NAG Library Routine Document

f08gqf (zhpevd)

1

Purpose

2

Specification

3

Description

4

References

5

Arguments

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

On entry: indicates whether eigenvectors are computed.

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

Only eigenvalues are computed.

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

Eigenvalues and eigenvectors are computed.

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

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

On entry: indicates whether the upper or lower triangular part of $A$ is stored.

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

The upper triangular part of $A$ is stored.

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

The lower triangular part of $A$ is stored.

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

3:     $\mathbf{n}$ – IntegerInput

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

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

4:     $\mathbf{ap}\left(*\right)$ – Complex (Kind=nag_wp) arrayInput/Output

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

On entry: the upper or lower triangle of the $n$ by $n$ Hermitian matrix $A$, packed by columns.

More precisely,

• if $\mathbf{uplo}=\text{'U'}$, the upper triangle of $A$ must be stored with element $A_{ij}$ in $\mathbf{ap}\left(i+j\left(j-1\right)/2\right)$ for $i\le j$;
• if $\mathbf{uplo}=\text{'L'}$, the lower triangle of $A$ must be stored with element $A_{ij}$ in $\mathbf{ap}\left(i+\left(2n-j\right)\left(j-1\right)/2\right)$ for $i\ge j$.

On exit: ap is overwritten by the values generated during the reduction to tridiagonal form. The elements of the diagonal and the off-diagonal of the tridiagonal matrix overwrite the corresponding elements of $A$.

5:     $\mathbf{w}\left(*\right)$ – 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 eigenvalues of the matrix $A$ in ascending order.

6:     $\mathbf{z}\left(\mathbf{ldz},*\right)$ – Complex (Kind=nag_wp) arrayOutput

Note: the second dimension of the array z must be at least $\max \left(1,\mathbf{n}\right)$ if $\mathbf{job}=\text{'V'}$ and at least $1$ if $\mathbf{job}=\text{'N'}$.

On exit: if $\mathbf{job}=\text{'V'}$, z is overwritten by the unitary matrix $Z$ which contains the eigenvectors of $A$.

If $\mathbf{job}=\text{'N'}$, z is not referenced.

7:     $\mathbf{ldz}$ – IntegerInput

On entry: the first dimension of the array z as declared in the (sub)program from which f08gqf (zhpevd) is called.

Constraints:

• if $\mathbf{job}=\text{'V'}$, $\mathbf{ldz}\ge \max \left(1,\mathbf{n}\right)$;
• if $\mathbf{job}=\text{'N'}$, $\mathbf{ldz}\ge 1$.

8:     $\mathbf{work}\left(\max \left(1,\mathbf{lwork}\right)\right)$ – Complex (Kind=nag_wp) arrayWorkspace

On exit: if $\mathbf{info}=\mathbf{0}$, $\mathbf{work}\left(1\right)$ contains the required minimal size of lwork.

9:     $\mathbf{lwork}$ – IntegerInput

On entry: the dimension of the array work as declared in the (sub)program from which f08gqf (zhpevd) is called.

If $\mathbf{lwork}=-1$, a workspace query is assumed; the routine only calculates the minimum dimension of the work array, returns this value as the first entry of the work array, and no error message related to lwork is issued.

Constraints:

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

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

On exit: if $\mathbf{info}=\mathbf{0}$, $\mathbf{rwork}\left(1\right)$ contains the required minimal size of $\mathbf{lrwork}$.

11:   $\mathbf{lrwork}$ – IntegerInput

On entry: the dimension of the array rwork as declared in the (sub)program from which f08gqf (zhpevd) is called.

If $\mathbf{lrwork}=-1$, a workspace query is assumed; the routine only calculates the minimum dimension of the rwork array, returns this value as the first entry of the rwork array, and no error message related to lrwork is issued.

Constraints:

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

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

On exit: if $\mathbf{info}=\mathbf{0}$, $\mathbf{iwork}\left(1\right)$ contains the required minimal size of liwork.

13:   $\mathbf{liwork}$ – IntegerInput

On entry: the dimension of the array iwork as declared in the (sub)program from which f08gqf (zhpevd) is called.

If $\mathbf{liwork}=-1$, a workspace query is assumed; the routine only calculates the minimum dimension of the iwork array, returns this value as the first entry of the iwork array, and no error message related to liwork is issued.

Constraints:

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

14:   $\mathbf{info}$ – 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.

$\mathbf{info}>0$

If $\mathbf{info}=〈\mathit{\text{value}}〉$ and $\mathbf{job}=\text{'N'}$, the algorithm failed to converge; $〈\mathit{\text{value}}〉$ elements of an intermediate tridiagonal form did not converge to zero; if $\mathbf{info}=〈\mathit{\text{value}}〉$ and $\mathbf{job}=\text{'V'}$, then the algorithm failed to compute an eigenvalue while working on the submatrix lying in rows and column $〈\mathit{\text{value}}〉/\left(\mathbf{n}+1\right)$ through $〈\mathit{\text{value}}〉\mathrm{ mod }\left(\mathbf{n}+1\right)$.

7

Accuracy

8

Parallelism and Performance

9

Further Comments

10

Example

10.1

Program Text

Program Text (f08gqfe.f90)

10.2

Program Data

Program Data (f08gqfe.d)

10.3

Program Results

Program Results (f08gqfe.r)