TVectorT.cxx

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// @(#)root/matrix:$Id$
// Authors: Fons Rademakers, Eddy Offermann  Nov 2003

/*************************************************************************
 * 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.             *
 *************************************************************************/

/** \class TVectorT
    \ingroup Matrix

 TVectorT

 Template class of Vectors in the linear algebra package

 Unless otherwise specified, vector indices always start with 0,
 spanning up to the specified limit-1.

 For (n) vectors where n <= kSizeMax (5 currently) storage space is
 available on the stack, thus avoiding expensive allocation/
 deallocation of heap space . However, this introduces of course
 kSizeMax overhead for each vector object . If this is an issue
 recompile with a new appropriate value (>=0) for kSizeMax

 Another way to assign and store vector data is through Use
 see for instance stressLinear.cxx file .

 Note that Constructors/assignments exists for all different matrix
 views

 For usage examples see $ROOTSYS/test/stressLinear.cxx

*/

#include "TVectorT.h"
#include "TClass.h"
#include "TMath.h"
#include "TROOT.h"
#include "Varargs.h"

templateClassImp(TVectorT)


////////////////////////////////////////////////////////////////////////////////
/// Delete data pointer m, if it was assigned on the heap

template<class Element>
void TVectorT<Element>::Delete_m(Int_t size,Element *&m)
{
   if (m) {
      if (size > kSizeMax)
         delete [] m;
      m = 0;
   }
}

////////////////////////////////////////////////////////////////////////////////
/// Return data pointer . if requested size <= kSizeMax, assign pointer
/// to the stack space

template<class Element>
Element* TVectorT<Element>::New_m(Int_t size)
{
   if (size == 0) return 0;
   else {
      if ( size <= kSizeMax )
         return fDataStack;
      else {
         Element *heap = new Element[size];
         return heap;
      }
   }
}

////////////////////////////////////////////////////////////////////////////////
/// Add vector v to this vector

template<class Element>
void TVectorT<Element>::Add(const TVectorT<Element> &v)
{
   if (gMatrixCheck && !AreCompatible(*this,v)) {
      Error("Add(TVectorT<Element> &)","vector's not compatible");
      return;
   }

   const Element *sp = v.GetMatrixArray();
         Element *tp = this->GetMatrixArray();
   const Element * const tp_last = tp+fNrows;
   while (tp < tp_last)
      *tp++ += *sp++;
}

////////////////////////////////////////////////////////////////////////////////
/// Set this vector to v1+v2

template<class Element>
void TVectorT<Element>::Add(const TVectorT<Element> &v1,const TVectorT<Element> &v2)
{
   if (gMatrixCheck) {
      if ( !AreCompatible(*this,v1) && !AreCompatible(*this,v2)) {
         Error("Add(TVectorT<Element> &)","vectors not compatible");
         return;
      }
   }

   const Element *sv1 = v1.GetMatrixArray();
   const Element *sv2 = v2.GetMatrixArray();
         Element *tp = this->GetMatrixArray();
   const Element * const tp_last = tp+fNrows;
   while (tp < tp_last)
      *tp++ = *sv1++ + *sv2++;
}

////////////////////////////////////////////////////////////////////////////////
/// Copy copySize doubles from *oldp to *newp . However take care of the
/// situation where both pointers are assigned to the same stack space

template<class Element>
Int_t TVectorT<Element>::Memcpy_m(Element *newp,const Element *oldp,Int_t copySize,
                                  Int_t newSize,Int_t oldSize)
{
   if (copySize == 0 || oldp == newp) return 0;
   else {
      if ( newSize <= kSizeMax && oldSize <= kSizeMax ) {
         // both pointers are inside fDataStack, be careful with copy direction !
         if (newp > oldp) {
            for (Int_t i = copySize-1; i >= 0; i--)
               newp[i] = oldp[i];
         } else {
            for (Int_t i = 0; i < copySize; i++)
               newp[i] = oldp[i];
         }
      }
      else {
         memcpy(newp,oldp,copySize*sizeof(Element));
      }
   }
   return 0;
}

////////////////////////////////////////////////////////////////////////////////
/// Allocate new vector. Arguments are number of rows and row
/// lowerbound (0 default).

template<class Element>
void TVectorT<Element>::Allocate(Int_t nrows,Int_t row_lwb,Int_t init)
{
   fIsOwner  = kTRUE;
   fNrows    = 0;
   fRowLwb   = 0;
   fElements = 0;

   if (nrows < 0)
   {
      Error("Allocate","nrows=%d",nrows);
      return;
   }

   MakeValid();
   fNrows   = nrows;
   fRowLwb  = row_lwb;

   fElements = New_m(fNrows);
   if (init)
      memset(fElements,0,fNrows*sizeof(Element));
}

////////////////////////////////////////////////////////////////////////////////
/// Constructor n-vector

template<class Element>
TVectorT<Element>::TVectorT(Int_t n)
{
   Allocate(n,0,1);
}

////////////////////////////////////////////////////////////////////////////////
/// Constructor [lwb..upb]-vector

template<class Element>
TVectorT<Element>::TVectorT(Int_t lwb,Int_t upb)
{
   Allocate(upb-lwb+1,lwb,1);
}

////////////////////////////////////////////////////////////////////////////////
/// Constructor n-vector with data copied from array elements

template<class Element>
TVectorT<Element>::TVectorT(Int_t n,const Element *elements)
{
   Allocate(n,0);
   SetElements(elements);
}

////////////////////////////////////////////////////////////////////////////////
/// Constructor [lwb..upb]-vector with data copied from array elements

template<class Element>
TVectorT<Element>::TVectorT(Int_t lwb,Int_t upb,const Element *elements)
{
   Allocate(upb-lwb+1,lwb);
   SetElements(elements);
}

////////////////////////////////////////////////////////////////////////////////
/// Copy constructor

template<class Element>
TVectorT<Element>::TVectorT(const TVectorT &another) : TObject(another)
{
   R__ASSERT(another.IsValid());
   Allocate(another.GetUpb()-another.GetLwb()+1,another.GetLwb());
   *this = another;
}

////////////////////////////////////////////////////////////////////////////////
/// Constructor : create vector from matrix row

template<class Element>
TVectorT<Element>::TVectorT(const TMatrixTRow_const<Element> &mr) : TObject()
{
   const TMatrixTBase<Element> *mt = mr.GetMatrix();
   R__ASSERT(mt->IsValid());
   Allocate(mt->GetColUpb()-mt->GetColLwb()+1,mt->GetColLwb());
   *this = mr;
}

////////////////////////////////////////////////////////////////////////////////
/// Constructor : create vector from matrix column

template<class Element>
TVectorT<Element>::TVectorT(const TMatrixTColumn_const<Element> &mc) : TObject()
{
   const TMatrixTBase<Element> *mt = mc.GetMatrix();
   R__ASSERT(mt->IsValid());
   Allocate(mt->GetRowUpb()-mt->GetRowLwb()+1,mt->GetRowLwb());
   *this = mc;
}

////////////////////////////////////////////////////////////////////////////////
/// Constructor : create vector from matrix diagonal

template<class Element>
TVectorT<Element>::TVectorT(const TMatrixTDiag_const<Element> &md) : TObject()
{
   const TMatrixTBase<Element> *mt = md.GetMatrix();
   R__ASSERT(mt->IsValid());
   Allocate(TMath::Min(mt->GetNrows(),mt->GetNcols()));
   *this = md;
}

////////////////////////////////////////////////////////////////////////////////
/// Make a vector and assign initial values. Argument list should contain
/// Element values to assign to vector elements. The list must be
/// terminated by the string "END". Example:
/// TVectorT foo(1,3,0.0,1.0,1.5,"END");

template<class Element>
TVectorT<Element>::TVectorT(Int_t lwb,Int_t upb,Double_t iv1, ...)
{
   if (upb < lwb) {
      Error("TVectorT(Int_t, Int_t, ...)","upb(%d) < lwb(%d)",upb,lwb);
      return;
   }

   Allocate(upb-lwb+1,lwb);

   va_list args;
   va_start(args,iv1);             // Init 'args' to the beginning of
                                        // the variable length list of args

   (*this)(lwb) = iv1;
   for (Int_t i = lwb+1; i <= upb; i++)
      (*this)(i) = (Element)va_arg(args,Double_t);

   if (strcmp((char *)va_arg(args,char *),"END"))
      Error("TVectorT(Int_t, Int_t, ...)","argument list must be terminated by \"END\"");

   va_end(args);
}

////////////////////////////////////////////////////////////////////////////////
/// Resize the vector to [lwb:upb] .
/// New dynamic elemenst are created, the overlapping part of the old ones are
/// copied to the new structures, then the old elements are deleted.

template<class Element>
TVectorT<Element> &TVectorT<Element>::ResizeTo(Int_t lwb,Int_t upb)
{
   R__ASSERT(IsValid());
   if (!fIsOwner) {
      Error("ResizeTo(lwb,upb)","Not owner of data array,cannot resize");
      return *this;
   }

   const Int_t new_nrows = upb-lwb+1;

   if (fNrows > 0) {

      if (fNrows == new_nrows && fRowLwb == lwb)
         return *this;
      else if (new_nrows == 0) {
         Clear();
         return *this;
      }

      Element    *elements_old = GetMatrixArray();
      const Int_t  nrows_old    = fNrows;
      const Int_t  rowLwb_old   = fRowLwb;

      Allocate(new_nrows,lwb);
      R__ASSERT(IsValid());
      if (fNrows > kSizeMax || nrows_old > kSizeMax)
         memset(GetMatrixArray(),0,fNrows*sizeof(Element));
      else if (fNrows > nrows_old)
         memset(GetMatrixArray()+nrows_old,0,(fNrows-nrows_old)*sizeof(Element));

