nag_zsyrk (f16zuc) (PDF version)
f16 Chapter Contents
f16 Chapter Introduction
NAG Library Manual

# NAG Library Function Documentnag_zsyrk (f16zuc)

## 1  Purpose

nag_zsyrk (f16zuc) performs a rank-$k$ update on a complex symmetric matrix.

## 2  Specification

 #include #include
 void nag_zsyrk (Nag_OrderType order, Nag_UploType uplo, Nag_TransType trans, Integer n, Integer k, Complex alpha, const Complex a[], Integer pda, Complex beta, Complex c[], Integer pdc, NagError *fail)

## 3  Description

nag_zsyrk (f16zuc) performs one of the symmetric rank-$k$ update operations
 $C←αAAT + βC or C←αATA + βC ,$
where $A$ is a complex matrix, $C$ is an $n$ by $n$ complex symmetric matrix, and $\alpha$ and $\beta$ 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:    $\mathbf{order}$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:    $\mathbf{uplo}$Nag_UploTypeInput
On entry: specifies whether the upper or lower triangular part of $C$ is stored.
${\mathbf{uplo}}=\mathrm{Nag_Upper}$
The upper triangular part of $C$ is stored.
${\mathbf{uplo}}=\mathrm{Nag_Lower}$
The lower triangular part of $C$ is stored.
Constraint: ${\mathbf{uplo}}=\mathrm{Nag_Upper}$ or $\mathrm{Nag_Lower}$.
3:    $\mathbf{trans}$Nag_TransTypeInput
On entry: specifies the operation to be performed.
${\mathbf{trans}}=\mathrm{Nag_NoTrans}$
$C←\alpha A{A}^{\mathrm{T}}+\beta C$.
${\mathbf{trans}}=\mathrm{Nag_Trans}$
$C←\alpha {A}^{\mathrm{T}}A+\beta C$.
Constraint: ${\mathbf{trans}}=\mathrm{Nag_NoTrans}$ or $\mathrm{Nag_Trans}$.
4:    $\mathbf{n}$IntegerInput
On entry: $n$, the order of the matrix $C$; the number of rows of $A$ if ${\mathbf{trans}}=\mathrm{Nag_NoTrans}$, or the number of columns of $A$ otherwise.
Constraint: ${\mathbf{n}}\ge 0$.
5:    $\mathbf{k}$IntegerInput
On entry: $k$, the number of columns of $A$ if ${\mathbf{trans}}=\mathrm{Nag_NoTrans}$, or the number of rows of $A$ otherwise.
Constraint: ${\mathbf{k}}\ge 0$.
6:    $\mathbf{alpha}$ComplexInput
On entry: the scalar $\alpha$.
7:    $\mathbf{a}\left[\mathit{dim}\right]$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 ${\mathbf{trans}}=\mathrm{Nag_Trans}$ and ${\mathbf{order}}=\mathrm{Nag_ColMajor}$;
• $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{k}}×{\mathbf{pda}}\right)$ when ${\mathbf{trans}}=\mathrm{Nag_Trans}$ and ${\mathbf{order}}=\mathrm{Nag_RowMajor}$.
If ${\mathbf{order}}=\mathrm{Nag_ColMajor}$, ${A}_{ij}$ is stored in ${\mathbf{a}}\left[\left(j-1\right)×{\mathbf{pda}}+i-1\right]$.
If ${\mathbf{order}}=\mathrm{Nag_RowMajor}$, ${A}_{ij}$ is stored in ${\mathbf{a}}\left[\left(i-1\right)×{\mathbf{pda}}+j-1\right]$.
On entry: the matrix $A$; $A$ is $n$ by $k$ if ${\mathbf{trans}}=\mathrm{Nag_NoTrans}$, or $k$ by $n$ otherwise.
8:    $\mathbf{pda}$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 ${\mathbf{trans}}=\mathrm{Nag_Trans}$, ${\mathbf{pda}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{k}}\right)$;
• 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 ${\mathbf{trans}}=\mathrm{Nag_Trans}$, ${\mathbf{pda}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
9:    $\mathbf{beta}$ComplexInput
On entry: the scalar $\beta$.
10:  $\mathbf{c}\left[\mathit{dim}\right]$ComplexInput/Output
