f16 Chapter Contents
f16 Chapter Introduction
NAG Library Manual

# NAG Library Function Documentnag_zhfrk (f16zqc)

## 1  Purpose

nag_zhfrk (f16zqc) performs one of the Hermitian rank-$k$ update operations
 $C←αAAH + βC or C←αAHA + βC ,$
where $A$ is a complex matrix, $C$ is an $n$ by $n$ complex Hermitian matrix stored in Rectangular Full Packed (RFP) format, and $\alpha$ and $\beta$ are real scalars.

## 2  Specification

 #include #include
 void nag_zhfrk (Nag_OrderType order, Nag_RFP_Store transr, Nag_UploType uplo, Nag_TransType trans, Integer n, Integer k, double alpha, const Complex a[], Integer pda, double beta, Complex cr[], NagError *fail)

## 3  Description

nag_zhfrk (f16zqc) performs one of the Hermitian rank-$k$ update operations
 $C←αAAH + βC or C←αAHA + βC ,$
where $A$ is a complex matrix, $C$ is an $n$ by $n$ complex Hermitian matrix stored in Rectangular Full Packed (RFP) format, and $\alpha$ and $\beta$ are real scalars. The RFP storage format is described in Section 3.3.3 in the f07 Chapter Introduction.
If $n=0$ or if $\beta =1.0$ and either $k=0$ or $\alpha =0.0$ then nag_zhfrk (f16zqc) returns immediately. If $\beta =0.0$ and either $k=0$ or $\alpha =0.0$ then $C$ is set to the zero matrix.

## 4  References

Gustavson F G, Waśniewski J, Dongarra J J and Langou J (2010) Rectangular full packed format for Cholesky's algorithm: factorization, solution, and inversion ACM Trans. Math. Software 37, 2

## 5  Arguments

1:    $\mathbf{order}$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 ${\mathbf{order}}=\mathrm{Nag_RowMajor}$. 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:    $\mathbf{transr}$Nag_RFP_StoreInput
On entry: specifies whether the normal RFP representation of $C$ or its conjugate transpose is stored.
${\mathbf{transr}}=\mathrm{Nag_RFP_Normal}$
The matrix $C$ is stored in normal RFP format.
${\mathbf{transr}}=\mathrm{Nag_RFP_ConjTrans}$
The conjugate transpose of the RFP representation of the matrix $C$ is stored.
Constraint: ${\mathbf{transr}}=\mathrm{Nag_RFP_Normal}$ or $\mathrm{Nag_RFP_ConjTrans}$.
3:    $\mathbf{uplo}$Nag_UploTypeInput
On entry: specifies whether the upper or lower triangular part of $C$ is stored in RFP format.
${\mathbf{uplo}}=\mathrm{Nag_Upper}$
The upper triangular part of $C$ is stored in RFP format.
${\mathbf{uplo}}=\mathrm{Nag_Lower}$
The lower triangular part of $C$ is stored in RFP format.
Constraint: ${\mathbf{uplo}}=\mathrm{Nag_Upper}$ or $\mathrm{Nag_Lower}$.
4:    $\mathbf{trans}$Nag_TransTypeInput
On entry: specifies the operation to be performed.
${\mathbf{trans}}=\mathrm{Nag_NoTrans}$
$C←\alpha A{A}^{\mathrm{H}}+\beta C$.
${\mathbf{trans}}=\mathrm{Nag_ConjTrans}$
$C←\alpha {A}^{\mathrm{H}}A+\beta C$.
Constraint: ${\mathbf{trans}}=\mathrm{Nag_NoTrans}$ or $\mathrm{Nag_ConjTrans}$.
5:    $\mathbf{n}$IntegerInput
On entry: $n$, the order of the matrix $C$.
Constraint: ${\mathbf{n}}\ge 0$.
6:    $\mathbf{k}$IntegerInput
On entry: $k$, the number of columns of $A$ if ${\mathbf{trans}}=\mathrm{Nag_NoTrans}$, or the number of rows of $A$ if ${\mathbf{trans}}=\mathrm{Nag_ConjTrans}$.
Constraint: ${\mathbf{k}}\ge 0$.
7:    $\mathbf{alpha}$doubleInput
On entry: the scalar $\alpha$.
8:    $\mathbf{a}\left[\mathit{dim}\right]$const ComplexInput
Note: the dimension, dim, of the array a must be at least
• $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{pda}}×{\mathbf{k}}\right)$ when ${\mathbf{trans}}=\mathrm{Nag_NoTrans}$ and ${\mathbf{order}}=\mathrm{Nag_ColMajor}$;
• $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}×{\mathbf{pda}}\right)$ when ${\mathbf{trans}}=\mathrm{Nag_NoTrans}$ and ${\mathbf{order}}=\mathrm{Nag_RowMajor}$;
• $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{pda}}×{\mathbf{n}}\right)$ when ${\mathbf{trans}}=\mathrm{Nag_ConjTrans}$ and ${\mathbf{order}}=\mathrm{Nag_ColMajor}$;
• $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{k}}×{\mathbf{pda}}\right)$ when ${\mathbf{trans}}=\mathrm{Nag_ConjTrans}$ and ${\mathbf{order}}=\mathrm{Nag_RowMajor}$.
On entry: the matrix $A$; $A$ is $n$ by $k$ if ${\mathbf{trans}}=\mathrm{Nag_NoTrans}$, or $k$ by $n$ if ${\mathbf{trans}}=\mathrm{Nag_ConjTrans}$. If ${\mathbf{alpha}}=0.0$, a is not referenced.
9:    $\mathbf{pda}$IntegerInput
On entry: the stride separating row or column elements (depending on the value of order) in the array a.
Constraints:
• if ${\mathbf{order}}=\mathrm{Nag_ColMajor}$,
• if ${\mathbf{trans}}=\mathrm{Nag_NoTrans}$, ${\mathbf{pda}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$;
• if ${\mathbf{trans}}=\mathrm{Nag_ConjTrans}$, ${\mathbf{pda}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{k}}\right)$;
• if ${\mathbf{order}}=\mathrm{Nag_RowMajor}$,
• if ${\mathbf{trans}}=\mathrm{Nag_NoTrans}$, ${\mathbf{pda}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{k}}\right)$;
• if ${\mathbf{trans}}=\mathrm{Nag_ConjTrans}$, ${\mathbf{pda}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
10:  $\mathbf{beta}$doubleInput
On entry: the scalar $\beta$.
11:  $\mathbf{cr}\left[{\mathbf{n}}×\left({\mathbf{n}}+1\right)/2\right]$ComplexInput/Output
On entry: the upper or lower triangular part (as specified by uplo) of the $n$ by $n$ Hermitian matrix $C$, stored in RFP format (as specified by transr). The storage format is described in detail in Section 3.3.3 in the f07 Chapter Introduction. If $\beta =0.0$, cr need not be set on entry.
On exit: the updated matrix $C$, that is its upper or lower triangular part stored in RFP format.
12:  $\mathbf{fail}$NagError *Input/Output
The NAG error argument (see Section 3.6 in the Essential Introduction).