    // Copy overlap
      const Int_t rowLwb_copy = TMath::Max(fRowLwb,rowLwb_old);
      const Int_t rowUpb_copy = TMath::Min(fRowLwb+fNrows-1,rowLwb_old+nrows_old-1);
      const Int_t nrows_copy  = rowUpb_copy-rowLwb_copy+1;

      const Int_t nelems_new = fNrows;
      Element *elements_new = GetMatrixArray();
      if (nrows_copy > 0) {
         const Int_t rowOldOff = rowLwb_copy-rowLwb_old;
         const Int_t rowNewOff = rowLwb_copy-fRowLwb;
         Memcpy_m(elements_new+rowNewOff,elements_old+rowOldOff,nrows_copy,nelems_new,nrows_old);
      }

      Delete_m(nrows_old,elements_old);
   } else {
      Allocate(upb-lwb+1,lwb,1);
   }

   return *this;
}

////////////////////////////////////////////////////////////////////////////////
/// Use the array data to fill the vector lwb..upb]

template<class Element>
TVectorT<Element> &TVectorT<Element>::Use(Int_t lwb,Int_t upb,Element *data)
{
   if (upb < lwb) {
      Error("Use","upb(%d) < lwb(%d)",upb,lwb);
      return *this;
   }

   Clear();
   fNrows    = upb-lwb+1;
   fRowLwb   = lwb;
   fElements = data;
   fIsOwner  = kFALSE;

   return *this;
}

////////////////////////////////////////////////////////////////////////////////
/// Get subvector [row_lwb..row_upb]; The indexing range of the
/// returned vector depends on the argument option:
///
/// option == "S" : return [0..row_upb-row_lwb+1] (default)
/// else          : return [row_lwb..row_upb]

template<class Element>
TVectorT<Element> &TVectorT<Element>::GetSub(Int_t row_lwb,Int_t row_upb,TVectorT<Element> &target,Option_t *option) const
{
   if (gMatrixCheck) {
      R__ASSERT(IsValid());
      if (row_lwb < fRowLwb || row_lwb > fRowLwb+fNrows-1) {
         Error("GetSub","row_lwb out of bounds");
         return target;
      }
      if (row_upb < fRowLwb || row_upb > fRowLwb+fNrows-1) {
         Error("GetSub","row_upb out of bounds");
         return target;
      }
      if (row_upb < row_lwb) {
         Error("GetSub","row_upb < row_lwb");
         return target;
      }
   }

   TString opt(option);
   opt.ToUpper();
   const Int_t shift = (opt.Contains("S")) ? 1 : 0;

   Int_t row_lwb_sub;
   Int_t row_upb_sub;
   if (shift) {
      row_lwb_sub = 0;
      row_upb_sub = row_upb-row_lwb;
   } else {
      row_lwb_sub = row_lwb;
      row_upb_sub = row_upb;
   }

   target.ResizeTo(row_lwb_sub,row_upb_sub);
   const Int_t nrows_sub = row_upb_sub-row_lwb_sub+1;

   const Element *ap = this->GetMatrixArray()+(row_lwb-fRowLwb);
         Element *bp = target.GetMatrixArray();

   for (Int_t irow = 0; irow < nrows_sub; irow++)
      *bp++ = *ap++;

   return target;
}

////////////////////////////////////////////////////////////////////////////////
/// Insert vector source starting at [row_lwb], thereby overwriting the part
/// [row_lwb..row_lwb+nrows_source];

template<class Element>
TVectorT<Element> &TVectorT<Element>::SetSub(Int_t row_lwb,const TVectorT<Element> &source)
{
   if (gMatrixCheck) {
      R__ASSERT(IsValid());
      R__ASSERT(source.IsValid());

      if (row_lwb < fRowLwb && row_lwb > fRowLwb+fNrows-1) {
         Error("SetSub","row_lwb outof bounds");
         return *this;
      }
      if (row_lwb+source.GetNrows() > fRowLwb+fNrows) {
         Error("SetSub","source vector too large");
         return *this;
      }
   }

   const Int_t nRows_source = source.GetNrows();

   const Element *bp = source.GetMatrixArray();
         Element *ap = this->GetMatrixArray()+(row_lwb-fRowLwb);

   for (Int_t irow = 0; irow < nRows_source; irow++)
      *ap++ = *bp++;

   return *this;
}

////////////////////////////////////////////////////////////////////////////////
/// Set vector elements to zero

template<class Element>
TVectorT<Element> &TVectorT<Element>::Zero()
{
   R__ASSERT(IsValid());
   memset(this->GetMatrixArray(),0,fNrows*sizeof(Element));
   return *this;
}

////////////////////////////////////////////////////////////////////////////////
/// Take an absolute value of a vector, i.e. apply Abs() to each element.

template<class Element>
TVectorT<Element> &TVectorT<Element>::Abs()
{
   R__ASSERT(IsValid());

         Element *ep = this->GetMatrixArray();
   const Element * const fp = ep+fNrows;
   while (ep < fp) {
      *ep = TMath::Abs(*ep);
      ep++;
   }

   return *this;
}

////////////////////////////////////////////////////////////////////////////////
/// Square each element of the vector.

template<class Element>
TVectorT<Element> &TVectorT<Element>::Sqr()
{
   R__ASSERT(IsValid());

         Element *ep = this->GetMatrixArray();
   const Element * const fp = ep+fNrows;
   while (ep < fp) {
      *ep = (*ep) * (*ep);
      ep++;
   }

   return *this;
}

////////////////////////////////////////////////////////////////////////////////
/// Take square root of all elements.

template<class Element>
TVectorT<Element> &TVectorT<Element>::Sqrt()
{
   R__ASSERT(IsValid());

         Element *ep = this->GetMatrixArray();
   const Element * const fp = ep+fNrows;
   while (ep < fp) {
      R__ASSERT(*ep >= 0);
      if (*ep >= 0)
         *ep = TMath::Sqrt(*ep);
      else
         Error("Sqrt()","v(%ld) = %g < 0",Long_t(ep-this->GetMatrixArray()),(float)*ep);
      ep++;
   }

   return *this;
}

////////////////////////////////////////////////////////////////////////////////
/// v[i] = 1/v[i]

template<class Element>
TVectorT<Element> &TVectorT<Element>::Invert()
{
   R__ASSERT(IsValid());

         Element *ep = this->GetMatrixArray();
   const Element * const fp = ep+fNrows;
   while (ep < fp) {
      R__ASSERT(*ep != 0.0);
      if (*ep != 0.0)
         *ep = 1./ *ep;
      else
         Error("Invert()","v(%ld) = %g",Long_t(ep-this->GetMatrixArray()),(float)*ep);
      ep++;
   }

   return *this;
}

////////////////////////////////////////////////////////////////////////////////
/// Keep only element as selected through array select non-zero

template<class Element>
TVectorT<Element> &TVectorT<Element>::SelectNonZeros(const TVectorT<Element> &select)
{
   if (gMatrixCheck && !AreCompatible(*this,select)) {
      Error("SelectNonZeros(const TVectorT<Element> &","vector's not compatible");
      return *this;
   }

   const Element *sp = select.GetMatrixArray();
         Element *ep = this->GetMatrixArray();
   const Element * const fp = ep+fNrows;
   while (ep < fp) {
      if (*sp == 0.0)
         *ep = 0.0;
      sp++; ep++;
   }

   return *this;
}

////////////////////////////////////////////////////////////////////////////////
/// Compute the 1-norm of the vector SUM{ |v[i]| }.

template<class Element>
Element TVectorT<Element>::Norm1() const
{
   R__ASSERT(IsValid());

   Element norm = 0;
   const Element *ep = this->GetMatrixArray();
   const Element * const fp = ep+fNrows;
   while (ep < fp)
      norm += TMath::Abs(*ep++);

   return norm;
}

////////////////////////////////////////////////////////////////////////////////
/// Compute the square of the 2-norm SUM{ v[i]^2 }.

template<class Element>
Element TVectorT<Element>::Norm2Sqr() const
{
   R__ASSERT(IsValid());

   Element norm = 0;
   const Element *ep = this->GetMatrixArray();
   const Element * const fp = ep+fNrows;
   while (ep < fp) {
      norm += (*ep) * (*ep);
      ep++;
   }

   return norm;
}

////////////////////////////////////////////////////////////////////////////////
/// Compute the infinity-norm of the vector MAX{ |v[i]| }.

template<class Element>
Element TVectorT<Element>::NormInf() const
{
   R__ASSERT(IsValid());

   Element norm = 0;
   const Element *ep = this->GetMatrixArray();
   const Element * const fp = ep+fNrows;
   while (ep < fp)
      norm = TMath::Max(norm,TMath::Abs(*ep++));

   return norm;
}

////////////////////////////////////////////////////////////////////////////////
/// Compute the number of elements != 0.0

template<class Element>
Int_t TVectorT<Element>::NonZeros() const
{
   R__ASSERT(IsValid());

   Int_t nr_nonzeros = 0;
   const Element *ep = this->GetMatrixArray();
   const Element * const fp = ep+fNrows;
   while (ep < fp)
      if (*ep++) nr_nonzeros++;

   return nr_nonzeros;
}

////////////////////////////////////////////////////////////////////////////////
/// Compute sum of elements

template<class Element>
Element TVectorT<Element>::Sum() const
{
   R__ASSERT(IsValid());

   Element sum = 0.0;
   const Element *ep = this->GetMatrixArray();
   const Element * const fp = ep+fNrows;
   while (ep < fp)
      sum += *ep++;

   return sum;
}

////////////////////////////////////////////////////////////////////////////////
/// return minimum vector element value

template<class Element>
Element TVectorT<Element>::Min() const
{
   R__ASSERT(IsValid());

   const Int_t index = TMath::LocMin(fNrows,fElements);
   return fElements[index];
}

