testMinim.cxx

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// test of minimization usingnew minimizer classes

#include "Math/Minimizer.h"
#include "Math/Factory.h"

#include "TVirtualFitter.h"

#include "Math/IFunction.h"
#include "Math/Util.h"
#include <cmath>
#include <cassert>

#include <string>
#include <iostream>

#include "TStopwatch.h"
#include "TMatrixD.h"
#include "TVectorD.h"
#include "TRandom3.h"
#include "TMath.h"

#include "RVersion.h"
#include "RConfigure.h"

//#define DEBUG

int gNCall = 0;
int gNCall2 = 0;
int gNmin = 1;
int gVerbose = 1;


bool minos = false;

double gAbsTolerance = 5.E-6;   // otherwise gsl_conjugate_PR fails

// Rosenbrok function to be minimize

typedef void   (*FCN)(Int_t &npar, Double_t *gin, Double_t &f, Double_t *u, Int_t flag);



// ROSENBROCK function
////////////////////////////////////////////////////////////////////////////////

void RosenBrock(Int_t &, Double_t *, Double_t &f, Double_t *par, Int_t /*iflag*/)
{
  gNCall++;
  const Double_t x = par[0];
  const Double_t y = par[1];
  const Double_t tmp1 = y-x*x;
  const Double_t tmp2 = 1-x;
  f = 100*tmp1*tmp1+tmp2*tmp2;
}



class RosenBrockFunction : public ROOT::Math::IMultiGenFunction {

public :


   unsigned int NDim() const { return 2; }

   ROOT::Math::IMultiGenFunction * Clone() const {
      return new RosenBrockFunction();
   }

   const double *  TrueXMinimum() const {
      fTrueMin[0] = 1;
      fTrueMin[1] = 1;
      return fTrueMin;
   }

   double TrueMinimum() const {
      return 0;
   }

   private:

   inline double DoEval (const double * x) const {
#ifdef USE_FREE_FUNC
      double f = 0;
      int ierr = 0;
      int i = 0;
      RosenBrock(i,0,f,const_cast<double *>(x),ierr);
      return f;
#else
      gNCall++;
      const Double_t xx = x[0];
      const Double_t yy = x[1];
      const Double_t tmp1 = yy-xx*xx;
      const Double_t tmp2 = 1-xx;
      return 100*tmp1*tmp1+tmp2*tmp2;
#endif
   }


   mutable double fTrueMin[2];
};

// TRIGONOMETRIC FLETCHER FUNCTION

class TrigoFletcherFunction : public ROOT::Math::IMultiGradFunction {

public :


   TrigoFletcherFunction(unsigned int dim) : fDim(dim) {
      double seed = 3;
      A.ResizeTo(dim,dim);
      B.ResizeTo(dim,dim);
      x0.ResizeTo(dim);
      sx0.ResizeTo(dim);
      cx0.ResizeTo(dim);
      sx.ResizeTo(dim);
      cx.ResizeTo(dim);
      v0.ResizeTo(dim);
      v.ResizeTo(dim);
      r.ResizeTo(dim);
      A.Randomize(-100.,100,seed);
      B.Randomize(-100.,100,seed);
      for (unsigned int i = 0; i < dim; i++) {
         for (unsigned int j = 0; j < dim; j++) {
            A(i,j) = int(A(i,j));
            B(i,j) = int(B(i,j));
         }
      }
      x0.Randomize(-TMath::Pi(),TMath::Pi(),seed);
      // calculate vector Ei
      for (unsigned int i = 0; i < fDim ; ++i) {
         cx0[i] = std::cos(x0[i]);
         sx0[i] = std::sin(x0[i]);
      }
      v0 = A*sx0+B*cx0;
   }


   unsigned int NDim() const { return fDim; }

   ROOT::Math::IMultiGenFunction * Clone() const {
      TrigoFletcherFunction * f = new TrigoFletcherFunction(*this);
//       std::cerr <<"cannot clone this function" << std::endl;
//       assert(0);
      return f;
   }


   void StartPoints(double * x, double * s) {
      TRandom3 rndm;
      const double stepSize = 0.01;
      const double deltaAmp = 0.1;
      const double pi = TMath::Pi();
      for (unsigned int i = 0; i < fDim; ++i) {
         double delta = rndm.Uniform(-deltaAmp*pi,deltaAmp*pi);
         x[i] = x0(i) + 0.1*delta;
         if (x[i] <=  - pi) x[i] += 2.*pi;
         if (x[i] >     pi) x[i] -= 2.*pi;
         s[i] = stepSize;
      }
   }


