TSpline.cxx

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// @(#)root/hist:$Id: TSpline.cxx 36247 2010-10-10 10:16:42Z brun $
// Author: Federico Carminati   28/02/2000

/*************************************************************************
 * Copyright (C) 1995-2000, Rene Brun and Fons Rademakers.               *
 * All rights reserved.                                                  *
 *                                                                       *
 * For the licensing terms see $ROOTSYS/LICENSE.                         *
 * For the list of contributors see $ROOTSYS/README/CREDITS.             *
 *************************************************************************/

//////////////////////////////////////////////////////////////////////////
//                                                                      //
// TSpline                                                              //
//                                                                      //
// Base class for spline implementation containing the Draw/Paint       //
// methods                                                              //
//                                                                      //
//////////////////////////////////////////////////////////////////////////

#include "TROOT.h"
#include "TSpline.h"
#include "TVirtualPad.h"
#include "TH1.h"
#include "TF1.h"
#include "TSystem.h"
#include "Riostream.h"
#include "TClass.h"
#include "TMath.h"

ClassImp(TSplinePoly)
ClassImp(TSplinePoly3)
ClassImp(TSplinePoly5)
ClassImp(TSpline3)
ClassImp(TSpline5)
ClassImp(TSpline)


//______________________________________________________________________________
TSpline::TSpline(const TSpline &sp) :
  TNamed(sp),
  TAttLine(sp),
  TAttFill(sp),
  TAttMarker(sp),
  fDelta(sp.fDelta),
  fXmin(sp.fXmin),
  fXmax(sp.fXmax),
  fNp(sp.fNp),
  fKstep(sp.fKstep),
  fHistogram(0),
  fGraph(0),
  fNpx(sp.fNpx)
{
   //copy constructor
}


//______________________________________________________________________________
TSpline::~TSpline()
{
   //destructor
   if(fHistogram) delete fHistogram;
   if(fGraph) delete fGraph;
}


//______________________________________________________________________________
TSpline& TSpline::operator=(const TSpline &sp)
{
   //assignment operator
   if(this!=&sp) {
      TNamed::operator=(sp);
      TAttLine::operator=(sp);
      TAttFill::operator=(sp);
      TAttMarker::operator=(sp);
      fDelta=sp.fDelta;
      fXmin=sp.fXmin;
      fXmax=sp.fXmax;
      fNp=sp.fNp;
      fKstep=sp.fKstep;
      fHistogram=0;
      fGraph=0;
      fNpx=sp.fNpx;
   }
   return *this;
}


//______________________________________________________________________________
void TSpline::Draw(Option_t *option)
{
   // Draw this function with its current attributes.
   //
   // Possible option values are:
   //   "SAME"  superimpose on top of existing picture
   //   "L"     connect all computed points with a straight line
   //   "C"     connect all computed points with a smooth curve.
   //   "P"     add a polymarker at each knot
   //
   // Note that the default value is "L". Therefore to draw on top
   // of an existing picture, specify option "LSAME"

   TString opt = option;
   opt.ToLower();
   if (gPad && !opt.Contains("same")) gPad->Clear();

   AppendPad(option);
}


//______________________________________________________________________________
Int_t TSpline::DistancetoPrimitive(Int_t px, Int_t py)
{
   // Compute distance from point px,py to a spline.

   if (!fHistogram) return 999;
   return fHistogram->DistancetoPrimitive(px, py);
}


//______________________________________________________________________________
void TSpline::ExecuteEvent(Int_t event, Int_t px, Int_t py)
{
   // Execute action corresponding to one event.

   if (!fHistogram) return;
   fHistogram->ExecuteEvent(event, px, py);
}


//______________________________________________________________________________
void TSpline::Paint(Option_t *option)
{
   // Paint this function with its current attributes.

   Int_t i;
   Double_t xv;

   TString opt = option;
   opt.ToLower();
   Double_t xmin, xmax, pmin, pmax;
   pmin = gPad->PadtoX(gPad->GetUxmin());
   pmax = gPad->PadtoX(gPad->GetUxmax());
   xmin = fXmin;
   xmax = fXmax;
   if (opt.Contains("same")) {
      if (xmax < pmin) return;  // Otto: completely outside
      if (xmin > pmax) return;
      if (xmin < pmin) xmin = pmin;
      if (xmax > pmax) xmax = pmax;
   } else {
      gPad->Clear();
   }

   //  Create a temporary histogram and fill each channel with the function value
   if (fHistogram)
      if ((!gPad->GetLogx() && fHistogram->TestBit(TH1::kLogX)) ||
          (gPad->GetLogx() && !fHistogram->TestBit(TH1::kLogX)))
         { delete fHistogram; fHistogram = 0;}

   if (fHistogram) {
      //if (xmin != fXmin || xmax != fXmax)
      fHistogram->GetXaxis()->SetLimits(xmin,xmax);
   } else {
   //      if logx, we must bin in logx and not in x !!!
   //      otherwise if several decades, one gets crazy results
      if (xmin > 0 && gPad->GetLogx()) {
         Double_t *xbins  = new Double_t[fNpx+1];
         Double_t xlogmin = TMath::Log10(xmin);
         Double_t xlogmax = TMath::Log10(xmax);
         Double_t dlogx   = (xlogmax-xlogmin)/((Double_t)fNpx);
         for (i=0;i<=fNpx;i++) {
            xbins[i] = gPad->PadtoX(xlogmin+ i*dlogx);
         }
         fHistogram = new TH1F("Spline",GetTitle(),fNpx,xbins);
         fHistogram->SetBit(TH1::kLogX);
         delete [] xbins;
      } else {
         fHistogram = new TH1F("Spline",GetTitle(),fNpx,xmin,xmax);
      }
      if (!fHistogram) return;
      fHistogram->SetDirectory(0);
   }
   for (i=1;i<=fNpx;i++) {
      xv = fHistogram->GetBinCenter(i);
      fHistogram->SetBinContent(i,this->Eval(xv));
   }

   // Copy Function attributes to histogram attributes
   fHistogram->SetBit(TH1::kNoStats);
   fHistogram->SetLineColor(GetLineColor());
   fHistogram->SetLineStyle(GetLineStyle());
   fHistogram->SetLineWidth(GetLineWidth());
   fHistogram->SetFillColor(GetFillColor());
   fHistogram->SetFillStyle(GetFillStyle());
   fHistogram->SetMarkerColor(GetMarkerColor());
   fHistogram->SetMarkerStyle(GetMarkerStyle());
   fHistogram->SetMarkerSize(GetMarkerSize());

   //  Draw the histogram
   //  but first strip off the 'p' option if any
   char *o = (char *) opt.Data();
   Int_t j=0;
   i=0;
   Bool_t graph=kFALSE;
   do
      if(o[i]=='p') graph=kTRUE ; else o[j++]=o[i];
   while(o[i++]);
   if (opt.Length() == 0 ) fHistogram->Paint("lf");
   else if (opt == "same") fHistogram->Paint("lfsame");
   else                    fHistogram->Paint(opt.Data());

   // Think about the graph, if demanded
   if(graph) {
      if(!fGraph) {
         Double_t *xx = new Double_t[fNp];
         Double_t *yy = new Double_t[fNp];
         for(i=0; i<fNp; ++i)
            GetKnot(i,xx[i],yy[i]);
         fGraph=new TGraph(fNp,xx,yy);
         delete [] xx;
         delete [] yy;
      }
      fGraph->SetMarkerColor(GetMarkerColor());
      fGraph->SetMarkerStyle(GetMarkerStyle());
      fGraph->SetMarkerSize(GetMarkerSize());
      fGraph->Paint("p");
   }
}


//______________________________________________________________________________
void TSpline::Streamer(TBuffer &R__b)
{
   // Stream an object of class TSpline.

   if (R__b.IsReading()) {
      UInt_t R__s, R__c;
      Version_t R__v = R__b.ReadVersion(&R__s, &R__c);
      if (R__v > 1) {
         R__b.ReadClassBuffer(TSpline::Class(), this, R__v, R__s, R__c);
         return;
      }
      //====process old versions before automatic schema evolution
      TNamed::Streamer(R__b);
      TAttLine::Streamer(R__b);
      TAttFill::Streamer(R__b);
      TAttMarker::Streamer(R__b);

      fNp = 0;
      /*
      R__b >> fDelta;
      R__b >> fXmin;
      R__b >> fXmax;
      R__b >> fNp;
      R__b >> fKstep;
      R__b >> fHistogram;
      R__b >> fGraph;
      R__b >> fNpx;
      */
      R__b.CheckByteCount(R__s, R__c, TSpline::IsA());
      //====end of old versions

   } else {
      R__b.WriteClassBuffer(TSpline::Class(),this);
   }
}


//////////////////////////////////////////////////////////////////////////
//                                                                      //
// TSplinePoly                                                          //
//                                                                      //
// Base class for TSpline knot                                          //
//                                                                      //
//////////////////////////////////////////////////////////////////////////


//______________________________________________________________________________
TSplinePoly &TSplinePoly::operator=(TSplinePoly const &other)
{
   //assignment operator
   if(this != &other) {
      TObject::operator=(other);
      CopyPoly(other);
   }
   return *this;
}

//______________________________________________________________________________
void TSplinePoly::CopyPoly(TSplinePoly const &other)
{
   //utility called by the copy constructors and = operator
   fX = other.fX;
   fY = other.fY;
}

//////////////////////////////////////////////////////////////////////////
//                                                                      //
// TSplinePoly3                                                         //
//                                                                      //
// Class for TSpline3 knot                                              //
//                                                                      //
//////////////////////////////////////////////////////////////////////////


//______________________________________________________________________________
TSplinePoly3 &TSplinePoly3::operator=(TSplinePoly3 const &other)
{
   //assignment operator
   if(this != &other) {
      TSplinePoly::operator=(other);
      CopyPoly(other);
   }
   return *this;
}

//______________________________________________________________________________
void TSplinePoly3::CopyPoly(TSplinePoly3 const &other)
{
   //utility called by the copy constructors and = operator
   fB = other.fB;
   fC = other.fC;
   fD = other.fD;
}

//////////////////////////////////////////////////////////////////////////
//                                                                      //
// TSplinePoly5                                                         //
//                                                                      //
// Class for TSpline5 knot                                              //
//                                                                      //
//////////////////////////////////////////////////////////////////////////


//______________________________________________________________________________
TSplinePoly5 &TSplinePoly5::operator=(TSplinePoly5 const &other)
{
   //assignment operator
   if(this != &other) {
      TSplinePoly::operator=(other);
      CopyPoly(other);
   }
   return *this;
}

//______________________________________________________________________________
void TSplinePoly5::CopyPoly(TSplinePoly5 const &other)
{
   //utility called by the copy constructors and = operator
   fB = other.fB;
   fC = other.fC;
   fD = other.fD;
   fE = other.fE;
   fF = other.fF;
}



//////////////////////////////////////////////////////////////////////////
//                                                                      //
// TSpline3                                                             //
//                                                                      //
// Class to create third splines to interpolate knots                   //
// Arbitrary conditions can be introduced for first and second          //
// derivatives at beginning and ending points                           //
//                                                                      //
//////////////////////////////////////////////////////////////////////////


//______________________________________________________________________________
TSpline3::TSpline3(const char *title,
                   Double_t x[], Double_t y[], Int_t n, const char *opt,
                   Double_t valbeg, Double_t valend) :
  TSpline(title,-1,x[0],x[n-1],n,kFALSE),
  fValBeg(valbeg), fValEnd(valend), fBegCond(0), fEndCond(0)
{
   // Third spline creator given an array of
   // arbitrary knots in increasing abscissa order and
   // possibly end point conditions

   fName="Spline3";

   // Set endpoint conditions
   if(opt) SetCond(opt);

   // Create the plynomial terms and fill
   // them with node information
   fPoly = new TSplinePoly3[n];
   for (Int_t i=0; i<n; ++i) {
      fPoly[i].X() = x[i];
      fPoly[i].Y() = y[i];
   }

   // Build the spline coefficients
   BuildCoeff();
}

//______________________________________________________________________________
TSpline3::TSpline3(const char *title,
                   Double_t xmin, Double_t xmax,
                   Double_t y[], Int_t n, const char *opt,
                   Double_t valbeg, Double_t valend) :
  TSpline(title,(xmax-xmin)/(n-1), xmin, xmax, n, kTRUE),
  fValBeg(valbeg), fValEnd(valend),
  fBegCond(0), fEndCond(0)
{
   // Third spline creator given an array of
   // arbitrary function values on equidistant n abscissa
   // values from xmin to xmax and possibly end point conditions

   fName="Spline3";

