NAG FL Interface
f08frf (zheevr)

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

1: $\mathbf{jobz}$ – 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: $\mathbf{range}$ – 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 $(\mathbf{vl},\mathbf{vu}]$ 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 and f08jxf are called.

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

3: $\mathbf{uplo}$ – 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: $\mathbf{n}$ – Integer Input

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

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

5: $\mathbf{a}(\mathbf{lda},*)$ – 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 $n\times n$ 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: $\mathbf{lda}$ – Integer Input

On entry: the first dimension of the array a as declared in the (sub)program from which f08frf is called.

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

7: $\mathbf{vl}$ – Real (Kind=nag_wp) Input

8: $\mathbf{vu}$ – 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: $\mathbf{il}$ – Integer Input

10: $\mathbf{iu}$ – Integer Input

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

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: $\mathbf{abstol}$ – 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 $[a,b]$ of width less than or equal to

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

where $\epsilon$ is the machine precision. If abstol is less than or equal to zero, then $\epsilon {\Vert T\Vert }_1$ 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 $\mathbf{x02amf}( )$, 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: $\mathbf{m}$ – Integer Output

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: $\mathbf{w}\left(*\right)$ – Real (Kind=nag_wp) array Output

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

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

14: $\mathbf{z}(\mathbf{ldz},*)$ – Complex (Kind=nag_wp) array Output

Note: the second dimension of the array z must be at least $\max (1,\mathbf{m})$ 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 $\max (1,\mathbf{m})$ 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: $\mathbf{ldz}$ – Integer Input

On entry: the first dimension of the array z as declared in the (sub)program from which f08frf is called.

Constraints:

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

16: $\mathbf{isuppz}\left(*\right)$ – Integer array Output

Note: the dimension of the array isuppz must be at least $\max (1,2\times \mathbf{m})$.

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 $\mathbf{range}=\text{'I'}$ and $\mathbf{iu}-\mathbf{il}=\mathbf{n}-1$.

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

18: $\mathbf{lwork}$ – Integer Input

On entry: the dimension of the array work as declared in the (sub)program from which f08frf is called.

If $\mathbf{lwork}=\mathrm{-1}$, 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, $\mathbf{lwork}\ge (\mathit{nb}+1)\times \mathbf{n}$, where $\mathit{nb}$ is the largest optimal block size for f08fsf and for f08fuf.

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

19: $\mathbf{rwork}\left(\max (1,\mathbf{lrwork})\right)$ – Real (Kind=nag_wp) array Workspace

On exit: if $\mathbf{info}=\mathbf{0}$, $\mathbf{rwork}\left(1\right)$ returns the optimal (and minimal) lrwork.

20: $\mathbf{lrwork}$ – Integer Input

On entry: the dimension of the array rwork as declared in the (sub)program from which f08frf is called.

If $\mathbf{lrwork}=\mathrm{-1}$, 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: $\mathbf{lrwork}\ge \max (1,24\times \mathbf{n})$ or $\mathbf{lrwork}=\mathrm{-1}$.

21: $\mathbf{iwork}\left(\max (1,\mathbf{liwork})\right)$ – Integer array Workspace

On exit: if $\mathbf{info}=\mathbf{0}$, $\mathbf{iwork}\left(1\right)$ returns the optimal (and minimal) liwork.

22: $\mathbf{liwork}$ – Integer Input

On entry: the dimension of the array iwork as declared in the (sub)program from which f08frf is called.

If $\mathbf{liwork}=\mathrm{-1}$, 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: $\mathbf{liwork}\ge \max (1,10\times \mathbf{n})$ or $\mathbf{liwork}=\mathrm{-1}$.

23: $\mathbf{info}$ – 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.

$\mathbf{info}>0$

f08frf failed to converge.

7 Accuracy

8 Parallelism and Performance

9 Further Comments

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results