NAG Library Routine Document

1Purpose

f06pbf (dgbmv) computes the matrix-vector product for a real general band matrix or its transpose.

2Specification

Fortran Interface
 Subroutine f06pbf ( m, n, kl, ku, a, lda, x, incx, beta, y, incy)
 Integer, Intent (In) :: m, n, kl, ku, 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) :: trans
#include nagmk26.h
 void f06pbf_ (const char *trans, const Integer *m, const Integer *n, const Integer *kl, const Integer *ku, 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_trans)
The routine may be called by its BLAS name dgbmv.

3Description

f06pbf (dgbmv) performs one of the matrix-vector operations
 $y←αAx+βy or y←αATx+βy ,$
where $A$ is an $m$ by $n$ real band matrix with ${k}_{l}$ subdiagonals and ${k}_{u}$ superdiagonals, $x$ and $y$ are real vectors, and $\alpha$ and $\beta$ are real scalars.
If $m=0$ or $n=0$, no operation is performed.

None.

5Arguments

1:     $\mathbf{trans}$ – Character(1)Input
On entry: specifies the operation to be performed.
${\mathbf{trans}}=\text{'N'}$
$y←\alpha Ax+\beta y$.
${\mathbf{trans}}=\text{'T'}$ or $\text{'C'}$
$y←\alpha {A}^{\mathrm{T}}x+\beta y$.
Constraint: ${\mathbf{trans}}=\text{'N'}$, $\text{'T'}$ or $\text{'C'}$.
2:     $\mathbf{m}$ – IntegerInput
On entry: $m$, the number of rows of the matrix $A$.
Constraint: ${\mathbf{m}}\ge 0$.
3:     $\mathbf{n}$ – IntegerInput
On entry: $n$, the number of columns of the matrix $A$.
Constraint: ${\mathbf{n}}\ge 0$.
4:     $\mathbf{kl}$ – IntegerInput
On entry: ${k}_{l}$, the number of subdiagonals within the band of $A$.
Constraint: ${\mathbf{kl}}\ge 0$.
5:     $\mathbf{ku}$ – IntegerInput
On entry: ${k}_{u}$, the number of superdiagonals within the band of $A$.
Constraint: ${\mathbf{ku}}\ge 0$.
6:     $\mathbf{alpha}$ – Real (Kind=nag_wp)Input
On entry: the scalar $\alpha$.
7:     $\mathbf{a}\left({\mathbf{lda}},*\right)$ – Real (Kind=nag_wp) arrayInput
Note: the second dimension of the array a must be at least ${\mathbf{n}}$.
On entry: the $m$ 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
 $aku+1+i-jj for ​max1,j-ku≤i≤minm,j+kl.$
8:     $\mathbf{lda}$ – IntegerInput
On entry: the first dimension of the array a as declared in the (sub)program from which f06pbf (dgbmv) is called.
Constraint: ${\mathbf{lda}}\ge {\mathbf{kl}}+{\mathbf{ku}}+1$.
9:     $\mathbf{x}\left(*\right)$ – Real (Kind=nag_wp) arrayInput
Note: the dimension of the array x must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,1+\left({\mathbf{n}}-1\right)×\left|{\mathbf{incx}}\right|\right)$ if ${\mathbf{trans}}=\text{'N'}$ and at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,1+\left({\mathbf{m}}-1\right)×\left|{\mathbf{incx}}\right|\right)$ 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 ${\mathbf{x}}\left(1+\left(\mathit{i}-1\right)×{\mathbf{incx}}\right)$, 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 $\mathit{i}=1,2,\dots ,{\mathbf{n}}$.
If ${\mathbf{trans}}=\text{'T'}$ or $\text{'C'}$,
• if ${\mathbf{incx}}>0$, ${x}_{\mathit{i}}$ must be stored in ${\mathbf{x}}\left(1+\left(\mathit{i}-1\right)×{\mathbf{incx}}\right)$, for $\mathit{i}=1,2,\dots ,{\mathbf{m}}$;
• if ${\mathbf{incx}}<0$, ${x}_{\mathit{i}}$ must be stored in ${\mathbf{x}}\left(1-\left({\mathbf{m}}-\mathit{i}\right)×{\mathbf{incx}}\right)$, for $\mathit{i}=1,2,\dots ,{\mathbf{m}}$.
10:   $\mathbf{incx}$ – IntegerInput
On entry: the increment in the subscripts of x between successive elements of $x$.
Constraint: ${\mathbf{incx}}\ne 0$.
11:   $\mathbf{beta}$ – Real (Kind=nag_wp)Input
On entry: the scalar $\beta$.
12:   $\mathbf{y}\left(*\right)$ – Real (Kind=nag_wp) arrayInput/Output
Note: the dimension of the array y must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,1+\left({\mathbf{m}}-1\right)×\left|{\mathbf{incy}}\right|\right)$ if ${\mathbf{trans}}=\text{'N'}$ and at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,1+\left({\mathbf{n}}-1\right)×\left|{\mathbf{incy}}\right|\right)$ 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 ${\mathbf{y}}\left(1+\left(\mathit{i}-1\right)×{\mathbf{incy}}\right)$, for $\mathit{i}=1,2,\dots ,{\mathbf{m}}$;
• if ${\mathbf{incy}}<0$, ${y}_{\mathit{i}}$ must be stored in ${\mathbf{y}}\left(1-\left({\mathbf{m}}-\mathit{i}\right)×{\mathbf{incy}}\right)$, 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 ${\mathbf{y}}\left(1+\left(\mathit{i}-1\right)×{\mathbf{incy}}\right)$, 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 $y$.
13:   $\mathbf{incy}$ – 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

f06pbf (dgbmv) is not threaded in any implementation.