# NAG CL Interfacef16sac (zgemv)

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## 1Purpose

f16sac performs matrix-vector multiplication for a complex general matrix.

## 2Specification

 #include
 void f16sac (Nag_OrderType order, Nag_TransType trans, Integer m, Integer n, Complex alpha, const Complex a[], Integer pda, const Complex x[], Integer incx, Complex beta, Complex y[], Integer incy, NagError *fail)
The function may be called by the names: f16sac, nag_blast_zgemv or nag_zgemv.

## 3Description

f16sac performs one of the matrix-vector operations
 $y←αAx+βy, y←αATx+βy or y←αAHx+βy$
where $A$ is an $m×n$ complex matrix, $x$ and $y$ are complex vectors, and $\alpha$ and $\beta$ are complex scalars.
If $m=0$ or $n=0$, no operation is performed.

## 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: $\mathbf{trans}$Nag_TransType Input
On entry: specifies the operation to be performed.
${\mathbf{trans}}=\mathrm{Nag_NoTrans}$
$y←\alpha Ax+\beta y$.
${\mathbf{trans}}=\mathrm{Nag_Trans}$
$y←\alpha {A}^{\mathrm{T}}x+\beta y$.
${\mathbf{trans}}=\mathrm{Nag_ConjTrans}$
$y←\alpha {A}^{\mathrm{H}}x+\beta y$.
Constraint: ${\mathbf{trans}}=\mathrm{Nag_NoTrans}$, $\mathrm{Nag_Trans}$ or $\mathrm{Nag_ConjTrans}$.
3: $\mathbf{m}$Integer Input
On entry: $m$, the number of rows of the matrix $A$.
Constraint: ${\mathbf{m}}\ge 0$.
4: $\mathbf{n}$Integer Input
On entry: $n$, the number of columns of the matrix $A$.
Constraint: ${\mathbf{n}}\ge 0$.
5: $\mathbf{alpha}$Complex Input
On entry: the scalar $\alpha$.
6: $\mathbf{a}\left[\mathit{dim}\right]$const Complex Input
Note: the dimension, dim, of the array a must be at least
• $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{pda}}×{\mathbf{n}}\right)$ when ${\mathbf{order}}=\mathrm{Nag_ColMajor}$;
• $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}}×{\mathbf{pda}}\right)$ when ${\mathbf{order}}=\mathrm{Nag_RowMajor}$.
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 $m×n$ matrix $A$.
7: $\mathbf{pda}$Integer Input
On entry: the stride separating row or column elements (depending on the value of order) in the array a.
Constraints:
• if ${\mathbf{order}}=\mathrm{Nag_ColMajor}$, ${\mathbf{pda}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}}\right)$;
• if ${\mathbf{order}}=\mathrm{Nag_RowMajor}$, ${\mathbf{pda}}\ge {\mathbf{n}}$.
8: $\mathbf{x}\left[\mathit{dim}\right]$const Complex Input
Note: the dimension, dim, of the array x must be at least
• $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,1+\left({\mathbf{n}}-1\right)|{\mathbf{incx}}|\right)$ when ${\mathbf{trans}}=\mathrm{Nag_NoTrans}$;
• $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,1+\left({\mathbf{m}}-1\right)|{\mathbf{incx}}|\right)$ when ${\mathbf{trans}}=\mathrm{Nag_Trans}$ or $\mathrm{Nag_ConjTrans}$.
On entry: the vector $x$.
If ${\mathbf{trans}}=\mathrm{Nag_NoTrans}$, then $x$ is an $n$-element vector.
• If ${\mathbf{incx}}>0$, ${x}_{\mathit{i}}$ must be stored in ${\mathbf{x}}\left[\left(\mathit{i}-1\right)×{\mathbf{incx}}\right]$, for $\mathit{i}=1,2,\dots ,{\mathbf{n}}$.
• If ${\mathbf{incx}}<0$, ${x}_{\mathit{i}}$ must be stored in ${\mathbf{x}}\left[\left({\mathbf{n}}-\mathit{i}\right)×|{\mathbf{incx}}|\right]$, for $\mathit{i}=1,2,\dots ,{\mathbf{n}}$.
• Intermediate elements of x are not referenced. If ${\mathbf{n}}=0$, x is not referenced and may be NULL.
Otherwise, $x$ is an $m$-element vector.
• If ${\mathbf{incx}}>0$, ${x}_{\mathit{i}}$ must be stored in ${\mathbf{x}}\left[\left(\mathit{i}-1\right)×{\mathbf{incx}}\right]$, for $\mathit{i}=1,2,\dots ,{\mathbf{m}}$.
• If ${\mathbf{incx}}<0$, ${x}_{\mathit{i}}$ must be stored in ${\mathbf{x}}\left[\left({\mathbf{m}}-\mathit{i}\right)×|{\mathbf{incx}}|\right]$, for $\mathit{i}=1,2,\dots ,{\mathbf{m}}$.
• Intermediate elements of x are not referenced. If ${\mathbf{m}}=0$, x is not referenced and may be NULL.
9: $\mathbf{incx}$Integer Input
On entry: the increment in the subscripts of x between successive elements of $x$.
Constraint: ${\mathbf{incx}}\ne 0$.
10: $\mathbf{beta}$Complex Input
On entry: the scalar $\beta$.
11: $\mathbf{y}\left[\mathit{dim}\right]$Complex Input/Output
Note: the dimension, dim, of the array y must be at least
• $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,1+\left({\mathbf{m}}-1\right)|{\mathbf{incy}}|\right)$ when ${\mathbf{trans}}=\mathrm{Nag_NoTrans}$;
• $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,1+\left({\mathbf{n}}-1\right)|{\mathbf{incy}}|\right)$ when ${\mathbf{trans}}=\mathrm{Nag_Trans}$ or $\mathrm{Nag_ConjTrans}$.
On entry: the vector $y$. See x for details of storage.
If ${\mathbf{beta}}=0$, y need not be set.
On exit: the updated vector $y$.
12: $\mathbf{incy}$Integer Input
On entry: the increment in the subscripts of y between successive elements of $y$.
Constraint: ${\mathbf{incy}}\ne 0$.
13: $\mathbf{fail}$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_INT
On entry, ${\mathbf{incx}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{incx}}\ne 0$.
On entry, ${\mathbf{incy}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{incy}}\ne 0$.
On entry, ${\mathbf{m}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{m}}\ge 0$.
On entry, ${\mathbf{n}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{n}}\ge 0$.
NE_INT_2
On entry, ${\mathbf{pda}}=⟨\mathit{\text{value}}⟩$, ${\mathbf{m}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{pda}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}}\right)$.
On entry, ${\mathbf{pda}}=⟨\mathit{\text{value}}⟩$ and ${\mathbf{n}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{pda}}\ge {\mathbf{n}}$.
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

f16sac is not threaded in any implementation.

None.

## 10Example

This example computes the matrix-vector product
 $y=αAx+βy$
where
 $A = ( 1.0+1.0i 1.0+2.0i 2.0+1.0i 2.0+2.0i 3.0+1.0i 3.0+2.0i ) ,$
 $x = ( 1.0-1.0i 2.0-2.0i ) ,$
 $y = ( -3.5-0.5i -4.5+1.5i -5.5+3.5i ) ,$
 $α=1.0+0.0i and β=2.0+0.0i .$

### 10.1Program Text

Program Text (f16sace.c)

### 10.2Program Data

Program Data (f16sace.d)

### 10.3Program Results

Program Results (f16sace.r)