% demonstration showing global error for 1-step integrators

 f = inline('x','t','x');
 t0 = 0;
 x0 = 1; 
 tfinal = 5;

 e_err = [];
 h_err = [];
 rk_err = [];
 h = [];
 for j=1:10
    h(j) = 2^(-j+1);
    [t,x] = euler(f,[t0 tfinal],x0,h(j));
     e_err(j) = abs(exp(t(end))*x0-x(end));
    [t,x] = heun(f,[t0 tfinal],x0,h(j));
     h_err(j) = abs(exp(t(end))*x0-x(end));
    [t,x] = rk4(f,[t0 tfinal],x0,h(j));
     rk_err(j) = abs(exp(t(end))*x0-x(end));
 end

 figure(1), clf
 loglog(h,e_err, 'r.','markersize',15); hold on
 loglog(h,e_err,'r-', 'linewidth',1.5);
 loglog(h,h_err, 'b.','markersize',15); hold on
 loglog(h,h_err,'b-', 'linewidth',1.5);
 loglog(h,rk_err, 'k.','markersize',15); hold on
 loglog(h,rk_err,'k-', 'linewidth',1.5);
      
 xlabel('step size, h','fontsize',16)
 ylabel('error at t=5 ','fontsize',16)
 title('error in one-step methods for various step-sizes, h','fontsize',16)
 set(gca,'fontsize',16)
 set(gca,'Xdir','reverse')