////////////////////////////////////////////////////////////////////////////////
/// return maximum vector element value

template<class Element>
Element TVectorT<Element>::Max() const
{
   R__ASSERT(IsValid());

   const Int_t index = TMath::LocMax(fNrows,fElements);
   return fElements[index];
}

////////////////////////////////////////////////////////////////////////////////
/// Notice that this assignment does NOT change the ownership :
/// if the storage space was adopted, source is copied to
/// this space .

template<class Element>
TVectorT<Element> &TVectorT<Element>::operator=(const TVectorT<Element> &source)
{
   if (gMatrixCheck && !AreCompatible(*this,source)) {
      Error("operator=(const TVectorT<Element> &)","vectors not compatible");
      return *this;
   }

   if (this->GetMatrixArray() != source.GetMatrixArray()) {
      TObject::operator=(source);
      memcpy(fElements,source.GetMatrixArray(),fNrows*sizeof(Element));
   }
   return *this;
}

////////////////////////////////////////////////////////////////////////////////
/// Assign a matrix row to a vector.

template<class Element>
TVectorT<Element> &TVectorT<Element>::operator=(const TMatrixTRow_const<Element> &mr)
{
   const TMatrixTBase<Element> *mt = mr.GetMatrix();

   if (gMatrixCheck) {
      R__ASSERT(IsValid());
      R__ASSERT(mt->IsValid());
      if (mt->GetColLwb() != fRowLwb || mt->GetNcols() != fNrows) {
         Error("operator=(const TMatrixTRow_const &)","vector and row not compatible");
         return *this;
      }
   }

   const Int_t inc   = mr.GetInc();
   const Element *rp = mr.GetPtr();              // Row ptr
         Element *ep = this->GetMatrixArray();   // Vector ptr
   const Element * const fp = ep+fNrows;
   while (ep < fp) {
      *ep++ = *rp;
       rp += inc;
   }

   R__ASSERT(rp == mr.GetPtr()+mt->GetNcols());

   return *this;
}

////////////////////////////////////////////////////////////////////////////////
/// Assign a matrix column to a vector.

template<class Element>
TVectorT<Element> &TVectorT<Element>::operator=(const TMatrixTColumn_const<Element> &mc)
{
   const TMatrixTBase<Element> *mt = mc.GetMatrix();

   if (gMatrixCheck) {
      R__ASSERT(IsValid());
      R__ASSERT(mt->IsValid());
      if (mt->GetRowLwb() != fRowLwb || mt->GetNrows() != fNrows) {
         Error("operator=(const TMatrixTColumn_const &)","vector and column not compatible");
         return *this;
      }
   }

   const Int_t inc    = mc.GetInc();
   const Element *cp = mc.GetPtr();              // Column ptr
         Element *ep = this->GetMatrixArray();   // Vector ptr
   const Element * const fp = ep+fNrows;
   while (ep < fp) {
      *ep++ = *cp;
       cp += inc;
   }

   R__ASSERT(cp == mc.GetPtr()+mt->GetNoElements());

   return *this;
}

////////////////////////////////////////////////////////////////////////////////
/// Assign the matrix diagonal to a vector.

template<class Element>
TVectorT<Element> &TVectorT<Element>::operator=(const TMatrixTDiag_const<Element> &md)
{
   const TMatrixTBase<Element> *mt = md.GetMatrix();

   if (gMatrixCheck) {
      R__ASSERT(IsValid());
      R__ASSERT(mt->IsValid());
      if (md.GetNdiags() != fNrows) {
         Error("operator=(const TMatrixTDiag_const &)","vector and matrix-diagonal not compatible");
        return *this;
      }
   }

   const Int_t    inc = md.GetInc();
   const Element *dp = md.GetPtr();              // Diag ptr
         Element *ep = this->GetMatrixArray();   // Vector ptr
   const Element * const fp = ep+fNrows;
   while (ep < fp) {
      *ep++ = *dp;
       dp += inc;
   }

   R__ASSERT(dp < md.GetPtr()+mt->GetNoElements()+inc);

   return *this;
}

////////////////////////////////////////////////////////////////////////////////
/// Assign a sparse matrix row to a vector. The matrix row is implicitly transposed
/// to allow the assignment in the strict sense.

template<class Element>
TVectorT<Element> &TVectorT<Element>::operator=(const TMatrixTSparseRow_const<Element> &mr)
{
   const TMatrixTBase<Element> *mt = mr.GetMatrix();

   if (gMatrixCheck) {
      R__ASSERT(IsValid());
      R__ASSERT(mt->IsValid());
      if (mt->GetColLwb() != fRowLwb || mt->GetNcols() != fNrows) {
         Error("operator=(const TMatrixTSparseRow_const &)","vector and row not compatible");
         return *this;
      }
   }

   const Int_t nIndex = mr.GetNindex();
   const Element * const prData = mr.GetDataPtr();          // Row Data ptr
   const Int_t    * const prCol  = mr.GetColPtr();           // Col ptr
         Element * const pvData = this->GetMatrixArray();   // Vector ptr

   memset(pvData,0,fNrows*sizeof(Element));
   for (Int_t index = 0; index < nIndex; index++) {
      const Int_t icol = prCol[index];
      pvData[icol] = prData[index];
   }

   return *this;
}

////////////////////////////////////////////////////////////////////////////////
/// Assign a sparse matrix diagonal to a vector.

template<class Element>
TVectorT<Element> &TVectorT<Element>::operator=(const TMatrixTSparseDiag_const<Element> &md)
{
  const TMatrixTBase<Element> *mt = md.GetMatrix();

   if (gMatrixCheck) {
      R__ASSERT(IsValid());
      R__ASSERT(mt->IsValid());
      if (md.GetNdiags() != fNrows) {
         Error("operator=(const TMatrixTSparseDiag_const &)","vector and matrix-diagonal not compatible");
         return *this;
      }
   }

   Element * const pvData = this->GetMatrixArray();
   for (Int_t idiag = 0; idiag < fNrows; idiag++)
      pvData[idiag] = md(idiag);

   return *this;
}

////////////////////////////////////////////////////////////////////////////////
/// Assign val to every element of the vector.

template<class Element>
TVectorT<Element> &TVectorT<Element>::operator=(Element val)
{
   R__ASSERT(IsValid());

         Element *ep = this->GetMatrixArray();
   const Element * const fp = ep+fNrows;
   while (ep < fp)
      *ep++ = val;

   return *this;
}

////////////////////////////////////////////////////////////////////////////////
/// Add val to every element of the vector.

template<class Element>
TVectorT<Element> &TVectorT<Element>::operator+=(Element val)
{
   R__ASSERT(IsValid());

         Element *ep = this->GetMatrixArray();
   const Element * const fp = ep+fNrows;
   while (ep < fp)
      *ep++ += val;

   return *this;
}

////////////////////////////////////////////////////////////////////////////////
/// Subtract val from every element of the vector.

template<class Element>
TVectorT<Element> &TVectorT<Element>::operator-=(Element val)
{
   R__ASSERT(IsValid());

         Element *ep = this->GetMatrixArray();
   const Element * const fp = ep+fNrows;
   while (ep < fp)
      *ep++ -= val;

   return *this;
}

////////////////////////////////////////////////////////////////////////////////
/// Multiply every element of the vector with val.

template<class Element>
TVectorT<Element> &TVectorT<Element>::operator*=(Element val)
{
   R__ASSERT(IsValid());

         Element *ep = this->GetMatrixArray();
   const Element * const fp = ep+fNrows;
   while (ep < fp)
      *ep++ *= val;

   return *this;
}

////////////////////////////////////////////////////////////////////////////////
/// Add vector source

template<class Element>
TVectorT<Element> &TVectorT<Element>::operator+=(const TVectorT<Element> &source)
{
   if (gMatrixCheck && !AreCompatible(*this,source)) {
      Error("operator+=(const TVectorT<Element> &)","vector's not compatible");
      return *this;
   }

   const Element *sp = source.GetMatrixArray();
         Element *tp = this->GetMatrixArray();
   const Element * const tp_last = tp+fNrows;
   while (tp < tp_last)
      *tp++ += *sp++;

   return *this;
}

////////////////////////////////////////////////////////////////////////////////
/// Subtract vector source

template<class Element>
TVectorT<Element> &TVectorT<Element>::operator-=(const TVectorT<Element> &source)
{
   if (gMatrixCheck && !AreCompatible(*this,source)) {
      Error("operator-=(const TVectorT<Element> &)","vector's not compatible");
      return *this;
   }

   const Element *sp = source.GetMatrixArray();
         Element *tp = this->GetMatrixArray();
   const Element * const tp_last = tp+fNrows;
   while (tp < tp_last)
      *tp++ -= *sp++;

   return *this;
}

////////////////////////////////////////////////////////////////////////////////
/// "Inplace" multiplication target = A*target. A needn't be a square one
/// If target has to be resized, it should own the storage: fIsOwner = kTRUE

template<class Element>
TVectorT<Element> &TVectorT<Element>::operator*=(const TMatrixT<Element> &a)
{
   if (gMatrixCheck) {
      R__ASSERT(IsValid());
      R__ASSERT(a.IsValid());
      if (a.GetNcols() != fNrows || a.GetColLwb() != fRowLwb) {
         Error("operator*=(const TMatrixT &)","vector and matrix incompatible");
         return *this;
      }
   }

   const Bool_t doResize = (fNrows != a.GetNrows() || fRowLwb != a.GetRowLwb());
   if (doResize && !fIsOwner) {
      Error("operator*=(const TMatrixT &)","vector has to be resized but not owner");
      return *this;
   }

