NAG FL Interface
f08uqf (zhbgvd)

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 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: $\mathbf{n}$ – Integer Input

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

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

4: $\mathbf{ka}$ – Integer Input

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: $\mathbf{kb}$ – Integer Input

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: $\mathbf{ab}\left(\mathbf{ldab},*\right)$ – Complex (Kind=nag_wp) array Input/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: $\mathbf{ldab}$ – Integer Input

On entry: the first dimension of the array ab as declared in the (sub)program from which f08uqf is called.

Constraint: $\mathbf{ldab}\ge \mathbf{ka}+1$.

8: $\mathbf{bb}\left(\mathbf{ldbb},*\right)$ – Complex (Kind=nag_wp) array Input/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.

9: $\mathbf{ldbb}$ – Integer Input

On entry: the first dimension of the array bb as declared in the (sub)program from which f08uqf is called.

Constraint: $\mathbf{ldbb}\ge \mathbf{kb}+1$.

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

On exit: the eigenvalues in ascending order.

11: $\mathbf{z}\left(\mathbf{ldz},*\right)$ – Complex (Kind=nag_wp) array Output

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: $\mathbf{ldz}$ – Integer Input

On entry: the first dimension of the array z as declared in the (sub)program from which f08uqf is called.

Constraints:

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

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

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.

14: $\mathbf{lwork}$ – Integer Input

On entry: the dimension of the array work as declared in the (sub)program from which f08uqf 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.

Constraints:

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

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

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

16: $\mathbf{lrwork}$ – Integer Input

On entry: the first dimension of the array rwork as declared in the (sub)program from which f08uqf 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 $\mathbf{n}>1$, $\mathbf{lrwork}\ge \max \left(1,\mathbf{n}\right)$;
• if $\mathbf{jobz}=\text{'V'}$ and $\mathbf{n}>1$, $\mathbf{lrwork}\ge 1+5\times \mathbf{n}+2\times {\mathbf{n}}^2$.

17: $\mathbf{iwork}\left(\max \left(1,\mathbf{liwork}\right)\right)$ – Integer array Workspace

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

18: $\mathbf{liwork}$ – Integer Input

On entry: the dimension of the array iwork as declared in the (sub)program from which f08uqf 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 $\mathbf{jobz}=\text{'N'}$ or $\mathbf{n}\le 1$, $\mathbf{liwork}\ge 1$;
• if $\mathbf{jobz}=\text{'V'}$ and $\mathbf{n}>1$, $\mathbf{liwork}\ge 3+5\times \mathbf{n}$.

19: $\mathbf{info}$ – Integer Output

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.

$\mathbf{info}=1,\dots ,\mathbf{n}$

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 $1\le 〈\mathit{\text{value}}〉\le \mathbf{n}$, f08utf returned $\mathbf{info}=〈\mathit{\text{value}}〉$: $B$ 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

10.2 Program Data

10.3 Program Results