NAG Library Routine Document

f08sqf (zhegvd)

1

Purpose

2

Specification

3

Description

4

References

5

Arguments

1:     $\mathbf{itype}$ – IntegerInput

On entry: specifies the problem type to be solved.

$\mathbf{itype}=1$

$Az=\lambda Bz$.

$\mathbf{itype}=2$

$ABz=\lambda z$.

$\mathbf{itype}=3$

$BAz=\lambda z$.

Constraint: $\mathbf{itype}=1$, $2$ or $3$.

2:     $\mathbf{jobz}$ – 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 $\text{'V'}$.

3:     $\mathbf{uplo}$ – Character(1)Input

On entry: if $\mathbf{uplo}=\text{'U'}$, the upper triangles of $A$ and $B$ are stored.

If $\mathbf{uplo}=\text{'L'}$, the lower triangles of $A$ and $B$ are stored.

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

4:     $\mathbf{n}$ – IntegerInput

On entry: $n$, the order of the matrices $A$ and $B$.

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

5:     $\mathbf{a}\left(\mathbf{lda},*\right)$ – 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 $n$ 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: if $\mathbf{jobz}=\text{'V'}$, a contains the matrix $Z$ of eigenvectors. The eigenvectors are normalized as follows:

• if $\mathbf{itype}=1$ or $2$, $Z^{\mathrm{H}}BZ=I$;
• if $\mathbf{itype}=3$, $Z^{\mathrm{H}}{B}^{-1}Z=I$.

If $\mathbf{jobz}=\text{'N'}$, the upper triangle (if $\mathbf{uplo}=\text{'U'}$) or the lower triangle (if $\mathbf{uplo}=\text{'L'}$) of a, including the diagonal, is overwritten.

6:     $\mathbf{lda}$ – IntegerInput

On entry: the first dimension of the array a as declared in the (sub)program from which f08sqf (zhegvd) is called.

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

7:     $\mathbf{b}\left(\mathbf{ldb},*\right)$ – Complex (Kind=nag_wp) arrayInput/Output

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

On entry: the $n$ by $n$ Hermitian matrix $B$.

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

On exit: the triangular factor $U$ or $L$ from the Cholesky factorization $B=U^{\mathrm{H}}U$ or $B=L{L}^{\mathrm{H}}$.

8:     $\mathbf{ldb}$ – IntegerInput

On entry: the first dimension of the array b as declared in the (sub)program from which f08sqf (zhegvd) is called.

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

9:     $\mathbf{w}\left(\mathbf{n}\right)$ – Real (Kind=nag_wp) arrayOutput

On exit: the eigenvalues in ascending order.

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

11:   $\mathbf{lwork}$ – IntegerInput

On entry: the dimension of the array work as declared in the (sub)program from which f08sqf (zhegvd) 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, lwork should usually be larger than the minimum, try increasing by $\mathit{nb}\times \mathbf{n}$, where $\mathit{nb}$ is the optimal block size.

Constraints:

• if $\mathbf{n}\le 1$, $\mathbf{lwork}\ge 1$;
• if $\mathbf{jobz}=\text{'N'}$ and $\mathbf{n}>1$, $\mathbf{lwork}\ge \mathbf{n}+1$;
• if $\mathbf{jobz}=\text{'V'}$ and $\mathbf{n}>1$, $\mathbf{lwork}\ge 2\times \mathbf{n}+{\mathbf{n}}^2$.

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

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

13:   $\mathbf{lrwork}$ – IntegerInput

On entry: the dimension of the array rwork as declared in the (sub)program from which f08sqf (zhegvd) 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.

Constraints:

• if $\mathbf{n}\le 1$, $\mathbf{lrwork}\ge 1$;
• if $\mathbf{jobz}=\text{'N'}$ and $\mathbf{n}>1$, $\mathbf{lrwork}\ge \mathbf{n}$;
• if $\mathbf{jobz}=\text{'V'}$ and $\mathbf{n}>1$, $\mathbf{lrwork}\ge 1+5\times \mathbf{n}+2\times {\mathbf{n}}^2$.

14:   $\mathbf{iwork}\left(\max \left(1,\mathbf{liwork}\right)\right)$ – Integer arrayWorkspace

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

15:   $\mathbf{liwork}$ – IntegerInput

On entry: the dimension of the array iwork as declared in the (sub)program from which f08sqf (zhegvd) 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.

Constraints:

• if $\mathbf{n}\le 1$, $\mathbf{liwork}\ge 1$;
• if $\mathbf{jobz}=\text{'N'}$ and $\mathbf{n}>1$, $\mathbf{liwork}\ge 1$;
• if $\mathbf{jobz}=\text{'V'}$ and $\mathbf{n}>1$, $\mathbf{liwork}\ge 3+5\times \mathbf{n}$.

16:   $\mathbf{info}$ – 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.

$\mathbf{info}=1\hspace{0.17em}\text{to}\hspace{0.17em}\mathbf{n}$

The algorithm failed to converge; $〈\mathit{\text{value}}〉$ off-diagonal elements of an intermediate tridiagonal form did not converge to zero.

$\mathbf{info}>\mathbf{n}$

If $\mathbf{info}=\mathbf{n}+〈\mathit{\text{value}}〉$, for $1\le 〈\mathit{\text{value}}〉\le \mathbf{n}$, then the leading minor of order $〈\mathit{\text{value}}〉$ of $B$ is not positive definite. The factorization of $B$ could not be completed and no eigenvalues or eigenvectors were computed.

7

Accuracy

8

Parallelism and Performance

9

Further Comments

10

Example

10.1

Program Text

Program Text (f08sqfe.f90)

10.2

Program Data

Program Data (f08sqfe.d)

10.3

Program Results

Program Results (f08sqfe.r)