# Subroutine predictor corrector

Hello everyone!!! I'm writing because I should write this subroutine:

Given the Cauchy problem

[img=http://s22.postimg.org/y2vo838hd/Senza_titolo_3.jpg]

write a subroutine for the calculation of its solution using the predictor corrector method in 5 steps (k = 4), according to the PECE scheme, with fixed step.

for the initiation values use the Matlab routine ode45.

So, I have this program but I need of to do some tests but I don't understand what kind of functions I should use...so, you can give me an example of tests that can I do? This is the program:

[code]
% The function calculates the solution of the Cauchy problem
% y'=f(t,y)
% y()=y0

% takes as input a function, the initial condition, the extremes of the %integration interval and the width of interval

function [t,y]=pece(fty,y0,x0,xn,h)
t=0;
p=(xn-x0)/h;
n=length(y0);
for i=1:p+1
t(i)=x0+(i-1)*h;

y=zeros(1,n);
f=zeros(3,n);

for i=1:4
f(i,:)=feval(fty,t(i),y0);
[t0,m_i]=ode45(fty,[t(1) t(2) t(3) t(4)],y0);

y(1:4,:)=m_i(1:4,:);

mi=m_i(4,:);
for i = 4:p
fnew= feval (fty, t0(4), mi);
m_i0=mi+(h/24)*(55*fnew-59*f(3,:)+37*f(2,:)-9*f(1,:));
f_i0= feval (fty, t0(4)+h, m_i0);
mi= mi+(h/24)*(9*f_i0+19*fnew-5*f(3,:)+f(2,:));
f(1,:) = f(2,:);
f(2,:) = f(3,:);
f(3,:) = fnew;
t0 = t0 + h;
y(i+1,:)= mi;
end

end

[/code]

Sorry for my mistakes...but I am Italian!!!! Thanks in advance!!!!