chapter xii, example 4. quasi-interpolant with good knots.
c  from  * a practical guide to splines *  by c. de boor
calls bsplpp(bsplvb)
c
      integer i,irate,istep,j,l,m,mmk,n,nlow,nhigh,nm1
      real algerp,aloger,bcoef(22),break(20),c(4,20),decay,dg,ddg
     *    ,dtip1,dtip2,dx,errmax,g,h,pnatx,scrtch(4,4),step,t(26),taui
c     istep and step = float(istep) specify point density for error det-
c     ermination.
      data step, istep /20., 20/
c        g  is the function to be approximated,  dg  is its first, and
c        ddg  its second derivative .
      g(x) = sqrt(x+1.)
      dg(x) = .5/g(x)
      ddg(x) = -.5*dg(x)/(x+1.)
      decay = 0.
c      read in the exponent  irate  for the knot distribution and the
c      lower and upper limit for the number  n  .
      read 500,irate,nlow,nhigh
  500 format(3i3)
      print 600
  600 format(28h  n   max.error   decay exp./)
c                              loop over  n = dim( spline(4,t) ) - 2 .
c              n  is chosen as the parameter in order to afford compar-
c              ison with examples 2 and 3  in which cubic spline interp-
c              olation at  n  data points was used .
      do 40 n=nlow,nhigh,2
         nm1 = n-1
         h = 1./float(nm1)
         m = n+2
         mmk = m-4
         do 5 i=1,4
            t(i) = -1.
    5       t(m+i) = 1.
c                 interior knots are equidistributed with respect to the
c                 function  (x + 1)**(1/irate) .
         do 6 i=1,mmk
    6       t(i+4) = 2.*(float(i)*h)**irate - 1.
c                                           construct quasi-interpolant.
c                                                bcoef(1) = g(-1.) = 0.
         bcoef(1) = 0.
         dtip2 = t(5) - t(4)
         taui = t(5)
c                           special choice of  tau(2)  to avoid infinite
c                           derivatives of  g  at left endpoint .
         bcoef(2) = g(taui) - 2.*dtip2*dg(taui)/3.
     *             + dtip2**2*ddg(taui)/6.
         do 15 i=3,m
            taui = t(i+2)
            dtip1 = dtip2
            dtip2 = t(i+3) - t(i+2)
c                                      formula xii(30) of text is used .
   15       bcoef(i) = g(taui) + (dtip2-dtip1)*dg(taui)/3.
     *               - dtip1*dtip2*ddg(taui)/6.
c                                         convert to pp-representation .
         call bsplpp(t,bcoef,m,4,scrtch,break,c,l)
c                            estimate max.interpolation error on (-1,1).
         errmax = 0.
c                                             loop over cubic pieces ...
         do 30 i=1,l
            dx = (break(i+1)-break(i))/step
c                       error is calculated at  istep  points per 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(break(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