NAG Library Routine Document

f08gnf (zhpev)

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:     $\mathbf{uplo}$ – 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 $\text{'L'}$.

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

On exit: the eigenvalues 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 $\mathbf{jobz}=\text{'V'}$, and at least $1$ otherwise.

On exit: if $\mathbf{jobz}=\text{'V'}$, z contains the orthonormal eigenvectors of the matrix $A$, 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.

7:     $\mathbf{ldz}$ – IntegerInput

On entry: the first dimension of the array z as declared in the (sub)program from which f08gnf (zhpev) is called.

Constraints:

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

8:     $\mathbf{work}\left(2\times \mathbf{n}-1\right)$ – Complex (Kind=nag_wp) arrayWorkspace

9:     $\mathbf{rwork}\left(3\times \mathbf{n}-2\right)$ – Real (Kind=nag_wp) arrayWorkspace

10:   $\mathbf{info}$ – 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.

$\mathbf{info}>0$

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

7

Accuracy

8

Parallelism and Performance

9

Further Comments

10

Example

10.1

Program Text

Program Text (f08gnfe.f90)

10.2

Program Data

Program Data (f08gnfe.d)

10.3

Program Results

Program Results (f08gnfe.r)