f16 Chapter Contents
f16 Chapter Introduction
NAG C Library Manual

# NAG Library Function Documentnag_zsyr2k (f16zwc)

## 1  Purpose

nag_zsyr2k (f16zwc) performs a rank-$2k$ update on a complex symmetric matrix.

## 2  Specification

 #include #include
 void nag_zsyr2k (Nag_OrderType order, Nag_UploType uplo, Nag_TransType trans, Integer n, Integer k, Complex alpha, const Complex a[], Integer pda, const Complex b[], Integer pdb, Complex beta, Complex c[], Integer pdc, NagError *fail)

## 3  Description

nag_zsyr2k (f16zwc) performs one of the symmetric rank-$2k$ update operations
 $C←αABT + αBAT + βC or C←αATB + αBTA + βC ,$
where $A$ and $B$ are complex matrices, $C$ is an $n$ by $n$ complex symmetric matrix, and $\alpha$ and $\beta$ are complex scalars.

## 4  References

The BLAS Technical Forum Standard (2001) http://www.netlib.org/blas/blast-forum

## 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 Nag_ColMajor.
2:     uploNag_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:     transNag_TransTypeInput
On entry: specifies the operation to be performed.
${\mathbf{trans}}=\mathrm{Nag_NoTrans}$
$C←\alpha A{B}^{\mathrm{T}}+\alpha B{A}^{\mathrm{T}}+\beta C$.
${\mathbf{trans}}=\mathrm{Nag_Trans}$
$C←\alpha {A}^{\mathrm{T}}B+\alpha {B}^{\mathrm{T}}A+\beta C$.
Constraint: ${\mathbf{trans}}=\mathrm{Nag_NoTrans}$ or $\mathrm{Nag_Trans}$.
4:     nIntegerInput
On entry: $n$, the order of the matrix $C$; the number of rows of $A$ and $B$ if ${\mathbf{trans}}=\mathrm{Nag_NoTrans}$, or the number of columns of $A$ and $B$ otherwise.
Constraint: ${\mathbf{n}}\ge 0$.
5:     kIntegerInput
On entry: $k$, the number of columns of $A$ and $B$ if ${\mathbf{trans}}=\mathrm{Nag_NoTrans}$, or the number of rows of $A$ and $B$ otherwise.
Constraint: ${\mathbf{k}}\ge 0$.
6:     alphaComplexInput
On entry: the scalar $\alpha$.
7:     a[$\mathit{dim}$]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 ${\mathbf{trans}}=\mathrm{Nag_Trans}$ or $\mathrm{Nag_ConjTrans}$ and ${\mathbf{order}}=\mathrm{Nag_ColMajor}$;
• $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{k}}×{\mathbf{pda}}\right)$ when ${\mathbf{trans}}=\mathrm{Nag_Trans}$ or $\mathrm{Nag_ConjTrans}$ 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:     pdaIntegerInput
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 ${\mathbf{trans}}=\mathrm{Nag_Trans}$ or $\mathrm{Nag_ConjTrans}$, ${\mathbf{pda}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{k}}\right)$;
• 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 ${\mathbf{trans}}=\mathrm{Nag_Trans}$ or $\mathrm{Nag_ConjTrans}$, ${\mathbf{pda}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
9:     b[$\mathit{dim}$]const ComplexInput
Note: the dimension, dim, of the array b must be at least
• $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{pdb}}×{\mathbf{k}}\right)$ when ${\mathbf{trans}}=\mathrm{Nag_NoTrans}$ and ${\mathbf{order}}=\mathrm{Nag_ColMajor}$;
• $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}×{\mathbf{pdb}}\right)$ when ${\mathbf{trans}}=\mathrm{Nag_NoTrans}$ and ${\mathbf{order}}=\mathrm{Nag_RowMajor}$;
• $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{pdb}}×{\mathbf{n}}\right)$ when ${\mathbf{trans}}=\mathrm{Nag_Trans}$ or $\mathrm{Nag_ConjTrans}$ and ${\mathbf{order}}=\mathrm{Nag_ColMajor}$;
• $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{k}}×{\mathbf{pdb}}\right)$ when ${\mathbf{trans}}=\mathrm{Nag_Trans}$ or $\mathrm{Nag_ConjTrans}$ and ${\mathbf{order}}=\mathrm{Nag_RowMajor}$.
If ${\mathbf{order}}=\mathrm{Nag_ColMajor}$, ${B}_{ij}$ is stored in ${\mathbf{b}}\left[\left(j-1\right)×{\mathbf{pdb}}+i-1\right]$.
If ${\mathbf{order}}=\mathrm{Nag_RowMajor}$, ${B}_{ij}$ is stored in ${\mathbf{b}}\left[\left(i-1\right)×{\mathbf{pdb}}+j-1\right]$.
On entry: the matrix $B$; $B$ is $n$ by $k$ if ${\mathbf{trans}}=\mathrm{Nag_NoTrans}$, or $k$ by $n$ otherwise.
10:   pdbIntegerInput
On entry: the stride separating row or column elements (depending on the value of order) in the array b.
Constraints:
• if ${\mathbf{order}}=\mathrm{Nag_ColMajor}$,
• if ${\mathbf{trans}}=\mathrm{Nag_NoTrans}$, ${\mathbf{pdb}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$;
• if ${\mathbf{trans}}=\mathrm{Nag_Trans}$ or $\mathrm{Nag_ConjTrans}$, ${\mathbf{pdb}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{k}}\right)$;
• if ${\mathbf{order}}=\mathrm{Nag_RowMajor}$,
• if ${\mathbf{trans}}=\mathrm{Nag_NoTrans}$, ${\mathbf{pdb}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{k}}\right)$;
• if ${\mathbf{trans}}=\mathrm{Nag_Trans}$ or $\mathrm{Nag_ConjTrans}$, ${\mathbf{pdb}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
11:   betaComplexInput
On entry: the scalar $\beta$.
12:   c[$\mathit{dim}$]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 $C$.
13:   pdcIntegerInput
On entry: the stride separating row or column elements (depending on the value of order) of the matrix $C$ in the array c.
Constraint: ${\mathbf{pdc}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
14:   failNagError *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.
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 ${\mathbf{trans}}=\mathrm{Nag_Trans}$ or $\mathrm{Nag_ConjTrans}$, ${\mathbf{pda}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{k}}\right)$.
On entry, ${\mathbf{trans}}=〈\mathit{\text{value}}〉$, ${\mathbf{k}}=〈\mathit{\text{value}}〉$, ${\mathbf{pdb}}=〈\mathit{\text{value}}〉$.
Constraint: if ${\mathbf{trans}}=\mathrm{Nag_NoTrans}$, ${\mathbf{pdb}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{k}}\right)$.
On entry, ${\mathbf{trans}}=〈\mathit{\text{value}}〉$, ${\mathbf{k}}=〈\mathit{\text{value}}〉$, ${\mathbf{pdb}}=〈\mathit{\text{value}}〉$.
Constraint: if ${\mathbf{trans}}=\mathrm{Nag_Trans}$ or $\mathrm{Nag_ConjTrans}$, ${\mathbf{pdb}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{k}}\right)$.
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 ${\mathbf{trans}}=\mathrm{Nag_Trans}$ or $\mathrm{Nag_ConjTrans}$, ${\mathbf{pda}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
On entry, ${\mathbf{trans}}=〈\mathit{\text{value}}〉$, ${\mathbf{n}}=〈\mathit{\text{value}}〉$, ${\mathbf{pdb}}=〈\mathit{\text{value}}〉$.
Constraint: if ${\mathbf{trans}}=\mathrm{Nag_NoTrans}$, ${\mathbf{pdb}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
On entry, ${\mathbf{trans}}=〈\mathit{\text{value}}〉$, ${\mathbf{n}}=〈\mathit{\text{value}}〉$, ${\mathbf{pdb}}=〈\mathit{\text{value}}〉$.
Constraint: if ${\mathbf{trans}}=\mathrm{Nag_Trans}$ or $\mathrm{Nag_ConjTrans}$, ${\mathbf{pdb}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
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.

