# NAG Library Routine Document

## 1Purpose

f06pdf (dsbmv) computes the matrix-vector product for a real symmetric band matrix.

## 2Specification

Fortran Interface
 Subroutine f06pdf ( uplo, n, k, a, lda, x, incx, beta, y, incy)
 Integer, Intent (In) :: n, k, lda, incx, incy Real (Kind=nag_wp), Intent (In) :: alpha, a(lda,*), x(*), beta Real (Kind=nag_wp), Intent (Inout) :: y(*) Character (1), Intent (In) :: uplo
#include nagmk26.h
 void f06pdf_ ( const char *uplo, const Integer *n, const Integer *k, const double *alpha, const double a[], const Integer *lda, const double x[], const Integer *incx, const double *beta, double y[], const Integer *incy, const Charlen length_uplo)
The routine may be called by its BLAS name dsbmv.

## 3Description

f06pdf (dsbmv) performs the matrix-vector operation
 $y←αAx+βy ,$
where $A$ is an $n$ by $n$ real symmetric band matrix with $k$ subdiagonals and $k$ superdiagonals, $x$ and $y$ are $n$-element real vectors, and $\alpha$ and $\beta$ are real scalars.

None.

## 5Arguments

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:     $\mathbf{n}$ – IntegerInput
On entry: $n$, the order of the matrix $A$.
Constraint: ${\mathbf{n}}\ge 0$.
3:     $\mathbf{k}$ – IntegerInput
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:     $\mathbf{a}\left({\mathbf{lda}},*\right)$ – Real (Kind=nag_wp) arrayInput
Note: the second dimension of the array a must be at least ${\mathbf{n}}$.
On entry: the $n$ by $n$ 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 ${\mathbf{a}}\left(k+1+i-j,j\right)\text{​ for ​}\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,j-k\right)\le i\le j$;
• if ${\mathbf{uplo}}=\text{'L'}$, the elements of the lower triangle of $A$ within the band must be stored with element ${A}_{ij}$ in ${\mathbf{a}}\left(1+i-j,j\right)\text{​ for ​}j\le i\le \mathrm{min}\phantom{\rule{0.125em}{0ex}}\left(n,j+k\right)\text{.}$
6:     $\mathbf{lda}$ – IntegerInput
On entry: the first dimension of the array a as declared in the (sub)program from which f06pdf (dsbmv) is called.
Constraint: ${\mathbf{lda}}\ge {\mathbf{k}}+1$.
7:     $\mathbf{x}\left(*\right)$ – Real (Kind=nag_wp) arrayInput
Note: the dimension of the array x must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,1+\left({\mathbf{n}}-1\right)×\left|{\mathbf{incx}}\right|\right)$.
On entry: the $n$-element vector $x$.
If ${\mathbf{incx}}>0$, ${x}_{\mathit{i}}$ must be stored in ${\mathbf{x}}\left(1+\left(\mathit{i}-1\right)×{\mathbf{incx}}\right)$, for $\mathit{i}=1,2,\dots ,{\mathbf{n}}$.
If ${\mathbf{incx}}<0$, ${x}_{\mathit{i}}$ must be stored in ${\mathbf{x}}\left(1-\left({\mathbf{n}}-\mathit{i}\right)×{\mathbf{incx}}\right)$, for $\mathit{i}=1,2,\dots ,{\mathbf{n}}$.
Intermediate elements of x are not referenced.
8:     $\mathbf{incx}$ – IntegerInput
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) arrayInput/Output
Note: the dimension of the array y must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,1+\left({\mathbf{n}}-1\right)×\left|{\mathbf{incy}}\right|\right)$.
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 ${\mathbf{y}}\left(1+\left(\mathit{i}–1\right)×{\mathbf{incy}}\right)$, for $\mathit{i}=1,2,\dots ,{\mathbf{n}}$.
If ${\mathbf{incy}}<0$, ${y}_{\mathit{i}}$ must be stored in ${\mathbf{y}}\left(1–\left({\mathbf{n}}–\mathit{i}\right)×{\mathbf{incy}}\right)$, 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:   $\mathbf{incy}$ – IntegerInput
On entry: the increment in the subscripts of y between successive elements of $y$.
Constraint: ${\mathbf{incy}}\ne 0$.

None.

Not applicable.

## 8Parallelism and Performance

f06pdf (dsbmv) is not threaded in any implementation.

None.

## 10Example

None.
© The Numerical Algorithms Group Ltd, Oxford, UK. 2017