NAG Library Function Document

nag_dgbmv (f16pbc)

void	nag_dgbmv (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)

3 Description

nag_dgbmv (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

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 http://www.netlib.org/blas/blast-forum/blas-report.pdf

5 Arguments

1: 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

order = Nag_RowMajor

. See Section 3.2.1.3 in the Essential Introduction for a more detailed explanation of the use of this argument.

Constraint:

order = Nag_RowMajor

Nag_ColMajor

2: trans – Nag_TransTypeInput

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 – IntegerInput

On entry:

m

, the number of rows of the matrix

A

Constraint:

m \geq 0

4: n – IntegerInput

On entry:

n

, the number of columns of the matrix

A

Constraint:

n \geq 0

5: kl – IntegerInput

On entry:

k_{l}

, the number of subdiagonals within the band of

A

Constraint:

kl \geq 0

6: ku – IntegerInput

On entry:

k_{u}

, the number of superdiagonals within the band of

A

Constraint:

ku \geq 0

7: alpha – doubleInput

On entry: the scalar

α

8: ab[ $\dim$ ] – const doubleInput

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

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 – 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:

pdab \geq kl + ku + 1

10: x[ $\dim$ ] – const doubleInput

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 incremented array x must contain the vector

x

11: incx – IntegerInput

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

x

Constraint:

incx \neq 0

12: beta – doubleInput

On entry: the scalar

β

13: y[ $\dim$ ] – doubleInput/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 incremented array y must contain the vector

x

beta = 0

, y need not be set.

On exit: the updated vector

y

14: incy – IntegerInput

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 3.6 in the Essential Introduction).

6 Error Indicators and Warnings

NE_ALLOC_FAIL: Dynamic memory allocation failed.
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: $pdab \geq kl + ku + 1$ .

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

Not applicable.

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

4

banded matrix with two subdiagonals and one superdiagonal, and

x

is a vector of length

4

NAG Library Function Documentnag_dgbmv (f16pbc)

+− 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 Library Function Document

nag_dgbmv (f16pbc)