NAG FL Interface
f06pdf (dsbmv)

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

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

On entry: specifies 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 $\text{'L'}$.

2: $\mathbf{n}$ – Integer Input

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

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

3: $\mathbf{k}$ – Integer Input

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

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

4: $\mathbf{alpha}$ – Real (Kind=nag_wp) Input

On entry: the scalar $\alpha$.

5: $\mathbf{a}(\mathbf{lda},*)$ – Real (Kind=nag_wp) array Input

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

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

6: $\mathbf{lda}$ – Integer Input

On entry: the first dimension of the array a as declared in the (sub)program from which f06pdf is called.

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

7: $\mathbf{x}\left(*\right)$ – Real (Kind=nag_wp) array Input

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

On entry: the $n$-element vector $x$.

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

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

Intermediate elements of x are not referenced.

8: $\mathbf{incx}$ – Integer Input

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

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

9: $\mathbf{beta}$ – Real (Kind=nag_wp) Input

On entry: the scalar $\beta$.

10: $\mathbf{y}\left(*\right)$ – Real (Kind=nag_wp) array Input/Output

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

On entry: the $n$-element vector $y$, if $\mathbf{beta}=0.0$ y need not be set.

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

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

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

11: $\mathbf{incy}$ – Integer Input

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

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

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

10 Example