   const double *  TrueXMinimum() const {
      return x0.GetMatrixArray();
   }

   double TrueMinimum() const { return 0; }

   void  Gradient (const double * x, double * g) const {
      gNCall2++;

      for (unsigned int i = 0; i < fDim ; ++i) {
         cx [i] = std::cos(x[i]);
         sx [i] = std::sin(x[i]);
      }

      v  = A*sx +B*cx;
      r  = v0-v;


      // calculate the grad components
      for (unsigned int i = 0; i < fDim ; ++i) {
         g[i]  = 0;
         for (unsigned int k = 0; k < fDim ; ++k) {
            g[i] += 2. * r(k) * ( - A(k,i) * cx(i) + B(k,i) * sx(i) );
         }
      }

   }

#ifdef USE_FDF
   void FdF (const double * x, double & f, double * g) const {
      gNCall++;

      for (unsigned int i = 0; i < fDim ; ++i) {
         cx [i] = std::cos(x[i]);
         sx [i] = std::sin(x[i]);
      }

      v  = A*sx +B*cx;
      r  = v0-v;

      f = r * r;


      // calculate the grad components
      for (unsigned int i = 0; i < fDim ; ++i) {
         g[i]  = 0;
         for (unsigned int k = 0; k < fDim ; ++k) {
            g[i] += 2. * r(k) * ( - A(k,i) * cx(i) + B(k,i) * sx(i) );
         }
      }
   }
#endif

   private:

//    TrigoFletcherFunction(const TrigoFletcherFunction & ) {}
//    TrigoFletcherFunction & operator=(const TrigoFletcherFunction &) { return *this; }

   double DoEval (const double * x) const {
      gNCall++;


      for (unsigned int i = 0; i < fDim ; ++i) {
         cx [i] = std::cos(x[i]);
         sx [i] = std::sin(x[i]);
      }

      v  = A*sx +B*cx;
      r  = v0-v;

      return  r * r;
   }


   double DoDerivative (const double * x, unsigned int i ) const {
      std::vector<double> g(fDim);
      Gradient(x,&g[0]);
      return  g[i];
   }

private:

   unsigned int fDim;

   TMatrixD A;
   TMatrixD B;
   TVectorD x0;
   mutable TVectorD sx0;
   mutable TVectorD cx0;
   mutable TVectorD sx;
   mutable TVectorD cx;
   mutable TVectorD v0;
   mutable TVectorD v;
   mutable TVectorD r;


};


// CHEBYQUAD FUNCTION

class ChebyQuadFunction : public ROOT::Math::IMultiGradFunction {

public :

   ChebyQuadFunction(unsigned int n) :
      fDim(n),
      fvec(std::vector<double>(n) ),
      fTrueMin(std::vector<double>(n) )
   {
   }

   unsigned int NDim() const { return fDim; }

   ROOT::Math::IMultiGenFunction * Clone() const {
      return new ChebyQuadFunction(*this);
   }

   const double *  TrueXMinimum() const {
      return &fTrueMin[0];
   }

   double TrueMinimum() const { return 0; }

   // use equally spaced points
   void StartPoints(double * x, double * s) {
      for (unsigned int i = 0; i < fDim; ++i) {
         s[i] = 0.01;
         x[i] = double(i)/(double(fDim)+1.0);
      }
   }

   // compute gradient

   void Gradient(const double * x, double * g) const {
      gNCall2++;
      unsigned int n = fDim;
      // estimate first the fvec
      DoCalculatefi(x);

      for (unsigned int j = 0; j <  n; ++j) {
         g[j] = 0.0;
         double t1 = 1.0;
         double t2 = 2.0 * x[j] - 1.0;
         double t = 2.0 * t2;
         double s1 = 0.0;
         double s2 = 2.0;
         for (unsigned int i = 0; i <  n; ++i) {
            g[j] += fvec[i] * s2;
            double th = 4.0 * t2 + t * s2 - s1;
            s1 = s2;
            s2 = th;
            th = t * t2 - t1;
            t1 = t2;
            t2 = th;
         }
         g[j] = 2. * g[j] / double(n);
      }


