# NAG CL Interfacef16yrc (dsyr2k)

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## 1Purpose

f16yrc performs a rank-$2k$ update on a real symmetric matrix.

## 2Specification

 #include
 void f16yrc (Nag_OrderType order, Nag_UploType uplo, Nag_TransType trans, Integer n, Integer k, double alpha, const double a[], Integer pda, const double b[], Integer pdb, double beta, double c[], Integer pdc, NagError *fail)
The function may be called by the names: f16yrc, nag_blast_dsyr2k or nag_dsyr2k.

## 3Description

f16yrc performs one of the symmetric rank-$2k$ update operations
 $C←αABT + αBAT + βC or C←αATB + αBTA+βC ,$
where $A$ and $B$ are real matrices, $C$ is an $n×n$ real symmetric matrix, and $\alpha$ and $\beta$ are real scalars.

## 4References

Basic Linear Algebra Subprograms Technical (BLAST) Forum (2001) Basic Linear Algebra Subprograms Technical (BLAST) Forum Standard University of Tennessee, Knoxville, Tennessee https://www.netlib.org/blas/blast-forum/blas-report.pdf

## 5Arguments

1: $\mathbf{order}$Nag_OrderType Input
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 3.1.3 in the Introduction to the NAG Library CL Interface 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_UploType Input
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_TransType Input
On entry: specifies the operation to be performed.
${\mathbf{trans}}=\mathrm{Nag_NoTrans}$
$C←\alpha A{B}^{\mathrm{T}}+\alpha B{A}^{\mathrm{T}}+\beta C$.
${\mathbf{trans}}=\mathrm{Nag_Trans}$ or $\mathrm{Nag_ConjTrans}$
$C←\alpha {A}^{\mathrm{T}}B+\alpha {B}^{\mathrm{T}}A+\beta C$.
Constraint: ${\mathbf{trans}}=\mathrm{Nag_NoTrans}$, $\mathrm{Nag_Trans}$ or $\mathrm{Nag_ConjTrans}$.
4: $\mathbf{n}$Integer Input
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: $\mathbf{k}$Integer Input
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: $\mathbf{alpha}$double Input
On entry: the scalar $\alpha$.
7: $\mathbf{a}\left[\mathit{dim}\right]$const double Input
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 $n×k$ if ${\mathbf{trans}}=\mathrm{Nag_NoTrans}$, or $k×n$ otherwise.
8: $\mathbf{pda}$Integer Input
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: $\mathbf{b}\left[\mathit{dim}\right]$const double Input
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 $n×k$ if ${\mathbf{trans}}=\mathrm{Nag_NoTrans}$, or $k×n$ otherwise.
10: $\mathbf{pdb}$Integer Input
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: $\mathbf{beta}$double Input
On entry: the scalar $\beta$.
12: $\mathbf{c}\left[\mathit{dim}\right]$double Input/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 $n×n$ 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: $\mathbf{pdc}$Integer Input
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: $\mathbf{fail}$NagError * Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

## 6Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface 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 ${\mathbf{trans}}=\mathrm{Nag_NoTrans}$, ${\mathbf{pda}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{k}}\right)$.
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.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.

## 7Accuracy

The BLAS standard requires accurate implementations which avoid unnecessary over/underflow (see Section 2.7 of Basic Linear Algebra Subprograms Technical (BLAST) Forum (2001)).

## 8Parallelism and Performance

f16yrc is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f16yrc 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.

## 10Example

Perform rank-$2k$ update of real symmetric $4×4$ matrix $C$ using $4×2$ matrices $A$ and $B$, $C=C-A{B}^{\mathrm{T}}-B{A}^{\mathrm{T}}$, where
 $C = ( 4.30 -3.96 0.40 -0.27 -3.96 -4.87 0.31 0.07 0.40 0.31 -8.02 -5.95 -0.27 0.07 -5.95 0.12 ) ,$
 $A = ( -3.0 -5.0 -1.0 1.0 2.0 -1.0 1.0 1.0 )$
and
 $B = ( 3.0 -2.0 -1.0 1.0 2.0 -1.0 1.0 0.0 ) .$

### 10.1Program Text

Program Text (f16yrce.c)

### 10.2Program Data

Program Data (f16yrce.d)

### 10.3Program Results

Program Results (f16yrce.r)