NAG FL Interface
f08wnf (zggev)

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

1: $\mathbf{jobvl}$ – Character(1) Input

On entry: if $\mathbf{jobvl}=\text{'N'}$, do not compute the left generalized eigenvectors.

If $\mathbf{jobvl}=\text{'V'}$, compute the left generalized eigenvectors.

Constraint: $\mathbf{jobvl}=\text{'N'}$ or $\text{'V'}$.

2: $\mathbf{jobvr}$ – Character(1) Input

On entry: if $\mathbf{jobvr}=\text{'N'}$, do not compute the right generalized eigenvectors.

If $\mathbf{jobvr}=\text{'V'}$, compute the right generalized eigenvectors.

Constraint: $\mathbf{jobvr}=\text{'N'}$ or $\text{'V'}$.

3: $\mathbf{n}$ – Integer Input

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

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

4: $\mathbf{a}(\mathbf{lda},*)$ – Complex (Kind=nag_wp) array Input/Output

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

On entry: the matrix $A$ in the pair $(A,B)$.

On exit: a has been overwritten.

5: $\mathbf{lda}$ – Integer Input

On entry: the first dimension of the array a as declared in the (sub)program from which f08wnf is called.

Constraint: $\mathbf{lda}\ge \max (1,\mathbf{n})$.

6: $\mathbf{b}(\mathbf{ldb},*)$ – Complex (Kind=nag_wp) array Input/Output

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

On entry: the matrix $B$ in the pair $(A,B)$.

On exit: b has been overwritten.

7: $\mathbf{ldb}$ – Integer Input

On entry: the first dimension of the array b as declared in the (sub)program from which f08wnf is called.

Constraint: $\mathbf{ldb}\ge \max (1,\mathbf{n})$.

8: $\mathbf{alpha}\left(\mathbf{n}\right)$ – Complex (Kind=nag_wp) array Output

On exit: see the description of beta.

9: $\mathbf{beta}\left(\mathbf{n}\right)$ – Complex (Kind=nag_wp) array Output

On exit: $\mathbf{alpha}\left(\mathit{j}\right)/\mathbf{beta}\left(\mathit{j}\right)$, for $\mathit{j}=1,2,\dots ,\mathbf{n}$, will be the generalized eigenvalues.

Note:  the quotients $\mathbf{alpha}\left(j\right)/\mathbf{beta}\left(j\right)$ may easily overflow or underflow, and $\mathbf{beta}\left(j\right)$ may even be zero. Thus, you should avoid naively computing the ratio ${\alpha }_j/{\beta }_j$. However, $\max \left|{\alpha }_j\right|$ will always be less than and usually comparable with ${\Vert A\Vert }_2$ in magnitude, and $\max \left|{\beta }_j\right|$ will always be less than and usually comparable with ${\Vert B\Vert }_2$.

10: $\mathbf{vl}(\mathbf{ldvl},*)$ – Complex (Kind=nag_wp) array Output

Note: the second dimension of the array vl must be at least $\max (1,\mathbf{n})$ if $\mathbf{jobvl}=\text{'V'}$, and at least $1$ otherwise.

On exit: if $\mathbf{jobvl}=\text{'V'}$, the left generalized eigenvectors $u_j$ are stored one after another in the columns of vl, in the same order as the corresponding eigenvalues. Each eigenvector will be scaled so the largest component will have $\left|\text{real part}\right|+\left|\text{imag. part}\right|=1$.

If $\mathbf{jobvl}=\text{'N'}$, vl is not referenced.

11: $\mathbf{ldvl}$ – Integer Input

On entry: the first dimension of the array vl as declared in the (sub)program from which f08wnf is called.

Constraints:

• if $\mathbf{jobvl}=\text{'V'}$, $\mathbf{ldvl}\ge \max (1,\mathbf{n})$;
• otherwise $\mathbf{ldvl}\ge 1$.

12: $\mathbf{vr}(\mathbf{ldvr},*)$ – Complex (Kind=nag_wp) array Output

Note: the second dimension of the array vr must be at least $\max (1,\mathbf{n})$ if $\mathbf{jobvr}=\text{'V'}$, and at least $1$ otherwise.

On exit: if $\mathbf{jobvr}=\text{'V'}$, the right generalized eigenvectors $v_j$ are stored one after another in the columns of vr, in the same order as the corresponding eigenvalues. Each eigenvector will be scaled so the largest component will have $\left|\text{real part}\right|+\left|\text{imag. part}\right|=1$.

If $\mathbf{jobvr}=\text{'N'}$, vr is not referenced.

13: $\mathbf{ldvr}$ – Integer Input

On entry: the first dimension of the array vr as declared in the (sub)program from which f08wnf is called.

Constraints:

• if $\mathbf{jobvr}=\text{'V'}$, $\mathbf{ldvr}\ge \max (1,\mathbf{n})$;
• otherwise $\mathbf{ldvr}\ge 1$.

14: $\mathbf{work}\left(\max (1,\mathbf{lwork})\right)$ – Complex (Kind=nag_wp) array Workspace

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.

15: $\mathbf{lwork}$ – Integer Input

On entry: the dimension of the array work as declared in the (sub)program from which f08wnf is called.

If $\mathbf{lwork}=\mathrm{-1}$, a workspace query is assumed; the routine only calculates the optimal size of the work array, returns this value as the first entry of the work array, and no error message related to lwork is issued.

Suggested value: for optimal performance, lwork must generally be larger than the minimum; increase workspace by, say, $\mathit{nb}\times \mathbf{n}$, where $\mathit{nb}$ is the optimal block size.

Constraint: $\mathbf{lwork}\ge \max (1,2\times \mathbf{n})$ or $\mathbf{lwork}=\mathrm{-1}$.

16: $\mathbf{rwork}\left(\max (1,8\times \mathbf{n})\right)$ – Real (Kind=nag_wp) array Workspace

17: $\mathbf{info}$ – Integer Output

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.

$\mathbf{info}=1,\dots ,\mathbf{n}$

The $QZ$ iteration failed. No eigenvectors have been calculated but alpha and beta should be correct from element $⟨\mathit{\text{value}}⟩$.

$\mathbf{info}=\mathbf{n}+1$

The $QZ$ iteration failed with an unexpected error, please contact NAG.

$\mathbf{info}=\mathbf{n}+2$

A failure occurred in f08yxf while computing generalized eigenvectors.

7 Accuracy

8 Parallelism and Performance

9 Further Comments

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results