Exercise "adaptive integration"

1. Implement a recursive adaptive integrator which estimates the integral with a required absolute (acc) and relative (eps) accuracy:
   function adapt24(f,a,b,acc,eps,oldfs){ var x=[1/6,2/6,4/6,5/6];// abscissas var w=[2/6,1/6,1/6,2/6];// high order var p=[1,0,0,1];// new points? var v=[1/2,1/2,1/2,1/2];// low order var n=x.length, h=b-a; if(typeof(oldfs)=="undefined")// first call fs=[f(a+x[i]*h) for(i in x)]; else for(let k=0,i=0;i<n;i++){ if(p[i])fs[i]=f(a+x[i]*h); else fs[i]=oldfs[k++];} for(var q4=q2=i=0;i<n;i++){ q4+=w[i]*fs[i]*h; q2+=v[i]*fs[i]*h/2;} var tol=acc+eps*Math.abs(q4) var err=Math.abs(q4-q2)/3 if(err<tol) return [q4, err] else{ acc/=Math.sqrt(2.) var mid=(a+b)/2 var left=[fs[i]for(i in fs)if(i< n/2)] var rght=[fs[i]for(i in fs)if(i>=n/2)] var [ql,el]=adapt24(f,a,mid,eps,acc,left) var [qr,er]=adapt24(f,mid,b,eps,acc,rght) return [ql+qr, Math.sqrt(el*el+er*er)] } }
2. Calculate ∫₀^{1   ln(x)}/_√[x] dx = -4 with acc=eps=0.001 and estimate the number of integrand evaluations.
3. Test your implementation on some other interesting integrals.
4. Implement your own classical quadrature with your favourite set of functions, eg {1/√x,ln(x),√x,1,x,...}, and a corresponding adaptive integrator.
5. A definite integral ∫_a^bf(x)dx can be refurmulated as an ODE, y'=f(x), y(a)=0, y(b)=?, which can be solved with your adaptive ODE solver. Pick an interesing f(x) and compare the effectiveness of your ODE drivers with your adaptive integrator.