NAG Library Routine Document

F08PKF (DHSEIN)

1  Purpose

2  Specification

3  Description

4  References

5  Parameters

1:     JOB – CHARACTER(1)Input

On entry: indicates whether left and/or right eigenvectors are to be computed.

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

Only right eigenvectors are computed.

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

Only left eigenvectors are computed.

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

Both left and right eigenvectors are computed.

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

2:     EIGSRC – CHARACTER(1)Input

On entry: indicates whether the eigenvalues of $H$ (stored in WR and WI) were found using F08PEF (DHSEQR).

$\mathbf{EIGSRC}=\text{'Q'}$

The eigenvalues of $H$ were found using F08PEF (DHSEQR); thus if $H$ has any zero subdiagonal elements (and so is block triangular), then the $j$th eigenvalue can be assumed to be an eigenvalue of the block containing the $j$th row/column. This property allows the routine to perform inverse iteration on just one diagonal block.

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

No such assumption is made and the routine performs inverse iteration using the whole matrix.

Constraint: $\mathbf{EIGSRC}=\text{'Q'}$ or $\text{'N'}$.

3:     INITV – CHARACTER(1)Input

On entry: indicates whether you are supplying initial estimates for the selected eigenvectors.

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

No initial estimates are supplied.

$\mathbf{INITV}=\text{'U'}$

Initial estimates are supplied in VL and/or VR.

Constraint: $\mathbf{INITV}=\text{'N'}$ or $\text{'U'}$.

4:     SELECT($*$) – LOGICAL arrayInput/Output

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

On entry: specifies which eigenvectors are to be computed. To obtain the real eigenvector corresponding to the real eigenvalue $\mathbf{WR}\left(j\right)$, $\mathbf{SELECT}\left(j\right)$ must be set .TRUE.. To select the complex eigenvector corresponding to the complex eigenvalue $\left(\mathbf{WR}\left(j\right),\mathbf{WI}\left(j\right)\right)$ with complex conjugate ($\mathbf{WR}\left(j+1\right),\mathbf{WI}\left(j+1\right)$), $\mathbf{SELECT}\left(j\right)$ and/or $\mathbf{SELECT}\left(j+1\right)$ must be set .TRUE.; the eigenvector corresponding to the first eigenvalue in the pair is computed.

On exit: if a complex eigenvector was selected as specified above, then $\mathbf{SELECT}\left(j\right)$ is set to .TRUE. and $\mathbf{SELECT}\left(j+1\right)$ to .FALSE..

5:     N – INTEGERInput

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

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

6:     H(LDH,$*$) – REAL (KIND=nag_wp) arrayInput

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

On entry: the $n$ by $n$ upper Hessenberg matrix $H$.

7:     LDH – INTEGERInput

On entry: the first dimension of the array H as declared in the (sub)program from which F08PKF (DHSEIN) is called.

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

8:     WR($*$) – REAL (KIND=nag_wp) arrayInput/Output

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

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

On entry: the real and imaginary parts, respectively, of the eigenvalues of the matrix $H$. Complex conjugate pairs of values must be stored in consecutive elements of the arrays. If $\mathbf{EIGSRC}=\text{'Q'}$, the arrays must be exactly as returned by F08PEF (DHSEQR).

On exit: some elements of WR may be modified, as close eigenvalues are perturbed slightly in searching for independent eigenvectors.

10:   VL(LDVL,$*$) – 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 $\mathbf{JOB}=\text{'L'}$ or $\text{'B'}$ and at least $1$ if $\mathbf{JOB}=\text{'R'}$.

On entry: if $\mathbf{INITV}=\text{'U'}$ and $\mathbf{JOB}=\text{'L'}$ or $\text{'B'}$, VL must contain starting vectors for inverse iteration for the left eigenvectors. Each starting vector must be stored in the same column or columns as will be used to store the corresponding eigenvector (see below).

If $\mathbf{INITV}=\text{'N'}$, VL need not be set.

On exit: if $\mathbf{JOB}=\text{'L'}$ or $\text{'B'}$, VL contains the computed left eigenvectors (as specified by SELECT). The eigenvectors are stored consecutively in the columns of the array, in the same order as their eigenvalues. Corresponding to each selected real eigenvalue is a real eigenvector, occupying one column. Corresponding to each selected complex eigenvalue is a complex eigenvector, occupying two columns: the first column holds the real part and the second column holds the imaginary part.

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

11:   LDVL – INTEGERInput

On entry: the first dimension of the array VL as declared in the (sub)program from which F08PKF (DHSEIN) is called.

