# NAG Library Routine Document

## 1Purpose

f06sqf (zhpr) computes the rank-1 update of a complex Hermitian matrix stored in packed form.

## 2Specification

Fortran Interface
 Subroutine f06sqf ( uplo, n, x, incx, ap)
 Integer, Intent (In) :: n, incx Real (Kind=nag_wp), Intent (In) :: alpha Complex (Kind=nag_wp), Intent (In) :: x(*) Complex (Kind=nag_wp), Intent (Inout) :: ap(*) Character (1), Intent (In) :: uplo
#include nagmk26.h
 void f06sqf_ (const char *uplo, const Integer *n, const double *alpha, const Complex x[], const Integer *incx, Complex ap[], const Charlen length_uplo)
The routine may be called by its BLAS name zhpr.

## 3Description

f06sqf (zhpr) performs the Hermitian rank-1 update operation
 $A←αxxH + A ,$
where $A$ is an $n$ by $n$ complex Hermitian matrix, stored in packed form, $x$ is an $n$-element complex vector, and $\alpha$ is a real scalar.
None.

## 5Arguments

1:     $\mathbf{uplo}$ – Character(1)Input
On entry: specifies whether the upper or lower triangular part of $A$ is stored.
${\mathbf{uplo}}=\text{'U'}$
The upper triangular part of $A$ is stored.
${\mathbf{uplo}}=\text{'L'}$
The lower triangular part of $A$ is stored.
Constraint: ${\mathbf{uplo}}=\text{'U'}$ or $\text{'L'}$.
2:     $\mathbf{n}$ – IntegerInput
On entry: $n$, the order of the matrix $A$.
Constraint: ${\mathbf{n}}\ge 0$.
3:     $\mathbf{alpha}$ – Real (Kind=nag_wp)Input
On entry: the scalar $\alpha$.
4:     $\mathbf{x}\left(*\right)$ – Complex (Kind=nag_wp) arrayInput
Note: the dimension 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 $n$-element vector $x$.
If ${\mathbf{incx}}>0$, ${x}_{\mathit{i}}$ must be stored in ${\mathbf{x}}\left(1+\left(\mathit{i}-1\right)×{\mathbf{incx}}\right)$, for $\mathit{i}=1,2,\dots ,{\mathbf{n}}$.
If ${\mathbf{incx}}<0$, ${x}_{\mathit{i}}$ must be stored in ${\mathbf{x}}\left(1-\left({\mathbf{n}}-\mathit{i}\right)×{\mathbf{incx}}\right)$, for $\mathit{i}=1,2,\dots ,{\mathbf{n}}$.
Intermediate elements of x are not referenced.
5:     $\mathbf{incx}$ – IntegerInput
On entry: the increment in the subscripts of x between successive elements of $x$.
Constraint: ${\mathbf{incx}}\ne 0$.
6:     $\mathbf{ap}\left(*\right)$ – Complex (Kind=nag_wp) arrayInput/Output
Note: the dimension of the array ap must be at least ${\mathbf{n}}×\left({\mathbf{n}}+1\right)/2$.
On entry: the $n$ by $n$ Hermitian matrix $A$, packed by columns.
More precisely,
• if ${\mathbf{uplo}}=\text{'U'}$, the upper triangle of $A$ must be stored with element ${A}_{ij}$ in ${\mathbf{ap}}\left(i+j\left(j-1\right)/2\right)$ for $i\le j$;
• if ${\mathbf{uplo}}=\text{'L'}$, the lower triangle of $A$ must be stored with element ${A}_{ij}$ in ${\mathbf{ap}}\left(i+\left(2n-j\right)\left(j-1\right)/2\right)$ for $i\ge j$.
On exit: the updated matrix $A$. The imaginary parts of the diagonal elements are set to zero.

None.

Not applicable.

## 8Parallelism and Performance

f06sqf (zhpr) is not threaded in any implementation.