void	nag_zhpr2 (Nag_OrderType order, Nag_UploType uplo, Integer n, Complex alpha, const Complex x[], Integer incx, const Complex y[], Integer incy, double beta, Complex ap[], NagError *fail)

3

Description

nag_zhpr2 (f16ssc) performs the Hermitian rank-2 update operation

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

where

A

is an

n

n

complex Hermitian matrix, stored in packed form,

x

and

y

are

n

-element complex vectors, while

α

is a complex scalar and

β

is a real scalar.

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: $uplo$ – 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: $alpha$ – ComplexInput

On entry: the scalar

α

5: $x [\dim]$ – const ComplexInput

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

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

On entry: the

n

-element vector

x

incx > 0

x_{i}

must be stored in

x [(i - 1) \times incx]

, for

i = 1, 2, \dots, n

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.

6: $incx$ – IntegerInput

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

x

Constraint:

incx \neq 0

7: $y [\dim]$ – const ComplexInput

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

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

On entry: the

n

-element vector

y

incy > 0

y_{i}

must be stored in

y [(i - 1) \times incy]

, for

i = 1, 2, \dots, n

incy < 0

y_{i}

must be stored in

y [(n - i) \times |incy|]

, for

i = 1, 2, \dots, n

Intermediate elements of y are not referenced. If

α = 0.0

n = 0

, y is not referenced and may be NULL.

8: $incy$ – IntegerInput

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

y

Constraint:

incy \neq 0

9: $beta$ – doubleInput

On entry: the scalar

β

10: $ap [\dim]$ – ComplexInput/Output

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

\max (1, n \times (n + 1) / 2)

On entry: the

n

n

Hermitian matrix

A

, packed by rows or columns.

The storage of elements

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 $ap [(j - 1) \times j / 2 + i - 1]$ , for $i \leq j$ ;
if $order = Nag_ColMajor$ and $uplo = Nag_Lower$ ,
$A_{i j}$ is stored in $ap [(2 n - j) \times (j - 1) / 2 + i - 1]$ , for $i \geq j$ ;
if $order = Nag_RowMajor$ and $uplo = Nag_Upper$ ,
$A_{i j}$ is stored in $ap [(2 n - i) \times (i - 1) / 2 + j - 1]$ , for $i \leq j$ ;
if $order = Nag_RowMajor$ and $uplo = Nag_Lower$ ,
$A_{i j}$ is stored in $ap [(i - 1) \times i / 2 + j - 1]$ , for $i \geq j$ .

On exit: the updated matrix

A

. The imaginary parts of the diagonal elements are set to zero.

11: $fail$ – 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, $n = 〈value〉$ .
Constraint: $n \geq 0$ .
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_zhpr2 (f16ssc) is not threaded in any implementation.

9

Further Comments

None.

10

Example

Perform rank-2 update of complex Hermitian matrix

A

, stored using packed storage format, using vectors

x

and

y

A \leftarrow A - x y^{H} - y x^{H},

where

A

is the

4

4

matrix given by

A = (\begin{array}{r} 23.0 + 0.0 i & 10.0 - 17.0 i & 13.0 + 14.2 i & - 19.0 + 8.0 i \\ 10.0 + 17.0 i & 1.0 + 0.0 i & 0.3 + 1.2 i & - 4.7 - 2.1 i \\ 13.0 - 14.2 i & 0.3 - 1.2 i & 1.0 + 0.0 i & - 5.9 - 0.1 i \\ - 19.0 - 8.0 i & - 4.7 + 2.1 i & - 5.9 + 0.1 i & 1.0 + 0.0 i \end{array}),

and where

x = (\begin{array}{r} 2.0 + 1.0 i \\ 2.0 + 3.0 i \\ 0.2 - 1.0 i \\ - 1.0 - 2.0 i \end{array})

and

y = (\begin{array}{r} 5.0 + 1.0 i \\ - 2.0 + 1.0 i \\ 7.0 - 1.0 i \\ - 5.0 - 2.0 i \end{array}) .

The vector

y

is stored in every second element of array y (

incy = 2

NAG Library Function Document

nag_zhpr2 (f16ssc)

▸▿ 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