# NAG Library Routine Document

## 1Purpose

x03abf calculates the value of a complex scalar product using basic precision or additional precision and adds it to a complex initial value.

## 2Specification

Fortran Interface
 Subroutine x03abf ( a, b, n, cx, dx, sw,
 Integer, Intent (In) :: isizea, isizeb, n, istepa, istepb Integer, Intent (Inout) :: ifail Complex (Kind=nag_wp), Intent (In) :: a(isizea), b(isizeb), cx Complex (Kind=nag_wp), Intent (Out) :: dx Logical, Intent (In) :: sw
#include nagmk26.h
 void x03abf_ (const Complex a[], const Integer *isizea, const Complex b[], const Integer *isizeb, const Integer *n, const Integer *istepa, const Integer *istepb, const Complex *cx, Complex *dx, const logical *sw, Integer *ifail)

## 3Description

x03abf calculates the scalar product of two complex vectors and adds it to an initial value $c$ to give a correctly rounded result $d$:
 $d=c+∑i=1naibi.$
If $n<1$, $d=c$.
The vector elements ${a}_{i}$ and ${b}_{i}$ are stored in selected elements of the one-dimensional array arguments a and b, which in the subroutine from which x03abf is called may be identified with parts of possibly multidimensional arrays according to the standard Fortran rules. For example, the vectors may be parts of a row or column of a matrix. See Section 5 for details, and Section 10 for an example.
The products are accumulated in basic precision or additional precision depending on the argument sw.
This routine has been designed primarily for use as an auxiliary routine by other routines in the NAG Library, especially those in the chapters on Linear Algebra.

None.

## 5Arguments

1:     $\mathbf{a}\left({\mathbf{isizea}}\right)$ – Complex (Kind=nag_wp) arrayInput
On entry: the elements of the first vector.
The $i$th vector element is stored in the array element ${\mathbf{a}}\left(\left(i-1\right)×{\mathbf{istepa}}+1\right)$. In your subroutine from which x03abf is called, a can be part of a multidimensional array and the actual argument must be the array element containing the first vector element.
2:     $\mathbf{isizea}$ – IntegerInput
On entry: the dimension of the array a as declared in the (sub)program from which x03abf is called.
The upper bound for isizea is found by multiplying together the dimensions of a as declared in your subroutine from which x03abf is called, subtracting the starting position and adding $1$.
Constraint: ${\mathbf{isizea}}\ge \left({\mathbf{n}}-1\right)×{\mathbf{istepa}}+1$.
3:     $\mathbf{b}\left({\mathbf{isizeb}}\right)$ – Complex (Kind=nag_wp) arrayInput
On entry: the elements of the second vector.
The $i$th vector element is stored in the array element ${\mathbf{b}}\left(\left(i-1\right)×{\mathbf{istepb}}+1\right)$. In your subroutine from which x03abf is called, b can be part of a multidimensional array and the actual argument must be the array element containing the first vector element.
4:     $\mathbf{isizeb}$ – IntegerInput
On entry: the dimension of the array b as declared in the (sub)program from which x03abf is called.
The upper bound for isizeb is found by multiplying together the dimensions of b as declared in your subroutine from which x03abf is called, subtracting the starting position and adding $1$.
Constraint: ${\mathbf{isizeb}}\ge \left({\mathbf{n}}-1\right)×{\mathbf{istepb}}+1$.
5:     $\mathbf{n}$ – IntegerInput
On entry: $n$, the number of elements in the scalar product.
6:     $\mathbf{istepa}$ – IntegerInput
On entry: the step length between elements of the first vector in array a.
Constraint: ${\mathbf{istepa}}>0$.
7:     $\mathbf{istepb}$ – IntegerInput
On entry: the step length between elements of the second vector in array b.
Constraint: ${\mathbf{istepb}}>0$.
8:     $\mathbf{cx}$ – Complex (Kind=nag_wp)Input
On entry: the initial value $c$.
9:     $\mathbf{dx}$ – Complex (Kind=nag_wp)Output
On exit: the result $d$.
10:   $\mathbf{sw}$ – LogicalInput
On entry: the precision to be used in the calculation.
${\mathbf{sw}}=\mathrm{.TRUE.}$
${\mathbf{sw}}=\mathrm{.FALSE.}$
basic precision.
11:   $\mathbf{ifail}$ – IntegerInput/Output
On entry: ifail must be set to $0$, $-1\text{​ or ​}1$. If you are unfamiliar with this argument you should refer to Section 3.4 in How to Use the NAG Library and its Documentation for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value $-1\text{​ or ​}1$ is recommended. If the output of error messages is undesirable, then the value $1$ is recommended. Otherwise, if you are not familiar with this argument, the recommended value is $0$. When the value $-\mathbf{1}\text{​ or ​}\mathbf{1}$ is used it is essential to test the value of ifail on exit.
On exit: ${\mathbf{ifail}}={\mathbf{0}}$ unless the routine detects an error or a warning has been flagged (see Section 6).

## 6Error Indicators and Warnings

If on entry ${\mathbf{ifail}}=0$ or $-1$, explanatory error messages are output on the current error message unit (as defined by x04aaf).
Errors or warnings detected by the routine:
${\mathbf{ifail}}=1$
 On entry, ${\mathbf{istepa}}\le 0$, or ${\mathbf{istepb}}\le 0$.
${\mathbf{ifail}}=2$
 On entry, ${\mathbf{isizea}}<\left({\mathbf{n}}-1\right)×{\mathbf{istepa}}+1$, or ${\mathbf{isizeb}}<\left({\mathbf{n}}-1\right)×{\mathbf{istepb}}+1$.
${\mathbf{ifail}}=-99$
See Section 3.9 in How to Use the NAG Library and its Documentation for further information.
${\mathbf{ifail}}=-399$
Your licence key may have expired or may not have been installed correctly.
See Section 3.8 in How to Use the NAG Library and its Documentation for further information.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.
See Section 3.7 in How to Use the NAG Library and its Documentation for further information.

## 7Accuracy

If the calculation is in additional precision, the result is correct to full implementation accuracy provided that exceptionally severe cancellation does not occur in the summation. If the calculation is in basic precision, such accuracy cannot be guaranteed.

## 8Parallelism and Performance

x03abf is not threaded in any implementation.

The time taken by x03abf is approximately proportional to $n$ and also depends on whether basic precision or additional precision is used.

## 10Example

This example calculates the scalar product of the second column of the matrix $A$ and the vector ${\mathbf{b}}$, and add it to an initial value of $1+i$, where
 $A= -1 -i 1 -2+3i i 2i -0 -1-i 1-2i , b= i 1-i -i .$

### 10.1Program Text

Program Text (x03abfe.f90)

### 10.2Program Data

Program Data (x03abfe.d)

### 10.3Program Results

Program Results (x03abfe.r)

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