   }

   private:

   double DoEval (const double * x) const {

      gNCall++;
      DoCalculatefi(x);
      double f = 0;
      for (unsigned int i = 0; i < fDim; ++i)
         f += fvec[i] * fvec[i];

      return f;

   }

   double DoDerivative (const double * x, unsigned int i ) const {
      std::vector<double> g(fDim);
      Gradient(x,&g[0]);
      return  g[i];
   }

   void DoCalculatefi(const double * x) const {
      // calculate the i- element ; F(X) = Sum {fi]
      unsigned int n = fDim;
      for (unsigned int i = 0; i < n; ++i)
         fvec[i] = 0;

      for (unsigned int  j = 0; j <  n; ++j) {
         double t1 = 1.0;
         double t2 = 2.0 * x[j] - 1.0;
         double t = 2.0 * t2;
         for (unsigned int  i = 0; i <  n; ++i) {
            fvec[i] +=  t2;
            double th = t * t2 - t1;
            t1 = t2;
            t2 = th;
         }
      }

      // sum with the integral (integral is zero for odd Cheb polynomial and = 1/(i**2 -1) for the even ones
      for (unsigned int  i = 1; i <=  n; ++i) {
         int l = i-1;
         fvec[l] /= double ( n );
         if ( ( i % 2 ) == 0 ) {
            fvec[l] += 1.0 / ( double ( i*i ) - 1.0 );
         }
      }
   }

   unsigned int fDim;
   mutable std::vector<double> fvec;
   mutable std::vector<double> fTrueMin;
};



const double *  TrueXMinimum(const  ROOT::Math::IMultiGenFunction & func) {

   const RosenBrockFunction * fRB = dynamic_cast< const RosenBrockFunction *> (&func);
   if (fRB != 0) return fRB->TrueXMinimum();
   const TrigoFletcherFunction * fTF = dynamic_cast< const TrigoFletcherFunction *> (&func);
   if (fTF != 0) return fTF->TrueXMinimum();
//    const ChebyQuadFunction * fCQ = dynamic_cast< const ChebyQuadFunction *> (&func);
//    if (fCQ != 0) return fCQ->TrueXMinimum();
   return 0;
}
double TrueMinimum(const  ROOT::Math::IMultiGenFunction & func) {

   const RosenBrockFunction * fRB = dynamic_cast< const RosenBrockFunction *> (&func);
   if (fRB != 0) return fRB->TrueMinimum();
   const TrigoFletcherFunction * fTF = dynamic_cast< const TrigoFletcherFunction *> (&func);
   if (fTF != 0) return fTF->TrueMinimum();
//    const ChebyQuadFunction * fCQ = dynamic_cast< const ChebyQuadFunction *> (&func);
//    if (fCQ != 0) return fCQ->TrueXMinimum();
   return 0;
}

void printMinimum(const std::vector<double> & x) {
   std::cout << "Minimum X values\n";
   std::cout << "\t";
   int pr = std::cout.precision(12);
   unsigned int n = x.size();
   for (unsigned int i = 0; i < n; ++i) {
      std::cout << x[i];
      if ( i != n-1 ) std::cout << " , ";
      if ( i > 0 && i % 6 == 0 ) std::cout << "\n\t";
   }
   std::cout << std::endl;
   std::cout.precision(pr);
}

int DoNewMinimization( const ROOT::Math::IMultiGenFunction & func, const double * x0, const double * s0, ROOT::Math::Minimizer * min, double &minval, double &edm, double * minx) {

   int iret = 0;

   if (func.NDim() >= 10) {
      min->SetMaxFunctionCalls(1000000);
      min->SetMaxIterations(100000);
      min->SetTolerance(0.01);
   }
   else
      min->SetTolerance(0.001);


   min->SetPrintLevel(gVerbose);
   // check if func provides gradient
   const ROOT::Math::IMultiGradFunction * gfunc = dynamic_cast<const  ROOT::Math::IMultiGradFunction *>(&func);
   if (gfunc != 0)
      min->SetFunction(*gfunc);
   else
      min->SetFunction(func);

   for (unsigned int i = 0; i < func.NDim(); ++i) {
      min->SetVariable(i,"x" + ROOT::Math::Util::ToString(i),x0[i], s0[i]);
   }

   bool ret = min->Minimize();
   minval = min->MinValue();
   edm = min->Edm();

   if (!ret) {
      delete min;
      return -1;
   }

   const double * xmin = min->X();

   bool ok = true;
   const double *  trueMin = TrueXMinimum(func);
   if (trueMin != 0) {
      ok &= (std::fabs(minval - TrueMinimum(func) ) < gAbsTolerance );
      for (unsigned int i = 0; i < func.NDim(); ++i)
         ok &= (std::fabs(xmin[i]-trueMin[i] ) < sqrt(gAbsTolerance));
   }

   if (!ok) iret = -2;

