NAG Library Function Document

nag_dgbmv (f16pbc)

1

Purpose

2

Specification

3

Description

4

References

5

Arguments

1:    $\mathbf{order}$ – Nag_OrderTypeInput

On entry: the order argument specifies the two-dimensional storage scheme being used, i.e., row-major ordering or column-major ordering. C language defined storage is specified by $\mathbf{order}=\mathrm{Nag\_RowMajor}$. See Section 3.3.1.3 in How to Use the NAG Library and its Documentation for a more detailed explanation of the use of this argument.

Constraint: $\mathbf{order}=\mathrm{Nag\_RowMajor}$ or $\mathrm{Nag\_ColMajor}$.

2:    $\mathbf{trans}$ – Nag_TransTypeInput

On entry: specifies the operation to be performed.

$\mathbf{trans}=\mathrm{Nag\_NoTrans}$

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

$\mathbf{trans}=\mathrm{Nag\_Trans}$ or $\mathrm{Nag\_ConjTrans}$

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

Constraint: $\mathbf{trans}=\mathrm{Nag\_NoTrans}$, $\mathrm{Nag\_Trans}$ or $\mathrm{Nag\_ConjTrans}$.

3:    $\mathbf{m}$ – IntegerInput

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

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

4:    $\mathbf{n}$ – IntegerInput

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

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

5:    $\mathbf{kl}$ – IntegerInput

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

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

6:    $\mathbf{ku}$ – IntegerInput

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

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

7:    $\mathbf{alpha}$ – doubleInput

On entry: the scalar $\alpha$.

8:    $\mathbf{ab}\left[\mathit{dim}\right]$ – const doubleInput

Note: the dimension, dim, of the array ab must be at least

• $\max \left(1,\mathbf{pdab}\times \mathbf{n}\right)$ when $\mathbf{order}=\mathrm{Nag\_ColMajor}$;
• $\max \left(1,\mathbf{m}\times \mathbf{pdab}\right)$ when $\mathbf{order}=\mathrm{Nag\_RowMajor}$.

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

This is stored as a notional two-dimensional array with row elements or column elements stored contiguously. The storage of elements $A_{ij}$, for row $i=1,\dots ,m$ and column $j=\max \left(1,i-k_l\right),\dots ,\min \left(n,i+k_u\right)$, depends on the order argument as follows:

• if $\mathbf{order}=\mathrm{Nag\_ColMajor}$, $A_{ij}$ is stored as $\mathbf{ab}\left[\left(j-1\right)\times \mathbf{pdab}+\mathbf{ku}+i-j\right]$;
• if $\mathbf{order}=\mathrm{Nag\_RowMajor}$, $A_{ij}$ is stored as $\mathbf{ab}\left[\left(i-1\right)\times \mathbf{pdab}+\mathbf{kl}+j-i\right]$.

9:    $\mathbf{pdab}$ – IntegerInput

On entry: the stride separating row or column elements (depending on the value of order) of the matrix $A$ in the array ab.

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

10:  $\mathbf{x}\left[\mathit{dim}\right]$ – const doubleInput

Note: the dimension, dim, of the array x must be at least

• $\max \left(1,1+\left(\mathbf{n}-1\right)\left|\mathbf{incx}\right|\right)$ when $\mathbf{trans}=\mathrm{Nag\_NoTrans}$;
• $\max \left(1,1+\left(\mathbf{m}-1\right)\left|\mathbf{incx}\right|\right)$ when $\mathbf{trans}=\mathrm{Nag\_Trans}$ or $\mathrm{Nag\_ConjTrans}$.

On entry: the vector $x$.

If $\mathbf{trans}=\mathrm{Nag\_NoTrans}$, then $x$ is an $n$-element vector.

• If $\mathbf{incx}>0$, $x_{\mathit{i}}$ must be stored in $\mathbf{x}\left[\left(\mathit{i}-1\right)\times \mathbf{incx}\right]$, for $\mathit{i}=1,2,\dots ,\mathbf{n}$.
• If $\mathbf{incx}<0$, $x_{\mathit{i}}$ must be stored in $\mathbf{x}\left[\left(\mathbf{n}-\mathit{i}\right)\times \left|\mathbf{incx}\right|\right]$, for $\mathit{i}=1,2,\dots ,\mathbf{n}$.
• Intermediate elements of x are not referenced. If $\mathbf{n}=0$, x is not referenced and may be NULL.

Otherwise, $x$ is an $m$-element vector.

• If $\mathbf{incx}>0$, $x_{\mathit{i}}$ must be stored in $\mathbf{x}\left[\left(\mathit{i}-1\right)\times \mathbf{incx}\right]$, for $\mathit{i}=1,2,\dots ,\mathbf{m}$.
• If $\mathbf{incx}<0$, $x_{\mathit{i}}$ must be stored in $\mathbf{x}\left[\left(\mathbf{m}-\mathit{i}\right)\times \left|\mathbf{incx}\right|\right]$, for $\mathit{i}=1,2,\dots ,\mathbf{m}$.
• Intermediate elements of x are not referenced. If $\mathbf{m}=0$, x is not referenced and may be NULL.

11:  $\mathbf{incx}$ – IntegerInput

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

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

12:  $\mathbf{beta}$ – doubleInput

On entry: the scalar $\beta$.

13:  $\mathbf{y}\left[\mathit{dim}\right]$ – doubleInput/Output

Note: the dimension, dim, of the array y must be at least

• $\max \left(1,1+\left(\mathbf{m}-1\right)\left|\mathbf{incy}\right|\right)$ when $\mathbf{trans}=\mathrm{Nag\_NoTrans}$;
• $\max \left(1,1+\left(\mathbf{n}-1\right)\left|\mathbf{incy}\right|\right)$ when $\mathbf{trans}=\mathrm{Nag\_Trans}$ or $\mathrm{Nag\_ConjTrans}$.

On entry: the vector $y$. See x for details of storage.

If $\mathbf{beta}=0$, y need not be set.

On exit: the updated vector $y$.

14:  $\mathbf{incy}$ – IntegerInput

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

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

15:  $\mathbf{fail}$ – NagError *Input/Output

The NAG error argument (see Section 3.7 in How to Use the NAG Library and its Documentation).

6

Error Indicators and Warnings

NE_ALLOC_FAIL

Dynamic memory allocation failed.
See Section 2.3.1.2 in How to Use the NAG Library and its Documentation for further information.

NE_BAD_PARAM

On entry, argument $〈\mathit{\text{value}}〉$ had an illegal value.

NE_INT

On entry, $\mathbf{incx}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{incx}\ne 0$.

On entry, $\mathbf{incy}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{incy}\ne 0$.

On entry, $\mathbf{kl}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{kl}\ge 0$.

On entry, $\mathbf{ku}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{ku}\ge 0$.

On entry, $\mathbf{m}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{m}\ge 0$.

On entry, $\mathbf{n}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{n}\ge 0$.

NE_INT_3

On entry, $\mathbf{pdab}=〈\mathit{\text{value}}〉$, $\mathbf{kl}=〈\mathit{\text{value}}〉$, $\mathbf{ku}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{pdab}\ge \mathbf{kl}+\mathbf{ku}+1$.

NE_NO_LICENCE

Your licence key may have expired or may not have been installed correctly.
See Section 2.7.5 in How to Use the NAG Library and its Documentation for further information.

7

Accuracy

8

Parallelism and Performance

9

Further Comments

10

Example

10.1

Program Text

Program Text (f16pbce.c)

10.2

Program Data

Program Data (f16pbce.d)

10.3

Program Results

Program Results (f16pbce.r)