void	f16pbc (Nag_OrderType order, Nag_TransType trans, Integer m, Integer n, Integer kl, Integer ku, double alpha, const double ab[], Integer pdab, const double x[], Integer incx, double beta, double y[], Integer incy, NagError *fail)

The function may be called by the names: f16pbc, nag_blast_dgbmv or nag_dgbmv.

3 Description

f16pbc performs one of the matrix-vector operations

y \leftarrow α A x + β y,   or y \leftarrow α A^{T} x + β y,

where

A

is an

m \times n

real band matrix with

k_{l}

subdiagonals and

k_{u}

superdiagonals,

x

and

y

are real vectors, and

α

and

β

are real scalars.

m = 0

n = 0

, no operation is performed.

4 References

Basic Linear Algebra Subprograms Technical (BLAST) Forum (2001) Basic Linear Algebra Subprograms Technical (BLAST) Forum Standard University of Tennessee, Knoxville, Tennessee https://www.netlib.org/blas/blast-forum/blas-report.pdf

5 Arguments

1: $order$ – Nag_OrderType Input

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

order = Nag_RowMajor

. See Section 3.1.3 in the Introduction to the NAG Library CL Interface for a more detailed explanation of the use of this argument.

Constraint:

order = Nag_RowMajor

Nag_ColMajor

2: $trans$ – Nag_TransType Input

On entry: specifies the operation to be performed.

$trans = Nag_NoTrans$: $y \leftarrow α A x + β y$ .
$trans = Nag_Trans$ or $Nag_ConjTrans$: $y \leftarrow α A^{T} x + β y$ .

Constraint:

trans = Nag_NoTrans

Nag_Trans

Nag_ConjTrans

3: $m$ – Integer Input

On entry:

m

, the number of rows of the matrix

A

Constraint:

m \geq 0

4: $n$ – Integer Input

On entry:

n

, the number of columns of the matrix

A

Constraint:

n \geq 0

5: $kl$ – Integer Input

On entry:

k_{l}

, the number of subdiagonals within the band of

A

Constraint:

kl \geq 0

6: $ku$ – Integer Input

On entry:

k_{u}

, the number of superdiagonals within the band of

A

Constraint:

ku \geq 0

7: $alpha$ – double Input

On entry: the scalar

α

8: $ab [\dim]$ – const double Input

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

$\max (1, pdab \times n)$ when $order = Nag_ColMajor$ ;
$\max (1, m \times pdab)$ when $order = Nag_RowMajor$ .

On entry: the

m \times 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_{i j}

, for row

i = 1, \dots, m

and column

j = \max (1, i - k_{l}), \dots, \min (n, i + k_{u})

, depends on the order argument as follows:

if $order = Nag_ColMajor$ , $A_{i j}$ is stored as $ab [(j - 1) \times pdab + ku + i - j]$ ;
if $order = Nag_RowMajor$ , $A_{i j}$ is stored as $ab [(i - 1) \times pdab + kl + j - i]$ .

9: $pdab$ – Integer Input

On entry: the stride separating row or column elements (depending on the value of order) of the matrix

A

in the array ab.

Constraint:

pdab \geq kl + ku + 1

10: $x [\dim]$ – const double Input

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

$\max (1, 1 + (n - 1) | incx |)$ when $trans = Nag_NoTrans$ ;
$\max (1, 1 + (m - 1) | incx |)$ when $trans = Nag_Trans$ or $Nag_ConjTrans$ .

On entry: the vector

x

trans = Nag_NoTrans

, then

x

is an

n

-element vector.

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

Otherwise,

x

is an

m

-element vector.

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

11: $incx$ – Integer Input

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

x

Constraint:

incx \neq 0

12: $beta$ – double Input

On entry: the scalar

β

13: $y [\dim]$ – double Input/Output

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

$\max (1, 1 + (m - 1) | incy |)$ when $trans = Nag_NoTrans$ ;
$\max (1, 1 + (n - 1) | incy |)$ when $trans = Nag_Trans$ or $Nag_ConjTrans$ .

On entry: the vector

y

. See x for details of storage.

beta = 0

, y need not be set.

On exit: the updated vector

y

14: $incy$ – Integer Input

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

y

Constraint:

incy \neq 0

15: $fail$ – NagError * Input/Output

The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

6 Error Indicators and Warnings

NE_ALLOC_FAIL: Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
NE_BAD_PARAM: On entry, argument $⟨ value ⟩$ had an illegal value.
NE_INT: On entry, $incx = ⟨ value ⟩$ .
Constraint: $incx \neq 0$ .

On entry, $incy = ⟨ value ⟩$ .
Constraint: $incy \neq 0$ .

On entry, $kl = ⟨ value ⟩$ .
Constraint: $kl \geq 0$ .

On entry, $ku = ⟨ value ⟩$ .
Constraint: $ku \geq 0$ .

On entry, $m = ⟨ value ⟩$ .
Constraint: $m \geq 0$ .

On entry, $n = ⟨ value ⟩$ .
Constraint: $n \geq 0$ .
NE_INT_3: On entry, $pdab = ⟨ value ⟩$ , $kl = ⟨ value ⟩$ , $ku = ⟨ value ⟩$ .
Constraint: $pdab \geq kl + ku + 1$ .
NE_NO_LICENCE: Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.

7 Accuracy

The BLAS standard requires accurate implementations which avoid unnecessary over/underflow (see Section 2.7 of Basic Linear Algebra Subprograms Technical (BLAST) Forum (2001)).

8 Parallelism and Performance

Background information to multithreading can be found in the Multithreading documentation.

f16pbc is not threaded in any implementation.

9 Further Comments

None.

10 Example

A vector

y

, of length

6

, is updated using

y \leftarrow 2 y + A x

, where

A

is a

6 \times 4

banded matrix with two subdiagonals and one superdiagonal, and

x

is a vector of length

4

f16pb: FL CL CPP AD PY MB

NAG CL Interfacef16pbc (dgbmv)

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

NAG CL Interface
f16pbc (dgbmv)