   // test Minos (use the default up of 1)
   if (minos) {

      double el,eu;
      for (unsigned int i = 0; i < func.NDim(); ++i) {
         ret  = min->GetMinosError(i,el,eu);
         if (ret) std::cout << "MINOS error  for " << i  << " = " << el << "   " << eu << std::endl;
         else     std::cout << "MINOS failed for " << i << std::endl;
      }
   }

#ifdef DEBUG
   std::cout << "ncalls = " << min->NCalls() << std::endl;
#endif

//   std::cout << "function at the minimum " << func(xmin) << std::endl;
   std::copy(xmin,xmin+func.NDim(),minx);
   min->Clear();

   return iret;
}

int DoOldMinimization( FCN  func, TVirtualFitter * min, double &minval, double &edm) {

  int iret = 0;

  assert(min != 0);
  min->SetFCN( func );

  Double_t arglist[100];
  arglist[0] = gVerbose-1;
  min->ExecuteCommand("SET PRINT",arglist,1);

  min->SetParameter(0,"x",-1.2,0.01,0,0);
  min->SetParameter(1,"y", 1.0,0.01,0,0);

  arglist[0] = 0;
  min->ExecuteCommand("SET NOW",arglist,0);
  arglist[0] = 1000; // number of function calls
  arglist[1] = 0.001; // tolerance
  //min->ExecuteCommand("MIGRAD",arglist,0);
  min->ExecuteCommand("MIGRAD",arglist,2);

  if (minos) min->ExecuteCommand("MINOS",arglist,0);

  Double_t parx,pary;
  Double_t we,al,bl;
  Char_t parName[32];
  min->GetParameter(0,parName,parx,we,al,bl);
  min->GetParameter(1,parName,pary,we,al,bl);

  bool ok = ( TMath::Abs(parx-1.) < sqrt(gAbsTolerance) &&
              TMath::Abs(pary-1.) < sqrt(gAbsTolerance) );



  double errdef = 0;
  int nvpar, nparx;
  min->GetStats(minval,edm,errdef,nvpar,nparx);

  if (!ok) iret = -2;

  min->Clear(); // for further use
  return iret;

}


int testNewMinimizer( const ROOT::Math::IMultiGenFunction & func, const double * x0, const double * s0, const std::string & minimizer, const std::string & algoType) {

   std::cout << "\n************************************************************\n";
   std::cout << "\tTest new ROOT::Math::Minimizer\n";
   std::cout << "\tMinimizer is " << minimizer << "  " << algoType << std::endl;

   int iret = 0;
   double minval = 0., edm = 0.;
   std::vector<double> xmin(func.NDim() );

   TStopwatch w;
   w.Start();

   ROOT::Math::Minimizer * min = ROOT::Math::Factory::CreateMinimizer(minimizer, algoType);
   if (min == 0) {
      std::cout << "Error using minimizer " << minimizer << "  " << algoType << std::endl;
      return -1;
   }

   for (int i = 0; i < gNmin; ++i) {
      gNCall = 0; gNCall2 = 0;
      iret |= DoNewMinimization(func, x0, s0, min,minval,edm,&xmin[0]);
   }

   w.Stop();
   if (iret != 0) std::cout << "\n****** ERROR:   Minimization FAILED ! \n";
   int pr = std::cout.precision(18);
   std::cout << "\nNCalls: \t" << gNCall << " , " << gNCall2
             << "\tMinValue: \t" << minval << "\tEdm: \t" << edm;  std::cout.precision(pr);
   std::cout << "\nTime:   \t" << w.RealTime() << " , " << w.CpuTime() << std::endl;
   printMinimum(xmin );
   std::cout << "\n************************************************************\n";