Note: the dimension, dim, of the array c must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{pdc}}×{\mathbf{n}}\right)$.
On entry: the $n$ by $n$ symmetric matrix $C$.
If ${\mathbf{order}}=\mathrm{Nag_ColMajor}$, ${C}_{ij}$ is stored in ${\mathbf{c}}\left[\left(j-1\right)×{\mathbf{pdc}}+i-1\right]$.
If ${\mathbf{order}}=\mathrm{Nag_RowMajor}$, ${C}_{ij}$ is stored in ${\mathbf{c}}\left[\left(i-1\right)×{\mathbf{pdc}}+j-1\right]$.
If ${\mathbf{uplo}}=\mathrm{Nag_Upper}$, the upper triangular part of $C$ must be stored and the elements of the array below the diagonal are not referenced.
If ${\mathbf{uplo}}=\mathrm{Nag_Lower}$, the lower triangular part of $C$ must be stored and the elements of the array above the diagonal are not referenced.
On exit: the updated matrix $C$.
11:  $\mathbf{pdc}$IntegerInput
On entry: the stride separating row or column elements (depending on the value of order) of the matrix $C$ in the array c.
Constraint: ${\mathbf{pdc}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
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.
NE_BAD_PARAM
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 ${\mathbf{trans}}=\mathrm{Nag_NoTrans}$, ${\mathbf{pda}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{k}}\right)$.
On entry, ${\mathbf{trans}}=〈\mathit{\text{value}}〉$, ${\mathbf{k}}=〈\mathit{\text{value}}〉$, ${\mathbf{pda}}=〈\mathit{\text{value}}〉$.
Constraint: if ${\mathbf{trans}}=\mathrm{Nag_Trans}$, ${\mathbf{pda}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{k}}\right)$.
On entry, ${\mathbf{trans}}=〈\mathit{\text{value}}〉$, ${\mathbf{n}}=〈\mathit{\text{value}}〉$, ${\mathbf{pda}}=〈\mathit{\text{value}}〉$.
Constraint: if ${\mathbf{trans}}=\mathrm{Nag_NoTrans}$, ${\mathbf{pda}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
On entry, ${\mathbf{trans}}=〈\mathit{\text{value}}〉$, ${\mathbf{n}}=〈\mathit{\text{value}}〉$, ${\mathbf{pda}}=〈\mathit{\text{value}}〉$.
Constraint: if ${\mathbf{trans}}=\mathrm{Nag_Trans}$, ${\mathbf{pda}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
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_INT_2
On entry, ${\mathbf{pdc}}=〈\mathit{\text{value}}〉$, ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{pdc}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
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.
An unexpected error has been triggered by this function. Please contact NAG.
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.

## 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-$k$ update of complex symmetric $4$ by $4$ matrix $C$ using $4$ by $2$ matrix $A$ ($k=2$), $C=C-\left(1.0-1.0i\right)A{A}^{\mathrm{T}}$, where
 $C = 4.78+1.03i 2.00-0.30i 2.89-1.34i -1.89+1.15i 2.00-0.30i -4.11-2.30i 2.36-4.25i 0.04-3.69i 2.89-1.34i 2.36-4.25i 4.15+0.57i -0.02+0.46i -1.89+1.15i 0.04-3.69i -0.02+0.46i 0.33-1.91i$
and
 $A = 1.7-2.3i -1.8+2.4i 2.9-2.1i 1.2+1.4i -2.9+1.0i 0.6+0.8i 1.5+0.9i -1.4-1.7i .$

### 10.1  Program Text

Program Text (f16zuce.c)

### 10.2  Program Data

Program Data (f16zuce.d)

### 10.3  Program Results

Program Results (f16zuce.r)

nag_zsyrk (f16zuc) (PDF version)
f16 Chapter Contents
f16 Chapter Introduction
NAG Library Manual