specification

3

Description

4

References

5

Arguments

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

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

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

5:     $\mathbf{a}\left(\mathbf{lda},*\right)$ – 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:     $\mathbf{lda}$ – 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:     $\mathbf{b}\left(\mathbf{ldb},*\right)$ – 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:     $\mathbf{ldb}$ – 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:     $\mathbf{vl}\left(\mathbf{ldvl},*\right)$ – 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:   $\mathbf{ldvl}$ – 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:   $\mathbf{vr}\left(\mathbf{ldvr},*\right)$ – 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:   $\mathbf{ldvr}$ – 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:   $\mathbf{mm}$ – 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:   $\mathbf{m}$ – 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:   $\mathbf{work}\left(6\times \mathbf{n}\right)$ – Real (Kind=nag_wp) arrayWorkspace

16:   $\mathbf{info}$ – 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.

$\mathbf{info}>0$

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)