#ifdef CHECK_WITHMINUIT
   // do Minuit after BFGS
   if (minimizer == "GSL_BFGS") {
      std::cout << "DO Minuit2 from last point\n";
      gNCall = 0;
      iret |= DoNewMinimization(func, &xmin.front(), s0, "Minuit2","",minval,edm,&xmin[0]);
      int pr = std::cout.precision(18);
      std::cout << "\nNCalls: \t" << gNCall << "\tMinValue: \t" << minval << "\tEdm: \t" << edm;  std::cout.precision(pr);
      std::cout << std::endl;
   }
#endif

   delete min;

   return iret;
}


int testOldMinimizer( FCN  func, const std::string & fitter, int n=25) {

   std::cout << "\n************************************************************\n";
   std::cout << "\tTest using TVirtualFitter\n";
   std::cout << "\tFitter is " << fitter << std::endl;

   int iret = 0;
   double minval = 0.,edm = 0.;

   TStopwatch w;
   w.Start();

   TVirtualFitter::SetDefaultFitter(fitter.c_str());

   TVirtualFitter *min = TVirtualFitter::Fitter(0,n);

   //min->Dump();

   for (int i = 0; i < gNmin; ++i) {
      gNCall = 0;
      iret |= DoOldMinimization(func, min,minval,edm);
   }

   w.Stop();
   if (iret != 0) std::cout << "\n****** ERROR:   Minimization FAILED ! \n";
   int pr = std::cout.precision(18);
   std::cout << "\nNCalls: \t" << gNCall << "\tMinValue: \t" << minval << "\tEdm: \t" << edm;  std::cout.precision(pr);
   std::cout << "\nTime: \t" << w.RealTime() << " , " << w.CpuTime() << std::endl;
   std::cout << "\n************************************************************\n";

   return iret;
}

int testRosenBrock() {

   int iret = 0;


   std::cout << "\n************************************************************\n";
   std::cout << "\tROSENBROCK function test\n\n";

   double s0[2] = {0.01,0.01};

   // minimize using Rosenbrock Function
#ifndef DEBUG
   gNmin = 1;
#endif


#if ROOT_VERSION_CODE < ROOT_VERSION(5,99,00)
   iret |= testOldMinimizer(RosenBrock,"Minuit",2);
   iret |= testOldMinimizer(RosenBrock,"Minuit2",2);
#endif

   RosenBrockFunction fRB;
   double xRB[2] = { -1.,1.2};
   iret |= testNewMinimizer(fRB,xRB,s0,"Minuit","");
   iret |= testNewMinimizer(fRB,xRB,s0,"Minuit2","");
#ifdef R__HAS_MATHMORE
   iret |= testNewMinimizer(fRB,xRB,s0,"GSLMultiMin","ConjugateFR");
   iret |= testNewMinimizer(fRB,xRB,s0,"GSLMultiMin","ConjugatePR");
   iret |= testNewMinimizer(fRB,xRB,s0,"GSLMultiMin","BFGS");
   iret |= testNewMinimizer(fRB,xRB,s0,"GSLMultiMin","BFGS2");
   //iret |= testNewMinimizer(fRB,xRB,s0,"Genetic","");
#endif

   return iret;
}


int testTrigoFletcher() {

   int iret = 0;


   // test with fletcher trigonometric function
#ifndef DEBUG
   gNmin = 1;
#endif

   const int nT = 50;
   TrigoFletcherFunction fTrigo(nT);
   double sTrigo[nT];
   double xTrigo[nT];
   fTrigo.StartPoints(xTrigo,sTrigo);

   std::cout << "\n************************************************************\n";
   std::cout << "\tTRIGONOMETRIC Fletcher function test , n = " << nT << "\n\n";


   iret |= testNewMinimizer(fTrigo,xTrigo,sTrigo,"Minuit2","");
   iret |= testNewMinimizer(fTrigo,xTrigo,sTrigo,"Minuit","");
#ifdef R__HAS_MATHMORE
   iret |= testNewMinimizer(fTrigo,xTrigo,sTrigo,"GSLMultiMin","ConjugateFR");
   iret |= testNewMinimizer(fTrigo,xTrigo,sTrigo,"GSLMultiMin","ConjugatePR");
   iret |= testNewMinimizer(fTrigo,xTrigo,sTrigo,"GSLMultiMin","BFGS");
#endif

   return iret;
}

int testChebyQuad() {

   int iret = 0;

