NAG Library Routine Document

F08NBF (DGEEVX)

1  Purpose

2  Specification

3  Description

4  References

5  Parameters

1:     BALANC – CHARACTER(1)Input

On entry: indicates how the input matrix should be diagonally scaled and/or permuted to improve the conditioning of its eigenvalues.

$\mathbf{BALANC}=\text{'N'}$

Do not diagonally scale or permute.

$\mathbf{BALANC}=\text{'P'}$

Perform permutations to make the matrix more nearly upper triangular. Do not diagonally scale.

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

Diagonally scale the matrix, i.e., replace $A$ by $DA{D}^{-1}$, where $D$ is a diagonal matrix chosen to make the rows and columns of $A$ more equal in norm. Do not permute.

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

Both diagonally scale and permute $A$.

Computed reciprocal condition numbers will be for the matrix after balancing and/or permuting. Permuting does not change condition numbers (in exact arithmetic), but balancing does.

Constraint: $\mathbf{BALANC}=\text{'N'}$, $\text{'P'}$, $\text{'S'}$ or $\text{'B'}$.

2:     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.

If $\mathbf{SENSE}=\text{'E'}$ or $\text{'B'}$, JOBVL must be set to $\mathbf{JOBVL}=\text{'V'}$.

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

3:     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.

If $\mathbf{SENSE}=\text{'E'}$ or $\text{'B'}$, JOBVR must be set to $\mathbf{JOBVR}=\text{'V'}$.

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

4:     SENSE – CHARACTER(1)Input

On entry: determines which reciprocal condition numbers are computed.

$\mathbf{SENSE}=\text{'N'}$

None are computed.

$\mathbf{SENSE}=\text{'E'}$

Computed for eigenvalues only.

$\mathbf{SENSE}=\text{'V'}$

Computed for right eigenvectors only.

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

Computed for eigenvalues and right eigenvectors.

If $\mathbf{SENSE}=\text{'E'}$ or $\text{'B'}$, both left and right eigenvectors must also be computed ($\mathbf{JOBVL}=\text{'V'}$ and $\mathbf{JOBVR}=\text{'V'}$).

Constraint: $\mathbf{SENSE}=\text{'N'}$, $\text{'E'}$, $\text{'V'}$ or $\text{'B'}$.

5:     N – INTEGERInput

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

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

6:     A(LDA,$*$) – REAL (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. If $\mathbf{JOBVL}=\text{'V'}$ or $\mathbf{JOBVR}=\text{'V'}$, $A$ contains the real Schur form of the balanced version of the input matrix $A$.

7:     LDA – INTEGERInput

On entry: the first dimension of the array A as declared in the (sub)program from which F08NBF (DGEEVX) is called.

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

8:     WR($*$) – REAL (KIND=nag_wp) arrayOutput

9:     WI($*$) – REAL (KIND=nag_wp) arrayOutput

Note: the dimension of the arrays WR and WI must be at least $\max \left(1,\mathbf{N}\right)$.

On exit: WR and WI contain the real and imaginary parts, respectively, of the computed eigenvalues. Complex conjugate pairs of eigenvalues appear consecutively with the eigenvalue having the positive imaginary part first.

10:   VL(LDVL,$*$) – REAL (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. If the $j$th eigenvalue is real, then $u_j=\mathbf{VL}\left(:,j\right)$, the $j$th column of VL. If the $j$th and $\left(j+1\right)$st eigenvalues form a complex conjugate pair, then $u_j=\mathbf{VL}\left(:,j\right)+i\times \mathbf{VL}\left(:,j+1\right)$ and $u_{j+1}=\mathbf{VL}\left(:,j\right)-i\times \mathbf{VL}\left(:,j+1\right)$.

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

11:   LDVL – INTEGERInput

On entry: the first dimension of the array VL as declared in the (sub)program from which F08NBF (DGEEVX) is called.

Constraints:

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

12:   VR(LDVR,$*$) – REAL (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. If the $j$th eigenvalue is real, then $v_j=\mathbf{VR}\left(:,j\right)$, the $j$th column of VR. If the $j$th and $\left(j+1\right)$st eigenvalues form a complex conjugate pair, then $v_j=\mathbf{VR}\left(:,j\right)+i\times \mathbf{VR}\left(:,j+1\right)$ and $v_{j+1}=\mathbf{VR}\left(:,j\right)-i\times \mathbf{VR}\left(:,j+1\right)$.