   // Set endpoint conditions
   if(opt) SetCond(opt);

   // Create the plynomial terms and fill
   // them with node information
   fPoly = new TSplinePoly3[n];
   for (Int_t i=0; i<n; ++i) {
      fPoly[i].X() = fXmin+i*fDelta;
      fPoly[i].Y() = y[i];
   }

   // Build the spline coefficients
   BuildCoeff();
}


//______________________________________________________________________________
TSpline3::TSpline3(const char *title,
                   Double_t x[], const TF1 *func, Int_t n, const char *opt,
                   Double_t valbeg, Double_t valend) :
  TSpline(title,-1, x[0], x[n-1], n, kFALSE),
  fValBeg(valbeg), fValEnd(valend),
  fBegCond(0), fEndCond(0)
{
   // Third spline creator given an array of
   // arbitrary abscissas in increasing order and a function
   // to interpolate and possibly end point conditions

   fName="Spline3";

   // Set endpoint conditions
   if(opt) SetCond(opt);

   // Create the plynomial terms and fill
   // them with node information
   fPoly = new TSplinePoly3[n];
   for (Int_t i=0; i<n; ++i) {
      fPoly[i].X() = x[i];
      fPoly[i].Y() = ((TF1*)func)->Eval(x[i]);
   }

   // Build the spline coefficients
   BuildCoeff();
}


//______________________________________________________________________________
TSpline3::TSpline3(const char *title,
                   Double_t xmin, Double_t xmax,
                   const TF1 *func, Int_t n, const char *opt,
                   Double_t valbeg, Double_t valend) :
  TSpline(title,(xmax-xmin)/(n-1), xmin, xmax, n, kTRUE),
  fValBeg(valbeg), fValEnd(valend),
  fBegCond(0), fEndCond(0)
{
   // Third spline creator given a function to be
   // evaluated on n equidistand abscissa points between xmin
   // and xmax and possibly end point conditions

   fName="Spline3";

   // Set endpoint conditions
   if(opt) SetCond(opt);

   // Create the plynomial terms and fill
   // them with node information
   fPoly = new TSplinePoly3[n];
   //when func is null we return. In this case it is assumed that the spline
   //points will be given later via SetPoint and SetPointCoeff
   if (!func) {fKstep = kFALSE; fDelta = -1; return;}
   for (Int_t i=0; i<n; ++i) {
      Double_t x=fXmin+i*fDelta;
      fPoly[i].X() = x;
      fPoly[i].Y() = ((TF1*)func)->Eval(x);
   }

   // Build the spline coefficients
   BuildCoeff();
}


//______________________________________________________________________________
TSpline3::TSpline3(const char *title,
                   const TGraph *g, const char *opt,
                   Double_t valbeg, Double_t valend) :
  TSpline(title,-1,0,0,g->GetN(),kFALSE),
  fValBeg(valbeg), fValEnd(valend),
  fBegCond(0), fEndCond(0)
{
   // Third spline creator given a TGraph with
   // abscissa in increasing order and possibly end
   // point conditions

   fName="Spline3";

   // Set endpoint conditions
   if(opt) SetCond(opt);

   // Create the plynomial terms and fill
   // them with node information
   fPoly = new TSplinePoly3[fNp];
   for (Int_t i=0; i<fNp; ++i) {
      Double_t xx, yy;
      g->GetPoint(i,xx,yy);
      fPoly[i].X()=xx;
      fPoly[i].Y()=yy;
   }
   fXmin = fPoly[0].X();
   fXmax = fPoly[fNp-1].X();

   // Build the spline coefficients
   BuildCoeff();
}


//______________________________________________________________________________
TSpline3::TSpline3(const TH1 *h, const char *opt,
                   Double_t valbeg, Double_t valend) :
  TSpline(h->GetTitle(),-1,0,0,h->GetNbinsX(),kFALSE),
  fValBeg(valbeg), fValEnd(valend),
  fBegCond(0), fEndCond(0)
{
   // Third spline creator given a TH1 

   fName=h->GetName();

   // Set endpoint conditions
   if(opt) SetCond(opt);

   // Create the plynomial terms and fill
   // them with node information
   fPoly = new TSplinePoly3[fNp];
   for (Int_t i=0; i<fNp; ++i) {
      fPoly[i].X()=h->GetXaxis()->GetBinCenter(i+1);
      fPoly[i].Y()=h->GetBinContent(i+1);
   }
   fXmin = fPoly[0].X();
   fXmax = fPoly[fNp-1].X();

   // Build the spline coefficients
   BuildCoeff();
}


//______________________________________________________________________________
TSpline3::TSpline3(const TSpline3& sp3) :
  TSpline(sp3),
  fPoly(0),
  fValBeg(sp3.fValBeg),
  fValEnd(sp3.fValEnd),
  fBegCond(sp3.fBegCond),
  fEndCond(sp3.fEndCond)
{
   //copy constructor
   if (fNp > 0) fPoly = new TSplinePoly3[fNp];
   for (Int_t i=0; i<fNp; ++i) 
      fPoly[i] = sp3.fPoly[i];
}


//______________________________________________________________________________
TSpline3& TSpline3::operator=(const TSpline3& sp3)
{
   //assignment operator
   if(this!=&sp3) {
      TSpline::operator=(sp3);
      fPoly= 0;
      if (fNp > 0) fPoly = new TSplinePoly3[fNp];
      for (Int_t i=0; i<fNp; ++i) 
         fPoly[i] = sp3.fPoly[i];
      
      fValBeg=sp3.fValBeg;
      fValEnd=sp3.fValEnd;
      fBegCond=sp3.fBegCond;
      fEndCond=sp3.fEndCond;
   }
   return *this;
}


//______________________________________________________________________________
void TSpline3::SetCond(const char *opt)
{
   // Check the boundary conditions

   const char *b1 = strstr(opt,"b1");
   const char *e1 = strstr(opt,"e1");
   const char *b2 = strstr(opt,"b2");
   const char *e2 = strstr(opt,"e2");
   if (b1 && b2)
      Error("SetCond","Cannot specify first and second derivative at first point");
   if (e1 && e2)
      Error("SetCond","Cannot specify first and second derivative at last point");
   if (b1) fBegCond=1;
   else if (b2) fBegCond=2;
   if (e1) fEndCond=1;
   else if (e2) fEndCond=2;
}


//______________________________________________________________________________
void TSpline3::Test()
{
   // Test method for TSpline5
   //
   //   n          number of data points.
   //   m          2*m-1 is order of spline.
   //                 m = 2 always for third spline.
   //   nn,nm1,mm,
   //   mm1,i,k,
   //   j,jj       temporary integer variables.
   //   z,p        temporary double precision variables.
   //   x[n]       the sequence of knots.
   //   y[n]       the prescribed function values at the knots.
   //   a[200][4]  two dimensional array whose columns are
   //                 the computed spline coefficients
   //   diff[3]    maximum values of differences of values and
   //                 derivatives to right and left of knots.
   //   com[3]     maximum values of coefficients.
   //
   //
   //   test of TSpline3 with nonequidistant knots and
   //      equidistant knots follows.

   Double_t hx;
   Double_t diff[3];
   Double_t a[800], c[4];
   Int_t i, j, k, m, n;
   Double_t x[200], y[200], z;
   Int_t jj, mm;
   Int_t mm1, nm1;
   Double_t com[3];
   printf("1         TEST OF TSpline3 WITH NONEQUIDISTANT KNOTS\n");
   n = 5;
   x[0] = -3;
   x[1] = -1;
   x[2] = 0;
   x[3] = 3;
   x[4] = 4;
   y[0] = 7;
   y[1] = 11;
   y[2] = 26;
   y[3] = 56;
   y[4] = 29;
   m = 2;
   mm = m << 1;
   mm1 = mm-1;
   printf("\n-N = %3d    M =%2d\n",n,m);
   TSpline3 *spline = new TSpline3("Test",x,y,n);
   for (i = 0; i < n; ++i)
      spline->GetCoeff(i,hx, a[i],a[i+200],a[i+400],a[i+600]);
   delete spline;
   for (i = 0; i < mm1; ++i) diff[i] = com[i] = 0;
   for (k = 0; k < n; ++k) {
      for (i = 0; i < mm; ++i) c[i] = a[k+i*200];
      printf(" ---------------------------------------%3d --------------------------------------------\n",k+1);
      printf("%12.8f\n",x[k]);
      if (k == n-1) {
         printf("%16.8f\n",c[0]);
      } else {
         for (i = 0; i < mm; ++i) printf("%16.8f",c[i]);
         printf("\n");
         for (i = 0; i < mm1; ++i)
            if ((z=TMath::Abs(a[k+i*200])) > com[i]) com[i] = z;
         z = x[k+1]-x[k];
         for (i = 1; i < mm; ++i)
            for (jj = i; jj < mm; ++jj) {
               j = mm+i-jj;
               c[j-2] = c[j-1]*z+c[j-2];
            }
         for (i = 0; i < mm; ++i) printf("%16.8f",c[i]);
         printf("\n");
         for (i = 0; i < mm1; ++i)
            if (!(k >= n-2 && i != 0))
               if((z = TMath::Abs(c[i]-a[k+1+i*200]))
                  > diff[i]) diff[i] = z;
      }
   }
   printf("  MAXIMUM ABSOLUTE VALUES OF DIFFERENCES \n");
   for (i = 0; i < mm1; ++i) printf("%18.9E",diff[i]);
   printf("\n");
   printf("  MAXIMUM ABSOLUTE VALUES OF COEFFICIENTS \n");
   if (TMath::Abs(c[0]) > com[0])
      com[0] = TMath::Abs(c[0]);
   for (i = 0; i < mm1; ++i) printf("%16.8f",com[i]);
   printf("\n");
   m = 2;
   for (n = 10; n <= 100; n += 10) {
      mm = m << 1;
      mm1 = mm-1;
      nm1 = n-1;
      for (i = 0; i < nm1; i += 2) {
         x[i] = i+1;
         x[i+1] = i+2;
         y[i] = 1;
         y[i+1] = 0;
      }
      if (n % 2 != 0) {
         x[n-1] = n;
         y[n-1] = 1;
      }
      printf("\n-N = %3d    M =%2d\n",n,m);
      spline = new TSpline3("Test",x,y,n);
      for (i = 0; i < n; ++i)
         spline->GetCoeff(i,hx,a[i],a[i+200],a[i+400],a[i+600]);
      delete spline;
      for (i = 0; i < mm1; ++i)
         diff[i] = com[i] = 0;
      for (k = 0; k < n; ++k) {
         for (i = 0; i < mm; ++i)
            c[i] = a[k+i*200];
         if (n < 11) {
            printf(" ---------------------------------------%3d --------------------------------------------\n",k+1);
            printf("%12.8f\n",x[k]);
            if (k == n-1) printf("%16.8f\n",c[0]);
         }
         if (k == n-1) break;
         if (n <= 10) {
            for (i = 0; i < mm; ++i) printf("%16.8f",c[i]);
            printf("\n");
         }
         for (i = 0; i < mm1; ++i)
         if ((z=TMath::Abs(a[k+i*200])) > com[i])
            com[i] = z;
         z = x[k+1]-x[k];
         for (i = 1; i < mm; ++i)
            for (jj = i; jj < mm; ++jj) {
               j = mm+i-jj;
               c[j-2] = c[j-1]*z+c[j-2];
            }
         if (n <= 10) {
            for (i = 0; i < mm; ++i) printf("%16.8f",c[i]);
            printf("\n");
         }
         for (i = 0; i < mm1; ++i)
         if (!(k >= n-2 && i != 0))
            if ((z = TMath::Abs(c[i]-a[k+1+i*200]))
               > diff[i]) diff[i] = z;
      }
      printf("  MAXIMUM ABSOLUTE VALUES OF DIFFERENCES \n");
      for (i = 0; i < mm1; ++i) printf("%18.9E",diff[i]);
      printf("\n");
      printf("  MAXIMUM ABSOLUTE VALUES OF COEFFICIENTS \n");
      if (TMath::Abs(c[0]) > com[0])
         com[0] = TMath::Abs(c[0]);
      for (i = 0; i < mm1; ++i) printf("%16.8E",com[i]);
         printf("\n");
   }
}


