Bisection 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.m

%% Newtonian vs. Relativistic Energy Accuracy
% 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;            

% Bisection parameters
maxiters = 100;
hi = c;
lo = 0;
% How accurate we are in our calculation.                       
% Not to be confused with tolerance of energy calculation (aka error).
TOL = .0001; 

% Use bisection to determine max velocity to keep energy within
% the specified tolerance.
for i=1:maxiters  
    % "Guess" the velocity
    v = (hi - lo) / 2 + lo;    
    
    % 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
    
    % Calculate energy error
    err = abs(K - E) / E;
    fprintf('%4d:  %14.4f  %8.6fc  %10.8f\n', i, v, v/c, err)
    
    % Error is too big, try a lower velocity
    if err > tolerance
        hi = v;
    % Error is within tolerance
    else  
        % Not close enough to target value, increase velocity
        if abs(err - tolerance) > TOL
            lo = v;
        % Error is within accepted tolerance, we're done. 
        else
            break
        end
    end
end