   Element work[kWorkMax];
   Bool_t isAllocated = kFALSE;
   Element *elements_old = work;
   const Int_t nrows_old = fNrows;
   if (nrows_old > kWorkMax) {
      isAllocated = kTRUE;
      elements_old = new Element[nrows_old];
   }
   memcpy(elements_old,fElements,nrows_old*sizeof(Element));

   if (doResize) {
      const Int_t rowlwb_new = a.GetRowLwb();
      const Int_t nrows_new  = a.GetNrows();
      ResizeTo(rowlwb_new,rowlwb_new+nrows_new-1);
   }
   memset(fElements,0,fNrows*sizeof(Element));

   const Element *mp = a.GetMatrixArray();     // Matrix row ptr
         Element *tp = this->GetMatrixArray(); // Target vector ptr
#ifdef CBLAS
   if (typeid(Element) == typeid(Double_t))
      cblas_dgemv(CblasRowMajor,CblasNoTrans,a.GetNrows(),a.GetNcols(),1.0,mp,
                  a.GetNcols(),elements_old,1,0.0,tp,1);
   else if (typeid(Element) != typeid(Float_t))
      cblas_sgemv(CblasRowMajor,CblasNoTrans,a.GetNrows(),a.GetNcols(),1.0,mp,
                  a.GetNcols(),elements_old,1,0.0,tp,1);
   else
      Error("operator*=","type %s not implemented in BLAS library",typeid(Element));
#else
   const Element * const tp_last = tp+fNrows;
   while (tp < tp_last) {
      Element sum = 0;
      for (const Element *sp = elements_old; sp < elements_old+nrows_old; )
         sum += *sp++ * *mp++;
      *tp++ = sum;
   }
   R__ASSERT(mp == a.GetMatrixArray()+a.GetNoElements());
#endif

   if (isAllocated)
      delete [] elements_old;

   return *this;
}

////////////////////////////////////////////////////////////////////////////////
/// "Inplace" multiplication target = A*target. A needn't be a square one
/// If target has to be resized, it should own the storage: fIsOwner = kTRUE

template<class Element>
TVectorT<Element> &TVectorT<Element>::operator*=(const TMatrixTSparse<Element> &a)
{
   if (gMatrixCheck) {
      R__ASSERT(IsValid());
      R__ASSERT(a.IsValid());
      if (a.GetNcols() != fNrows || a.GetColLwb() != fRowLwb) {
         Error("operator*=(const TMatrixTSparse &)","vector and matrix incompatible");
         return *this;
      }
   }

   const Bool_t doResize = (fNrows != a.GetNrows() || fRowLwb != a.GetRowLwb());
   if (doResize && !fIsOwner) {
      Error("operator*=(const TMatrixTSparse &)","vector has to be resized but not owner");
      return *this;
   }

   Element work[kWorkMax];
   Bool_t isAllocated = kFALSE;
   Element *elements_old = work;
   const Int_t nrows_old = fNrows;
   if (nrows_old > kWorkMax) {
      isAllocated = kTRUE;
      elements_old = new Element[nrows_old];
   }
   memcpy(elements_old,fElements,nrows_old*sizeof(Element));

   if (doResize) {
      const Int_t rowlwb_new = a.GetRowLwb();
      const Int_t nrows_new  = a.GetNrows();
      ResizeTo(rowlwb_new,rowlwb_new+nrows_new-1);
   }
   memset(fElements,0,fNrows*sizeof(Element));

   const Int_t   * const pRowIndex = a.GetRowIndexArray();
   const Int_t   * const pColIndex = a.GetColIndexArray();
   const Element * const mp        = a.GetMatrixArray();     // Matrix row ptr

   const Element * const sp = elements_old;
         Element *       tp = this->GetMatrixArray(); // Target vector ptr

   for (Int_t irow = 0; irow < fNrows; irow++) {
      const Int_t sIndex = pRowIndex[irow];
      const Int_t eIndex = pRowIndex[irow+1];
      Element sum = 0.0;
      for (Int_t index = sIndex; index < eIndex; index++) {
         const Int_t icol = pColIndex[index];
         sum += mp[index]*sp[icol];
      }
      tp[irow] = sum;
   }

   if (isAllocated)
      delete [] elements_old;

   return *this;
}

////////////////////////////////////////////////////////////////////////////////
/// "Inplace" multiplication target = A*target. A is symmetric .
/// vector size will not change

template<class Element>
TVectorT<Element> &TVectorT<Element>::operator*=(const TMatrixTSym<Element> &a)
{
  if (gMatrixCheck) {
     R__ASSERT(IsValid());
     R__ASSERT(a.IsValid());
     if (a.GetNcols() != fNrows || a.GetColLwb() != fRowLwb) {
        Error("operator*=(const TMatrixTSym &)","vector and matrix incompatible");
        return *this;
     }
  }

   Element work[kWorkMax];
   Bool_t isAllocated = kFALSE;
   Element *elements_old = work;
   const Int_t nrows_old = fNrows;
   if (nrows_old > kWorkMax) {
      isAllocated = kTRUE;
      elements_old = new Element[nrows_old];
   }
   memcpy(elements_old,fElements,nrows_old*sizeof(Element));
   memset(fElements,0,fNrows*sizeof(Element));

   const Element *mp = a.GetMatrixArray();     // Matrix row ptr
         Element *tp = this->GetMatrixArray(); // Target vector ptr
#ifdef CBLAS
   if (typeid(Element) == typeid(Double_t))
      cblas_dsymv(CblasRowMajor,CblasUpper,fNrows,1.0,mp,
                  fNrows,elements_old,1,0.0,tp,1);
   else if (typeid(Element) != typeid(Float_t))
      cblas_ssymv(CblasRowMajor,CblasUpper,fNrows,1.0,mp,
                  fNrows,elements_old,1,0.0,tp,1);
   else
      Error("operator*=","type %s not implemented in BLAS library",typeid(Element));
#else
   const Element * const tp_last = tp+fNrows;
   while (tp < tp_last) {
      Element sum = 0;
      for (const Element *sp = elements_old; sp < elements_old+nrows_old; )
         sum += *sp++ * *mp++;
      *tp++ = sum;
   }
   R__ASSERT(mp == a.GetMatrixArray()+a.GetNoElements());
#endif

   if (isAllocated)
      delete [] elements_old;

   return *this;
}

////////////////////////////////////////////////////////////////////////////////
/// Are all vector elements equal to val?

template<class Element>
Bool_t TVectorT<Element>::operator==(Element val) const
{
   R__ASSERT(IsValid());

   const Element *ep = this->GetMatrixArray();
   const Element * const fp = ep+fNrows;
   while (ep < fp)
      if (!(*ep++ == val))
         return kFALSE;

   return kTRUE;
}

////////////////////////////////////////////////////////////////////////////////
/// Are all vector elements not equal to val?

template<class Element>
Bool_t TVectorT<Element>::operator!=(Element val) const
{
   R__ASSERT(IsValid());

   const Element *ep = this->GetMatrixArray();
   const Element * const fp = ep+fNrows;
   while (ep < fp)
      if (!(*ep++ != val))
         return kFALSE;

   return kTRUE;
}

////////////////////////////////////////////////////////////////////////////////
/// Are all vector elements < val?

template<class Element>
Bool_t TVectorT<Element>::operator<(Element val) const
{
   R__ASSERT(IsValid());

   const Element *ep = this->GetMatrixArray();
   const Element * const fp = ep+fNrows;
   while (ep < fp)
      if (!(*ep++ < val))
         return kFALSE;

   return kTRUE;
}

////////////////////////////////////////////////////////////////////////////////
/// Are all vector elements <= val?

template<class Element>
Bool_t TVectorT<Element>::operator<=(Element val) const
{
   R__ASSERT(IsValid());

   const Element *ep = this->GetMatrixArray();
   const Element * const fp = ep+fNrows;
   while (ep < fp)
      if (!(*ep++ <= val))
         return kFALSE;

   return kTRUE;
}

////////////////////////////////////////////////////////////////////////////////
/// Are all vector elements > val?

template<class Element>
Bool_t TVectorT<Element>::operator>(Element val) const
{
   R__ASSERT(IsValid());

   const Element *ep = this->GetMatrixArray();
   const Element * const fp = ep+fNrows;
   while (ep < fp)
      if (!(*ep++ > val))
         return kFALSE;

   return kTRUE;
}

////////////////////////////////////////////////////////////////////////////////
/// Are all vector elements >= val?

template<class Element>
Bool_t TVectorT<Element>::operator>=(Element val) const
{
   R__ASSERT(IsValid());

   const Element *ep = this->GetMatrixArray();
   const Element * const fp = ep+fNrows;
   while (ep < fp)
      if (!(*ep++ >= val))
         return kFALSE;

   return kTRUE;
}

////////////////////////////////////////////////////////////////////////////////
/// Check if vector elements as selected through array select are non-zero

template<class Element>
Bool_t TVectorT<Element>::MatchesNonZeroPattern(const TVectorT<Element> &select)
{
   if (gMatrixCheck && !AreCompatible(*this,select)) {
      Error("MatchesNonZeroPattern(const TVectorT&)","vector's not compatible");
      return kFALSE;
   }

   const Element *sp = select.GetMatrixArray();
   const Element *ep = this->GetMatrixArray();
   const Element * const fp = ep+fNrows;
   while (ep < fp) {
      if (*sp == 0.0 && *ep != 0.0)
         return kFALSE;
      sp++; ep++;
   }

   return kTRUE;
}

////////////////////////////////////////////////////////////////////////////////
/// Check if vector elements as selected through array select are all positive

template<class Element>
Bool_t TVectorT<Element>::SomePositive(const TVectorT<Element> &select)
{
   if (gMatrixCheck && !AreCompatible(*this,select)) {
      Error("SomePositive(const TVectorT&)","vector's not compatible");
      return kFALSE;
   }

   const Element *sp = select.GetMatrixArray();
   const Element *ep = this->GetMatrixArray();
   const Element * const fp = ep+fNrows;
   while (ep < fp) {
      if (*sp != 0.0 && *ep <= 0.0)
         return kFALSE;
      sp++; ep++;
   }

   return kTRUE;
}

////////////////////////////////////////////////////////////////////////////////
/// Add to vector elements as selected through array select the value val

template<class Element>
void TVectorT<Element>::AddSomeConstant(Element val,const TVectorT<Element> &select)
{
   if (gMatrixCheck && !AreCompatible(*this,select))
      Error("AddSomeConstant(Element,const TVectorT&)(const TVectorT&)","vector's not compatible");

   const Element *sp = select.GetMatrixArray();
         Element *ep = this->GetMatrixArray();
   const Element * const fp = ep+fNrows;
   while (ep < fp) {
      if (*sp)
         *ep += val;
      sp++; ep++;
   }
}

extern Double_t Drand(Double_t &ix);

