# NAG CL Interfacef16yqc (dsfrk)

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## 1Purpose

f16yqc performs one of the symmetric rank-$k$ update operations
 $C←αAAT + βC or C←αATA + βC ,$
where $A$ is a real matrix, $C$ is an $n×n$ real symmetric matrix stored in Rectangular Full Packed (RFP) format, and $\alpha$ and $\beta$ are real scalars.

## 2Specification

 #include
 void f16yqc (Nag_OrderType order, Nag_RFP_Store transr, Nag_UploType uplo, Nag_TransType trans, Integer n, Integer k, double alpha, const double a[], Integer pda, double beta, double cr[], NagError *fail)
The function may be called by the names: f16yqc, nag_blast_dsfrk or nag_dsfrk.

## 3Description

f16yqc performs one of the symmetric rank-$k$ update operations
 $C←αAAT + βC or C←αATA + βC ,$
where $A$ is a real matrix, $C$ is an $n×n$ real symmetric matrix stored in Rectangular Full Packed (RFP) format, and $\alpha$ and $\beta$ are real scalars. The RFP storage format is described in Section 3.4.3 in the F07 Chapter Introduction.
If $n=0$ or if $\beta =1.0$ and either $k=0$ or $\alpha =0.0$ then f16yqc returns immediately. If $\beta =0.0$ and either $k=0$ or $\alpha =0.0$ then $C$ is set to the zero matrix.

## 4References

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

## 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: $\mathbf{transr}$Nag_RFP_Store Input
On entry: specifies whether the RFP representation of $C$ is normal or transposed.
${\mathbf{transr}}=\mathrm{Nag_RFP_Normal}$
The matrix $C$ is stored in normal RFP format.
${\mathbf{transr}}=\mathrm{Nag_RFP_Trans}$
The matrix $C$ is stored in transposed RFP format.
Constraint: ${\mathbf{transr}}=\mathrm{Nag_RFP_Normal}$ or $\mathrm{Nag_RFP_Trans}$.
3: $\mathbf{uplo}$Nag_UploType Input
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_TransType Input
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 $\mathrm{Nag_Trans}$.
5: $\mathbf{n}$Integer Input
On entry: $n$, the order of the matrix $C$.
Constraint: ${\mathbf{n}}\ge 0$.
6: $\mathbf{k}$Integer Input
On entry: $k$, the number of columns of $A$ if ${\mathbf{trans}}=\mathrm{Nag_NoTrans}$, or the number of rows of $A$ if ${\mathbf{trans}}=\mathrm{Nag_Trans}$.
Constraint: ${\mathbf{k}}\ge 0$.
7: $\mathbf{alpha}$double Input
On entry: the scalar $\alpha$.
8: $\mathbf{a}\left[\mathit{dim}\right]$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 ${\mathbf{trans}}=\mathrm{Nag_Trans}$ 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 ${\mathbf{order}}=\mathrm{Nag_RowMajor}$.
On entry: the matrix $A$; $A$ is $n×k$ if ${\mathbf{trans}}=\mathrm{Nag_NoTrans}$, or $k×n$ if ${\mathbf{trans}}=\mathrm{Nag_Trans}$. If ${\mathbf{alpha}}=0.0$, a is not referenced.
9: $\mathbf{pda}$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 ${\mathbf{trans}}=\mathrm{Nag_Trans}$, ${\mathbf{pda}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{k}}\right)$;
• 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 ${\mathbf{trans}}=\mathrm{Nag_Trans}$, ${\mathbf{pda}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
10: $\mathbf{beta}$double Input
On entry: the scalar $\beta$.
11: $\mathbf{cr}\left[{\mathbf{n}}×\left({\mathbf{n}}+1\right)/2\right]$double Input/Output
On entry: the upper or lower triangular part (as specified by uplo) of the $n×n$ symmetric matrix $C$, stored in RFP format (as specified by transr). The storage format is described in detail in Section 3.4.3 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 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 ${\mathbf{trans}}=\mathrm{Nag_Trans}$, ${\mathbf{pda}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{k}}\right)$.
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_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 7.5 in the Introduction to the NAG Library CL Interface for further information.
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.

Not applicable.

## 8Parallelism and Performance

f16yqc 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

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 $6×4$ matrix $A$ and then performs the symmetric rank-$4$ update $C←\alpha A{A}^{\mathrm{T}}+\beta C$.

### 10.1Program Text

Program Text (f16yqce.c)

### 10.2Program Data

Program Data (f16yqce.d)

### 10.3Program Results

Program Results (f16yqce.r)