SIMPLE AND MODIFIED EULER METHOD

##Usage:Call Octave from terminal
##and then call EulerMethodUmitAlkus.m
##from octave and finally
##press enter. That's all.
##Simple Euler Method
##Constants and initializations
x=[]; ## initial empty vector for x
y=[]; ## initial empty vector for y
x(1)=1; ## initial value of x
y(1)=1; ## initial value of y
h=1E-3; ## increment in x
dery=[]; ## 1st derivative of y wrt x
dery(1)=0;## 1st entry of dery
n=1; ## inital loop index for while
## enter the while loop for the interval x=[1,2]
while (x(n)<=2)
x(n+1)=x(n)+h;
dery(n+1)=x(n)*x(n)-2*y(n)/x(n); ##given
y(n+1)=y(n)+h*dery(n+1); ##Euler method
n++;
endwhile
##exit from the 1st while loop
##Modified Euler Method
##Constant and initializations
x(1)=1; ## beginnig of the interval [1,2]
ymod(1)=1; ## inital value for modified y.
ymid=[]; ## empty vector function evaluated at x midpoint
xmid=[]; ## empty vector func. of midpoints of the interval h in x-axis.
derymod=[]; ## modified derivatives of ymod wrt x.
derymid=[]; ## derivative of ymid wrt xmid.
derymid(1)=0; ##1st entry of der. of y wrt x at midpoints.
##Enter the 2nd while loop
n=1;
while (x(n)<=2) ## x(n)'s are the beginning values of the interval h.
xmid(n)=x(n)+h/2;
derymod(n)=x(n)*x(n)-2*ymod(n)/x(n); ##given equation
ymid(n)=ymod(n)+derymod(n)*h/2;
derymid(n+1)=xmid(n)*xmid(n)-2*ymid(n)/xmid(n);
ymod(n+1)=ymod(n)+derymid(n)*h; ##modified Euler Method
x(n+1)=x(n)+h;
n++;
endwhile
## Plot for Simple Euler Method
subplot(2,1,1)
hold on
plot(x,dery,'c-');
plot(x,ymod,'r-');
legend('dy/dx','y')
title('Simple Euler Method');
xlabel('x');
hold off
##Plot for Modified Euler Method
subplot(2,1,2)
hold on
plot(x,derymid,'m-');
plot(x,ymod,'g-');
title('Modified Euler Method');
xlabel('x');
legend('dy/dx(mid)','y-mod');
hold off
print('-dpsc','SIMMODEULER.ps')
save -text SIMMODEULER.dat