////////////////////////////////////////////////////////////////////////////////
/// randomize vector elements value

template<class Element>
void TVectorT<Element>::Randomize(Element alpha,Element beta,Double_t &seed)
{
   R__ASSERT(IsValid());

   const Element scale = beta-alpha;
   const Element shift = alpha/scale;

         Element *       ep = GetMatrixArray();
   const Element * const fp = ep+fNrows;
   while (ep < fp)
      *ep++ = scale*(Drand(seed)+shift);
}

////////////////////////////////////////////////////////////////////////////////
/// Apply action to each element of the vector.

template<class Element>
TVectorT<Element> &TVectorT<Element>::Apply(const TElementActionT<Element> &action)
{
   R__ASSERT(IsValid());
   for (Element *ep = fElements; ep < fElements+fNrows; ep++)
      action.Operation(*ep);
   return *this;
}

////////////////////////////////////////////////////////////////////////////////
/// Apply action to each element of the vector. In action the location
/// of the current element is known.

template<class Element>
TVectorT<Element> &TVectorT<Element>::Apply(const TElementPosActionT<Element> &action)
{
   R__ASSERT(IsValid());

   Element *ep = fElements;
   for (action.fI = fRowLwb; action.fI < fRowLwb+fNrows; action.fI++)
      action.Operation(*ep++);

   R__ASSERT(ep == fElements+fNrows);

   return *this;
}

////////////////////////////////////////////////////////////////////////////////
/// Draw this vector
/// The histogram is named "TVectorT" by default and no title

template<class Element>
void TVectorT<Element>::Draw(Option_t *option)
{
   gROOT->ProcessLine(Form("THistPainter::PaintSpecialObjects((TObject*)0x%lx,\"%s\");",
                           (ULong_t)this, option));
}

////////////////////////////////////////////////////////////////////////////////
/// Print the vector as a list of elements.

template<class Element>
void TVectorT<Element>::Print(Option_t *flag) const
{
  if (!IsValid()) {
      Error("Print","Vector is invalid");
      return;
   }

   printf("\nVector (%d) %s is as follows",fNrows,flag);

   printf("\n\n     |   %6d  |", 1);
   printf("\n%s\n", "------------------");
   for (Int_t i = 0; i < fNrows; i++) {
      printf("%4d |",i+fRowLwb);
      //printf("%11.4g \n",(*this)(i+fRowLwb));
      printf("%g \n",(*this)(i+fRowLwb));
   }
   printf("\n");
}

////////////////////////////////////////////////////////////////////////////////
/// Check to see if two vectors are identical.

template<class Element>
Bool_t operator==(const TVectorT<Element> &v1,const TVectorT<Element> &v2)
{
   if (!AreCompatible(v1,v2)) return kFALSE;
   return (memcmp(v1.GetMatrixArray(),v2.GetMatrixArray(),v1.GetNrows()*sizeof(Element)) == 0);
}

////////////////////////////////////////////////////////////////////////////////
/// Compute the scalar product.

template<class Element>
Element operator*(const TVectorT<Element> &v1,const TVectorT<Element> &v2)
{
   if (gMatrixCheck) {
      if (!AreCompatible(v1,v2)) {
         Error("operator*(const TVectorT<Element> &,const TVectorT<Element> &)","vector's are incompatible");
         return 0.0;
      }
   }

   return Dot(v1,v2);
}

////////////////////////////////////////////////////////////////////////////////
/// Return source1+source2

template<class Element>
TVectorT<Element> operator+(const TVectorT<Element> &source1,const TVectorT<Element> &source2)
{
   TVectorT<Element> target = source1;
   target += source2;
   return target;
}

////////////////////////////////////////////////////////////////////////////////
/// Return source1-source2

template<class Element>
TVectorT<Element> operator-(const TVectorT<Element> &source1,const TVectorT<Element> &source2)
{
   TVectorT<Element> target = source1;
   target -= source2;
   return target;
}

////////////////////////////////////////////////////////////////////////////////
/// return A * source

template<class Element>
TVectorT<Element> operator*(const TMatrixT<Element> &a,const TVectorT<Element> &source)
{
   R__ASSERT(a.IsValid());
   TVectorT<Element> target(a.GetRowLwb(),a.GetRowUpb());
   return Add(target,Element(1.0),a,source);
}

////////////////////////////////////////////////////////////////////////////////
/// return A * source

template<class Element>
TVectorT<Element> operator*(const TMatrixTSym<Element> &a,const TVectorT<Element> &source)
{
   R__ASSERT(a.IsValid());
   TVectorT<Element> target(a.GetRowLwb(),a.GetRowUpb());
   return Add(target,Element(1.0),a,source);
}

////////////////////////////////////////////////////////////////////////////////
/// return A * source

template<class Element>
TVectorT<Element> operator*(const TMatrixTSparse<Element> &a,const TVectorT<Element> &source)
{
   R__ASSERT(a.IsValid());
   TVectorT<Element> target(a.GetRowLwb(),a.GetRowUpb());
   return Add(target,Element(1.0),a,source);
}

////////////////////////////////////////////////////////////////////////////////
/// return val * source

template<class Element>
TVectorT<Element> operator*(Element val,const TVectorT<Element> &source)
{
   TVectorT<Element> target = source;
   target *= val;
   return target;
}

////////////////////////////////////////////////////////////////////////////////
/// return inner-produvt v1 . v2

template<class Element>
Element Dot(const TVectorT<Element> &v1,const TVectorT<Element> &v2)
{
   const Element *v1p = v1.GetMatrixArray();
   const Element *v2p = v2.GetMatrixArray();
   Element sum = 0.0;
   const Element * const fv1p = v1p+v1.GetNrows();
   while (v1p < fv1p)
      sum += *v1p++ * *v2p++;

   return sum;
}

////////////////////////////////////////////////////////////////////////////////
/// Return the matrix M = v1 * v2'

template <class Element1, class Element2>
TMatrixT<Element1>
OuterProduct(const TVectorT<Element1> &v1,const TVectorT<Element2> &v2)
{
   // TMatrixD::GetSub does:
   //   TMatrixT tmp;
   // Doesn't compile here, because we are outside the class?
   // So we'll be explicit:
   TMatrixT<Element1> target;

   return OuterProduct(target,v1,v2);
}

////////////////////////////////////////////////////////////////////////////////
/// Return the matrix M = v1 * v2'

template <class Element1,class Element2,class Element3>
TMatrixT<Element1>
&OuterProduct(TMatrixT<Element1> &target,const TVectorT<Element2> &v1,const TVectorT<Element3> &v2)
{
   target.ResizeTo(v1.GetLwb(), v1.GetUpb(), v2.GetLwb(), v2.GetUpb());

         Element1 *       mp      = target.GetMatrixArray();
   const Element1 * const m_last  = mp + target.GetNoElements();

   const Element2 *       v1p     = v1.GetMatrixArray();
   const Element2 * const v1_last = v1p + v1.GetNrows();

   const Element3 * const v20     = v2.GetMatrixArray();
   const Element3 *       v2p     = v20;
   const Element3 * const v2_last = v2p + v2.GetNrows();

   while (v1p < v1_last) {
      v2p = v20;
      while (v2p < v2_last) {
         *mp++ = *v1p * *v2p++ ;
      }
      v1p++;
  }

  R__ASSERT(v1p == v1_last && mp == m_last && v2p == v2_last);

  return target;
}

////////////////////////////////////////////////////////////////////////////////
/// Perform v1 * M * v2, a scalar result

template <class Element1, class Element2, class Element3>
Element1 Mult(const TVectorT<Element1> &v1,const TMatrixT<Element2> &m,
              const TVectorT<Element3> &v2)
{
   if (gMatrixCheck) {
      if (!AreCompatible(v1, m)) {
         ::Error("Mult", "Vector v1 and matrix m incompatible");
         return 0;
      }
      if (!AreCompatible(m, v2)) {
         ::Error("Mult", "Matrix m and vector v2 incompatible");
         return 0;
      }
   }

   const Element1 *       v1p     = v1.GetMatrixArray();    // first of v1
   const Element1 * const v1_last = v1p + v1.GetNrows();    // last of  v1

   const Element2 *       mp      = m.GetMatrixArray();     // first of m
   const Element2 * const m_last  = mp + m.GetNoElements(); // last of  m

   const Element3 * const v20     = v2.GetMatrixArray();    // first of v2
   const Element3 *       v2p     = v20;                    // running  v2
   const Element3 * const v2_last = v2p + v2.GetNrows();    // last of  v2

   Element1 sum     = 0;      // scalar result accumulator
   Element3 dot     = 0;      // M_row * v2 dot product accumulator

   while (v1p < v1_last) {
      v2p  = v20;               // at beginning of v2
      while (v2p < v2_last) {   // compute (M[i] * v2) dot product
         dot += *mp++ * *v2p++;
      }
      sum += *v1p++ * dot;      // v1[i] * (M[i] * v2)
      dot  = 0;                 // start next dot product
   }

