chapter xii, example 2. cubic spline interpolation with good knots
c  from  * a practical guide to splines *  by c. de boor
calls  cubspl, newnot
      integer i,istep,iter,itermx,j,n,nhigh,nlow,nmax,nm1
      parameter (nmax=20)
      real algerp,aloger,decay,dx,errmax,c(4,nmax),g,h,pnatx
     *    ,scrtch(2,nmax),step,tau(nmax),taunew(nmax)
C     real algerp,aloger,decay,dx,errmax,c(4,20),g,h,pnatx
C    *    ,scrtch(2,20),step,tau(20),taunew(20)
c     istep and step = float(istep) specify point density for error det-
c     ermination.
      data step, istep /20., 20/
c                               the function  g  is to be interpolated .
      g(x) = sqrt(x+1.)
      decay = 0.
c      read in the number of iterations to be carried out and the lower
c      and upper limit for the number  n  of data points to be used.
      read 500,itermx,nlow,nhigh
  500 format(3i3)
      print 600, itermx
  600 format(i4,22h cycles through newnot//
     *      28h  n   max.error   decay exp./)
c                                 loop over  n = number of data points .
      do 40 n=nlow,nhigh,2
c                                        knots are initially equispaced.
         nm1 = n-1
         h = 2./float(nm1)
         do 10 i=1,n
   10       tau(i) = float(i-1)*h - 1.
         iter = 1
c               construct cubic spline interpolant. then, itermx  times,
c                determine new knots from it and find a new interpolant.
   11       do 15 i=1,n
   15          c(1,i) = g(tau(i))
            call cubspl ( tau, c, n, 0, 0 )
            if (iter .gt. itermx)       go to 19
            iter = iter+1
            call newnot(tau,c,nm1,4,taunew,nm1,scrtch)
            do 18 i=1,n
   18          tau(i) = taunew(i)
                                        go to 11
   19    continue
c                            estimate max.interpolation error on (-1,1).
         errmax = 0.
c                           loop over polynomial pieces of interpolant .
         do 30 i=1,nm1
            dx = (tau(i+1)-tau(i))/step
c                interpolation error is calculated at  istep  points per
c                    polynomial piece .
            do 30 j=1,istep
               h = float(j)*dx
               pnatx = c(1,i)+h*(c(2,i)+h*(c(3,i)+h*c(4,i)/3.)/2.)
   30       errmax = amax1(errmax,abs(g(tau(i)+h)-pnatx))
c                                             calculate decay exponent .
         aloger = alog(errmax)
         if (n .gt. nlow)  decay =
     *       (aloger - algerp)/alog(float(n)/float(n-2))
         algerp = aloger
c
   40    print 640,n,errmax,decay
  640 format(i3,e12.4,f11.2)
                                        stop
      end