NAG Library Routine Document

F06SBF (ZGBMV)

1  Purpose

2  Specification

3  Description

4  References

5  Parameters

1:     $\mathrm{TRANS}$ – 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:     $\mathrm{M}$ – INTEGERInput

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

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

3:     $\mathrm{N}$ – INTEGERInput

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

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

4:     $\mathrm{KL}$ – INTEGERInput

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

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

5:     $\mathrm{KU}$ – INTEGERInput

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

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

6:     $\mathrm{ALPHA}$ – COMPLEX (KIND=nag_wp)Input

On entry: the scalar $\alpha$.

7:     $\mathrm{A}\left(\mathbf{LDA},*\right)$ – 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:     $\mathrm{LDA}$ – 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:     $\mathrm{X}\left(*\right)$ – 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:   $\mathrm{INCX}$ – INTEGERInput

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

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

11:   $\mathrm{BETA}$ – COMPLEX (KIND=nag_wp)Input

On entry: the scalar $\beta$.

12:   $\mathrm{Y}\left(*\right)$ – 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:   $\mathrm{INCY}$ – 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