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# NAG Toolbox: nag_blast_zaxpby (f16gc)

## Purpose

nag_blast_zaxpby (f16gc) computes the sum of two scaled vectors, for complex scalars and vectors.

## Syntax

[y] = f16gc(n, alpha, x, incx, beta, y, incy)
[y] = nag_blast_zaxpby(n, alpha, x, incx, beta, y, incy)

## Description

nag_blast_zaxpby (f16gc) 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 function returns immediately.

## References

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

## Parameters

### Compulsory Input Parameters

1:     $\mathrm{n}$int64int32nag_int scalar
$n$, the number of elements in $x$ and $y$.
2:     $\mathrm{alpha}$ – complex scalar
The scalar $\alpha$.
3:     $\mathrm{x}\left(1+\left({\mathbf{n}}-1\right)×\left|{\mathbf{incx}}\right|\right)$ – complex array
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.
4:     $\mathrm{incx}$int64int32nag_int scalar
The increment in the subscripts of x between successive elements of $x$.
Constraint: ${\mathbf{incx}}\ne 0$.
5:     $\mathrm{beta}$ – complex scalar
The scalar $\beta$.
6:     $\mathrm{y}\left(1+\left({\mathbf{n}}-1\right)×\left|{\mathbf{incy}}\right|\right)$ – complex array
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.
7:     $\mathrm{incy}$int64int32nag_int scalar
The increment in the subscripts of y between successive elements of $y$.
Constraint: ${\mathbf{incy}}\ne 0$.

None.

### Output Parameters

1:     $\mathrm{y}\left(1+\left({\mathbf{n}}-1\right)×\left|{\mathbf{incy}}\right|\right)$ – complex array
The updated vector $y$ stored in the array elements used to supply the original vector $y$.
Intermediate elements of y are unchanged.

## Error Indicators and Warnings

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

## Accuracy

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

None.

## Example

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.
```function f16gc_example

fprintf('f16gc example results\n\n');

n = int64(3);
x = [ -4 + 2.1i    3.7 + 4.5i     -6   + 1.2i];
y = [ -3 - 2.4i    6.4 - 5.0i     -5.1 + 0.0i];

% z = alpha*x +beta*y;
alpha =  3 + 2i;
beta  =  0 - 1i;

incx = int64(-1);
incy = int64(-1);
[z] = f16gc( ...
n, alpha, x, incx, beta, y, incy);

disp('x, y:');
fprintf('  x = ');
for j = 1:n
fprintf('%11.4f %+ 8.4fi',real(x(j)),imag(x(j)));
end
fprintf('\n  y = ');
for j = 1:n
fprintf('%11.4f %+ 8.4fi',real(y(j)),imag(y(j)));
end

fprintf('\n\nalpha = %5.1f%+5.1fi      beta = %5.1f%+5.1fi\n', ...
real(alpha), imag(alpha), real(beta), imag(beta));
fprintf('\nalpha*x + beta*y = \n      ');
for j = 1:n
fprintf('%11.4f %+ 8.4fi',real(z(j)),imag(z(j)));
end
fprintf('\n');

```
```f16gc example results

x, y:
x =     -4.0000  +2.1000i     3.7000  +4.5000i    -6.0000  +1.2000i
y =     -3.0000  -2.4000i     6.4000  -5.0000i    -5.1000  +0.0000i

alpha =   3.0 +2.0i      beta =   0.0 -1.0i

alpha*x + beta*y =
-18.6000  +1.3000i    -2.9000 +14.5000i   -20.4000  -3.3000i
```

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