## Computing the Nth Roots of a Number

In Matlab when we want to compute the square root of a number we use the 'sqrt()' command. For example, if *z* = 9 we find the square root of *z* by typing the

>> sqrt(z)

command giving us the result

ans =

3

That result, however, provides only the 'principle' square root of *z*. As we know the number *z* = 9 has two square roots, i.e., +3 and -3. Likewise, if we want the cube root of *z* (also called the "3rd root of *z*") we type

>> z^(1/3)

command giving us the principle result

ans =

2.0801

Again, the 3rd root of *z* has three solutions. They are:

2.0801

-1.0400 + 1.8014i

-1.0400 - 1.8014i.

(1)

The following Matlab code computes all of the *n* *n*th roots of the number(s) *Num*. If input argument Num has more than one element, then it must be a row vector. The elements in *Num* can be real-valued of complex-valued. Argument *n* is the order of the roots. If you want a complex-plane plot of the roots on Num that option, using the string 'Y', is also available. In such a plot the principle root is plotted as a blue square and the remaining roots are plotted with red circles.

For example, using the commands

Num = 9;

n=3;

[nth_Roots] = roots_nth(Num, n, 'Y')

produces the results in Eq. (1) and the plot shown below.

Using the command

[nth_Roots] = roots_nth(Num, n)

computes the *n* roots of *Num*, but no complex-plane plot is generated.

```
function [nth_Roots] = roots_nth(Num, n, Plot)
% ROOTS_NTH computes the nth roots of Num.
%
% Inputs:
% Num is the number(s) for which the nth roots will be computed.
% Num can be real-valued or complex-valued.
% If Num has more than one element, then Num must be a row vector.
% n must be a positive integer.
% Plot is a string equal to 'Y' if a complex plane plot of
% the n complex roots of each element in Num is desired.
% Principle root is plotted as a blue square. Remaining roots
% are red dots. If input Plot string does not exist, then no
% plot is generated.
%
% Output:
% nth_Roots is a column vector (with n rows) giving the nth roots of Num.
%
% Example:
% clear, clc % clear all variables
% Num = [2+j*11, 3*exp(j*pi)]; % Two numbers
% n = 3; % find the 3rd roots
% [nth_Roots] = roots_nth(Num, n, 'Y')
%
% Results are:
% nth_Roots =
%
% 2.0000 + 1.0000i 0.7211 + 1.2490i
% -1.8660 + 1.2321i -1.4422 + 0.0000i
% -0.1340 - 2.2321i 0.7211 - 1.2490i
%
% [R. Lyons, Jan. 2013]
% Input argument check
if (nargin<2),
disp('Not enough input arguments!!')
return
end;
if (nargin==2)
Plot == 'N'
end
Num_of_Columns = max(size(Num)); % How many elements are in Num?
for Column_Loop = 1:Num_of_Columns;
Mag(Column_Loop) = abs(Num(Column_Loop));
Angle_Degrees(Column_Loop) = angle(Num(Column_Loop))*180/pi;
for k = 1:n
Root_Mag = Mag(Column_Loop)^(1/n);
Root_Angle_Deg(k) = Angle_Degrees(Column_Loop)/n + (k-1)*360/n;
nth_Roots(k,Column_Loop) = Root_Mag*exp(j*Root_Angle_Deg(k)*pi/180);
end
% Plot roots, on complex plane(s), if necessary
if Plot == 'Y'
figure
% Plot unit circle
Num_Pts = 200; % Number of circle points
Index = 0 : Num_Pts-1;
Alpha = 1/Num_Pts;
Points = Root_Mag*exp(j*2*pi*Alpha*Index);
plot(Points, ':k','markersize',5)
grid on
hold on
% Plot principle root (square blue dot) and list results in plot
axis([-1.1*Root_Mag, 1.1*Root_Mag, -1.1*Root_Mag, 1.1*Root_Mag]);
plot(Root_Mag*cos(Root_Angle_Deg(1)*pi/180), ...
Root_Mag*sin(Root_Angle_Deg(1)*pi/180), 'bs');
text(-Root_Mag/2, 3*Root_Mag/4, ['Magnitudes = ',num2str(Root_Mag)]);
text(-Root_Mag/2, 3*Root_Mag/4-Root_Mag/10, ...
['Principle Angle (deg.) = ',num2str(Root_Angle_Deg(1))]);
% Plot remaining roots as red dots
for k = 2:n
plot(Root_Mag*cos(Root_Angle_Deg(k)*pi/180), ...
Root_Mag*sin(Root_Angle_Deg(k)*pi/180), 'ro');
text(-Root_Mag/2, 3*Root_Mag/4-k*Root_Mag/10, ...
[' Angle ',num2str(k),' (deg.) = ', ...
num2str(Root_Angle_Deg(k))]);
end % End k loop
xlabel('Real'); ylabel('Imag.');
hold off
end % End 'if Plot == 'Y''
end % End Column_Loop
```

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