Constraints:

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

12:   VR(LDVR,$*$) – 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 $\mathbf{JOB}=\text{'R'}$ or $\text{'B'}$ and at least $1$ if $\mathbf{JOB}=\text{'L'}$.

On entry: if $\mathbf{INITV}=\text{'U'}$ and $\mathbf{JOB}=\text{'R'}$ or $\text{'B'}$, VR must contain starting vectors for inverse iteration for the right eigenvectors. Each starting vector must be stored in the same column or columns as will be used to store the corresponding eigenvector (see below).

If $\mathbf{INITV}=\text{'N'}$, VR need not be set.

On exit: if $\mathbf{JOB}=\text{'R'}$ or $\text{'B'}$, VR contains the computed right eigenvectors (as specified by SELECT). The eigenvectors are stored consecutively in the columns of the array, in the same order as their eigenvalues. Corresponding to each selected real eigenvalue is a real eigenvector, occupying one column. Corresponding to each selected complex eigenvalue is a complex eigenvector, occupying two columns: the first column holds the real part and the second column holds the imaginary part.

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

13:   LDVR – INTEGERInput

On entry: the first dimension of the array VR as declared in the (sub)program from which F08PKF (DHSEIN) is called.

Constraints:

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

14:   MM – INTEGERInput

On entry: the number of columns in the arrays VL and/or VR . The actual number of columns required, $\mathit{m}$, is obtained by counting $1$ for each selected real eigenvector and $2$ for each selected complex eigenvector (see SELECT); $0\le \mathit{m}\le n$.

Constraint: $\mathbf{MM}\ge \mathit{m}$.

15:   M – INTEGEROutput

On exit: $\mathit{m}$, the number of columns of VL and/or VR required to store the selected eigenvectors.

16:   WORK($\left(\mathbf{N}+2\right)\times \mathbf{N}$) – REAL (KIND=nag_wp) arrayWorkspace

17:   IFAILL($*$) – INTEGER arrayOutput

Note: the dimension of the array IFAILL must be at least $\max \left(1,\mathbf{MM}\right)$ if $\mathbf{JOB}=\text{'L'}$ or $\text{'B'}$ and at least $1$ if $\mathbf{JOB}=\text{'R'}$.

On exit: if $\mathbf{JOB}=\text{'L'}$ or $\text{'B'}$, then $\mathbf{IFAILL}\left(i\right)=0$ if the selected left eigenvector converged and $\mathbf{IFAILL}\left(i\right)=j>0$ if the eigenvector stored in the $i$th column of VL (corresponding to the $j$th eigenvalue as held in $\left(\mathbf{WR}\left(j\right),\mathbf{WI}\left(j\right)\right)$ failed to converge. If the $i$th and $\left(i+1\right)$th columns of VL contain a selected complex eigenvector, then $\mathbf{IFAILL}\left(i\right)$ and $\mathbf{IFAILL}\left(i+1\right)$ are set to the same value.

If $\mathbf{JOB}=\text{'R'}$, IFAILL is not referenced.

18:   IFAILR($*$) – INTEGER arrayOutput

Note: the dimension of the array IFAILR must be at least $\max \left(1,\mathbf{MM}\right)$ if $\mathbf{JOB}=\text{'R'}$ or $\text{'B'}$ and at least $1$ if $\mathbf{JOB}=\text{'L'}$.

On exit: if $\mathbf{JOB}=\text{'R'}$ or $\text{'B'}$, then $\mathbf{IFAILR}\left(i\right)=0$ if the selected right eigenvector converged and $\mathbf{IFAILR}\left(i\right)=j>0$ if the eigenvector stored in the $i$th row or column of VR (corresponding to the $j$th eigenvalue as held in $\left(\mathbf{WR}\left(j\right),\mathbf{WI}\left(j\right)\right)$) failed to converge. If the $i$th and $\left(i+1\right)$th rows or columns of VR contain a selected complex eigenvector, then $\mathbf{IFAILR}\left(i\right)$ and $\mathbf{IFAILR}\left(i+1\right)$ are set to the same value.

If $\mathbf{JOB}=\text{'L'}$, IFAILR is not referenced.

19:   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 $\mathbf{INFO}=i$, then $i$ eigenvectors (as indicated by the parameters IFAILL and/or IFAILR above) failed to converge. The corresponding columns of VL and/or VR contain no useful information.

7  Accuracy

8  Further Comments

9  Example