NAG Library Routine Document

F08UNF (ZHBGV)

1  Purpose

2  Specification

3  Description

4  References

5  Parameters

1:     $\mathrm{JOBZ}$ – 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:     $\mathrm{UPLO}$ – 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 $\text{'L'}$.

3:     $\mathrm{N}$ – INTEGERInput

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

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

4:     $\mathrm{KA}$ – INTEGERInput

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

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

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

5:     $\mathrm{KB}$ – INTEGERInput

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

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

Constraint: $\mathbf{KA}\ge \mathbf{KB}\ge 0$.

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

On exit: the contents of AB are overwritten.

7:     $\mathrm{LDAB}$ – INTEGERInput

On entry: the first dimension of the array AB as declared in the (sub)program from which F08UNF (ZHBGV) is called.

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

8:     $\mathrm{BB}\left(\mathbf{LDBB},*\right)$ – COMPLEX (KIND=nag_wp) arrayInput/Output

Note: the second dimension of the array BB 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 $B$.

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

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

On exit: the factor $S$ from the split Cholesky factorization $B=S^{\mathrm{H}}S$, as returned by F08UTF (ZPBSTF).

9:     $\mathrm{LDBB}$ – INTEGERInput

On entry: the first dimension of the array BB as declared in the (sub)program from which F08UNF (ZHBGV) is called.

Constraint: $\mathbf{LDBB}\ge \mathbf{KB}+1$.

10:   $\mathrm{W}\left(\mathbf{N}\right)$ – REAL (KIND=nag_wp) arrayOutput

On exit: the eigenvalues in ascending order.

11:   $\mathrm{Z}\left(\mathbf{LDZ},*\right)$ – 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 matrix $Z$ of eigenvectors, with the $i$th column of $Z$ holding the eigenvector associated with $\mathbf{W}\left(i\right)$. The eigenvectors are normalized so that $Z^{\mathrm{H}}BZ=I$.

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

12:   $\mathrm{LDZ}$ – INTEGERInput

On entry: the first dimension of the array Z as declared in the (sub)program from which F08UNF (ZHBGV) is called.

Constraints:

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

13:   $\mathrm{WORK}\left(\mathbf{N}\right)$ – COMPLEX (KIND=nag_wp) arrayWorkspace

14:   $\mathrm{RWORK}\left(3\times \mathbf{N}\right)$ – REAL (KIND=nag_wp) arrayWorkspace

15:   $\mathrm{INFO}$ – 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 $i\le \mathbf{N}$, the algorithm failed to converge; $i$ off-diagonal elements of an intermediate tridiagonal form did not converge to zero.

If $\mathbf{INFO}=i$ and $i>\mathbf{N}$, if $\mathbf{INFO}=\mathbf{N}+i$, for $1\le i\le \mathbf{N}$, then F08UTF (ZPBSTF) returned $\mathbf{INFO}=i$: $B$ 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 (f08unfe.f90)

10.2  Program Data

Program Data (f08unfe.d)

10.3  Program Results

Program Results (f08unfe.r)