//______________________________________________________________________________
Int_t TSpline3::FindX(Double_t x) const
{
   // Find X

   Int_t klow=0, khig=fNp-1;
   //
   // If out of boundaries, extrapolate
   // It may be badly wrong
   if(x<=fXmin) klow=0;
   else if(x>=fXmax) klow=khig;
   else {
      if(fKstep) {
         //
         // Equidistant knots, use histogramming
         klow = TMath::FloorNint((x-fXmin)/fDelta);
         // Correction for rounding errors
         if (x < fPoly[klow].X())
            klow = TMath::Max(klow-1,0);
         else if (klow < khig) {
            if (x > fPoly[klow+1].X()) ++klow;
         }
      } else {
         Int_t khalf;
         //
         // Non equidistant knots, binary search
         while(khig-klow>1)
            if(x>fPoly[khalf=(klow+khig)/2].X())
               klow=khalf;
            else
               khig=khalf;
         //
         // This could be removed, sanity check
         if(!(fPoly[klow].X()<=x && x<=fPoly[klow+1].X()))
            Error("Eval",
                  "Binary search failed x(%d) = %f < x= %f < x(%d) = %f\n",
                  klow,fPoly[klow].X(),x,klow+1,fPoly[klow+1].X());
      }
   }
   return klow;
}


//______________________________________________________________________________
Double_t TSpline3::Eval(Double_t x) const
{
   // Eval this spline at x

   Int_t klow=FindX(x);
   if (klow >= fNp-1) klow = fNp-2; //see: https://savannah.cern.ch/bugs/?71651
   return fPoly[klow].Eval(x);
}


//______________________________________________________________________________
Double_t TSpline3::Derivative(Double_t x) const
{
   // Derivative

   Int_t klow=FindX(x);
   if (klow >= fNp-1) klow = fNp-2; //see: https://savannah.cern.ch/bugs/?71651
   return fPoly[klow].Derivative(x);
}


//______________________________________________________________________________
void TSpline3::SaveAs(const char *filename, Option_t * /*option*/) const
{
   // write this spline as a C++ function that can be executed without ROOT
   // the name of the function is the name of the file up to the "." if any

   //open the file
   ofstream *f = new ofstream(filename,ios::out);
   if (f == 0 || gSystem->AccessPathName(filename,kWritePermission)) {
      Error("SaveAs","Cannot open file:%s\n",filename);
      return;
   }

   //write the function name and the spline constants
   char buffer[512];
   Int_t nch = strlen(filename);
   snprintf(buffer,512,"double %s",filename);
   char *dot = strstr(buffer,".");
   if (dot) *dot = 0;
   strlcat(buffer,"(double x) {\n",512);
   nch = strlen(buffer); f->write(buffer,nch);
   snprintf(buffer,512,"   const int fNp = %d, fKstep = %d;\n",fNp,fKstep);
   nch = strlen(buffer); f->write(buffer,nch);
   snprintf(buffer,512,"   const double fDelta = %g, fXmin = %g, fXmax = %g;\n",fDelta,fXmin,fXmax);
   nch = strlen(buffer); f->write(buffer,nch);

   //write the spline coefficients
   //array fX
   snprintf(buffer,512,"   const double fX[%d] = {",fNp);
   nch = strlen(buffer); f->write(buffer,nch);
   buffer[0] = 0;
   Int_t i;
   char numb[20];
   for (i=0;i<fNp;i++) {
      snprintf(numb,20," %g,",fPoly[i].X());
      nch = strlen(numb);
      if (i == fNp-1) numb[nch-1]=0;
      strlcat(buffer,numb,512);
      if (i%5 == 4 || i == fNp-1) {
         nch = strlen(buffer); f->write(buffer,nch);
         if (i != fNp-1) snprintf(buffer,512,"\n                       ");
      }
   }
   snprintf(buffer,512," };\n");
   nch = strlen(buffer); f->write(buffer,nch);
   //array fY
   snprintf(buffer,512,"   const double fY[%d] = {",fNp);
   nch = strlen(buffer); f->write(buffer,nch);
   buffer[0] = 0;
   for (i=0;i<fNp;i++) {
      snprintf(numb,20," %g,",fPoly[i].Y());
      nch = strlen(numb);
      if (i == fNp-1) numb[nch-1]=0;
      strlcat(buffer,numb,512);
      if (i%5 == 4 || i == fNp-1) {
         nch = strlen(buffer); f->write(buffer,nch);
         if (i != fNp-1) snprintf(buffer,512,"\n                       ");
      }
   }
   snprintf(buffer,512," };\n");
   nch = strlen(buffer); f->write(buffer,nch);
   //array fB
   snprintf(buffer,512,"   const double fB[%d] = {",fNp);
   nch = strlen(buffer); f->write(buffer,nch);
   buffer[0] = 0;
   for (i=0;i<fNp;i++) {
      snprintf(numb,20," %g,",fPoly[i].B());
      nch = strlen(numb);
      if (i == fNp-1) numb[nch-1]=0;
      strlcat(buffer,numb,512);
      if (i%5 == 4 || i == fNp-1) {
         nch = strlen(buffer); f->write(buffer,nch);
         if (i != fNp-1) snprintf(buffer,512,"\n                       ");
      }
   }
   snprintf(buffer,512," };\n");
   nch = strlen(buffer); f->write(buffer,nch);
   //array fC
   snprintf(buffer,512,"   const double fC[%d] = {",fNp);
   nch = strlen(buffer); f->write(buffer,nch);
   buffer[0] = 0;
   for (i=0;i<fNp;i++) {
      snprintf(numb,20," %g,",fPoly[i].C());
      nch = strlen(numb);
      if (i == fNp-1) numb[nch-1]=0;
      strlcat(buffer,numb,512);
      if (i%5 == 4 || i == fNp-1) {
         nch = strlen(buffer); f->write(buffer,nch);
         if (i != fNp-1) snprintf(buffer,512,"\n                       ");
      }
   }
   snprintf(buffer,512," };\n");
   nch = strlen(buffer); f->write(buffer,nch);
    //array fD
   snprintf(buffer,512,"   const double fD[%d] = {",fNp);
   nch = strlen(buffer); f->write(buffer,nch);
   buffer[0] = 0;
   for (i=0;i<fNp;i++) {
      snprintf(numb,20," %g,",fPoly[i].D());
      nch = strlen(numb);
      if (i == fNp-1) numb[nch-1]=0;
      strlcat(buffer,numb,512);
      if (i%5 == 4 || i == fNp-1) {
         nch = strlen(buffer); f->write(buffer,nch);
         if (i != fNp-1) snprintf(buffer,512,"\n                       ");
      }
   }
   snprintf(buffer,512," };\n");
   nch = strlen(buffer); f->write(buffer,nch);

   //generate code for the spline evaluation
   snprintf(buffer,512,"   int klow=0;\n");
   nch = strlen(buffer); f->write(buffer,nch);

   snprintf(buffer,512,"   // If out of boundaries, extrapolate. It may be badly wrong\n");
   snprintf(buffer,512,"   if(x<=fXmin) klow=0;\n");
   nch = strlen(buffer); f->write(buffer,nch);
   snprintf(buffer,512,"   else if(x>=fXmax) klow=fNp-1;\n");
   nch = strlen(buffer); f->write(buffer,nch);
   snprintf(buffer,512,"   else {\n");
   nch = strlen(buffer); f->write(buffer,nch);
   snprintf(buffer,512,"     if(fKstep) {\n");
   nch = strlen(buffer); f->write(buffer,nch);

   snprintf(buffer,512,"       // Equidistant knots, use histogramming\n");
   nch = strlen(buffer); f->write(buffer,nch);
   snprintf(buffer,512,"       klow = int((x-fXmin)/fDelta);\n");
   nch = strlen(buffer); f->write(buffer,nch);
   snprintf(buffer,512,"       if (klow < fNp-1) klow = fNp-1;\n");
   nch = strlen(buffer); f->write(buffer,nch);
   snprintf(buffer,512,"     } else {\n");
   nch = strlen(buffer); f->write(buffer,nch);
   snprintf(buffer,512,"       int khig=fNp-1, khalf;\n");
   nch = strlen(buffer); f->write(buffer,nch);

   snprintf(buffer,512,"       // Non equidistant knots, binary search\n");
   nch = strlen(buffer); f->write(buffer,nch);
   snprintf(buffer,512,"       while(khig-klow>1)\n");
   nch = strlen(buffer); f->write(buffer,nch);
   snprintf(buffer,512,"         if(x>fX[khalf=(klow+khig)/2]) klow=khalf;\n");
   nch = strlen(buffer); f->write(buffer,nch);
   snprintf(buffer,512,"         else khig=khalf;\n");
   nch = strlen(buffer); f->write(buffer,nch);
   snprintf(buffer,512,"     }\n");
   nch = strlen(buffer); f->write(buffer,nch);
   snprintf(buffer,512,"   }\n");
   nch = strlen(buffer); f->write(buffer,nch);
   snprintf(buffer,512,"   // Evaluate now\n");
   nch = strlen(buffer); f->write(buffer,nch);
   snprintf(buffer,512,"   double dx=x-fX[klow];\n");
   nch = strlen(buffer); f->write(buffer,nch);
   snprintf(buffer,512,"   return (fY[klow]+dx*(fB[klow]+dx*(fC[klow]+dx*fD[klow])));\n");
   nch = strlen(buffer); f->write(buffer,nch);

   //close file
   f->write("}\n",2);

   if (f) { f->close(); delete f;}
}


//______________________________________________________________________________
void TSpline3::SavePrimitive(ostream &out, Option_t *option /*= ""*/)
{
    // Save primitive as a C++ statement(s) on output stream out

   char quote = '"';
   out<<"   "<<endl;
   if (gROOT->ClassSaved(TSpline3::Class())) {
      out<<"   ";
   } else {
      out<<"   TSpline3 *";
   }
   out<<"spline3 = new TSpline3("<<quote<<GetTitle()<<quote<<","
      <<fXmin<<","<<fXmax<<",(TF1*)0,"<<fNp<<","<<quote<<quote<<","
      <<fValBeg<<","<<fValEnd<<");"<<endl;
   out<<"   spline3->SetName("<<quote<<GetName()<<quote<<");"<<endl;

   SaveFillAttributes(out,"spline3",0,1001);
   SaveLineAttributes(out,"spline3",1,1,1);
   SaveMarkerAttributes(out,"spline3",1,1,1);
   if (fNpx != 100) out<<"   spline3->SetNpx("<<fNpx<<");"<<endl;

   for (Int_t i=0;i<fNp;i++) {
      out<<"   spline3->SetPoint("<<i<<","<<fPoly[i].X()<<","<<fPoly[i].Y()<<");"<<endl;
      out<<"   spline3->SetPointCoeff("<<i<<","<<fPoly[i].B()<<","<<fPoly[i].C()<<","<<fPoly[i].D()<<");"<<endl;
   }
   out<<"   spline3->Draw("<<quote<<option<<quote<<");"<<endl;
}


//______________________________________________________________________________
void TSpline3::SetPoint(Int_t i, Double_t x, Double_t y)
{
   //set point number i.
   
   if (i < 0 || i >= fNp) return;
   fPoly[i].X()= x;
   fPoly[i].Y()= y;
}

//______________________________________________________________________________
void TSpline3::SetPointCoeff(Int_t i, Double_t b, Double_t c, Double_t d)
{
   // set point coefficient number i
 
   if (i < 0 || i >= fNp) return;
   fPoly[i].B()= b;
   fPoly[i].C()= c;
   fPoly[i].D()= d;
}

//______________________________________________________________________________
void TSpline3::BuildCoeff()
{
   //      subroutine cubspl ( tau, c, n, ibcbeg, ibcend )
   //  from  * a practical guide to splines *  by c. de boor
   //     ************************  input  ***************************
   //     n = number of data points. assumed to be .ge. 2.
   //     (tau(i), c(1,i), i=1,...,n) = abscissae and ordinates of the
   //        data points. tau is assumed to be strictly increasing.
   //     ibcbeg, ibcend = boundary condition indicators, and
   //     c(2,1), c(2,n) = boundary condition information. specifically,
   //        ibcbeg = 0  means no boundary condition at tau(1) is given.
   //           in this case, the not-a-knot condition is used, i.e. the
   //           jump in the third derivative across tau(2) is forced to
   //           zero, thus the first and the second cubic polynomial pieces
   //           are made to coincide.)
   //        ibcbeg = 1  means that the slope at tau(1) is made to equal
   //           c(2,1), supplied by input.
   //        ibcbeg = 2  means that the second derivative at tau(1) is
   //           made to equal c(2,1), supplied by input.
   //        ibcend = 0, 1, or 2 has analogous meaning concerning the
   //           boundary condition at tau(n), with the additional infor-
   //           mation taken from c(2,n).
   //     ***********************  output  **************************
   //     c(j,i), j=1,...,4; i=1,...,l (= n-1) = the polynomial coefficients
   //        of the cubic interpolating spline with interior knots (or
   //        joints) tau(2), ..., tau(n-1). precisely, in the interval
   //        (tau(i), tau(i+1)), the spline f is given by
   //           f(x) = c(1,i)+h*(c(2,i)+h*(c(3,i)+h*c(4,i)/3.)/2.)
   //        where h = x - tau(i). the function program *ppvalu* may be
   //        used to evaluate f or its derivatives from tau,c, l = n-1,
   //        and k=4.

