NAG Library Routine Document

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:     $\mathbf{n}$ – IntegerInput

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

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

4:     $\mathbf{ka}$ – 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:     $\mathbf{kb}$ – 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:     $\mathbf{ab}\left(\mathbf{ldab},*\right)$ – 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:     $\mathbf{ldab}$ – IntegerInput

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

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

8:     $\mathbf{bb}\left(\mathbf{ldbb},*\right)$ – 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:     $\mathbf{ldbb}$ – IntegerInput

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

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

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

On exit: the eigenvalues in ascending order.

11:   $\mathbf{z}\left(\mathbf{ldz},*\right)$ – 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 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:   $\mathbf{ldz}$ – IntegerInput

On entry: the first dimension of the array z as declared in the (sub)program from which f08uqf (zhbgvd) 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) arrayWorkspace

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:   $\mathbf{lwork}$ – IntegerInput

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

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

16:   $\mathbf{lrwork}$ – IntegerInput

On entry: the first dimension of the array rwork as declared in the (sub)program from which f08uqf (zhbgvd) 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 arrayWorkspace

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

18:   $\mathbf{liwork}$ – IntegerInput

On entry: the dimension of the array iwork as declared in the (sub)program from which f08uqf (zhbgvd) 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:   $\mathbf{info}$ – 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 (f08uqfe.f90)

10.2

Program Data

Program Data (f08uqfe.d)

10.3

Program Results

Program Results (f08uqfe.r)