# NAG Library Routine Document

## 1Purpose

f16gcf (blas_zaxpby) computes the sum of two scaled vectors, for complex scalars and vectors.

## 2Specification

Fortran Interface
 Subroutine f16gcf ( n, x, incx, beta, y, incy)
 Integer, Intent (In) :: n, incx, incy Complex (Kind=nag_wp), Intent (In) :: alpha, x(1+(n-1)*ABS(incx)), beta Complex (Kind=nag_wp), Intent (Inout) :: y(1+(n-1)*ABS(incy))
#include nagmk26.h
 void f16gcf_ (const Integer *n, const Complex *alpha, const Complex x[], const Integer *incx, const Complex *beta, Complex y[], const Integer *incy)
The routine may be called by its BLAST name blas_zaxpby.

## 3Description

f16gcf (blas_zaxpby) performs the operation
 $y ← αx+βy,$
where $x$ and $y$ are $n$-element complex vectors, and $\alpha$ and $\beta$ are complex scalars. If $n$ is less than or equal to zero, or if $\alpha$ is equal to zero and $\beta$ is equal to $1$, this routine returns immediately.

## 4References

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

## 5Arguments

1:     $\mathbf{n}$ – IntegerInput
On entry: $n$, the number of elements in $x$ and $y$.
2:     $\mathbf{alpha}$ – Complex (Kind=nag_wp)Input
On entry: the scalar $\alpha$.
3:     $\mathbf{x}\left(1+\left({\mathbf{n}}-1\right)×\left|{\mathbf{incx}}\right|\right)$ – Complex (Kind=nag_wp) arrayInput
On entry: the $n$-element vector $x$.
If ${\mathbf{incx}}>0$, ${x}_{\mathit{i}}$ must be stored in ${\mathbf{x}}\left(\left(\mathit{i}-1\right)×{\mathbf{incx}}+1\right)$, for $\mathit{i}=1,2,\dots ,{\mathbf{n}}$.
If ${\mathbf{incx}}<0$, ${x}_{\mathit{i}}$ must be stored in ${\mathbf{x}}\left(\left({\mathbf{n}}-\mathit{i}\right)×\left|{\mathbf{incx}}\right|+1\right)$, for $\mathit{i}=1,2,\dots ,{\mathbf{n}}$.
Intermediate elements of x are not referenced. If ${\mathbf{n}}=0$, x is not referenced.
4:     $\mathbf{incx}$ – IntegerInput
On entry: the increment in the subscripts of x between successive elements of $x$.
Constraint: ${\mathbf{incx}}\ne 0$.
5:     $\mathbf{beta}$ – Complex (Kind=nag_wp)Input
On entry: the scalar $\beta$.
6:     $\mathbf{y}\left(1+\left({\mathbf{n}}-1\right)×\left|{\mathbf{incy}}\right|\right)$ – Complex (Kind=nag_wp) arrayInput/Output
On entry: the $n$-element vector $y$.
If ${\mathbf{incy}}>0$, ${y}_{\mathit{i}}$ must be stored in ${\mathbf{y}}\left(\left(\mathit{i}-1\right)×{\mathbf{incy}}+1\right)$, for $\mathit{i}=1,2,\dots ,{\mathbf{n}}$.
If ${\mathbf{incy}}<0$, ${y}_{\mathit{i}}$ must be stored in ${\mathbf{y}}\left(\left({\mathbf{n}}-\mathit{i}\right)×\left|{\mathbf{incy}}\right|+1\right)$, for $\mathit{i}=1,2,\dots ,{\mathbf{n}}$.
Intermediate elements of y are not referenced.
On exit: the updated vector $y$ stored in the array elements used to supply the original vector $y$.
Intermediate elements of y are unchanged.
7:     $\mathbf{incy}$ – IntegerInput
On entry: the increment in the subscripts of y between successive elements of $y$.
Constraint: ${\mathbf{incy}}\ne 0$.

## 6Error Indicators and Warnings

If ${\mathbf{incx}}=0$ or ${\mathbf{incy}}=0$, an error message is printed and program execution is terminated.

## 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

f16gcf (blas_zaxpby) 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 routine. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

None.

## 10Example

This example computes the result of a scaled vector accumulation for
 $α=3+2i, x = -6+1.2i,3.7+4.5i,-4+2.1iT , β=-i, y = -5.1,6.4-5i,-3-2.4iT .$
$x$ and $y$ are stored in reverse order.

### 10.1Program Text

Program Text (f16gcfe.f90)

### 10.2Program Data

Program Data (f16gcfe.d)

### 10.3Program Results

Program Results (f16gcfe.r)

© The Numerical Algorithms Group Ltd, Oxford, UK. 2017