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Chapter Contents

Chapter Introduction

NAG Toolbox: nag_complex_sqrt (a02aa)

▸▿ Contents

1 Purpose

2 Syntax

3 Description

4 References

▸▿ 5 Parameters

5.1 Compulsory Input Parameters

5.2 Optional Input Parameters

5.3 Output Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

9 Example

Purpose

nag_complex_sqrt (a02aa) evaluates the square root of the complex number

x = (x_{r}, x_{i})

.

Syntax

[yr, yi] = a02aa(xr, xi)

[yr, yi] = nag_complex_sqrt(xr, xi)

Description

The method of evaluating

y = \sqrt{x}

depends on the value of

x_{r}

.

For

x_{r} \geq 0

,

y_{r} = \sqrt{\frac{x_{r} + \sqrt{x_{r}^{2} + x_{i}^{2}}}{2}}, y_{i} = \frac{x_{i}}{2 y_{r}} .

For

x_{r} < 0

,

y_{i} = sign (x_{i}) \times \sqrt{\frac{|x_{r}| + \sqrt{x_{r}^{2} + x_{i}^{2}}}{2}}, y_{r} = \frac{x_{i}}{2 y_{i}} .

Overflow is avoided when squaring

x_{i}

and

x_{r}

by calling nag_complex_abs (a02ab) to evaluate

\sqrt{x_{r}^{2} + x_{i}^{2}}

.

References

Wilkinson J H and Reinsch C (1971) Handbook for Automatic Computation II, Linear Algebra Springer–Verlag

Parameters

Compulsory Input Parameters

1: $xr$ – double scalar
2: $xi$ – double scalar: $x_{r}$ and $x_{i}$ , the real and imaginary parts of $x$ , respectively.

Optional Input Parameters

None.

Output Parameters

1: $yr$ – double scalar
2: $yi$ – double scalar: $y_{r}$ and $y_{i}$ , the real and imaginary parts of $y$ , respectively.

Error Indicators and Warnings

None.

Accuracy

The result should be correct to machine precision.

Further Comments

The time taken by nag_complex_sqrt (a02aa) is negligible.

Example

This example finds the square root of

- 1.7 + 2.6 i

.

Open in the MATLAB editor: a02aa_example

function a02aa_example


fprintf('a02aa example results\n\n');

xr = -1.7;
xi = 2.6;
x  = xr + i*xi;

[yr, yi] = a02aa(xr, xi);
y = yr+i*yi;
fprintf('The square root of ');
disp(x);
fprintf('                is ');
disp(y);

a02aa example results

The square root of   -1.7000 + 2.6000i

                is    0.8386 + 1.5502i

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Chapter Contents

Chapter Introduction

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