Ralston's method function

I am trying to write a function for Ralston's method, a second order Runge Kutta method. My function does not appear to be graphing correctly in another script. Is there something wrong with my function?

function [t, y] = ralston( f, tspan, y0, n)

% ralston implement ralston method

a = tspan(1);

b = tspan(2);

h = (b - a)/n;

t = a + h*(0:n)';

y = 0*t; %vector w all elements zero

y(1) = y0;

for i = 1:n

k1 = h*f(t(i), y(i));

k2 = h*f(t(i) + 3*h/4, y(i) + 3*k1/4);

y(i+1) = y(i) + k1/3 + 2*k2/3;

end

end

[darova]

Can you attach function f?

Answer 1

[Jim Riggs]

There is an error in your function in the k2 term. For Ralstons method, it should be:

k1 = h*f(t(i), y(i));
k2 = h*( f( t(i) + 2*h/3, y(i) + 2*k1/3 ) );

[Jim Riggs]

After some further review, there is also an error in y(i+1). Ralston's method should be:

k1 = h*f( t(i), y(i) );
k2 = h*f( t(i) + 2*h/3,  y(i) + 2*k1/3 );
y(i+1) = y(i) + k1/4 + 3*k2/4;

or

k1 = f( t(i), y(i) );
k2 = f( t(i) + 2*h/3, y(i) + 2*h*k1/3 );
y(i+1) = y(i) + h*(k1/4 + 3*k2/4);

Also note that in order for the function to graph as expected, you must supply the correct starting value, y0.