function TotalNumericalError
% My implementation of example 4.5
% Find the first derivative of f(x) at
% x = 0.5 using central difference methods

x = 0.5;
h = 1;
n = 10;

dftrue = -0.4*x^3-0.45*x^2-x-0.25;
df = zeros(n,1);
step = zeros(n,1);
error = zeros(n,1);

% Use second order central difference derivative equation
% for repeatedly smaller values of the stepsize h
for i = 1:n
    df(i,1) = (f(x+h)-f(x-h))/(2*h);
    step(i,1) = h;
    error(i,1) = abs(dftrue-df(i,1));
    
    h = 0.1*h;
end

loglog(step,error,'ro-')
xlabel('stepsize')
ylabel('error')
legend('2nd order approx')
hold on
pause

% Now use fourth order central difference derivative equation
% for repeatedly smaller values of the stepsize h
h = 1;
for i = 1:n
    df(i,1) = (-f(x+2*h)+8*f(x+h)-8*f(x-h)+f(x-2*h))/(12*h);
    step(i,1) = h;
    error(i,1) = abs(dftrue-df(i,1));
    
    h = 0.1*h;
end

loglog(step,error,'bo-')
legend('2nd order approx','4th order approx')

function y = f(x)
y = -0.1*x^4-0.15*x^3-0.5*x^2-0.25*x+1.2;