## 7  Accuracy

The BLAS standard requires accurate implementations which avoid unnecessary over/underflow (see Section 2.7 of The BLAS Technical Forum Standard (2001)).

None.

## 9  Example

Perform rank-$2k$ update of complex symmetric $4$ by $4$ matrix $C$ using $4$ by $2$ matrices $A$ and $B$, $C=C+\left(-0.5+0.5i\right)A{B}^{\mathrm{T}}+\left(-0.5-0.5i\right)B{A}^{\mathrm{T}}$, where
 $C = 4.78+0.00i 2.00-0.30i 2.89-1.34i -1.89+1.15i 2.00-0.30i -4.11+0.00i 2.36-4.25i 0.04-3.69i 2.89-1.34i 2.36-4.25i 4.15+0.00i -0.02+0.46i -1.89+1.15i 0.04-3.69i -0.02+0.46i 0.33+0.00i ,$
 $A = 1.7-2.3i -1.8+2.4i 2.9-2.1i 1.2+1.4i -2.9+1.0i 0.6+0.8i 1.5+0.9i -1.4-1.7i$
and
 $B = -0.3-1.9i 2.1-1.1i -2.4+1.4i 0.6-2.9i -0.2-2.9i -1.5+0.1i 3.5+0.8i 2.2+3.7i .$

### 9.1  Program Text

Program Text (f16zwce.c)

### 9.2  Program Data

Program Data (f16zwce.d)

### 9.3  Program Results

Program Results (f16zwce.r)