## 6  Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 3.2.1.2 in the Essential Introduction for further information.
On entry, argument $〈\mathit{\text{value}}〉$ had an illegal value.
NE_ENUM_INT_2
On entry, ${\mathbf{trans}}=〈\mathit{\text{value}}〉$, ${\mathbf{k}}=〈\mathit{\text{value}}〉$, ${\mathbf{pda}}=〈\mathit{\text{value}}〉$.
Constraint: if ${\mathbf{trans}}=\mathrm{Nag_ConjTrans}$, ${\mathbf{pda}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{k}}\right)$.
On entry, ${\mathbf{trans}}=〈\mathit{\text{value}}〉$, ${\mathbf{k}}=〈\mathit{\text{value}}〉$, ${\mathbf{pda}}=〈\mathit{\text{value}}〉$.
Constraint: if ${\mathbf{trans}}=\mathrm{Nag_NoTrans}$, ${\mathbf{pda}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{k}}\right)$.
On entry, ${\mathbf{trans}}=〈\mathit{\text{value}}〉$, ${\mathbf{n}}=〈\mathit{\text{value}}〉$, ${\mathbf{pda}}=〈\mathit{\text{value}}〉$.
Constraint: if ${\mathbf{trans}}=\mathrm{Nag_ConjTrans}$, ${\mathbf{pda}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
On entry, ${\mathbf{trans}}=〈\mathit{\text{value}}〉$, ${\mathbf{n}}=〈\mathit{\text{value}}〉$, ${\mathbf{pda}}=〈\mathit{\text{value}}〉$.
Constraint: if ${\mathbf{trans}}=\mathrm{Nag_NoTrans}$, ${\mathbf{pda}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
NE_INT
On entry, ${\mathbf{k}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{k}}\ge 0$.
On entry, ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{n}}\ge 0$.
NE_INTERNAL_ERROR
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
See Section 3.6.6 in the Essential Introduction for further information.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 3.6.5 in the Essential Introduction for further information.

Not applicable.

## 8  Parallelism and Performance

nag_zhfrk (f16zqc) is not threaded by NAG in any implementation.
nag_zhfrk (f16zqc) makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.
Please consult the X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this function. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

None.

## 10  Example

This example reads in the lower triangular part of a symmetric matrix $C$ which it converts to RFP format. It also reads in $\alpha$, $\beta$ and a $4$ by $3$ matrix $A$ and then performs the Hermitian rank-$3$ update $C←\alpha A{A}^{\mathrm{H}}+\beta C$.

### 10.1  Program Text

Program Text (f16zqce.c)

### 10.2  Program Data

Program Data (f16zqce.d)

### 10.3  Program Results

Program Results (f16zqce.r)