Newton's Method. This will work with either MATLAB or Octave. Used to determine the maximum speed at which kinetic energy can be described by the Newtonian equation with an error with a specified tolerance of relativistic energy.

M03_Kinetic_Energy_v2.m

%% Newtonian vs. Relativistic Energy Accuracy - Using Newton's Method
% Determines the maximum speed a particle can have such that its 
% kinetic energy described as 0.5*m*v^2 with an error of no greater 
% than <tolerance>.
close all; clear all; clc;

% Input - The tolerance of the accuracy of the Newtonian kinetic energy 
% compared to the relativistic energy.
tolerance = .005;  

% Speed of light  [m/s]
c = 299792458;            

% Newton's method
maxiters = 1e5;
TOL = 1e-4; 
guess = .5*c;

% Use Newton's method to solve
for i=1:maxiters  
    % "Guess" the velocity
    v = guess;
    
    % Calculate our energies
    g = (1 - v^2/c^2)^(-1/2);   % Gamma
    K = 0.5*v^2;            	% Newtonian kinetic energy per unit mass
    E = g*c^2 - c^2;            % Relativistic kinetic energy per unit mass
    
    % Newton's method
    f = abs( (E - K) / E ) - tolerance;
    df = v / E;
    guess = v - f/df;
    
    fprintf('%4d:  %14.4f  %14.4f  %8.6fc  %10.8f\n', i, guess, v, v/c, f + .005)
    
    % If close enough, break
    if abs(guess - v) < TOL
        break
    end
end