   R__ASSERT(v1p == v1_last && mp == m_last && v2p == v2_last);

   return sum;
}

////////////////////////////////////////////////////////////////////////////////
/// Modify addition: target += scalar * source.

template<class Element>
TVectorT<Element> &Add(TVectorT<Element> &target,Element scalar,const TVectorT<Element> &source)
{
   if (gMatrixCheck && !AreCompatible(target,source)) {
      Error("Add(TVectorT<Element> &,Element,const TVectorT<Element> &)","vector's are incompatible");
      return target;
   }

   const Element *       sp  = source.GetMatrixArray();
         Element *       tp  = target.GetMatrixArray();
   const Element * const ftp = tp+target.GetNrows();
   if (scalar == 1.0 ) {
      while ( tp < ftp )
         *tp++ += *sp++;
   } else if (scalar == -1.0) {
      while ( tp < ftp )
         *tp++ -= *sp++;
   } else {
      while ( tp < ftp )
         *tp++ += scalar * *sp++;
   }

   return target;
}

////////////////////////////////////////////////////////////////////////////////
/// Modify addition: target += scalar * A * source.
/// NOTE: in case scalar=0, do  target = A * source.

template<class Element>
TVectorT<Element> &Add(TVectorT<Element> &target,Element scalar,
                       const TMatrixT<Element> &a,const TVectorT<Element> &source)
{
   if (gMatrixCheck) {
      R__ASSERT(target.IsValid());
      R__ASSERT(a.IsValid());
      if (a.GetNrows() != target.GetNrows() || a.GetRowLwb() != target.GetLwb()) {
         Error("Add(TVectorT &,const TMatrixT &,const TVectorT &)","target vector and matrix are incompatible");
         return target;
      }

      R__ASSERT(source.IsValid());
      if (a.GetNcols() != source.GetNrows() || a.GetColLwb() != source.GetLwb()) {
         Error("Add(TVectorT &,const TMatrixT &,const TVectorT &)","source vector and matrix are incompatible");
         return target;
      }
   }

   const Element * const sp = source.GetMatrixArray();  // sources vector ptr
   const Element *       mp = a.GetMatrixArray();       // Matrix row ptr
         Element *       tp = target.GetMatrixArray();  // Target vector ptr
#ifdef CBLAS
   if (typeid(Element) == typeid(Double_t))
      cblas_dsymv(CblasRowMajor,CblasUpper,fNrows,scalar,mp,
                  fNrows,sp,1,0.0,tp,1);
   else if (typeid(Element) != typeid(Float_t))
      cblas_ssymv(CblasRowMajor,CblasUpper,fNrows,scalar,mp,
                  fNrows,sp,1,0.0,tp,1);
   else
      Error("operator*=","type %s not implemented in BLAS library",typeid(Element));
#else
   const Element * const sp_last = sp+source.GetNrows();
   const Element * const tp_last = tp+target.GetNrows();
   if (scalar == 1.0) {
      while (tp < tp_last) {
         const Element *sp1 = sp;
         Element sum = 0;
         while (sp1 < sp_last)
            sum += *sp1++ * *mp++;
         *tp++ += sum;
      }
   } else if (scalar == 0.0) {
      while (tp < tp_last) {
         const Element *sp1 = sp;
         Element sum = 0;
         while (sp1 < sp_last)
            sum += *sp1++ * *mp++;
         *tp++  = sum;
      }
   } else if (scalar == -1.0) {
      while (tp < tp_last) {
         const Element *sp1 = sp;
         Element sum = 0;
         while (sp1 < sp_last)
            sum += *sp1++ * *mp++;
         *tp++ -= sum;
      }
   } else {
      while (tp < tp_last) {
        const Element *sp1 = sp;
        Element sum = 0;
        while (sp1 < sp_last)
           sum += *sp1++ * *mp++;
        *tp++ += scalar * sum;
      }
   }

   if (gMatrixCheck) R__ASSERT(mp == a.GetMatrixArray()+a.GetNoElements());
#endif

   return target;
}

////////////////////////////////////////////////////////////////////////////////
/// Modify addition: target += A * source.
/// NOTE: in case scalar=0, do  target = A * source.

template<class Element>
TVectorT<Element> &Add(TVectorT<Element> &target,Element scalar,
                       const TMatrixTSym<Element> &a,const TVectorT<Element> &source)
{
   if (gMatrixCheck) {
      R__ASSERT(target.IsValid());
      R__ASSERT(source.IsValid());
      R__ASSERT(a.IsValid());
      if (a.GetNrows() != target.GetNrows() || a.GetRowLwb() != target.GetLwb()) {
         Error("Add(TVectorT &,const TMatrixT &,const TVectorT &)","target vector and matrix are incompatible");
         return target;
      }
   }

   const Element * const sp = source.GetMatrixArray();  // sources vector ptr
   const Element *       mp = a.GetMatrixArray();       // Matrix row ptr
         Element *       tp = target.GetMatrixArray();  // Target vector ptr
#ifdef CBLAS
   if (typeid(Element) == typeid(Double_t))
      cblas_dsymv(CblasRowMajor,CblasUpper,fNrows,1.0,mp,
                  fNrows,sp,1,0.0,tp,1);
   else if (typeid(Element) != typeid(Float_t))
      cblas_ssymv(CblasRowMajor,CblasUpper,fNrows,1.0,mp,
                  fNrows,sp,1,0.0,tp,1);
   else
      Error("operator*=","type %s not implemented in BLAS library",typeid(Element));
#else
   const Element * const sp_last = sp+source.GetNrows();
   const Element * const tp_last = tp+target.GetNrows();
   if (scalar == 1.0) {
      while (tp < tp_last) {
         const Element *sp1 = sp;
         Element sum = 0;
         while (sp1 < sp_last)
            sum += *sp1++ * *mp++;
         *tp++ += sum;
      }
   } else if (scalar == 0.0) {
      while (tp < tp_last) {
         const Element *sp1 = sp;
         Element sum = 0;
         while (sp1 < sp_last)
            sum += *sp1++ * *mp++;
         *tp++  = sum;
      }
   } else if (scalar == -1.0) {
      while (tp < tp_last) {
         const Element *sp1 = sp;
         Element sum = 0;
         while (sp1 < sp_last)
            sum += *sp1++ * *mp++;
         *tp++ -= sum;
      }
   } else {
      while (tp < tp_last) {
         const Element *sp1 = sp;
         Element sum = 0;
         while (sp1 < sp_last)
            sum += *sp1++ * *mp++;
         *tp++ += scalar * sum;
      }
   }
   R__ASSERT(mp == a.GetMatrixArray()+a.GetNoElements());
#endif

   return target;
}

////////////////////////////////////////////////////////////////////////////////
/// Modify addition: target += A * source.
/// NOTE: in case scalar=0, do  target = A * source.

template<class Element>
TVectorT<Element> &Add(TVectorT<Element> &target,Element scalar,
                       const TMatrixTSparse<Element> &a,const TVectorT<Element> &source)
{
   if (gMatrixCheck) {
      R__ASSERT(target.IsValid());
      R__ASSERT(a.IsValid());
      if (a.GetNrows() != target.GetNrows() || a.GetRowLwb() != target.GetLwb()) {
         Error("Add(TVectorT &,const TMatrixT &,const TVectorT &)","target vector and matrix are incompatible");
         return target;
      }

      R__ASSERT(source.IsValid());
      if (a.GetNcols() != source.GetNrows() || a.GetColLwb() != source.GetLwb()) {
         Error("Add(TVectorT &,const TMatrixT &,const TVectorT &)","source vector and matrix are incompatible");
         return target;
      }
   }

   const Int_t   * const pRowIndex = a.GetRowIndexArray();
   const Int_t   * const pColIndex = a.GetColIndexArray();
   const Element * const mp        = a.GetMatrixArray();     // Matrix row ptr

   const Element * const sp = source.GetMatrixArray(); // Source vector ptr
         Element *       tp = target.GetMatrixArray(); // Target vector ptr

   if (scalar == 1.0) {
      for (Int_t irow = 0; irow < a.GetNrows(); irow++) {
         const Int_t sIndex = pRowIndex[irow];
         const Int_t eIndex = pRowIndex[irow+1];
         Element sum = 0.0;
         for (Int_t index = sIndex; index < eIndex; index++) {
            const Int_t icol = pColIndex[index];
            sum += mp[index]*sp[icol];
         }
         tp[irow] += sum;
      }
   } else if (scalar == 0.0) {
      for (Int_t irow = 0; irow < a.GetNrows(); irow++) {
         const Int_t sIndex = pRowIndex[irow];
         const Int_t eIndex = pRowIndex[irow+1];
         Element sum = 0.0;
         for (Int_t index = sIndex; index < eIndex; index++) {
            const Int_t icol = pColIndex[index];
            sum += mp[index]*sp[icol];
         }
         tp[irow]  = sum;
      }
   } else if (scalar == -1.0) {
     for (Int_t irow = 0; irow < a.GetNrows(); irow++) {
        const Int_t sIndex = pRowIndex[irow];
        const Int_t eIndex = pRowIndex[irow+1];
        Element sum = 0.0;
        for (Int_t index = sIndex; index < eIndex; index++) {
           const Int_t icol = pColIndex[index];
           sum += mp[index]*sp[icol];
        }
        tp[irow] -= sum;
      }
   } else {
      for (Int_t irow = 0; irow < a.GetNrows(); irow++) {
        const Int_t sIndex = pRowIndex[irow];
        const Int_t eIndex = pRowIndex[irow+1];
        Element sum = 0.0;
        for (Int_t index = sIndex; index < eIndex; index++) {
           const Int_t icol = pColIndex[index];
           sum += mp[index]*sp[icol];
        }
        tp[irow] += scalar * sum;
      }
   }

   return target;
}

////////////////////////////////////////////////////////////////////////////////
/// Modify addition: target += scalar * ElementMult(source1,source2) .

