f16 Chapter Contents
f16 Chapter Introduction
NAG Library Manual

# NAG Library Function Documentnag_zhpr2 (f16ssc)

## 1  Purpose

nag_zhpr2 (f16ssc) performs a Hermitian rank-2 update on a complex Hermitian matrix stored in packed form.

## 2  Specification

 #include #include
 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
 $A ← α x yH + α- y xH + β A ,$
where $A$ is an $n$ by $n$ complex Hermitian matrix, stored in packed form, $x$ and $y$ are $n$-element complex vectors, while $\alpha$ is a complex scalar and $\beta$ 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:     orderNag_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 ${\mathbf{order}}=\mathrm{Nag_RowMajor}$. See Section 3.2.1.3 in the Essential Introduction for a more detailed explanation of the use of this argument.
Constraint: ${\mathbf{order}}=\mathrm{Nag_RowMajor}$ or $\mathrm{Nag_ColMajor}$.
2:     uploNag_UploTypeInput
On entry: specifies whether the upper or lower triangular part of $A$ is stored.
${\mathbf{uplo}}=\mathrm{Nag_Upper}$
The upper triangular part of $A$ is stored.
${\mathbf{uplo}}=\mathrm{Nag_Lower}$
The lower triangular part of $A$ is stored.
Constraint: ${\mathbf{uplo}}=\mathrm{Nag_Upper}$ or $\mathrm{Nag_Lower}$.
3:     nIntegerInput
On entry: $n$, the order of the matrix $A$.
Constraint: ${\mathbf{n}}\ge 0$.
4:     alphaComplexInput
On entry: the scalar $\alpha$.
5:     x[$\mathit{dim}$]const ComplexInput
Note: the dimension, dim, of the array x must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,1+\left({\mathbf{n}}-1\right)\left|{\mathbf{incx}}\right|\right)$.
On entry: the vector $x$.
6:     incxIntegerInput
On entry: the increment in the subscripts of x between successive elements of $x$.
Constraint: ${\mathbf{incx}}\ne 0$.
7:     y[$\mathit{dim}$]const ComplexInput
Note: the dimension, dim, of the array y must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,1+\left({\mathbf{n}}-1\right)\left|{\mathbf{incy}}\right|\right)$.
On entry: the vector $y$.
8:     incyIntegerInput
On entry: the increment in the subscripts of y between successive elements of $y$.
Constraint: ${\mathbf{incy}}\ne 0$.
On entry: the scalar $\beta$.
10:   ap[$\mathit{dim}$]ComplexInput/Output
Note: the dimension, dim, of the array ap must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}×\left({\mathbf{n}}+1\right)/2\right)$.
On entry: the $n$ by $n$ Hermitian matrix $A$, packed by rows or columns.
The storage of elements ${A}_{ij}$ depends on the order and uplo arguments as follows:
• if ${\mathbf{order}}=\mathrm{Nag_ColMajor}$ and ${\mathbf{uplo}}=\mathrm{Nag_Upper}$,
${A}_{ij}$ is stored in ${\mathbf{ap}}\left[\left(j-1\right)×j/2+i-1\right]$, for $i\le j$;
• if ${\mathbf{order}}=\mathrm{Nag_ColMajor}$ and ${\mathbf{uplo}}=\mathrm{Nag_Lower}$,
${A}_{ij}$ is stored in ${\mathbf{ap}}\left[\left(2n-j\right)×\left(j-1\right)/2+i-1\right]$, for $i\ge j$;
• if ${\mathbf{order}}=\mathrm{Nag_RowMajor}$ and ${\mathbf{uplo}}=\mathrm{Nag_Upper}$,
${A}_{ij}$ is stored in ${\mathbf{ap}}\left[\left(2n-i\right)×\left(i-1\right)/2+j-1\right]$, for $i\le j$;
• if ${\mathbf{order}}=\mathrm{Nag_RowMajor}$ and ${\mathbf{uplo}}=\mathrm{Nag_Lower}$,
${A}_{ij}$ is stored in ${\mathbf{ap}}\left[\left(i-1\right)×i/2+j-1\right]$, for $i\ge j$.
On exit: the updated matrix $A$. The imaginary parts of the diagonal elements are set to zero.
11:   failNagError *Input/Output
The NAG error argument (see Section 3.6 in the Essential Introduction).

## 6  Error Indicators and Warnings

On entry, argument $⟨\mathit{\text{value}}⟩$ had an illegal value.
NE_INT
On entry, ${\mathbf{incx}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{incx}}\ne 0$.
On entry, ${\mathbf{incy}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{incy}}\ne 0$.
On entry, ${\mathbf{n}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{n}}\ge 0$.

## 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)).

Not applicable.

None.

## 10  Example

Perform rank-2 update of complex Hermitian matrix $A$, stored using packed storage format, using vectors $x$ and $y$:
 $A ← A - x yH - y xH ,$
where $A$ is the $4$ by $4$ matrix given by
 $A = 23.0+00.0i 10.0-17.0i 13.0+14.2i -19.0+8.0i 10.0+17.0i 1.0+00.0i 0.3+01.2i -4.7-2.1i 13.0-14.2i 0.3-01.2i 1.0+00.0i -5.9-0.1i -19.0-08.0i -4.7+02.1i -5.9+00.1i 1.0+0.0i ,$
and where
 $x = 2.0+1.0i 2.0+3.0i 0.2-1.0i -1.0-2.0i$
and
 $y = 5.0+1.0i -2.0+1.0i 7.0-1.0i -5.0-2.0i .$
The vector $y$ is stored in every second element of array y (${\mathbf{incy}}=2$).

### 10.1  Program Text

Program Text (f16ssce.c)

### 10.2  Program Data

Program Data (f16ssce.d)

### 10.3  Program Results

Program Results (f16ssce.r)