NAG Library Routine Document

F08FRF (ZHEEVR)

1  Purpose

2  Specification

3  Description

4  References

5  Parameters

1:     JOBZ – CHARACTER(1)Input

On entry: indicates whether eigenvectors are computed.

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

Only eigenvalues are computed.

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

Eigenvalues and eigenvectors are computed.

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

2:     RANGE – CHARACTER(1)Input

On entry: if $\mathbf{RANGE}=\text{'A'}$, all eigenvalues will be found.

If $\mathbf{RANGE}=\text{'V'}$, all eigenvalues in the half-open interval $\left(\mathbf{VL},\mathbf{VU}\right]$ will be found.

If $\mathbf{RANGE}=\text{'I'}$, the ILth to IUth eigenvalues will be found.

For $\mathbf{RANGE}=\text{'V'}$ or $\text{'I'}$ and $\mathbf{IU}-\mathbf{IL}<\mathbf{N}-1$, F08JJF (DSTEBZ) and F08JXF (ZSTEIN) are called.

Constraint: $\mathbf{RANGE}=\text{'A'}$, $\text{'V'}$ or $\text{'I'}$.

3:     UPLO – CHARACTER(1)Input

On entry: if $\mathbf{UPLO}=\text{'U'}$, the upper triangular part of $A$ is stored.

If $\mathbf{UPLO}=\text{'L'}$, the lower triangular part of $A$ is stored.

Constraint: $\mathbf{UPLO}=\text{'U'}$ or $\text{'L'}$.

4:     N – INTEGERInput

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

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

5:     A(LDA,$*$) – 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 $n$ Hermitian matrix $A$.

• If $\mathbf{UPLO}=\text{'U'}$, the upper triangular part of $A$ must be stored and the elements of the array below the diagonal are not referenced.
• If $\mathbf{UPLO}=\text{'L'}$, the lower triangular part of $A$ must be stored and the elements of the array above the diagonal are not referenced.

On exit: the lower triangle (if $\mathbf{UPLO}=\text{'L'}$) or the upper triangle (if $\mathbf{UPLO}=\text{'U'}$) of A, including the diagonal, is overwritten.

6:     LDA – INTEGERInput

On entry: the first dimension of the array A as declared in the (sub)program from which F08FRF (ZHEEVR) is called.

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

7:     VL – REAL (KIND=nag_wp)Input

8:     VU – REAL (KIND=nag_wp)Input

On entry: if $\mathbf{RANGE}=\text{'V'}$, the lower and upper bounds of the interval to be searched for eigenvalues.

If $\mathbf{RANGE}=\text{'A'}$ or $\text{'I'}$, VL and VU are not referenced.

Constraint: if $\mathbf{RANGE}=\text{'V'}$, $\mathbf{VL}<\mathbf{VU}$.

9:     IL – INTEGERInput

10:   IU – INTEGERInput

On entry: if $\mathbf{RANGE}=\text{'I'}$, the indices (in ascending order) of the smallest and largest eigenvalues to be returned.

If $\mathbf{RANGE}=\text{'A'}$ or $\text{'V'}$, IL and IU are not referenced.

Constraints:

• if $\mathbf{RANGE}=\text{'I'}$ and $\mathbf{N}=0$, $\mathbf{IL}=1$ and $\mathbf{IU}=0$;
• if $\mathbf{RANGE}=\text{'I'}$ and $\mathbf{N}>0$, $1\le \mathbf{IL}\le \mathbf{IU}\le \mathbf{N}$.

11:   ABSTOL – REAL (KIND=nag_wp)Input

On entry: the absolute error tolerance for the eigenvalues. An approximate eigenvalue is accepted as converged when it is determined to lie in an interval $\left[a,b\right]$ of width less than or equal to

$$\mathbf{ABSTOL}+\epsilon \max \left(\left|a\right|,\left|b\right|\right)\text{,}$$

where $\epsilon$ is the machine precision. If ABSTOL is less than or equal to zero, then $\epsilon {‖T‖}_1$ will be used in its place, where $T$ is the real tridiagonal matrix obtained by reducing $A$ to tridiagonal form. See Demmel and Kahan (1990).

If high relative accuracy is important, set ABSTOL to $\mathbf{X02AMF}\left( \right)$, although doing so does not currently guarantee that eigenvalues are computed to high relative accuracy. See Barlow and Demmel (1990) for a discussion of which matrices can define their eigenvalues to high relative accuracy.

12:   M – INTEGEROutput

On exit: the total number of eigenvalues found. $0\le \mathbf{M}\le \mathbf{N}$.

If $\mathbf{RANGE}=\text{'A'}$, $\mathbf{M}=\mathbf{N}$.

