NAG Library Routine Document

f06sbf (zgbmv)

1

Purpose

2

Specification

3

Description

4

References

5

Arguments

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

On entry: specifies the operation to be performed.

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

$y\leftarrow \alpha Ax+\beta y$.

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

$y\leftarrow \alpha A^{\mathrm{T}}x+\beta y$.

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

$y\leftarrow \alpha A^{\mathrm{H}}x+\beta y$.

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

2:     $\mathbf{m}$ – IntegerInput

On entry: $m$, the number of rows of the matrix $A$.

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

3:     $\mathbf{n}$ – IntegerInput

On entry: $n$, the number of columns of the matrix $A$.

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

4:     $\mathbf{kl}$ – IntegerInput

On entry: $k_l$, the number of subdiagonals within the band of $A$.

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

5:     $\mathbf{ku}$ – IntegerInput

On entry: $k_u$, the number of superdiagonals within the band of $A$.

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

6:     $\mathbf{alpha}$ – Complex (Kind=nag_wp)Input

On entry: the scalar $\alpha$.

7:     $\mathbf{a}\left(\mathbf{lda},*\right)$ – Complex (Kind=nag_wp) arrayInput

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

On entry: the $m$ by $n$ band matrix $A$.

The matrix is stored in rows $1$ to $k_l+k_u+1$, more precisely, the element $A_{ij}$ must be stored in

$$\mathbf{a}\left(k_u+1+i-j,j\right)\text{\hspace{1em} for }\max \left(1,j-k_u\right)\le i\le \min \left(m,j+k_l\right)\text{.}$$

8:     $\mathbf{lda}$ – IntegerInput

On entry: the first dimension of the array a as declared in the (sub)program from which f06sbf (zgbmv) is called.

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

9:     $\mathbf{x}\left(*\right)$ – Complex (Kind=nag_wp) arrayInput

Note: the dimension of the array x must be at least $\max \left(1,1+\left(\mathbf{n}-1\right)\times \left|\mathbf{incx}\right|\right)$ if $\mathbf{trans}=\text{'N'}$ and at least $\max \left(1,1+\left(\mathbf{m}-1\right)\times \left|\mathbf{incx}\right|\right)$ if $\mathbf{trans}=\text{'T'}$ or $\text{'C'}$.

On entry: the vector $x$.

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

• 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 $\mathit{i}=1,2,\dots ,\mathbf{n}$.

If $\mathbf{trans}=\text{'T'}$ or $\text{'C'}$,

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

10:   $\mathbf{incx}$ – IntegerInput

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

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

11:   $\mathbf{beta}$ – Complex (Kind=nag_wp)Input

On entry: the scalar $\beta$.

12:   $\mathbf{y}\left(*\right)$ – Complex (Kind=nag_wp) arrayInput/Output

Note: the dimension of the array y must be at least $\max \left(1,1+\left(\mathbf{m}-1\right)\times \left|\mathbf{incy}\right|\right)$ if $\mathbf{trans}=\text{'N'}$ and at least $\max \left(1,1+\left(\mathbf{n}-1\right)\times \left|\mathbf{incy}\right|\right)$ if $\mathbf{trans}=\text{'T'}$ or $\text{'C'}$.

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

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

• if $\mathbf{incy}>0$, $y_{\mathit{i}}$ must be stored in $\mathbf{y}\left(1+\left(\mathit{i}-1\right)\times \mathbf{incy}\right)$, for $\mathit{i}=1,2,\dots ,\mathbf{m}$;
• if $\mathbf{incy}<0$, $y_{\mathit{i}}$ must be stored in $\mathbf{y}\left(1-\left(\mathbf{m}-\mathit{i}\right)\times \mathbf{incy}\right)$, for $\mathit{i}=1,2,\dots ,\mathbf{m}$.

If $\mathbf{trans}=\text{'T'}$ or $\text{'C'}$,

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

13:   $\mathbf{incy}$ – IntegerInput

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