F06PDF (DSBMV) (PDF version)
F06 Chapter Contents
F06 Chapter Introduction
NAG Library Manual

# NAG Library Routine DocumentF06PDF (DSBMV)

Note:  before using this routine, please read the Users' Note for your implementation to check the interpretation of bold italicised terms and other implementation-dependent details.

## 1  Purpose

F06PDF (DSBMV) computes the matrix-vector product for a real symmetric band matrix.

## 2  Specification

 SUBROUTINE F06PDF ( UPLO, N, K, ALPHA, A, LDA, X, INCX, BETA, Y, INCY)
 INTEGER N, K, LDA, INCX, INCY REAL (KIND=nag_wp) ALPHA, A(LDA,*), X(*), BETA, Y(*) CHARACTER(1) UPLO
The routine may be called by its BLAS name dsbmv.

## 3  Description

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.

## 5  Parameters

1:     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:     N – INTEGERInput
On entry: $n$, the order of the matrix $A$.
Constraint: ${\mathbf{N}}\ge 0$.
3:     K – INTEGERInput
On entry: $k$, the number of subdiagonals or superdiagonals of the matrix $A$.
Constraint: ${\mathbf{K}}\ge 0$.
4:     ALPHA – REAL (KIND=nag_wp)Input
On entry: the scalar $\alpha$.
5:     A(LDA,$*$) – 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:     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:     X($*$) – 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:     INCX – INTEGERInput
On entry: the increment in the subscripts of X between successive elements of $x$.
Constraint: ${\mathbf{INCX}}\ne 0$.
9:     BETA – REAL (KIND=nag_wp)Input
On entry: the scalar $\beta$.
10:   Y($*$) – 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:   INCY – INTEGERInput
On entry: the increment in the subscripts of Y between successive elements of $y$.
Constraint: ${\mathbf{INCY}}\ne 0$.

None.

Not applicable.

None.

## 9  Example

None.

F06PDF (DSBMV) (PDF version)
F06 Chapter Contents
F06 Chapter Introduction
NAG Library Manual