   Int_t i, j, l, m;
   Double_t   divdf1,divdf3,dtau,g=0;
   //***** a tridiagonal linear system for the unknown slopes s(i) of
   //  f  at tau(i), i=1,...,n, is generated and then solved by gauss elim-
   //  ination, with s(i) ending up in c(2,i), all i.
   //     c(3,.) and c(4,.) are used initially for temporary storage.
   l = fNp-1;
   // compute first differences of x sequence and store in C also,
   // compute first divided difference of data and store in D.
   for (m=1; m<fNp ; ++m) {
      fPoly[m].C() = fPoly[m].X() - fPoly[m-1].X();
      fPoly[m].D() = (fPoly[m].Y() - fPoly[m-1].Y())/fPoly[m].C();
   }
   // construct first equation from the boundary condition, of the form
   //             D[0]*s[0] + C[0]*s[1] = B[0]
   if(fBegCond==0) {
      if(fNp == 2) {
         //     no condition at left end and n = 2.
         fPoly[0].D() = 1.;
         fPoly[0].C() = 1.;
         fPoly[0].B() = 2.*fPoly[1].D();
      } else {
         //     not-a-knot condition at left end and n .gt. 2.
         fPoly[0].D() = fPoly[2].C();
         fPoly[0].C() = fPoly[1].C() + fPoly[2].C();
         fPoly[0].B() =((fPoly[1].C()+2.*fPoly[0].C())*fPoly[1].D()*fPoly[2].C()+fPoly[1].C()*fPoly[1].C()*fPoly[2].D())/fPoly[0].C();
      }
   } else if (fBegCond==1) {
      //     slope prescribed at left end.
      fPoly[0].B() = fValBeg;
      fPoly[0].D() = 1.;
      fPoly[0].C() = 0.;
   } else if (fBegCond==2) {
      //     second derivative prescribed at left end.
      fPoly[0].D() = 2.;
      fPoly[0].C() = 1.;
      fPoly[0].B() = 3.*fPoly[1].D() - fPoly[1].C()/2.*fValBeg;
   }
   if(fNp > 2) {
      //  if there are interior knots, generate the corresp. equations and car-
      //  ry out the forward pass of gauss elimination, after which the m-th
      //  equation reads    D[m]*s[m] + C[m]*s[m+1] = B[m].
      for (m=1; m<l; ++m) {
         g = -fPoly[m+1].C()/fPoly[m-1].D();
         fPoly[m].B() = g*fPoly[m-1].B() + 3.*(fPoly[m].C()*fPoly[m+1].D()+fPoly[m+1].C()*fPoly[m].D());
         fPoly[m].D() = g*fPoly[m-1].C() + 2.*(fPoly[m].C() + fPoly[m+1].C());
      }
      // construct last equation from the second boundary condition, of the form
      //           (-g*D[n-2])*s[n-2] + D[n-1]*s[n-1] = B[n-1]
      //     if slope is prescribed at right end, one can go directly to back-
      //     substitution, since c array happens to be set up just right for it
      //     at this point.
      if(fEndCond == 0) {
         if (fNp > 3 || fBegCond != 0) {
            //     not-a-knot and n .ge. 3, and either n.gt.3 or  also not-a-knot at
            //     left end point.
            g = fPoly[fNp-2].C() + fPoly[fNp-1].C();
            fPoly[fNp-1].B() = ((fPoly[fNp-1].C()+2.*g)*fPoly[fNp-1].D()*fPoly[fNp-2].C()
                         + fPoly[fNp-1].C()*fPoly[fNp-1].C()*(fPoly[fNp-2].Y()-fPoly[fNp-3].Y())/fPoly[fNp-2].C())/g;
            g = -g/fPoly[fNp-2].D();
            fPoly[fNp-1].D() = fPoly[fNp-2].C();
         } else {
            //     either (n=3 and not-a-knot also at left) or (n=2 and not not-a-
            //     knot at left end point).
            fPoly[fNp-1].B() = 2.*fPoly[fNp-1].D();
            fPoly[fNp-1].D() = 1.;
            g = -1./fPoly[fNp-2].D();
         }
      } else if (fEndCond == 1) {
         fPoly[fNp-1].B() = fValEnd;
         goto L30;
      } else if (fEndCond == 2) {
         //     second derivative prescribed at right endpoint.
         fPoly[fNp-1].B() = 3.*fPoly[fNp-1].D() + fPoly[fNp-1].C()/2.*fValEnd;
         fPoly[fNp-1].D() = 2.;
         g = -1./fPoly[fNp-2].D();
      }
   } else {
      if(fEndCond == 0) {
         if (fBegCond > 0) {
            //     either (n=3 and not-a-knot also at left) or (n=2 and not not-a-
            //     knot at left end point).
            fPoly[fNp-1].B() = 2.*fPoly[fNp-1].D();
            fPoly[fNp-1].D() = 1.;
            g = -1./fPoly[fNp-2].D();
         } else {
            //     not-a-knot at right endpoint and at left endpoint and n = 2.
            fPoly[fNp-1].B() = fPoly[fNp-1].D();
            goto L30;
         }
      } else if(fEndCond == 1) {
         fPoly[fNp-1].B() = fValEnd;
         goto L30;
      } else if(fEndCond == 2) {
         //     second derivative prescribed at right endpoint.
         fPoly[fNp-1].B() = 3.*fPoly[fNp-1].D() + fPoly[fNp-1].C()/2.*fValEnd;
         fPoly[fNp-1].D() = 2.;
         g = -1./fPoly[fNp-2].D();
      }
   }
   // complete forward pass of gauss elimination.
   fPoly[fNp-1].D() = g*fPoly[fNp-2].C() + fPoly[fNp-1].D();
   fPoly[fNp-1].B() = (g*fPoly[fNp-2].B() + fPoly[fNp-1].B())/fPoly[fNp-1].D();
   // carry out back substitution
L30: j = l-1;
   do {
      fPoly[j].B() = (fPoly[j].B() - fPoly[j].C()*fPoly[j+1].B())/fPoly[j].D();
      --j;
   }  while (j>=0);
   //****** generate cubic coefficients in each interval, i.e., the deriv.s
   //  at its left endpoint, from value and slope at its endpoints.
   for (i=1; i<fNp; ++i) {
      dtau = fPoly[i].C();
      divdf1 = (fPoly[i].Y() - fPoly[i-1].Y())/dtau;
      divdf3 = fPoly[i-1].B() + fPoly[i].B() - 2.*divdf1;
      fPoly[i-1].C() = (divdf1 - fPoly[i-1].B() - divdf3)/dtau;
      fPoly[i-1].D() = (divdf3/dtau)/dtau;
   }
}


//______________________________________________________________________________
void TSpline3::Streamer(TBuffer &R__b)
{
   // Stream an object of class TSpline3.

   if (R__b.IsReading()) {
      UInt_t R__s, R__c;
      Version_t R__v = R__b.ReadVersion(&R__s, &R__c);
      if (R__v > 1) {
         R__b.ReadClassBuffer(TSpline3::Class(), this, R__v, R__s, R__c);
         return;
      }
      //====process old versions before automatic schema evolution
      TSpline::Streamer(R__b);
      if (fNp > 0) {
         fPoly = new TSplinePoly3[fNp];
         for(Int_t i=0; i<fNp; ++i) {
            fPoly[i].Streamer(R__b);
         }
      }
      //      R__b >> fPoly;
      R__b >> fValBeg;
      R__b >> fValEnd;
      R__b >> fBegCond;
      R__b >> fEndCond;
   } else {
      R__b.WriteClassBuffer(TSpline3::Class(),this);
   }
}


//////////////////////////////////////////////////////////////////////////
//                                                                      //
// TSpline5                                                             //
//                                                                      //
// Class to create quintic natural splines to interpolate knots         //
// Arbitrary conditions can be introduced for first and second          //
// derivatives using double knots (see BuildCoeff) for more on this.    //
// Double knots are automatically introduced at ending points           //
//                                                                      //
//////////////////////////////////////////////////////////////////////////


//______________________________________________________________________________
TSpline5::TSpline5(const char *title,
                   Double_t x[], Double_t y[], Int_t n,
                   const char *opt, Double_t b1, Double_t e1,
                   Double_t b2, Double_t e2) :
  TSpline(title,-1, x[0], x[n-1], n, kFALSE)
{
   // Quintic natural spline creator given an array of
   // arbitrary knots in increasing abscissa order and
   // possibly end point conditions

   Int_t beg, end;
   const char *cb1, *ce1, *cb2, *ce2;
   fName="Spline5";

   // Check endpoint conditions
   BoundaryConditions(opt,beg,end,cb1,ce1,cb2,ce2);

   // Create the plynomial terms and fill
   // them with node information
   fPoly = new TSplinePoly5[fNp];
   for (Int_t i=0; i<n; ++i) {
      fPoly[i+beg].X() = x[i];
      fPoly[i+beg].Y() = y[i];
   }

   // Set the double knots at boundaries
   SetBoundaries(b1,e1,b2,e2,cb1,ce1,cb2,ce2);

   // Build the spline coefficients
   BuildCoeff();
}


//______________________________________________________________________________
TSpline5::TSpline5(const char *title,
                   Double_t xmin, Double_t xmax,
                   Double_t y[], Int_t n,
                   const char *opt, Double_t b1, Double_t e1,
                   Double_t b2, Double_t e2) :
  TSpline(title,(xmax-xmin)/(n-1), xmin, xmax, n, kTRUE)
{
   // Quintic natural spline creator given an array of
   // arbitrary function values on equidistant n abscissa
   // values from xmin to xmax and possibly end point conditions

   Int_t beg, end;
   const char *cb1, *ce1, *cb2, *ce2;
   fName="Spline5";

   // Check endpoint conditions
   BoundaryConditions(opt,beg,end,cb1,ce1,cb2,ce2);

   // Create the plynomial terms and fill
   // them with node information
   fPoly = new TSplinePoly5[fNp];
   for (Int_t i=0; i<n; ++i) {
      fPoly[i+beg].X() = fXmin+i*fDelta;
      fPoly[i+beg].Y() = y[i];
   }

   // Set the double knots at boundaries
   SetBoundaries(b1,e1,b2,e2,cb1,ce1,cb2,ce2);

   // Build the spline coefficients
   BuildCoeff();
}


//______________________________________________________________________________
TSpline5::TSpline5(const char *title,
                   Double_t x[], const TF1 *func, Int_t n,
                   const char *opt, Double_t b1, Double_t e1,
                   Double_t b2, Double_t e2) :
  TSpline(title,-1, x[0], x[n-1], n, kFALSE)
{
   // Quintic natural spline creator given an array of
   // arbitrary abscissas in increasing order and a function
   // to interpolate and possibly end point conditions

   Int_t beg, end;
   const char *cb1, *ce1, *cb2, *ce2;
   fName="Spline5";

   // Check endpoint conditions
   BoundaryConditions(opt,beg,end,cb1,ce1,cb2,ce2);