If $\mathbf{RANGE}=\text{'I'}$, $\mathbf{M}=\mathbf{IU}-\mathbf{IL}+1$.

13:   W($*$) – REAL (KIND=nag_wp) arrayOutput

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

On exit: the first M elements contain the selected eigenvalues in ascending order.

14:   Z(LDZ,$*$) – COMPLEX (KIND=nag_wp) arrayOutput

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

On exit: if $\mathbf{JOBZ}=\text{'V'}$, the first M columns of $Z$ contain the orthonormal eigenvectors of the matrix $A$ corresponding to the selected eigenvalues, with the $i$th column of $Z$ holding the eigenvector associated with $\mathbf{W}\left(i\right)$.

If $\mathbf{JOBZ}=\text{'N'}$, Z is not referenced.

Note:  you must ensure that at least $\max \left(1,\mathbf{M}\right)$ columns are supplied in the array Z; if $\mathbf{RANGE}=\text{'V'}$, the exact value of M is not known in advance and an upper bound of at least N must be used.

15:   LDZ – INTEGERInput

On entry: the first dimension of the array Z as declared in the (sub)program from which F08FRF (ZHEEVR) is called.

Constraints:

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

16:   ISUPPZ($*$) – INTEGER arrayOutput

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

On exit: the support of the eigenvectors in Z, i.e., the indices indicating the nonzero elements in Z. The $i$th eigenvector is nonzero only in elements $\mathbf{ISUPPZ}\left(2\times i-1\right)$ through $\mathbf{ISUPPZ}\left(2\times i\right)$. Implemented only for $\mathbf{RANGE}=\text{'A'}$ or $\text{'I'}$ and $\mathbf{IU}-\mathbf{IL}=\mathbf{N}-1$.

17:   WORK($\max \left(1,\mathbf{LWORK}\right)$) – 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.

18:   LWORK – INTEGERInput

On entry: the dimension of the array WORK as declared in the (sub)program from which F08FRF (ZHEEVR) is called.

If $\mathbf{LWORK}=-1$, a workspace query is assumed; the routine only calculates the optimal sizes of the WORK, RWORK and IWORK arrays, returns these values as the first entries of the WORK, RWORK and IWORK arrays, and no error message related to LWORK, LRWORK or LIWORK is issued.

Suggested value: for optimal performance, $\mathbf{LWORK}\ge \left(\mathit{nb}+1\right)\times \mathbf{N}$, where $\mathit{nb}$ is the largest optimal block size for F08FSF (ZHETRD) and for F08FUF (ZUNMTR).

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

19:   RWORK($\max \left(1,\mathbf{LRWORK}\right)$) – REAL (KIND=nag_wp) arrayWorkspace

On exit: if $\mathbf{INFO}=\mathbf{0}$, $\mathbf{RWORK}\left(1\right)$ returns the optimal (and minimal) LRWORK.

20:   LRWORK – INTEGERInput

On entry: the dimension of the array RWORK as declared in the (sub)program from which F08FRF (ZHEEVR) is called.

If $\mathbf{LRWORK}=-1$, a workspace query is assumed; the routine only calculates the optimal sizes of the WORK, RWORK and IWORK arrays, returns these values as the first entries of the WORK, RWORK and IWORK arrays, and no error message related to LWORK, LRWORK or LIWORK is issued.

Constraint: $\mathbf{LRWORK}\ge \max \left(1,24\times \mathbf{N}\right)$.

21:   IWORK($\max \left(1,\mathbf{LIWORK}\right)$) – INTEGER arrayWorkspace

On exit: if $\mathbf{INFO}=\mathbf{0}$, $\mathbf{IWORK}\left(1\right)$ returns the optimal (and minimal) LIWORK.

22:   LIWORK – INTEGERInput

On entry: the dimension of the array IWORK as declared in the (sub)program from which F08FRF (ZHEEVR) is called.

If $\mathbf{LIWORK}=-1$, a workspace query is assumed; the routine only calculates the optimal sizes of the WORK, RWORK and IWORK arrays, returns these values as the first entries of the WORK, RWORK and IWORK arrays, and no error message related to LWORK, LRWORK or LIWORK is issued.

Constraint: $\mathbf{LIWORK}\ge \max \left(1,10\times \mathbf{N}\right)$.

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.

$\mathbf{INFO}>0$

F08FRF (ZHEEVR) failed to converge.

7  Accuracy

8  Further Comments

9  Example

9.1  Program Text

Program Text (f08frfe.f90)

9.2  Program Data

Program Data (f08frfe.d)

9.3  Program Results

Program Results (f08frfe.r)