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

3

Description

nag_zgbmv (f16sbc) performs one of the matrix-vector operations

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

where

A

is an

m

n

complex band matrix with

k_{l}

subdiagonals and

k_{u}

superdiagonals,

x

and

y

are complex vectors, and

α

and

β

are complex 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 3.3.1.3 in How to Use the NAG Library and its Documentation 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.

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

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: $alpha$ – ComplexInput

On entry: the scalar

α

8: $ab [\dim]$ – const ComplexInput

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 ComplexInput

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: $incx$ – IntegerInput

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

x

Constraint:

incx \neq 0

12: $beta$ – ComplexInput

On entry: the scalar

β

13: $y [\dim]$ – ComplexInput/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: $incy$ – 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 3.7 in How to Use the NAG Library and its Documentation).

6

Error Indicators and Warnings

NE_ALLOC_FAIL: Dynamic memory allocation failed.
See Section 2.3.1.2 in How to Use the NAG Library and its Documentation 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 2.7.5 in How to Use the NAG Library and its Documentation 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

nag_zgbmv (f16sbc) is not threaded in any implementation.

9

Further Comments

None.

10

Example

This example computes the matrix-vector product

y = α A x + β y

where

A = (\begin{array}{r} 1.0 + 1.0 i & 1.0 + 2.0 i & 0.0 + 0.0 i & 0.0 + 0.0 i \\ 2.0 + 1.0 i & 2.0 + 2.0 i & 2.0 + 3.0 i & 0.0 + 0.0 i \\ 3.0 + 1.0 i & 3.0 + 2.0 i & 3.0 + 3.0 i & 3.0 + 4.0 i \\ 0.0 + 0.0 i & 4.0 + 2.0 i & 4.0 + 3.0 i & 4.0 + 4.0 i \\ 0.0 + 0.0 i & 0.0 + 0.0 i & 5.0 + 3.0 i & 5.0 + 4.0 i \\ 0.0 + 0.0 i & 0.0 + 0.0 i & 0.0 + 0.0 i & 6.0 + 4.0 i \end{array}),

x = (\begin{array}{r} 1.0 - 1.0 i \\ 2.0 - 2.0 i \\ 3.0 - 3.0 i \\ 4.0 - 4.0 i \end{array}),

y = (\begin{array}{r} - 3.5 + 0.0 i \\ - 11.5 + 1.0 i \\ - 27.5 + 3.0 i \\ - 29.0 + 7.5 i \\ - 25.5 + 10.0 i \\ - 14.5 + 10.0 i \end{array}),

α = 1.0 + 0.0 i and β = 2.0 + 0.0 i .

NAG Library Function Document

nag_zgbmv (f16sbc)

▸▿ 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

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