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

13:   LDVR – INTEGERInput

On entry: the first dimension of the array VR as declared in the (sub)program from which F08NBF (DGEEVX) is called.

Constraints:

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

14:   ILO – INTEGEROutput

15:   IHI – INTEGEROutput

On exit: ILO and IHI are integer values determined when $A$ was balanced. The balanced $A$ has $a_{ij}=0$ if $i>j$ and $j=1,2,\dots ,\mathbf{ILO}-1$ or $i=\mathbf{IHI}+1,\dots ,\mathbf{N}$.

16:   SCALE($*$) – REAL (KIND=nag_wp) arrayOutput

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

On exit: details of the permutations and scaling factors applied when balancing $A$.

If $p_j$ is the index of the row and column interchanged with row and column $j$, and $d_j$ is the scaling factor applied to row and column $j$, then

• $\mathbf{SCALE}\left(\mathit{j}\right)=p_{\mathit{j}}$, for $\mathit{j}=1,2,\dots ,\mathbf{ILO}-1$;
• $\mathbf{SCALE}\left(\mathit{j}\right)=d_{\mathit{j}}$, for $\mathit{j}=\mathbf{ILO},\dots ,\mathbf{IHI}$;
• $\mathbf{SCALE}\left(\mathit{j}\right)=p_{\mathit{j}}$, for $\mathit{j}=\mathbf{IHI}+1,\dots ,\mathbf{N}$.

The order in which the interchanges are made is N to $\mathbf{IHI}+1$, then $1$ to $\mathbf{ILO}-1$.

17:   ABNRM – REAL (KIND=nag_wp)Output

On exit: the $1$-norm of the balanced matrix (the maximum of the sum of absolute values of elements of any column).

18:   RCONDE($*$) – REAL (KIND=nag_wp) arrayOutput

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

On exit: $\mathbf{RCONDE}\left(j\right)$ is the reciprocal condition number of the $j$th eigenvalue.

19:   RCONDV($*$) – REAL (KIND=nag_wp) arrayOutput

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

On exit: $\mathbf{RCONDV}\left(j\right)$ is the reciprocal condition number of the $j$th right eigenvector.

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

On exit: if $\mathbf{INFO}=\mathbf{0}$, $\mathbf{WORK}\left(1\right)$ contains the minimum value of LWORK required for optimal performance.

21:   LWORK – INTEGERInput

On entry: the dimension of the array WORK as declared in the (sub)program from which F08NBF (DGEEVX) 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 must generally be larger than the minimum, increase LWORK by, say, $\mathbf{N}\times \mathit{nb}$, where $\mathit{nb}$ is the optimal block size for F08NEF (DGEHRD).

Constraints:

• if $\mathbf{JOBVL}=\text{'N'}$ and $\mathbf{JOBVR}=\text{'N'}$,
  • if $\mathbf{SENSE}=\text{'N'}$, $\mathbf{LWORK}\ge \max \left(1,2\times \mathbf{N}\right)$;
  • otherwise $\mathbf{LWORK}\ge \max \left(1,{\mathbf{N}}^2+6\times \mathbf{N}\right)$;
• if $\mathbf{JOBVL}=\text{'V'}$ or $\mathbf{JOBVR}=\text{'V'}$,
  • if $\mathbf{SENSE}=\text{'N'}$ or $\text{'E'}$, $\mathbf{LWORK}\ge \max \left(1,3\times \mathbf{N}\right)$;
  • otherwise $\mathbf{LWORK}\ge \max \left(1,{\mathbf{N}}^2+6\times \mathbf{N}\right)$.

22:   IWORK($*$) – INTEGER arrayWorkspace

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

If $\mathbf{SENSE}=\text{'N'}$ or $\text{'E'}$, IWORK is not referenced.

23:   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 or condition numbers have been computed; elements $1:\mathbf{ILO}-1$ and $i+1:\mathbf{N}$ of WRWI contain eigenvalues which have converged.

7  Accuracy

8  Further Comments

9  Example

9.1  Program Text

Program Text (f08nbfe.f90)

9.2  Program Data

Program Data (f08nbfe.d)

9.3  Program Results

Program Results (f08nbfe.r)