NAG Library Routine Document

F08HQF (ZHBEVD)

1  Purpose

2  Specification

3  Description

4  References

5  Parameters

1:     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:     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:     N – INTEGERInput

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

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

4:     KD – INTEGERInput

On entry: if $\mathbf{UPLO}=\text{'U'}$, the number of superdiagonals, $k_d$, of the matrix $A$.

If $\mathbf{UPLO}=\text{'L'}$, the number of subdiagonals, $k_d$, of the matrix $A$.

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

5:     AB(LDAB,$*$) – COMPLEX (KIND=nag_wp) arrayInput/Output

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

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

The matrix is stored in rows $1$ to $k_d+1$, more precisely,

• if $\mathbf{UPLO}=\text{'U'}$, the elements of the upper triangle of $A$ within the band must be stored with element $A_{ij}$ in $\mathbf{AB}\left(k_d+1+i-j,j\right)\text{ for }\max \left(1,j-k_d\right)\le i\le j$;
• if $\mathbf{UPLO}=\text{'L'}$, the elements of the lower triangle of $A$ within the band must be stored with element $A_{ij}$ in $\mathbf{AB}\left(1+i-j,j\right)\text{ for }j\le i\le \min \left(n,j+k_d\right)\text{.}$

On exit: AB is overwritten by values generated during the reduction to tridiagonal form.

The first superdiagonal or subdiagonal and the diagonal of the tridiagonal matrix $T$ are returned in AB using the same storage format as described above.

6:     LDAB – INTEGERInput

On entry: the first dimension of the array AB as declared in the (sub)program from which F08HQF (ZHBEVD) is called.

Constraint: $\mathbf{LDAB}\ge \mathbf{KD}+1$.

7:     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 eigenvalues of the matrix $A$ in ascending order.

8:     Z(LDZ,$*$) – 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 $A$. The $i$th column of $Z$ contains the eigenvector which corresponds to the eigenvalue $\mathbf{W}\left(i\right)$.

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

9:     LDZ – INTEGERInput

On entry: the first dimension of the array Z as declared in the (sub)program from which F08HQF (ZHBEVD) is called.

Constraints:

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

10:   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 required minimal size of LWORK.

11:   LWORK – INTEGERInput

On entry: the dimension of the array WORK as declared in the (sub)program from which F08HQF (ZHBEVD) 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 $\mathbf{N}>1$, $\mathbf{LWORK}\ge 2\times {\mathbf{N}}^2$ or $\mathbf{LWORK}=-1$.

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

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

13:   LRWORK – INTEGERInput

On entry: the dimension of the array RWORK as declared in the (sub)program from which F08HQF (ZHBEVD) 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 $\mathbf{LRWORK}=-1$.

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

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

15:   LIWORK – INTEGERInput

On entry: the dimension of the array IWORK as declared in the (sub)program from which F08HQF (ZHBEVD) 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 $\mathbf{LIWORK}=-1$.

16:   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 $\mathbf{INFO}=i$ and $\mathbf{JOB}=\text{'N'}$, the algorithm failed to converge; $i$ elements of an intermediate tridiagonal form did not converge to zero; if $\mathbf{INFO}=i$ and $\mathbf{JOB}=\text{'V'}$, then the algorithm failed to compute an eigenvalue while working on the submatrix lying in rows and column $i/\left(\mathbf{N}+1\right)$ through $i\mathrm{ mod }\left(\mathbf{N}+1\right)$.

7  Accuracy

8  Further Comments

9  Example

9.1  Program Text

Program Text (f08hqfe.f90)

9.2  Program Data

Program Data (f08hqfe.d)

9.3  Program Results

Program Results (f08hqfe.r)