   // test with ChebyQuad function

   const int n = 8;
   ChebyQuadFunction func(n);

#ifndef DEBUG
   gNmin = std::max(1, int(1000/n/n) );
#endif


   double s0[n];
   double x0[n];
   func.StartPoints(x0,s0);

   std::cout << "\n************************************************************\n";
   std::cout << "\tCHEBYQUAD function test , n = " << n << "\n\n";


//    double x[8] = {0.043153E+00, 0.193091E+00, 0.266329E+00, 0.500000E+00,
//                   0.500000E+00, 0.733671E+00, 0.806910E+00, 0.956847E+00 };
//    double x[2] = {0.5, 0.5001};
//    std::cout << "FUNC " << func(x) << std::endl;
   double x1[100] = { 0.00712780070646 , 0.0123441993113 , 0.0195428378255 , 0.0283679084192 , 0.0385291131289 , 0.0492202424892 , 0.0591277976178 ,
        0.0689433195252 , 0.0791293590525 , 0.088794974369 , 0.0988949579193 , 0.108607151294 , 0.118571075831 ,
        0.128605446238 , 0.137918291068 , 0.149177761352 , 0.156665324587 , 0.170851061982 , 0.174688134016 ,
        0.192838903364 , 0.193078270803 , 0.209255377225 , 0.217740096779 , 0.225888518345 , 0.241031047421 ,
        0.244253844041 , 0.257830449676 , 0.269467652526 , 0.274286498012 , 0.288877029988 , 0.297549406597 ,
        0.304950954529 , 0.319230811642 , 0.326387092206 , 0.335229058731 , 0.349178359226 , 0.355905988048 ,
        0.365197862755 , 0.379068092603 , 0.385826036925 , 0.394978252826 , 0.408974425717 , 0.415968185065 ,
        0.424621041584 , 0.438837361714 , 0.446214149031 , 0.454242324351 , 0.468614308013 , 0.476506553416 ,
        0.483916944941 , 0.498229247409 , 0.506794629616 , 0.513736742474 , 0.527712475478 , 0.537073277673 ,
        0.543731917673 , 0.557187513963 , 0.567346279639 , 0.57379846397 , 0.586691058785 , 0.597561941009 ,
        0.60382873461 , 0.616316037506 , 0.627719652101 , 0.633760038662 , 0.646175283836 , 0.657809344891 ,
        0.663569004722 , 0.676314563639 , 0.687674566849 , 0.69332205923 , 0.706839545953 , 0.716907408637 ,
        0.723407327715 , 0.738019389561 , 0.744806584048 , 0.754657613362 , 0.769181875619 , 0.772250323489 ,
        0.787104833193 , 0.795856360905 , 0.804099304478 , 0.82142178741 , 0.819589601284 , 0.839024540481 ,
        0.842457233039 , 0.857393475964 , 0.86408033345 , 0.876329840525 , 0.884867318008 , 0.895744532071 ,
        0.905113958163 , 0.915445338697 , 0.925148068352 , 0.935344457785 , 0.945127838313 , 0.955272197168 ,
                      0.965687518559 , 0.975129521484 , 0.982662007764 };

   std::cout << "FUNC " << func(x1) << std::endl;


   iret |= testNewMinimizer(func,x0,s0, "Minuit2","");
   iret |= testNewMinimizer(func,x0,s0, "Minuit","");
#ifdef R__HAS_MATHMORE
   iret |= testNewMinimizer(func,x0,s0, "GSLMultiMin","ConjugateFR");
   iret |= testNewMinimizer(func,x0,s0, "GSLMultiMin","ConjugatePR");
   iret |= testNewMinimizer(func,x0,s0, "GSLMultiMin","BFGS");
#endif

   return iret;
}

int main() {

   int iret = 0;

#ifdef DEBUG
   gVerbose = 3;
   gNmin = 1;
#endif

   iret |=  testRosenBrock();
//    iret |=  testChebyQuad();
//    iret |=  testTrigoFletcher();



   if (iret != 0)
      std::cerr << "testMinim :\t FAILED " << std::endl;
   else
      std::cerr << "testMinim :\t OK " << std::endl;
   return iret;

}