   // Create the plynomial terms and fill
   // them with node information
   fPoly = new TSplinePoly5[fNp];
   for (Int_t i=0; i<n; i++) {
      fPoly[i+beg].X() = x[i];
      fPoly[i+beg].Y() = ((TF1*)func)->Eval(x[i]);
   }

   // Set the double knots at boundaries
   SetBoundaries(b1,e1,b2,e2,cb1,ce1,cb2,ce2);

   // Build the spline coefficients
   BuildCoeff();
}


//______________________________________________________________________________
TSpline5::TSpline5(const char *title,
                   Double_t xmin, Double_t xmax,
                   const TF1 *func, Int_t n,
                   const char *opt, Double_t b1, Double_t e1,
                   Double_t b2, Double_t e2) :
  TSpline(title,(xmax-xmin)/(n-1), xmin, xmax, n, kTRUE)
{
   // Quintic natural spline creator given a function to be
   // evaluated on n equidistand abscissa points between xmin
   // and xmax and possibly end point conditions

   Int_t beg, end;
   const char *cb1, *ce1, *cb2, *ce2;
   fName="Spline5";

   // Check endpoint conditions
   BoundaryConditions(opt,beg,end,cb1,ce1,cb2,ce2);

   // Create the plynomial terms and fill
   // them with node information
   fPoly = new TSplinePoly5[fNp];
   for (Int_t i=0; i<n; ++i) {
      Double_t x=fXmin+i*fDelta;
      fPoly[i+beg].X() = x;
      if (func) fPoly[i+beg].Y() = ((TF1*)func)->Eval(x);
   }
   if (!func) {fDelta = -1; fKstep = kFALSE;}

   // Set the double knots at boundaries
   SetBoundaries(b1,e1,b2,e2,cb1,ce1,cb2,ce2);

   // Build the spline coefficients
   if (func) BuildCoeff();
}


//______________________________________________________________________________
TSpline5::TSpline5(const char *title,
                   const TGraph *g,
                   const char *opt, Double_t b1, Double_t e1,
                   Double_t b2, Double_t e2) :
  TSpline(title,-1,0,0,g->GetN(),kFALSE)
{
   // Quintic natural spline creator given a TGraph with
   // abscissa in increasing order and possibly end
   // point conditions

   Int_t beg, end;
   const char *cb1, *ce1, *cb2, *ce2;
   fName="Spline5";

   // Check endpoint conditions
   BoundaryConditions(opt,beg,end,cb1,ce1,cb2,ce2);

   // Create the plynomial terms and fill
   // them with node information
   fPoly = new TSplinePoly5[fNp];
   for (Int_t i=0; i<fNp-beg; ++i) {
      Double_t xx, yy;
      g->GetPoint(i,xx,yy);
      fPoly[i+beg].X()=xx;
      fPoly[i+beg].Y()=yy;
   }

   // Set the double knots at boundaries
   SetBoundaries(b1,e1,b2,e2,cb1,ce1,cb2,ce2);
   fXmin = fPoly[0].X();
   fXmax = fPoly[fNp-1].X();

   // Build the spline coefficients
   BuildCoeff();
}


//______________________________________________________________________________
TSpline5::TSpline5(const TH1 *h,
                   const char *opt, Double_t b1, Double_t e1,
                   Double_t b2, Double_t e2) :
  TSpline(h->GetTitle(),-1,0,0,h->GetNbinsX(),kFALSE)
{
   // Quintic natural spline creator given a TH1

   Int_t beg, end;
   const char *cb1, *ce1, *cb2, *ce2;
   fName=h->GetName();

   // Check endpoint conditions
   BoundaryConditions(opt,beg,end,cb1,ce1,cb2,ce2);

   // Create the plynomial terms and fill
   // them with node information
   fPoly = new TSplinePoly5[fNp];
   for (Int_t i=0; i<fNp-beg; ++i) {
      fPoly[i+beg].X()=h->GetXaxis()->GetBinCenter(i+1);
      fPoly[i+beg].Y()=h->GetBinContent(i+1);
   }

   // Set the double knots at boundaries
   SetBoundaries(b1,e1,b2,e2,cb1,ce1,cb2,ce2);
   fXmin = fPoly[0].X();
   fXmax = fPoly[fNp-1].X();

   // Build the spline coefficients
   BuildCoeff();
}


//______________________________________________________________________________
TSpline5::TSpline5(const TSpline5& sp5) :
  TSpline(sp5),
  fPoly(0)
{
   //copy constructor
   if (fNp > 0) fPoly = new TSplinePoly5[fNp];
   for (Int_t i=0; i<fNp; ++i) {
      fPoly[i] = sp5.fPoly[i];
   }
}


//______________________________________________________________________________
TSpline5& TSpline5::operator=(const TSpline5& sp5)
{
   //assignment operator
   if(this!=&sp5) {
      TSpline::operator=(sp5);
      fPoly=0;
      if (fNp > 0) fPoly = new TSplinePoly5[fNp];
      for (Int_t i=0; i<fNp; ++i) {
         fPoly[i] = sp5.fPoly[i];
      }
   }
   return *this;
}


//______________________________________________________________________________
void TSpline5::BoundaryConditions(const char *opt,Int_t &beg,Int_t &end,
                                  const char *&cb1,const char *&ce1,
                                  const char *&cb2,const char *&ce2)
{
   // Check the boundary conditions and the
   // amount of extra double knots needed

   cb1=ce1=cb2=ce2=0;
   beg=end=0;
   if(opt) {
      cb1 = strstr(opt,"b1");
      ce1 = strstr(opt,"e1");
      cb2 = strstr(opt,"b2");
      ce2 = strstr(opt,"e2");
      if(cb2) {
         fNp=fNp+2;
         beg=2;
      } else if(cb1) {
         fNp=fNp+1;
         beg=1;
      }
      if(ce2) {
         fNp=fNp+2;
         end=2;
      } else if(ce1) {
         fNp=fNp+1;
         end=1;
      }
   }
}


//______________________________________________________________________________
void TSpline5::SetBoundaries(Double_t b1, Double_t e1, Double_t b2, Double_t e2,
                             const char *cb1, const char *ce1, const char *cb2,
                             const char *ce2)
{
   // Set the boundary conditions at double/triple knots

   if(cb2) {

      // Second derivative at the beginning
      fPoly[0].X() = fPoly[1].X() = fPoly[2].X();
      fPoly[0].Y() = fPoly[2].Y();
      fPoly[2].Y()=b2;

      // If first derivative not given, we take the finite
      // difference from first and second point... not recommended
      if(cb1)
         fPoly[1].Y()=b1;
      else
         fPoly[1].Y()=(fPoly[3].Y()-fPoly[0].Y())/(fPoly[3].X()-fPoly[2].X());
   } else if(cb1) {

      // First derivative at the end
      fPoly[0].X() = fPoly[1].X();
      fPoly[0].Y() = fPoly[1].Y();
      fPoly[1].Y()=b1;
   }
   if(ce2) {

      // Second derivative at the end
      fPoly[fNp-1].X() = fPoly[fNp-2].X() = fPoly[fNp-3].X();
      fPoly[fNp-1].Y()=e2;

      // If first derivative not given, we take the finite
      // difference from first and second point... not recommended
      if(ce1)
         fPoly[fNp-2].Y()=e1;
      else
         fPoly[fNp-2].Y()=
         (fPoly[fNp-3].Y()-fPoly[fNp-4].Y())
         /(fPoly[fNp-3].X()-fPoly[fNp-4].X());
   } else if(ce1) {

      // First derivative at the end
      fPoly[fNp-1].X() = fPoly[fNp-2].X();
      fPoly[fNp-1].Y()=e1;
   }
}


//______________________________________________________________________________
Int_t TSpline5::FindX(Double_t x) const
{
   // Find X

   Int_t klow=0;

   // If out of boundaries, extrapolate
   // It may be badly wrong
   if(x<=fXmin) klow=0;
   else if(x>=fXmax) klow=fNp-1;
   else {
      if(fKstep) {

         // Equidistant knots, use histogramming
         klow = TMath::Min(Int_t((x-fXmin)/fDelta),fNp-1);
      } else {
         Int_t khig=fNp-1, khalf;

         // Non equidistant knots, binary search
         while(khig-klow>1)
            if(x>fPoly[khalf=(klow+khig)/2].X())
               klow=khalf;
            else
               khig=khalf;
      }

      // This could be removed, sanity check
      if(!(fPoly[klow].X()<=x && x<=fPoly[klow+1].X()))
         Error("Eval",
               "Binary search failed x(%d) = %f < x(%d) = %f\n",
                klow,fPoly[klow].X(),klow+1,fPoly[klow+1].X());
   }
   return klow;
}


//______________________________________________________________________________
Double_t TSpline5::Eval(Double_t x) const
{
   // Eval this spline at x

   Int_t klow=FindX(x);
   return fPoly[klow].Eval(x);
}


//______________________________________________________________________________
Double_t TSpline5::Derivative(Double_t x) const
{
   // Derivative

   Int_t klow=FindX(x);
   return fPoly[klow].Derivative(x);
}


//______________________________________________________________________________
void TSpline5::SaveAs(const char *filename, Option_t * /*option*/) const
{
   // write this spline as a C++ function that can be executed without ROOT
   // the name of the function is the name of the file up to the "." if any

   //open the file
   ofstream *f = new ofstream(filename,ios::out);
   if (f == 0 || gSystem->AccessPathName(filename,kWritePermission)) {
      Error("SaveAs","Cannot open file:%s\n",filename);
      return;
   }

   //write the function name and the spline constants
   char buffer[512];
   Int_t nch = strlen(filename);
   snprintf(buffer,512,"double %s",filename);
   char *dot = strstr(buffer,".");
   if (dot) *dot = 0;
   strlcat(buffer,"(double x) {\n",512);
   nch = strlen(buffer); f->write(buffer,nch);
   snprintf(buffer,512,"   const int fNp = %d, fKstep = %d;\n",fNp,fKstep);
   nch = strlen(buffer); f->write(buffer,nch);
   snprintf(buffer,512,"   const double fDelta = %g, fXmin = %g, fXmax = %g;\n",fDelta,fXmin,fXmax);
   nch = strlen(buffer); f->write(buffer,nch);

