NAG Library Routine Document

f08hsf (zhbtrd)

1

Purpose

2

Specification

3

Description

4

References

5

Arguments

1:     $\mathbf{vect}$ – Character(1)Input

On entry: indicates whether $Q$ is to be returned.

$\mathbf{vect}=\text{'V'}$

$Q$ is returned.

$\mathbf{vect}=\text{'U'}$

$Q$ is updated (and the array q must contain a matrix on entry).

$\mathbf{vect}=\text{'N'}$

$Q$ is not required.

Constraint: $\mathbf{vect}=\text{'V'}$, $\text{'U'}$ or $\text{'N'}$.

2:     $\mathbf{uplo}$ – 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:     $\mathbf{n}$ – IntegerInput

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

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

4:     $\mathbf{kd}$ – 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:     $\mathbf{ab}\left(\mathbf{ldab},*\right)$ – 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 $k_d+1$, 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:     $\mathbf{ldab}$ – IntegerInput

On entry: the first dimension of the array ab as declared in the (sub)program from which f08hsf (zhbtrd) is called.

Constraint: $\mathbf{ldab}\ge \max \left(1,\mathbf{kd}+1\right)$.

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

On exit: the diagonal elements of the tridiagonal matrix $T$.

8:     $\mathbf{e}\left(\mathbf{n}-1\right)$ – Real (Kind=nag_wp) arrayOutput

On exit: the off-diagonal elements of the tridiagonal matrix $T$.

9:     $\mathbf{q}\left(\mathbf{ldq},*\right)$ – Complex (Kind=nag_wp) arrayInput/Output

Note: the second dimension of the array q must be at least $\max \left(1,\mathbf{n}\right)$ if $\mathbf{vect}=\text{'V'}$ or $\text{'U'}$ and at least $1$ if $\mathbf{vect}=\text{'N'}$.

On entry: if $\mathbf{vect}=\text{'U'}$, q must contain the matrix formed in a previous stage of the reduction (for example, the reduction of a banded Hermitian-definite generalized eigenproblem); otherwise q need not be set.

On exit: if $\mathbf{vect}=\text{'V'}$ or $\text{'U'}$, the $n$ by $n$ matrix $Q$.

If $\mathbf{vect}=\text{'N'}$, q is not referenced.

10:   $\mathbf{ldq}$ – IntegerInput

On entry: the first dimension of the array q as declared in the (sub)program from which f08hsf (zhbtrd) is called.

Constraints:

• if $\mathbf{vect}=\text{'V'}$ or $\text{'U'}$, $\mathbf{ldq}\ge \max \left(1,\mathbf{n}\right)$;
• if $\mathbf{vect}=\text{'N'}$, $\mathbf{ldq}\ge 1$.

11:   $\mathbf{work}\left(\mathbf{n}\right)$ – Complex (Kind=nag_wp) arrayWorkspace

12:   $\mathbf{info}$ – 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.

7

Accuracy

8

Parallelism and Performance

9

Further Comments

10

Example

10.1

Program Text

Program Text (f08hsfe.f90)

10.2

Program Data

Program Data (f08hsfe.d)

10.3

Program Results

Program Results (f08hsfe.r)