NAG FL Interface
f08hef (dsbtrd)

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: $\mathbf{n}$ – Integer Input

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

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

4: $\mathbf{kd}$ – Integer Input

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: $\mathbf{ab}(\mathbf{ldab},*)$ – Real (Kind=nag_wp) array Input/Output

Note: the second dimension of the array ab must be at least $\max (1,\mathbf{n})$.

On entry: the upper or lower triangle of the $n\times n$ symmetric 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 $\mathbf{ab}(k_d+1+i-j,j)\text{ for }\max (1,j-k_d)\le i\le j$;
• if $\mathbf{uplo}=\text{'L'}$, the elements of the lower triangle of $A$ within the band must be stored with element $A_{ij}$ in $\mathbf{ab}(1+i-j,j)\text{ for }j\le i\le \min (n,j+k_d)\text{.}$

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: $\mathbf{ldab}$ – Integer Input

On entry: the first dimension of the array ab as declared in the (sub)program from which f08hef is called.

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

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

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

8: $\mathbf{e}\left(*\right)$ – Real (Kind=nag_wp) array Output

Note: the dimension of the array e must be at least $\max (1,\mathbf{n}-1)$.

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

9: $\mathbf{q}(\mathbf{ldq},*)$ – Real (Kind=nag_wp) array Input/Output

Note: the second dimension of the array q must be at least $\max (1,\mathbf{n})$ 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 symmetric-definite generalized eigenproblem); otherwise q need not be set.

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

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

10: $\mathbf{ldq}$ – Integer Input

On entry: the first dimension of the array q as declared in the (sub)program from which f08hef is called.

Constraints:

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

11: $\mathbf{work}\left(\mathbf{n}\right)$ – Real (Kind=nag_wp) array Workspace

12: $\mathbf{info}$ – 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.

7 Accuracy

8 Parallelism and Performance

9 Further Comments

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results