eulers improved method code error

tried to solve ode using eulers improved method for a function of F=2xy

with step size h=0.1,y0=1,x0=1 and seeking y(1.5)

The code is as follows

function yout=improvedeuler3(F,x0,h,xfinal,y0);
y=y0;
yout=y;

for x=x0:h:xfinal-h;
    s1=F(x,y);            
    s2=F(x+h/2,y+h*s1/2); %% is it correct for modified euler
    y=y+h*s2;
    yout=[yout,y];
end
end

initial conditions and conecting script is

F=@(x,y)2*x*y;
x0=1;
h=0.1;
xfinal=1.5;
y0=1;
yout=improvedeuler(F,x0,h,xfinal,y0);

%comparision with exact
x=x0:h:xfinal;
y_exact=exp((x.^2)-1);
plot(x,yout,'r',x,y_exact,'k'),legend('Improved','Exact');

I am getting a matlab output of y values different from that calculated manually

It seems there is an error in the code,unable to makeout.

1 1.2310 1.5452 1.9779 2.5814 3.4348 ------ matlaboutput

1 1.2320 1.5479 1.9832 2.5908 3.4509 ------ manual calculation

[darova]

Try to reduce step and report if something changes

[Syed Haque]

1 1.205 1.4754 1.8352 2.3193 2.9776 with h=0.09

1 1.231 1.5453 1.978 2.5814 3.4348 with h=0.10

1 1.2576 1.6193 2.1345 2.88 with h=0.11 (y1.5 with this step is missing)

something wrong with the code

[darova]

Looks good as for me. What do you think?

Answer 1

[James Tursa]

s2=F(x+h/2,y+h*s1/2); %% is it correct for modified euler

If "Improved" Euler's Method means "Modified" Euler's Method as your comment indicates, then no it is not correct.

Modified Euler is basically finding the derivative at the current point, finding the derivative one full step ahead of the current point, and then using the average of the two derivatives to make your integration step. So this:

    s1=F(x,y);            
    s2=F(x+h/2,y+h*s1/2); %% is it correct for modified euler
    y=y+h*s2;

should look something like this instead (inserting spaces for reading clarity):

    s1 = F(x  , y     ); % derivative at current point            
    s2 = F(x+h, y+h*s1); % derivative at one full step ahead (using predicted point from standard Euler Method)
    y = y + h*(s1 + s2)/2; % integrate forward using the average of the two derivatives

Did you really mean to use the "Midpoint" Method? What is the actual assignment you were given?

[Syed Haque]

Getting an error,assignment is to calculate y1.5 and compare error at all points upto 1.5 with actual.

[James Tursa]

Can you post your current code and the assignment text? Many contributors here (myself included) are not willing to open active documents like Excel or Word etc.

Answer 2

[Syed Haque]

function yout=improveuler(F,x0,h,xfinal,y0);
y=y0;
yout=y;

for x=x0:h:xfinal-h;
    s1=F(x,y);
    s2=F(x+h,y+h*s1);
    y=y+h*(s1+s2)/2;
    yout=[yout,y];
end
end


F=@(x,y)2*x*y;
x0=1;
h=0.1;
xfinal=1.5;
y0=1;
yout=improveuler(F,x0,h,xfinal,y0);

%comparision with exact
x=x0:h:xfinal;
y_exact=exp((x.^2)-1);
plot(x,yout,'r',x,y_exact,'k'),legend('Improved','Exact');

% COMMAND WINDOW ERROR PROMPT
>> improveuler
Not enough input arguments.

Error in improveuler (line 2)
y=y0;

develop improved euler method to solve y1.5 given y'=2xy, y1=1

[Daniel M]

This should not be posted as an answer. And I did not receive an error when running this code.

[James Tursa]

Same for me. When I put this code in a file called improveuler.m

function yout=improveuler(F,x0,h,xfinal,y0);
y=y0;
yout=y;
for x=x0:h:xfinal-h;
    s1=F(x,y);
    s2=F(x+h,y+h*s1);
    y=y+h*(s1+s2)/2;
    yout=[yout,y];
end
end

and this code in a separate file called improveuler_test.m

F=@(x,y)2*x*y;
x0=1;
h=0.1;
xfinal=1.5;
y0=1;
yout=improveuler(F,x0,h,xfinal,y0);
%comparision with exact
x=x0:h:xfinal;
y_exact=exp((x.^2)-1);
plot(x,yout,'r',x,y_exact,'k'),legend('Improved','Exact');

and then type the following at the command line

improveuler_test

I get it to run without errors and produce what looks like a reasonable plot.

[Syed Haque]

Thanks James,could finally achieve the end result,y values are matching with calculated ones.seems all those multiple files of the same assignment was playing havoc,deleted them,pasted the script again from the forum(debugging cropped up-shouldn't have done that) and finally after three/four hits I could see the code run successfully.

Also sorry for posting earlier comment as an answer.