NAG Library Routine Document

F08NNF (ZGEEV)

1  Purpose

2  Specification

3  Description

4  References

5  Parameters

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

On entry: if $\mathbf{JOBVL}=\text{'N'}$, the left eigenvectors of $A$ are not computed.

If $\mathbf{JOBVL}=\text{'V'}$, the left eigenvectors of $A$ are computed.

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

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

On entry: if $\mathbf{JOBVR}=\text{'N'}$, the right eigenvectors of $A$ are not computed.

If $\mathbf{JOBVR}=\text{'V'}$, the right eigenvectors of $A$ are computed.

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

3:     $\mathrm{N}$ – INTEGERInput

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

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

4:     $\mathrm{A}\left(\mathbf{LDA},*\right)$ – COMPLEX (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 $n$ by $n$ matrix $A$.

On exit: A has been overwritten.

5:     $\mathrm{LDA}$ – INTEGERInput

On entry: the first dimension of the array A as declared in the (sub)program from which F08NNF (ZGEEV) is called.

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

6:     $\mathrm{W}\left(*\right)$ – COMPLEX (KIND=nag_wp) arrayOutput

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

On exit: contains the computed eigenvalues.

7:     $\mathrm{VL}\left(\mathbf{LDVL},*\right)$ – COMPLEX (KIND=nag_wp) arrayOutput

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

On exit: if $\mathbf{JOBVL}=\text{'V'}$, the left eigenvectors $u_j$ are stored one after another in the columns of VL, in the same order as their corresponding eigenvalues; that is $u_j=\mathbf{VL}\left(:,j\right)$, the $j$th column of VL.

If $\mathbf{JOBVL}=\text{'N'}$, VL is not referenced.

8:     $\mathrm{LDVL}$ – INTEGERInput

On entry: the first dimension of the array VL as declared in the (sub)program from which F08NNF (ZGEEV) is called.

Constraints:

• if $\mathbf{JOBVL}=\text{'V'}$, $\mathbf{LDVL}\ge \max \left(1,\mathbf{N}\right)$;
• otherwise $\mathbf{LDVL}\ge 1$.

9:     $\mathrm{VR}\left(\mathbf{LDVR},*\right)$ – COMPLEX (KIND=nag_wp) arrayOutput

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

On exit: if $\mathbf{JOBVR}=\text{'V'}$, the right eigenvectors $v_j$ are stored one after another in the columns of VR, in the same order as their corresponding eigenvalues; that is $v_j=\mathbf{VR}\left(:,j\right)$, the $j$th column of VR.

If $\mathbf{JOBVR}=\text{'N'}$, VR is not referenced.

10:   $\mathrm{LDVR}$ – INTEGERInput

On entry: the first dimension of the array VR as declared in the (sub)program from which F08NNF (ZGEEV) is called.

Constraints:

• if $\mathbf{JOBVR}=\text{'V'}$, $\mathbf{LDVR}\ge \max \left(1,\mathbf{N}\right)$;
• otherwise $\mathbf{LDVR}\ge 1$.

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

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.

12:   $\mathrm{LWORK}$ – INTEGERInput

On entry: the dimension of the array WORK as declared in the (sub)program from which F08NNF (ZGEEV) is called.

If $\mathbf{LWORK}=-1$, 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 should be generally larger than the minimum, say $\mathbf{N}+\mathit{nb}\times \mathbf{N}$, where $\mathit{nb}$ is the optimal block size for F08NSF (ZGEHRD).

Constraint: $\mathbf{LWORK}\ge \max \left(1,2\times \mathbf{N}\right)$.

13:   $\mathrm{RWORK}\left(*\right)$ – REAL (KIND=nag_wp) arrayWorkspace

Note: the dimension of the array RWORK must be at least $\max \left(1,2\times \mathbf{N}\right)$.

14:   $\mathrm{INFO}$ – 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 $QR$ algorithm failed to compute all the eigenvalues, and no eigenvectors have been computed; elements $i+1:\mathbf{N}$ of W contain eigenvalues which have converged.

7  Accuracy

8  Parallelism and Performance

9  Further Comments

10  Example

10.1  Program Text

Program Text (f08nnfe.f90)

10.2  Program Data

Program Data (f08nnfe.d)

10.3  Program Results

Program Results (f08nnfe.r)