Exercise "adaptive integration"

1. Implement an open quadrature open24 which calculates 2- and 4-point quadratures and returns the estimates of the integral and the error:

   function open24(f,a,b){
   	h=b-a;
   	f1=f(a+h*1/6); f2=f(a+h*2/6); f3=f(a+h*4/6); f4=f(a+h*5/6);
   	i2=(f2+f3)/2*h;
   	i4=(2*f1+f2+f3+2*f4)/6*h;
   	i=i4; err=abs(i4-i2); return {i:i,err:err};}
   

2. Implement an adaptive integrator which estimates the integral with a given absolute (acc) and relative (eps) accuracy:

   function adapt(f,a,b,acc,eps){
   	y=open24(f,a,b); tol = acc+eps*abs(y.i);
   	if (y.err < tol) then return y;
   	else {
   		y1=adapt(f,a,(a+b)/2,acc/√[2],eps);
   		y2=adapt(f,(a+b)/2,b,acc/√[2],eps);
   		return {i:y1.i+y2.i, err:√[y1.err2 + y2.err2]}
   		}
   

3. Reuse points.
4. Calculate ∫₀^{1   ln(x)}/_√[x] dx = -4 with acc=eps=0.001 and estimate the number of need integrand evaluations.
5. Test your implementation on some interesting integrals.