NAG FL Interface
f06sgf (ztbmv)

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

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

On entry: specifies whether $A$ is upper or lower triangular.

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

$A$ is upper triangular.

$\mathbf{uplo}=\text{'L'}$

$A$ is lower triangular.

Constraint: $\mathbf{uplo}=\text{'U'}$ or $\text{'L'}$.

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

On entry: specifies the operation to be performed.

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

$x\leftarrow Ax$.

$\mathbf{trans}=\text{'T'}$

$x\leftarrow A^{\mathrm{T}}x$.

$\mathbf{trans}=\text{'C'}$

$x\leftarrow A^{\mathrm{H}}x$.

Constraint: $\mathbf{trans}=\text{'N'}$, $\text{'T'}$ or $\text{'C'}$.

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

On entry: specifies whether $A$ has nonunit or unit diagonal elements.

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

The diagonal elements are stored explicitly.

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

The diagonal elements are assumed to be $1$, and are not referenced.

Constraint: $\mathbf{diag}=\text{'N'}$ or $\text{'U'}$.

4: $\mathbf{n}$ – Integer Input

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

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

5: $\mathbf{k}$ – Integer Input

On entry: $k$, the number of subdiagonals or superdiagonals of the matrix $A$.

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

6: $\mathbf{a}\left(\mathbf{lda},*\right)$ – Complex (Kind=nag_wp) array Input

Note: the second dimension of the array a must be at least $\mathbf{n}$.

On entry: the $n$ by $n$ triangular band matrix $A$

The matrix is stored in rows $1$ to $k+1$, 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 $\mathbf{a}\left(k+1+i-j,j\right)\text{ for }\max \left(1,j-k\right)\le i\le j$;
• if $\mathbf{uplo}=\text{'L'}$, the elements of the lower triangle of $A$ within the band must be stored with element $A_{ij}$ in $\mathbf{a}\left(1+i-j,j\right)\text{ for }j\le i\le \min \left(n,j+k\right)\text{.}$

If $\mathbf{diag}=\text{'U'}$, the diagonal elements of $A$ are assumed to be $1$, and are not referenced.

7: $\mathbf{lda}$ – Integer Input

On entry: the first dimension of the array a as declared in the (sub)program from which f06sgf is called.

Constraint: $\mathbf{lda}\ge \mathbf{k}+1$.

8: $\mathbf{x}\left(*\right)$ – Complex (Kind=nag_wp) array Input/Output

Note: the dimension of the array x must be at least $\max \left(1,1+\left(\mathbf{n}-1\right)\times \left|\mathbf{incx}\right|\right)$.

On entry: the vector $x$.

If $\mathbf{incx}>0$, $x_{\mathit{i}}$ must be stored in $\mathbf{x}\left(1+\left(\mathit{i}–1\right)\times \mathbf{incx}\right)$, for $\mathit{i}=1,2,\dots ,\mathbf{n}$.

If $\mathbf{incx}<0$, $x_{\mathit{i}}$ must be stored in $\mathbf{x}\left(1–\left(\mathbf{n}–\mathit{i}\right)\times \mathbf{incx}\right)$, for $\mathit{i}=1,2,\dots ,\mathbf{n}$.

On exit: the updated vector $x$ stored in the array elements used to supply the original vector $x$.

9: $\mathbf{incx}$ – Integer Input

On entry: the increment in the subscripts of x between successive elements of $x$.

Constraint: $\mathbf{incx}\ne 0$.

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

10 Example