   //write the spline coefficients
   //array fX
   snprintf(buffer,512,"   const double fX[%d] = {",fNp);
   nch = strlen(buffer); f->write(buffer,nch);
   buffer[0] = 0;
   Int_t i;
   char numb[20];
   for (i=0;i<fNp;i++) {
      snprintf(numb,20," %g,",fPoly[i].X());
      nch = strlen(numb);
      if (i == fNp-1) numb[nch-1]=0;
      strlcat(buffer,numb,512);
      if (i%5 == 4 || i == fNp-1) {
         nch = strlen(buffer); f->write(buffer,nch);
         if (i != fNp-1) snprintf(buffer,512,"\n                       ");
      }
   }
   snprintf(buffer,512," };\n");
   nch = strlen(buffer); f->write(buffer,nch);
   //array fY
   snprintf(buffer,512,"   const double fY[%d] = {",fNp);
   nch = strlen(buffer); f->write(buffer,nch);
   buffer[0] = 0;
   for (i=0;i<fNp;i++) {
      snprintf(numb,20," %g,",fPoly[i].Y());
      nch = strlen(numb);
      if (i == fNp-1) numb[nch-1]=0;
      strlcat(buffer,numb,512);
      if (i%5 == 4 || i == fNp-1) {
         nch = strlen(buffer); f->write(buffer,nch);
         if (i != fNp-1) snprintf(buffer,512,"\n                       ");
      }
   }
   snprintf(buffer,512," };\n");
   nch = strlen(buffer); f->write(buffer,nch);
   //array fB
   snprintf(buffer,512,"   const double fB[%d] = {",fNp);
   nch = strlen(buffer); f->write(buffer,nch);
   buffer[0] = 0;
   for (i=0;i<fNp;i++) {
      snprintf(numb,20," %g,",fPoly[i].B());
      nch = strlen(numb);
      if (i == fNp-1) numb[nch-1]=0;
      strlcat(buffer,numb,512);
      if (i%5 == 4 || i == fNp-1) {
         nch = strlen(buffer); f->write(buffer,nch);
         if (i != fNp-1) snprintf(buffer,512,"\n                       ");
      }
   }
   snprintf(buffer,512," };\n");
   nch = strlen(buffer); f->write(buffer,nch);
   //array fC
   snprintf(buffer,512,"   const double fC[%d] = {",fNp);
   nch = strlen(buffer); f->write(buffer,nch);
   buffer[0] = 0;
   for (i=0;i<fNp;i++) {
      snprintf(numb,20," %g,",fPoly[i].C());
      nch = strlen(numb);
      if (i == fNp-1) numb[nch-1]=0;
      strlcat(buffer,numb,512);
      if (i%5 == 4 || i == fNp-1) {
         nch = strlen(buffer); f->write(buffer,nch);
         if (i != fNp-1) snprintf(buffer,512,"\n                       ");
      }
   }
   snprintf(buffer,512," };\n");
   nch = strlen(buffer); f->write(buffer,nch);
    //array fD
   snprintf(buffer,512,"   const double fD[%d] = {",fNp);
   nch = strlen(buffer); f->write(buffer,nch);
   buffer[0] = 0;
   for (i=0;i<fNp;i++) {
      snprintf(numb,20," %g,",fPoly[i].D());
      nch = strlen(numb);
      if (i == fNp-1) numb[nch-1]=0;
      strlcat(buffer,numb,512);
      if (i%5 == 4 || i == fNp-1) {
         nch = strlen(buffer); f->write(buffer,nch);
         if (i != fNp-1) snprintf(buffer,512,"\n                       ");
      }
   }
   snprintf(buffer,512," };\n");
   nch = strlen(buffer); f->write(buffer,nch);
    //array fE
   snprintf(buffer,512,"   const double fE[%d] = {",fNp);
   nch = strlen(buffer); f->write(buffer,nch);
   buffer[0] = 0;
   for (i=0;i<fNp;i++) {
      snprintf(numb,20," %g,",fPoly[i].E());
      nch = strlen(numb);
      if (i == fNp-1) numb[nch-1]=0;
      strlcat(buffer,numb,512);
      if (i%5 == 4 || i == fNp-1) {
         nch = strlen(buffer); f->write(buffer,nch);
         if (i != fNp-1) snprintf(buffer,512,"\n                       ");
      }
   }
   snprintf(buffer,512," };\n");
   nch = strlen(buffer); f->write(buffer,nch);
    //array fF
   snprintf(buffer,512,"   const double fF[%d] = {",fNp);
   nch = strlen(buffer); f->write(buffer,nch);
   buffer[0] = 0;
   for (i=0;i<fNp;i++) {
      snprintf(numb,20," %g,",fPoly[i].F());
      nch = strlen(numb);
      if (i == fNp-1) numb[nch-1]=0;
      strlcat(buffer,numb,512);
      if (i%5 == 4 || i == fNp-1) {
         nch = strlen(buffer); f->write(buffer,nch);
         if (i != fNp-1) snprintf(buffer,512,"\n                       ");
      }
   }
   snprintf(buffer,512," };\n");
   nch = strlen(buffer); f->write(buffer,nch);

   //generate code for the spline evaluation
   snprintf(buffer,512,"   int klow=0;\n");
   nch = strlen(buffer); f->write(buffer,nch);

   snprintf(buffer,512,"   // If out of boundaries, extrapolate. It may be badly wrong\n");
   snprintf(buffer,512,"   if(x<=fXmin) klow=0;\n");
   nch = strlen(buffer); f->write(buffer,nch);
   snprintf(buffer,512,"   else if(x>=fXmax) klow=fNp-1;\n");
   nch = strlen(buffer); f->write(buffer,nch);
   snprintf(buffer,512,"   else {\n");
   nch = strlen(buffer); f->write(buffer,nch);
   snprintf(buffer,512,"     if(fKstep) {\n");
   nch = strlen(buffer); f->write(buffer,nch);

   snprintf(buffer,512,"       // Equidistant knots, use histogramming\n");
   nch = strlen(buffer); f->write(buffer,nch);
   snprintf(buffer,512,"       klow = int((x-fXmin)/fDelta);\n");
   nch = strlen(buffer); f->write(buffer,nch);
   snprintf(buffer,512,"       if (klow < fNp-1) klow = fNp-1;\n");
   nch = strlen(buffer); f->write(buffer,nch);
   snprintf(buffer,512,"     } else {\n");
   nch = strlen(buffer); f->write(buffer,nch);
   snprintf(buffer,512,"       int khig=fNp-1, khalf;\n");
   nch = strlen(buffer); f->write(buffer,nch);

   snprintf(buffer,512,"       // Non equidistant knots, binary search\n");
   nch = strlen(buffer); f->write(buffer,nch);
   snprintf(buffer,512,"       while(khig-klow>1)\n");
   nch = strlen(buffer); f->write(buffer,nch);
   snprintf(buffer,512,"         if(x>fX[khalf=(klow+khig)/2]) klow=khalf;\n");
   nch = strlen(buffer); f->write(buffer,nch);
   snprintf(buffer,512,"         else khig=khalf;\n");
   nch = strlen(buffer); f->write(buffer,nch);
   snprintf(buffer,512,"     }\n");
   nch = strlen(buffer); f->write(buffer,nch);
   snprintf(buffer,512,"   }\n");
   nch = strlen(buffer); f->write(buffer,nch);
   snprintf(buffer,512,"   // Evaluate now\n");
   nch = strlen(buffer); f->write(buffer,nch);
   snprintf(buffer,512,"   double dx=x-fX[klow];\n");
   nch = strlen(buffer); f->write(buffer,nch);
   snprintf(buffer,512,"   return (fY[klow]+dx*(fB[klow]+dx*(fC[klow]+dx*(fD[klow]+dx*(fE[klow]+dx*fF[klow])))));\n");
   nch = strlen(buffer); f->write(buffer,nch);

   //close file
   f->write("}\n",2);

   if (f) { f->close(); delete f;}
}


//______________________________________________________________________________
void TSpline5::SavePrimitive(ostream &out, Option_t *option /*= ""*/)
{
    // Save primitive as a C++ statement(s) on output stream out

   char quote = '"';
   out<<"   "<<endl;
   if (gROOT->ClassSaved(TSpline5::Class())) {
      out<<"   ";
   } else {
      out<<"   TSpline5 *";
   }
   Double_t b1 = fPoly[1].Y();
   Double_t e1 = fPoly[fNp-1].Y();
   Double_t b2 = fPoly[2].Y();
   Double_t e2 = fPoly[fNp-1].Y();
   out<<"spline5 = new TSpline5("<<quote<<GetTitle()<<quote<<","
      <<fXmin<<","<<fXmax<<",(TF1*)0,"<<fNp<<","<<quote<<quote<<","
      <<b1<<","<<e1<<","<<b2<<","<<e2<<");"<<endl;
   out<<"   spline5->SetName("<<quote<<GetName()<<quote<<");"<<endl;

   SaveFillAttributes(out,"spline5",0,1001);
   SaveLineAttributes(out,"spline5",1,1,1);
   SaveMarkerAttributes(out,"spline5",1,1,1);
   if (fNpx != 100) out<<"   spline5->SetNpx("<<fNpx<<");"<<endl;

   for (Int_t i=0;i<fNp;i++) {
      out<<"   spline5->SetPoint("<<i<<","<<fPoly[i].X()<<","<<fPoly[i].Y()<<");"<<endl;
      out<<"   spline5->SetPointCoeff("<<i<<","<<fPoly[i].B()<<","<<fPoly[i].C()<<","<<fPoly[i].D()<<","<<fPoly[i].E()<<","<<fPoly[i].F()<<");"<<endl;
   }
   out<<"   spline5->Draw("<<quote<<option<<quote<<");"<<endl;
}


//______________________________________________________________________________
void TSpline5::SetPoint(Int_t i, Double_t x, Double_t y)
{
   //set point number i.
   
   
   if (i < 0 || i >= fNp) return;
   fPoly[i].X()= x;
   fPoly[i].Y()= y;
}

//______________________________________________________________________________
void TSpline5::SetPointCoeff(Int_t i, Double_t b, Double_t c, Double_t d,
                             Double_t e, Double_t f)
{
   // set point coefficient number i
   
   if (i < 0 || i >= fNp) return;
   fPoly[i].B()= b;
   fPoly[i].C()= c;
   fPoly[i].D()= d;
   fPoly[i].E()= e;
   fPoly[i].F()= f;
}



//______________________________________________________________________________
void TSpline5::BuildCoeff()
{
   //     algorithm 600, collected algorithms from acm.
   //     algorithm appeared in acm-trans. math. software, vol.9, no. 2,
   //     jun., 1983, p. 258-259.
   //
   //     TSpline5 computes the coefficients of a quintic natural quintic spli
   //     s(x) with knots x(i) interpolating there to given function values:
   //               s(x(i)) = y(i)  for i = 1,2, ..., n.
   //     in each interval (x(i),x(i+1)) the spline function s(xx) is a
   //     polynomial of fifth degree:
   //     s(xx) = ((((f(i)*p+e(i))*p+d(i))*p+c(i))*p+b(i))*p+y(i)    (*)
   //           = ((((-f(i)*q+e(i+1))*q-d(i+1))*q+c(i+1))*q-b(i+1))*q+y(i+1)
   //     where  p = xx - x(i)  and  q = x(i+1) - xx.
   //     (note the first subscript in the second expression.)
   //     the different polynomials are pieced together so that s(x) and
   //     its derivatives up to s"" are continuous.
   //
   //        input:
   //
   //     n          number of data points, (at least three, i.e. n > 2)
   //     x(1:n)     the strictly increasing or decreasing sequence of
   //                knots.  the spacing must be such that the fifth power
   //                of x(i+1) - x(i) can be formed without overflow or
   //                underflow of exponents.
   //     y(1:n)     the prescribed function values at the knots.
   //
   //        output:
   //
   //     b,c,d,e,f  the computed spline coefficients as in (*).
   //         (1:n)  specifically
   //                b(i) = s'(x(i)), c(i) = s"(x(i))/2, d(i) = s"'(x(i))/6,
   //                e(i) = s""(x(i))/24,  f(i) = s""'(x(i))/120.
   //                f(n) is neither used nor altered.  the five arrays
   //                b,c,d,e,f must always be distinct.
   //
   //        option:
   //
   //     it is possible to specify values for the first and second
   //     derivatives of the spline function at arbitrarily many knots.
   //     this is done by relaxing the requirement that the sequence of
   //     knots be strictly increasing or decreasing.  specifically:
   //
   //     if x(j) = x(j+1) then s(x(j)) = y(j) and s'(x(j)) = y(j+1),
   //     if x(j) = x(j+1) = x(j+2) then in addition s"(x(j)) = y(j+2).
   //
   //     note that s""(x) is discontinuous at a double knot and, in
   //     addition, s"'(x) is discontinuous at a triple knot.  the
   //     subroutine assigns y(i) to y(i+1) in these cases and also to
   //     y(i+2) at a triple knot.  the representation (*) remains
   //     valid in each open interval (x(i),x(i+1)).  at a double knot,
   //     x(j) = x(j+1), the output coefficients have the following values:
   //       y(j) = s(x(j))          = y(j+1)
   //       b(j) = s'(x(j))         = b(j+1)
   //       c(j) = s"(x(j))/2       = c(j+1)
   //       d(j) = s"'(x(j))/6      = d(j+1)
   //       e(j) = s""(x(j)-0)/24     e(j+1) = s""(x(j)+0)/24
   //       f(j) = s""'(x(j)-0)/120   f(j+1) = s""'(x(j)+0)/120
   //     at a triple knot, x(j) = x(j+1) = x(j+2), the output
   //     coefficients have the following values:
   //       y(j) = s(x(j))         = y(j+1)    = y(j+2)
   //       b(j) = s'(x(j))        = b(j+1)    = b(j+2)
   //       c(j) = s"(x(j))/2      = c(j+1)    = c(j+2)
   //       d(j) = s"'((x(j)-0)/6    d(j+1) = 0  d(j+2) = s"'(x(j)+0)/6
   //       e(j) = s""(x(j)-0)/24    e(j+1) = 0  e(j+2) = s""(x(j)+0)/24
   //       f(j) = s""'(x(j)-0)/120  f(j+1) = 0  f(j+2) = s""'(x(j)+0)/120