template<class Element>
TVectorT<Element> &AddElemMult(TVectorT<Element> &target,Element scalar,
                      const TVectorT<Element> &source1,const TVectorT<Element> &source2)
{
   if (gMatrixCheck && !(AreCompatible(target,source1) && AreCompatible(target,source2))) {
      Error("AddElemMult(TVectorT<Element> &,Element,const TVectorT<Element> &,const TVectorT<Element> &)",
             "vector's are incompatible");
      return target;
   }

   const Element *       sp1 = source1.GetMatrixArray();
   const Element *       sp2 = source2.GetMatrixArray();
         Element *       tp  = target.GetMatrixArray();
   const Element * const ftp = tp+target.GetNrows();

   if (scalar == 1.0 ) {
      while ( tp < ftp )
         *tp++ += *sp1++ * *sp2++;
   } else if (scalar == -1.0) {
      while ( tp < ftp )
         *tp++ -= *sp1++ * *sp2++;
   } else {
      while ( tp < ftp )
         *tp++ += scalar * *sp1++ * *sp2++;
   }

   return target;
}

////////////////////////////////////////////////////////////////////////////////
/// Modify addition: target += scalar * ElementMult(source1,source2) only for those elements
/// where select[i] != 0.0

template<class Element>
TVectorT<Element> &AddElemMult(TVectorT<Element> &target,Element scalar,
                      const TVectorT<Element> &source1,const TVectorT<Element> &source2,const TVectorT<Element> &select)
{
   if (gMatrixCheck && !( AreCompatible(target,source1) && AreCompatible(target,source2) &&
          AreCompatible(target,select) )) {
      Error("AddElemMult(TVectorT<Element> &,Element,const TVectorT<Element> &,const TVectorT<Element> &,onst TVectorT<Element> &)",
             "vector's are incompatible");
      return target;
   }

   const Element *       sp1 = source1.GetMatrixArray();
   const Element *       sp2 = source2.GetMatrixArray();
   const Element *       mp  = select.GetMatrixArray();
         Element *       tp  = target.GetMatrixArray();
   const Element * const ftp = tp+target.GetNrows();

   if (scalar == 1.0 ) {
      while ( tp < ftp ) {
         if (*mp) *tp += *sp1 * *sp2;
         mp++; tp++; sp1++; sp2++;
      }
   } else if (scalar == -1.0) {
      while ( tp < ftp ) {
         if (*mp) *tp -= *sp1 * *sp2;
         mp++; tp++; sp1++; sp2++;
      }
   } else {
      while ( tp < ftp ) {
         if (*mp) *tp += scalar * *sp1 * *sp2;
         mp++; tp++; sp1++; sp2++;
      }
   }

   return target;
}

////////////////////////////////////////////////////////////////////////////////
/// Modify addition: target += scalar * ElementDiv(source1,source2) .

template<class Element>
TVectorT<Element> &AddElemDiv(TVectorT<Element> &target,Element scalar,
                     const TVectorT<Element> &source1,const TVectorT<Element> &source2)
{
   if (gMatrixCheck && !(AreCompatible(target,source1) && AreCompatible(target,source2))) {
      Error("AddElemDiv(TVectorT<Element> &,Element,const TVectorT<Element> &,const TVectorT<Element> &)",
             "vector's are incompatible");
      return target;
   }

   const Element *       sp1 = source1.GetMatrixArray();
   const Element *       sp2 = source2.GetMatrixArray();
         Element *       tp  = target.GetMatrixArray();
   const Element * const ftp = tp+target.GetNrows();

   if (scalar == 1.0 ) {
      while ( tp < ftp ) {
         if (*sp2 != 0.0)
            *tp += *sp1 / *sp2;
         else {
            const Int_t irow = (sp2-source2.GetMatrixArray())/source2.GetNrows();
            Error("AddElemDiv","source2 (%d) is zero",irow);
         }
         tp++; sp1++; sp2++;
      }
   } else if (scalar == -1.0) {
      while ( tp < ftp ) {
         if (*sp2 != 0.0)
            *tp -= *sp1 / *sp2;
         else {
            const Int_t irow = (sp2-source2.GetMatrixArray())/source2.GetNrows();
            Error("AddElemDiv","source2 (%d) is zero",irow);
         }
         tp++; sp1++; sp2++;
      }
   } else {
      while ( tp < ftp ) {
         if (*sp2 != 0.0)
            *tp += scalar * *sp1 / *sp2;
         else {
            const Int_t irow = (sp2-source2.GetMatrixArray())/source2.GetNrows();
            Error("AddElemDiv","source2 (%d) is zero",irow);
         }
         tp++; sp1++; sp2++;
      }
   }

   return target;
}

////////////////////////////////////////////////////////////////////////////////
/// Modify addition: target += scalar * ElementDiv(source1,source2) only for those elements
/// where select[i] != 0.0

template<class Element>
TVectorT<Element> &AddElemDiv(TVectorT<Element> &target,Element scalar,
                     const TVectorT<Element> &source1,const TVectorT<Element> &source2,const TVectorT<Element> &select)
{
   if (gMatrixCheck && !( AreCompatible(target,source1) && AreCompatible(target,source2) &&
          AreCompatible(target,select) )) {
      Error("AddElemDiv(TVectorT<Element> &,Element,const TVectorT<Element> &,const TVectorT<Element> &,onst TVectorT<Element> &)",
             "vector's are incompatible");
      return target;
   }

   const Element *       sp1 = source1.GetMatrixArray();
   const Element *       sp2 = source2.GetMatrixArray();
   const Element *       mp  = select.GetMatrixArray();
         Element *       tp  = target.GetMatrixArray();
   const Element * const ftp = tp+target.GetNrows();

   if (scalar == 1.0 ) {
      while ( tp < ftp ) {
         if (*mp) {
            if (*sp2 != 0.0)
               *tp += *sp1 / *sp2;
            else {
               const Int_t irow = (sp2-source2.GetMatrixArray())/source2.GetNrows();
               Error("AddElemDiv","source2 (%d) is zero",irow);
            }
         }
         mp++; tp++; sp1++; sp2++;
      }
   } else if (scalar == -1.0) {
      while ( tp < ftp ) {
         if (*mp) {
            if (*sp2 != 0.0)
               *tp -= *sp1 / *sp2;
            else {
               const Int_t irow = (sp2-source2.GetMatrixArray())/source2.GetNrows();
               Error("AddElemDiv","source2 (%d) is zero",irow);
            }
         }
         mp++; tp++; sp1++; sp2++;
      }
   } else {
      while ( tp < ftp ) {
         if (*mp) {
            if (*sp2 != 0.0)
               *tp += scalar * *sp1 / *sp2;
            else {
               const Int_t irow = (sp2-source2.GetMatrixArray())/source2.GetNrows();
               Error("AddElemDiv","source2 (%d) is zero",irow);
            }
         }
         mp++; tp++; sp1++; sp2++;
      }
   }

   return target;
}

////////////////////////////////////////////////////////////////////////////////
/// Multiply target by the source, element-by-element.

template<class Element>
TVectorT<Element> &ElementMult(TVectorT<Element> &target,const TVectorT<Element> &source)
{
   if (gMatrixCheck && !AreCompatible(target,source)) {
      Error("ElementMult(TVectorT<Element> &,const TVectorT<Element> &)","vector's are incompatible");
      return target;
   }

   const Element *       sp  = source.GetMatrixArray();
         Element *       tp  = target.GetMatrixArray();
   const Element * const ftp = tp+target.GetNrows();
   while ( tp < ftp )
      *tp++ *= *sp++;

   return target;
}

////////////////////////////////////////////////////////////////////////////////
/// Multiply target by the source, element-by-element only where select[i] != 0.0

template<class Element>
TVectorT<Element> &ElementMult(TVectorT<Element> &target,const TVectorT<Element> &source,const TVectorT<Element> &select)
{
   if (gMatrixCheck && !(AreCompatible(target,source) && AreCompatible(target,select))) {
      Error("ElementMult(TVectorT<Element> &,const TVectorT<Element> &,const TVectorT<Element> &)","vector's are incompatible");
      return target;
   }

   const Element *       sp  = source.GetMatrixArray();
   const Element *       mp  = select.GetMatrixArray();
         Element *       tp  = target.GetMatrixArray();
   const Element * const ftp = tp+target.GetNrows();
   while ( tp < ftp ) {
      if (*mp) *tp *= *sp;
      mp++; tp++; sp++;
   }

   return target;
}

////////////////////////////////////////////////////////////////////////////////
/// Divide target by the source, element-by-element.

