NAG Library Function Document

nag_dsbmv (f16pdc)

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:    $\mathbf{uplo}$ – Nag_UploTypeInput

On entry: specifies whether the upper or lower triangular part of $A$ is stored.

$\mathbf{uplo}=\mathrm{Nag\_Upper}$

The upper triangular part of $A$ is stored.

$\mathbf{uplo}=\mathrm{Nag\_Lower}$

The lower triangular part of $A$ is stored.

Constraint: $\mathbf{uplo}=\mathrm{Nag\_Upper}$ or $\mathrm{Nag\_Lower}$.

3:    $\mathbf{n}$ – IntegerInput

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

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

4:    $\mathbf{k}$ – IntegerInput

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

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

5:    $\mathbf{alpha}$ – doubleInput

On entry: the scalar $\alpha$.

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

Note: the dimension, dim, of the array ab must be at least $\max \left(1,\mathbf{pdab}\times \mathbf{n}\right)$.

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

This is stored as a notional two-dimensional array with row elements or column elements stored contiguously. The storage of elements of $A_{ij}$, depends on the order and uplo arguments as follows:

• if $\mathbf{order}=\mathrm{Nag\_ColMajor}$ and $\mathbf{uplo}=\mathrm{Nag\_Upper}$,
            $A_{ij}$ is stored in $\mathbf{ab}\left[k+i-j+\left(j-1\right)\times \mathbf{pdab}\right]$, for $j=1,\dots ,n$ and $i=\max \left(1,j-k\right),\dots ,j$;
• if $\mathbf{order}=\mathrm{Nag\_ColMajor}$ and $\mathbf{uplo}=\mathrm{Nag\_Lower}$,
            $A_{ij}$ is stored in $\mathbf{ab}\left[i-j+\left(j-1\right)\times \mathbf{pdab}\right]$, for $j=1,\dots ,n$ and $i=j,\dots ,\min \left(n,j+k\right)$;
• if $\mathbf{order}=\mathrm{Nag\_RowMajor}$ and $\mathbf{uplo}=\mathrm{Nag\_Upper}$,
            $A_{ij}$ is stored in $\mathbf{ab}\left[j-i+\left(i-1\right)\times \mathbf{pdab}\right]$, for $i=1,\dots ,n$ and $j=i,\dots ,\min \left(n,i+k\right)$;
• if $\mathbf{order}=\mathrm{Nag\_RowMajor}$ and $\mathbf{uplo}=\mathrm{Nag\_Lower}$,
            $A_{ij}$ is stored in $\mathbf{ab}\left[k+j-i+\left(i-1\right)\times \mathbf{pdab}\right]$, for $i=1,\dots ,n$ and $j=\max \left(1,i-k\right),\dots ,i$.

7:    $\mathbf{pdab}$ – 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: $\mathbf{pdab}\ge \mathbf{k}+1$.

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

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

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

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.

9:    $\mathbf{incx}$ – IntegerInput

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

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

10:  $\mathbf{beta}$ – doubleInput

On entry: the scalar $\beta$.

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

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

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 $y$.

12:  $\mathbf{incy}$ – IntegerInput

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

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

13:  $\mathbf{fail}$ – 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, $\mathbf{k}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{k}\ge 0$.

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

NE_INT_2

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

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 (f16pdce.c)

10.2

Program Data

Program Data (f16pdce.d)

10.3

Program Results

Program Results (f16pdce.r)