NAG Library Function Document

nag_zhbmv (f16sdc)

void	nag_zhbmv (Nag_OrderType order, Nag_UploType uplo, Integer n, Integer k, Complex alpha, const Complex ab[], Integer pdab, const Complex x[], Integer incx, Complex beta, Complex y[], Integer incy, NagError *fail)

3 Description

nag_zhbmv (f16sdc) performs the matrix-vector operation

y \leftarrow α A x + β y,

where

A

is an

n

n

complex Hermitian band matrix with

k

subdiagonals and

k

superdiagonals,

x

and

y

are

n

-element complex vectors, and

α

and

β

are complex scalars.

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: uplo – Nag_UploTypeInput

On entry: specifies whether the upper or lower triangular part of

A

is stored.

$uplo = Nag_Upper$: The upper triangular part of $A$ is stored.
$uplo = Nag_Lower$: The lower triangular part of $A$ is stored.

Constraint:

uplo = Nag_Upper

Nag_Lower

3: n – IntegerInput

On entry:

n

, the order of the matrix

A

Constraint:

n \geq 0

4: k – IntegerInput

On entry:

k

, the number of subdiagonals or superdiagonals of the matrix

A

Constraint:

k \geq 0

5: alpha – ComplexInput

On entry: the scalar

α

6: ab[ $\dim$ ] – const ComplexInput

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

\max (1, pdab \times n)

On entry: the

n

n

Hermitian band matrix

A

This is stored as a notional two-dimensional array with row elements or column elements stored contiguously. The storage of elements of

A_{i j}

, depends on the order and uplo arguments as follows:

if $order = Nag_ColMajor$ and $uplo = Nag_Upper$ ,
$A_{i j}$ is stored in $ab [k + i - j + (j - 1) \times pdab]$ , for $j = 1, \dots, n$ and $i = \max (1, j - k), \dots, j$ ;
if $order = Nag_ColMajor$ and $uplo = Nag_Lower$ ,
$A_{i j}$ is stored in $ab [i - j + (j - 1) \times pdab]$ , for $j = 1, \dots, n$ and $i = j, \dots, \min (n, j + k)$ ;
if $order = Nag_RowMajor$ and $uplo = Nag_Upper$ ,
$A_{i j}$ is stored in $ab [j - i + (i - 1) \times pdab]$ , for $i = 1, \dots, n$ and $j = i, \dots, \min (n, i + k)$ ;
if $order = Nag_RowMajor$ and $uplo = Nag_Lower$ ,
$A_{i j}$ is stored in $ab [k + j - i + (i - 1) \times pdab]$ , for $i = 1, \dots, n$ and $j = \max (1, i - k), \dots, i$ .

7: 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 k + 1

8: x[ $\dim$ ] – const ComplexInput

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

\max (1, 1 + (n - 1) |incx|)

On entry: the vector

x

9: incx – IntegerInput

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

x

Constraint:

incx \neq 0

10: beta – ComplexInput

On entry: the scalar

β

11: y[ $\dim$ ] – ComplexInput/Output

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

\max (1, 1 + (n - 1) |incy|)

On entry: the vector

y

beta = 0

, y need not be set.

On exit: the updated vector

y

12: incy – IntegerInput

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

y

Constraint:

incy \neq 0

13: fail – NagError *Input/Output

The NAG error argument (see Section 3.6 in the Essential Introduction).

6 Error Indicators and Warnings

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, $k = ⟨value⟩$ .
Constraint: $k \geq 0$ .
On entry, $n = ⟨value⟩$ .
Constraint: $n \geq 0$ .
NE_INT_2: On entry, $pdab = ⟨value⟩$ , $k = ⟨value⟩$ .
Constraint: $pdab \geq k + 1$ .

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

This example computes the matrix-vector product

y = α A x + β y

where

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

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

y = (\begin{array}{r} 3.0 - 0.5 i \\ - 0.5 - 6.0 i \\ 0.5 - 8.5 i \\ 2.5 - 6.0 i \\ 14.0 - 2.0 i \end{array}),

α = 1.0 + 0.0 i and β = 2.0 + 0.0 i .

NAG Library Function Documentnag_zhbmv (f16sdc)

+− 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_zhbmv (f16sdc)