   Int_t i, m;
   Double_t pqqr, p, q, r, s, t, u, v,
      b1, p2, p3, q2, q3, r2, pq, pr, qr;

   if (fNp <= 2) {
      return;
   }

   //     coefficients of a positive definite, pentadiagonal matrix,
   //     stored in D, E, F from 1 to n-3.
   m = fNp-2;
   q = fPoly[1].X()-fPoly[0].X();
   r = fPoly[2].X()-fPoly[1].X();
   q2 = q*q;
   r2 = r*r;
   qr = q+r;
   fPoly[0].D() = fPoly[0].E() = 0;
   if (q) fPoly[1].D() = q*6.*q2/(qr*qr);
   else fPoly[1].D() = 0;

   if (m > 1) {
      for (i = 1; i < m; ++i) {
         p = q;
         q = r;
         r = fPoly[i+2].X()-fPoly[i+1].X();
         p2 = q2;
         q2 = r2;
         r2 = r*r;
         pq = qr;
         qr = q+r;
         if (q) {
            q3 = q2*q;
            pr = p*r;
            pqqr = pq*qr;
            fPoly[i+1].D() = q3*6./(qr*qr);
            fPoly[i].D() += (q+q)*(pr*15.*pr+(p+r)*q
                                 *(pr* 20.+q2*7.)+q2*
                                 ((p2+r2)*8.+pr*21.+q2+q2))/(pqqr*pqqr);
            fPoly[i-1].D() += q3*6./(pq*pq);
            fPoly[i].E() = q2*(p*qr+pq*3.*(qr+r+r))/(pqqr*qr);
            fPoly[i-1].E() += q2*(r*pq+qr*3.*(pq+p+p))/(pqqr*pq);
            fPoly[i-1].F() = q3/pqqr;
         } else
            fPoly[i+1].D() = fPoly[i].E() = fPoly[i-1].F() = 0;
      }
   }
   if (r) fPoly[m-1].D() += r*6.*r2/(qr*qr);

   //     First and second order divided differences of the given function
   //     values, stored in b from 2 to n and in c from 3 to n
   //     respectively. care is taken of double and triple knots.
   for (i = 1; i < fNp; ++i) {
      if (fPoly[i].X() != fPoly[i-1].X()) {
         fPoly[i].B() =
            (fPoly[i].Y()-fPoly[i-1].Y())/(fPoly[i].X()-fPoly[i-1].X());
      } else {
         fPoly[i].B() = fPoly[i].Y();
         fPoly[i].Y() = fPoly[i-1].Y();
      }
   }
   for (i = 2; i < fNp; ++i) {
      if (fPoly[i].X() != fPoly[i-2].X()) {
         fPoly[i].C() =
            (fPoly[i].B()-fPoly[i-1].B())/(fPoly[i].X()-fPoly[i-2].X());
      } else {
         fPoly[i].C() = fPoly[i].B()*.5;
         fPoly[i].B() = fPoly[i-1].B();
      }
   }

   //     Solve the linear system with c(i+2) - c(i+1) as right-hand side. */
   if (m > 1) {
      p=fPoly[0].C()=fPoly[m-1].E()=fPoly[0].F()
         =fPoly[m-2].F()=fPoly[m-1].F()=0;
      fPoly[1].C() = fPoly[3].C()-fPoly[2].C();
      fPoly[1].D() = 1./fPoly[1].D();

      if (m > 2) {
         for (i = 2; i < m; ++i) {
            q = fPoly[i-1].D()*fPoly[i-1].E();
            fPoly[i].D() = 1./(fPoly[i].D()-p*fPoly[i-2].F()-q*fPoly[i-1].E());
            fPoly[i].E() -= q*fPoly[i-1].F();
            fPoly[i].C() = fPoly[i+2].C()-fPoly[i+1].C()-p*fPoly[i-2].C()
                           -q*fPoly[i-1].C();
            p = fPoly[i-1].D()*fPoly[i-1].F();
         }
      }
   }

   fPoly[fNp-2].C() = fPoly[fNp-1].C() = 0;
   if (fNp > 3)
      for (i=fNp-3; i > 0; --i)
         fPoly[i].C() = (fPoly[i].C()-fPoly[i].E()*fPoly[i+1].C()
                        -fPoly[i].F()*fPoly[i+2].C())*fPoly[i].D();

   //     Integrate the third derivative of s(x)
   m = fNp-1;
   q = fPoly[1].X()-fPoly[0].X();
   r = fPoly[2].X()-fPoly[1].X();
   b1 = fPoly[1].B();
   q3 = q*q*q;
   qr = q+r;
   if (qr) {
      v = fPoly[1].C()/qr;
      t = v;
   } else
      v = t = 0;
   if (q) fPoly[0].F() = v/q;
   else fPoly[0].F() = 0;
   for (i = 1; i < m; ++i) {
      p = q;
      q = r;
      if (i != m-1) r = fPoly[i+2].X()-fPoly[i+1].X();
      else r = 0;
      p3 = q3;
      q3 = q*q*q;
      pq = qr;
      qr = q+r;
      s = t;
      if (qr) t = (fPoly[i+1].C()-fPoly[i].C())/qr;
      else t = 0;
      u = v;
      v = t-s;
      if (pq) {
         fPoly[i].F() = fPoly[i-1].F();
         if (q) fPoly[i].F() = v/q;
         fPoly[i].E() = s*5.;
         fPoly[i].D() = (fPoly[i].C()-q*s)*10;
         fPoly[i].C() =
         fPoly[i].D()*(p-q)+(fPoly[i+1].B()-fPoly[i].B()+(u-fPoly[i].E())*
                            p3-(v+fPoly[i].E())*q3)/pq;
         fPoly[i].B() = (p*(fPoly[i+1].B()-v*q3)+q*(fPoly[i].B()-u*p3))/pq-p
         *q*(fPoly[i].D()+fPoly[i].E()*(q-p));
      } else {
         fPoly[i].C() = fPoly[i-1].C();
         fPoly[i].D() = fPoly[i].E() = fPoly[i].F() = 0;
      }
   }

   //     End points x(1) and x(n)
   p = fPoly[1].X()-fPoly[0].X();
   s = fPoly[0].F()*p*p*p;
   fPoly[0].E() = fPoly[0].D() = 0;
   fPoly[0].C() = fPoly[1].C()-s*10;
   fPoly[0].B() = b1-(fPoly[0].C()+s)*p;

   q = fPoly[fNp-1].X()-fPoly[fNp-2].X();
   t = fPoly[fNp-2].F()*q*q*q;
   fPoly[fNp-1].E() = fPoly[fNp-1].D() = 0;
   fPoly[fNp-1].C() = fPoly[fNp-2].C()+t*10;
   fPoly[fNp-1].B() += (fPoly[fNp-1].C()-t)*q;
}


//______________________________________________________________________________
void TSpline5::Test()
{
   // Test method for TSpline5
   //
   //
   //   n          number of data points.
   //   m          2*m-1 is order of spline.
   //                 m = 3 always for quintic spline.
   //   nn,nm1,mm,
   //   mm1,i,k,
   //   j,jj       temporary integer variables.
   //   z,p        temporary double precision variables.
   //   x[n]       the sequence of knots.
   //   y[n]       the prescribed function values at the knots.
   //   a[200][6]  two dimensional array whose columns are
   //                 the computed spline coefficients
   //   diff[5]    maximum values of differences of values and
   //                 derivatives to right and left of knots.
   //   com[5]     maximum values of coefficients.
   //
   //
   //   test of TSpline5 with nonequidistant knots and
   //      equidistant knots follows.

   Double_t hx;
   Double_t diff[5];
   Double_t a[1200], c[6];
   Int_t i, j, k, m, n;
   Double_t p, x[200], y[200], z;
   Int_t jj, mm, nn;
   Int_t mm1, nm1;
   Double_t com[5];

   printf("1         TEST OF TSpline5 WITH NONEQUIDISTANT KNOTS\n");
   n = 5;
   x[0] = -3;
   x[1] = -1;
   x[2] = 0;
   x[3] = 3;
   x[4] = 4;
   y[0] = 7;
   y[1] = 11;
   y[2] = 26;
   y[3] = 56;
   y[4] = 29;
   m = 3;
   mm = m << 1;
   mm1 = mm-1;
   printf("\n-N = %3d    M =%2d\n",n,m);
   TSpline5 *spline = new TSpline5("Test",x,y,n);
   for (i = 0; i < n; ++i)
      spline->GetCoeff(i,hx, a[i],a[i+200],a[i+400],
                       a[i+600],a[i+800],a[i+1000]);
   delete spline;
   for (i = 0; i < mm1; ++i) diff[i] = com[i] = 0;
   for (k = 0; k < n; ++k) {
      for (i = 0; i < mm; ++i) c[i] = a[k+i*200];
      printf(" ---------------------------------------%3d --------------------------------------------\n",k+1);
      printf("%12.8f\n",x[k]);
      if (k == n-1) {
         printf("%16.8f\n",c[0]);
      } else {
         for (i = 0; i < mm; ++i) printf("%16.8f",c[i]);
         printf("\n");
         for (i = 0; i < mm1; ++i)
            if ((z=TMath::Abs(a[k+i*200])) > com[i]) com[i] = z;
         z = x[k+1]-x[k];
         for (i = 1; i < mm; ++i)
            for (jj = i; jj < mm; ++jj) {
               j = mm+i-jj;
               c[j-2] = c[j-1]*z+c[j-2];
            }
         for (i = 0; i < mm; ++i) printf("%16.8f",c[i]);
         printf("\n");
         for (i = 0; i < mm1; ++i)
            if (!(k >= n-2 && i != 0))
               if((z = TMath::Abs(c[i]-a[k+1+i*200]))
                  > diff[i]) diff[i] = z;
      }
   }
   printf("  MAXIMUM ABSOLUTE VALUES OF DIFFERENCES \n");
   for (i = 0; i < mm1; ++i) printf("%18.9E",diff[i]);
   printf("\n");
   printf("  MAXIMUM ABSOLUTE VALUES OF COEFFICIENTS \n");
   if (TMath::Abs(c[0]) > com[0])
      com[0] = TMath::Abs(c[0]);
   for (i = 0; i < mm1; ++i) printf("%16.8f",com[i]);
   printf("\n");
   m = 3;
   for (n = 10; n <= 100; n += 10) {
      mm = m << 1;
      mm1 = mm-1;
      nm1 = n-1;
      for (i = 0; i < nm1; i += 2) {
         x[i] = i+1;
         x[i+1] = i+2;
         y[i] = 1;
         y[i+1] = 0;
      }
      if (n % 2 != 0) {
         x[n-1] = n;
         y[n-1] = 1;
      }
      printf("\n-N = %3d    M =%2d\n",n,m);
      spline = new TSpline5("Test",x,y,n);
      for (i = 0; i < n; ++i)
         spline->GetCoeff(i,hx,a[i],a[i+200],a[i+400],
                          a[i+600],a[i+800],a[i+1000]);
      delete spline;
      for (i = 0; i < mm1; ++i)
         diff[i] = com[i] = 0;
      for (k = 0; k < n; ++k) {
         for (i = 0; i < mm; ++i)
            c[i] = a[k+i*200];
         if (n < 11) {
            printf(" ---------------------------------------%3d --------------------------------------------\n",k+1);
            printf("%12.8f\n",x[k]);
            if (k == n-1) printf("%16.8f\n",c[0]);
         }
         if (k == n-1) break;
         if (n <= 10) {
            for (i = 0; i < mm; ++i) printf("%16.8f",c[i]);
            printf("\n");
         }
         for (i = 0; i < mm1; ++i)
            if ((z=TMath::Abs(a[k+i*200])) > com[i])
               com[i] = z;
         z = x[k+1]-x[k];
         for (i = 1; i < mm; ++i)
            for (jj = i; jj < mm; ++jj) {
               j = mm+i-jj;
               c[j-2] = c[j-1]*z+c[j-2];
            }
         if (n <= 10) {
            for (i = 0; i < mm; ++i) printf("%16.8f",c[i]);
            printf("\n");
         }
         for (i = 0; i < mm1; ++i)
            if (!(k >= n-2 && i != 0))
               if ((z = TMath::Abs(c[i]-a[k+1+i*200]))
                  > diff[i]) diff[i] = z;
      }
      printf("  MAXIMUM ABSOLUTE VALUES OF DIFFERENCES \n");
      for (i = 0; i < mm1; ++i) printf("%18.9E",diff[i]);
      printf("\n");
      printf("  MAXIMUM ABSOLUTE VALUES OF COEFFICIENTS \n");
      if (TMath::Abs(c[0]) > com[0])
         com[0] = TMath::Abs(c[0]);
      for (i = 0; i < mm1; ++i) printf("%16.8E",com[i]);
      printf("\n");
   }

   //     Test of TSpline5 with nonequidistant double knots follows
   printf("1  TEST OF TSpline5 WITH NONEQUIDISTANT DOUBLE KNOTS\n");
   n = 5;
   x[0] = -3;
   x[1] = -3;
   x[2] = -1;
   x[3] = -1;
   x[4] = 0;
   x[5] = 0;
   x[6] = 3;
   x[7] = 3;
   x[8] = 4;
   x[9] = 4;
   y[0] = 7;
   y[1] = 2;
   y[2] = 11;
   y[3] = 15;
   y[4] = 26;
   y[5] = 10;
   y[6] = 56;
   y[7] = -27;
   y[8] = 29;
   y[9] = -30;
   m = 3;
   nn = n << 1;
   mm = m << 1;
   mm1 = mm-1;
   printf("-N = %3d    M =%2d\n",n,m);
   spline = new TSpline5("Test",x,y,nn);
   for (i = 0; i < nn; ++i)
      spline->GetCoeff(i,hx,a[i],a[i+200],a[i+400],
                       a[i+600],a[i+800],a[i+1000]);
   delete spline;
   for (i = 0; i < mm1; ++i)
      diff[i] = com[i] = 0;
   for (k = 0; k < nn; ++k) {
      for (i = 0; i < mm; ++i)
         c[i] = a[k+i*200];
      printf(" ---------------------------------------%3d --------------------------------------------\n",k+1);
      printf("%12.8f\n",x[k]);
      if (k == nn-1) {
         printf("%16.8f\n",c[0]);
         break;
      }
      for (i = 0; i < mm; ++i) printf("%16.8f",c[i]);
      printf("\n");
      for (i = 0; i < mm1; ++i)
         if ((z=TMath::Abs(a[k+i*200])) > com[i]) com[i] = z;
      z = x[k+1]-x[k];
      for (i = 1; i < mm; ++i)
         for (jj = i; jj < mm; ++jj) {
            j = mm+i-jj;
            c[j-2] = c[j-1]*z+c[j-2];
         }
      for (i = 0; i < mm; ++i) printf("%16.8f",c[i]);
      printf("\n");
      for (i = 0; i < mm1; ++i)
         if (!(k >= nn-2 && i != 0))
            if ((z = TMath::Abs(c[i]-a[k+1+i*200]))
               > diff[i]) diff[i] = z;
   }
   printf("  MAXIMUM ABSOLUTE VALUES OF DIFFERENCES \n");
   for (i = 1; i <= mm1; ++i) {
      printf("%18.9E",diff[i-1]);
   }
   printf("\n");
   if (TMath::Abs(c[0]) > com[0])
      com[0] = TMath::Abs(c[0]);
   printf("  MAXIMUM ABSOLUTE VALUES OF COEFFICIENTS \n");
   for (i = 0; i < mm1; ++i) printf("%16.8f",com[i]);
   printf("\n");
   m = 3;
   for (n = 10; n <= 100; n += 10) {
      nn = n << 1;
      mm = m << 1;
      mm1 = mm-1;
      p = 0;
      for (i = 0; i < n; ++i) {
         p += TMath::Abs(TMath::Sin(i+1));
         x[(i << 1)] = p;
         x[(i << 1)+1] = p;
         y[(i << 1)] = TMath::Cos(i+1)-.5;
         y[(i << 1)+1] = TMath::Cos((i << 1)+2)-.5;
      }
      printf("-N = %3d    M =%2d\n",n,m);
      spline = new TSpline5("Test",x,y,nn);
      for (i = 0; i < nn; ++i)
         spline->GetCoeff(i,hx,a[i],a[i+200],a[i+400],
                          a[i+600],a[i+800],a[i+1000]);
      delete spline;
      for (i = 0; i < mm1; ++i)
         diff[i] = com[i] = 0;
      for (k = 0; k < nn; ++k) {
         for (i = 0; i < mm; ++i)
            c[i] = a[k+i*200];
         if (n < 11) {
            printf(" ---------------------------------------%3d --------------------------------------------\n",k+1);
            printf("%12.8f\n",x[k]);
            if (k == nn-1) printf("%16.8f\n",c[0]);
         }
         if (k == nn-1) break;
         if (n <= 10) {
            for (i = 0; i < mm; ++i) printf("%16.8f",c[i]);
            printf("\n");
         }
         for (i = 0; i < mm1; ++i)
            if ((z=TMath::Abs(a[k+i*200])) > com[i]) com[i] = z;
         z = x[k+1]-x[k];
         for (i = 1; i < mm; ++i) {
            for (jj = i; jj < mm; ++jj) {
               j = mm+i-jj;
               c[j-2] = c[j-1]*z+c[j-2];
            }
         }
         if (n <= 10) {
            for (i = 0; i < mm; ++i) printf("%16.8f",c[i]);
            printf("\n");
         }
         for (i = 0; i < mm1; ++i)
            if (!(k >= nn-2 && i != 0))
               if ((z = TMath::Abs(c[i]-a[k+1+i*200]))
                  > diff[i]) diff[i] = z;
      }
      printf("  MAXIMUM ABSOLUTE VALUES OF DIFFERENCES \n");
      for (i = 0; i < mm1; ++i) printf("%18.9E",diff[i]);
      printf("\n");
      printf("  MAXIMUM ABSOLUTE VALUES OF COEFFICIENTS \n");
      if (TMath::Abs(c[0]) > com[0])
         com[0] = TMath::Abs(c[0]);
      for (i = 0; i < mm1; ++i) printf("%18.9E",com[i]);
      printf("\n");
   }

   //     test of TSpline5 with nonequidistant knots, one double knot,
   //        one triple knot, follows.
   printf("1         TEST OF TSpline5 WITH NONEQUIDISTANT KNOTS,\n");
   printf("             ONE DOUBLE, ONE TRIPLE KNOT\n");
   n = 8;
   x[0] = -3;
   x[1] = -1;
   x[2] = -1;
   x[3] = 0;
   x[4] = 3;
   x[5] = 3;
   x[6] = 3;
   x[7] = 4;
   y[0] = 7;
   y[1] = 11;
   y[2] = 15;
   y[3] = 26;
   y[4] = 56;
   y[5] = -30;
   y[6] = -7;
   y[7] = 29;
   m = 3;
   mm = m << 1;
   mm1 = mm-1;
   printf("-N = %3d    M =%2d\n",n,m);
   spline=new TSpline5("Test",x,y,n);
   for (i = 0; i < n; ++i)
      spline->GetCoeff(i,hx,a[i],a[i+200],a[i+400],
                       a[i+600],a[i+800],a[i+1000]);
   delete spline;
   for (i = 0; i < mm1; ++i)
      diff[i] = com[i] = 0;
   for (k = 0; k < n; ++k) {
      for (i = 0; i < mm; ++i)
         c[i] = a[k+i*200];
      printf(" ---------------------------------------%3d --------------------------------------------\n",k+1);
      printf("%12.8f\n",x[k]);
      if (k == n-1) {
         printf("%16.8f\n",c[0]);
         break;
      }
      for (i = 0; i < mm; ++i) printf("%16.8f",c[i]);
      printf("\n");
      for (i = 0; i < mm1; ++i)
         if ((z=TMath::Abs(a[k+i*200])) > com[i]) com[i] = z;
      z = x[k+1]-x[k];
      for (i = 1; i < mm; ++i)
         for (jj = i; jj < mm; ++jj) {
            j = mm+i-jj;
            c[j-2] = c[j-1]*z+c[j-2];
         }
      for (i = 0; i < mm; ++i) printf("%16.8f",c[i]);
      printf("\n");
      for (i = 0; i < mm1; ++i)
         if (!(k >= n-2 && i != 0))
            if ((z = TMath::Abs(c[i]-a[k+1+i*200]))
               > diff[i]) diff[i] = z;
   }
   printf("  MAXIMUM ABSOLUTE VALUES OF DIFFERENCES \n");
   for (i = 0; i < mm1; ++i) printf("%18.9E",diff[i]);
   printf("\n");
   printf("  MAXIMUM ABSOLUTE VALUES OF COEFFICIENTS \n");
   if (TMath::Abs(c[0]) > com[0])
      com[0] = TMath::Abs(c[0]);
   for (i = 0; i < mm1; ++i) printf("%16.8f",com[i]);
   printf("\n");

   //     Test of TSpline5 with nonequidistant knots, two double knots,
   //        one triple knot,follows.
   printf("1         TEST OF TSpline5 WITH NONEQUIDISTANT KNOTS,\n");
   printf("             TWO DOUBLE, ONE TRIPLE KNOT\n");
   n = 10;
   x[0] = 0;
   x[1] = 2;
   x[2] = 2;
   x[3] = 3;
   x[4] = 3;
   x[5] = 3;
   x[6] = 5;
   x[7] = 8;
   x[8] = 9;
   x[9] = 9;
   y[0] = 163;
   y[1] = 237;
   y[2] = -127;
   y[3] = 119;
   y[4] = -65;
   y[5] = 192;
   y[6] = 293;
   y[7] = 326;
   y[8] = 0;
   y[9] = -414;
   m = 3;
   mm = m << 1;
   mm1 = mm-1;
   printf("-N = %3d    M =%2d\n",n,m);
   spline = new TSpline5("Test",x,y,n);
   for (i = 0; i < n; ++i)
      spline->GetCoeff(i,hx,a[i],a[i+200],a[i+400],
                       a[i+600],a[i+800],a[i+1000]);
   delete spline;
   for (i = 0; i < mm1; ++i)
      diff[i] = com[i] = 0;
   for (k = 0; k < n; ++k) {
      for (i = 0; i < mm; ++i)
         c[i] = a[k+i*200];
      printf(" ---------------------------------------%3d --------------------------------------------\n",k+1);
      printf("%12.8f\n",x[k]);
      if (k == n-1) {
         printf("%16.8f\n",c[0]);
         break;
      }
      for (i = 0; i < mm; ++i) printf("%16.8f",c[i]);
      printf("\n");
      for (i = 0; i < mm1; ++i)
         if ((z=TMath::Abs(a[k+i*200])) > com[i]) com[i] = z;
      z = x[k+1]-x[k];
      for (i = 1; i < mm; ++i)
         for (jj = i; jj < mm; ++jj) {
            j = mm+i-jj;
            c[j-2] = c[j-1]*z+c[j-2];
         }
      for (i = 0; i < mm; ++i) printf("%16.8f",c[i]);
      printf("\n");
      for (i = 0; i < mm1; ++i)
         if (!(k >= n-2 && i != 0))
            if((z = TMath::Abs(c[i]-a[k+1+i*200]))
               > diff[i]) diff[i] = z;
   }
   printf("  MAXIMUM ABSOLUTE VALUES OF DIFFERENCES \n");
   for (i = 0; i < mm1; ++i) printf("%18.9E",diff[i]);
   printf("\n");
   printf("  MAXIMUM ABSOLUTE VALUES OF COEFFICIENTS \n");
   if (TMath::Abs(c[0]) > com[0])
      com[0] = TMath::Abs(c[0]);
   for (i = 0; i < mm1; ++i) printf("%16.8f",com[i]);
   printf("\n");
}


//______________________________________________________________________________
void TSpline5::Streamer(TBuffer &R__b)
{
   // Stream an object of class TSpline5.

   if (R__b.IsReading()) {
      UInt_t R__s, R__c;
      Version_t R__v = R__b.ReadVersion(&R__s, &R__c);
      if (R__v > 1) {
         R__b.ReadClassBuffer(TSpline5::Class(), this, R__v, R__s, R__c);
         return;
      }
      //====process old versions before automatic schema evolution
      TSpline::Streamer(R__b);
      if (fNp > 0) {
         fPoly = new TSplinePoly5[fNp];
         for(Int_t i=0; i<fNp; ++i) {
            fPoly[i].Streamer(R__b);
         }
      }
      //      R__b >> fPoly;
   } else {
      R__b.WriteClassBuffer(TSpline5::Class(),this);
   }
}

---