template<class Element>
TVectorT<Element> &ElementDiv(TVectorT<Element> &target,const TVectorT<Element> &source)
{
   if (gMatrixCheck && !AreCompatible(target,source)) {
      Error("ElementDiv(TVectorT<Element> &,const TVectorT<Element> &)","vector's are incompatible");
      return target;
   }

   const Element *       sp  = source.GetMatrixArray();
         Element *       tp  = target.GetMatrixArray();
   const Element * const ftp = tp+target.GetNrows();
   while ( tp < ftp ) {
      if (*sp  != 0.0)
         *tp++ /= *sp++;
      else {
         const Int_t irow = (sp-source.GetMatrixArray())/source.GetNrows();
         Error("ElementDiv","source (%d) is zero",irow);
      }
   }

   return target;
}

////////////////////////////////////////////////////////////////////////////////
/// Divide target by the source, element-by-element only where select[i] != 0.0

template<class Element>
TVectorT<Element> &ElementDiv(TVectorT<Element> &target,const TVectorT<Element> &source,const TVectorT<Element> &select)
{
   if (gMatrixCheck && !AreCompatible(target,source)) {
      Error("ElementDiv(TVectorT<Element> &,const TVectorT<Element> &,const TVectorT<Element> &)","vector's are incompatible");
      return target;
   }

   const Element *       sp  = source.GetMatrixArray();
   const Element *       mp  = select.GetMatrixArray();
         Element *       tp  = target.GetMatrixArray();
   const Element * const ftp = tp+target.GetNrows();
   while ( tp < ftp ) {
      if (*mp) {
         if (*sp != 0.0)
            *tp /= *sp;
         else {
            const Int_t irow = (sp-source.GetMatrixArray())/source.GetNrows();
            Error("ElementDiv","source (%d) is zero",irow);
         }
      }
      mp++; tp++; sp++;
   }

   return target;
}

////////////////////////////////////////////////////////////////////////////////
/// Check if v1 and v2 are both valid and have the same shape

template<class Element1,class Element2>
Bool_t AreCompatible(const TVectorT<Element1> &v1,const TVectorT<Element2> &v2,Int_t verbose)
{
   if (!v1.IsValid()) {
      if (verbose)
         ::Error("AreCompatible", "vector 1 not valid");
      return kFALSE;
   }
   if (!v2.IsValid()) {
      if (verbose)
         ::Error("AreCompatible", "vector 2 not valid");
      return kFALSE;
   }

   if (v1.GetNrows() != v2.GetNrows() || v1.GetLwb() != v2.GetLwb()) {
      if (verbose)
         ::Error("AreCompatible", "matrices 1 and 2 not compatible");
      return kFALSE;
   }

   return kTRUE;
}

////////////////////////////////////////////////////////////////////////////////
/// Check if m and v are both valid and have compatible shapes for M * v

template<class Element1, class Element2>
Bool_t AreCompatible(const TMatrixT<Element1> &m,const TVectorT<Element2> &v,Int_t verbose)
{
   if (!m.IsValid()) {
      if (verbose)
         ::Error("AreCompatible", "Matrix not valid");
      return kFALSE;
   }
   if (!v.IsValid()) {
      if (verbose)
         ::Error("AreCompatible", "vector not valid");
      return kFALSE;
   }

   if (m.GetNcols() != v.GetNrows() ) {
      if (verbose)
         ::Error("AreCompatible", "matrix and vector not compatible");
      return kFALSE;
   }

   return kTRUE;
}

////////////////////////////////////////////////////////////////////////////////
/// Check if m and v are both valid and have compatible shapes for v * M

template<class Element1, class Element2>
Bool_t AreCompatible(const TVectorT<Element1> &v,const TMatrixT<Element2> &m,Int_t verbose)
{
   if (!m.IsValid()) {
      if (verbose)
         ::Error("AreCompatible", "Matrix not valid");
      return kFALSE;
   }
   if (!v.IsValid()) {
      if (verbose)
         ::Error("AreCompatible", "vector not valid");
      return kFALSE;
   }

   if (v.GetNrows() != m.GetNrows() ) {
      if (verbose)
         ::Error("AreCompatible", "vector and matrix not compatible");
      return kFALSE;
   }

   return kTRUE;
}

////////////////////////////////////////////////////////////////////////////////
/// Compare two vectors and print out the result of the comparison.

template<class Element>
void Compare(const TVectorT<Element> &v1,const TVectorT<Element> &v2)
{
   if (!AreCompatible(v1,v2)) {
      Error("Compare(const TVectorT<Element> &,const TVectorT<Element> &)","vectors are incompatible");
      return;
   }

   printf("\n\nComparison of two TVectorTs:\n");

   Element norm1  = 0;       // Norm of the Matrices
   Element norm2  = 0;       // Norm of the Matrices
   Element ndiff  = 0;       // Norm of the difference
   Int_t    imax   = 0;       // For the elements that differ most
   Element difmax = -1;
   const Element *mp1 = v1.GetMatrixArray();    // Vector element pointers
   const Element *mp2 = v2.GetMatrixArray();

   for (Int_t i = 0; i < v1.GetNrows(); i++) {
      const Element mv1  = *mp1++;
      const Element mv2  = *mp2++;
      const Element diff = TMath::Abs(mv1-mv2);

      if (diff > difmax) {
         difmax = diff;
         imax = i;
      }
      norm1 += TMath::Abs(mv1);
      norm2 += TMath::Abs(mv2);
      ndiff += TMath::Abs(diff);
   }

   imax += v1.GetLwb();
   printf("\nMaximal discrepancy    \t\t%g",difmax);
   printf("\n   occured at the point\t\t(%d)",imax);
   const Element mv1 = v1(imax);
   const Element mv2 = v2(imax);
   printf("\n Vector 1 element is    \t\t%g",mv1);
   printf("\n Vector 2 element is    \t\t%g",mv2);
   printf("\n Absolute error v2[i]-v1[i]\t\t%g",mv2-mv1);
   printf("\n Relative error\t\t\t\t%g\n",
          (mv2-mv1)/TMath::Max(TMath::Abs(mv2+mv1)/2,(Element)1e-7));

   printf("\n||Vector 1||   \t\t\t%g",norm1);
   printf("\n||Vector 2||   \t\t\t%g",norm2);
   printf("\n||Vector1-Vector2||\t\t\t\t%g",ndiff);
   printf("\n||Vector1-Vector2||/sqrt(||Vector1|| ||Vector2||)\t%g\n\n",
          ndiff/TMath::Max(TMath::Sqrt(norm1*norm2),1e-7));
}

////////////////////////////////////////////////////////////////////////////////
/// Validate that all elements of vector have value val within maxDevAllow .

template<class Element>
Bool_t VerifyVectorValue(const TVectorT<Element> &v,Element val,
                         Int_t verbose,Element maxDevAllow)
{
   Int_t   imax      = 0;
   Element maxDevObs = 0;

   if (TMath::Abs(maxDevAllow) <= 0.0)
      maxDevAllow = std::numeric_limits<Element>::epsilon();

   for (Int_t i = v.GetLwb(); i <= v.GetUpb(); i++) {
      const Element dev = TMath::Abs(v(i)-val);
      if (dev > maxDevObs) {
         imax      = i;
         maxDevObs = dev;
      }
   }

   if (maxDevObs == 0)
      return kTRUE;

   if (verbose) {
      printf("Largest dev for (%d); dev = |%g - %g| = %g\n",imax,v(imax),val,maxDevObs);
      if (maxDevObs > maxDevAllow)
         Error("VerifyVectorValue","Deviation > %g\n",maxDevAllow);
   }

   if (maxDevObs > maxDevAllow)
      return kFALSE;
   return kTRUE;
}

////////////////////////////////////////////////////////////////////////////////
/// Verify that elements of the two vectors are equal within maxDevAllow .

template<class Element>
Bool_t VerifyVectorIdentity(const TVectorT<Element> &v1,const TVectorT<Element> &v2,
                            Int_t verbose, Element maxDevAllow)
{
   Int_t   imax      = 0;
   Element maxDevObs = 0;

   if (!AreCompatible(v1,v2))
      return kFALSE;

   if (TMath::Abs(maxDevAllow) <= 0.0)
      maxDevAllow = std::numeric_limits<Element>::epsilon();

   for (Int_t i = v1.GetLwb(); i <= v1.GetUpb(); i++) {
      const Element dev = TMath::Abs(v1(i)-v2(i));
      if (dev > maxDevObs) {
         imax      = i;
         maxDevObs = dev;
      }
   }

   if (maxDevObs == 0)
      return kTRUE;

   if (verbose) {
      printf("Largest dev for (%d); dev = |%g - %g| = %g\n",imax,v1(imax),v2(imax),maxDevObs);
      if (maxDevObs > maxDevAllow)
         Error("VerifyVectorIdentity","Deviation > %g\n",maxDevAllow);
   }

   if (maxDevObs > maxDevAllow) {
      return kFALSE;
   }
   return kTRUE;
}

////////////////////////////////////////////////////////////////////////////////
/// Stream an object of class TVectorT.

template<class Element>
void TVectorT<Element>::Streamer(TBuffer &R__b)
{
   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) {
         Clear();
         R__b.ReadClassBuffer(TVectorT<Element>::Class(),this,R__v,R__s,R__c);
      } else { //====process old versions before automatic schema evolution
         TObject::Streamer(R__b);
         R__b >> fRowLwb;
         fNrows = R__b.ReadArray(fElements);
         R__b.CheckByteCount(R__s, R__c, TVectorT<Element>::IsA());
      }
      if (fNrows > 0 && fNrows <= kSizeMax) {
         memcpy(fDataStack,fElements,fNrows*sizeof(Element));
         delete [] fElements;
         fElements = fDataStack;
      } else if (fNrows < 0)
         Invalidate();

      if (R__v < 3)
         MakeValid();
    } else {
       R__b.WriteClassBuffer(TVectorT<Element>::Class(),this);
   }
}

#include "TMatrixFfwd.h"
#include "TMatrixFSymfwd.h"
#include "TMatrixFSparsefwd.h"

template class TVectorT<Float_t>;

template Bool_t    operator==          <Float_t>(const TVectorF       &source1,const TVectorF &source2);
template TVectorF  operator+           <Float_t>(const TVectorF       &source1,const TVectorF &source2);
template TVectorF  operator-           <Float_t>(const TVectorF       &source1,const TVectorF &source2);
template Float_t   operator*           <Float_t>(const TVectorF       &source1,const TVectorF &source2);
template TVectorF  operator*           <Float_t>(const TMatrixF       &a,      const TVectorF &source);
template TVectorF  operator*           <Float_t>(const TMatrixFSym    &a,      const TVectorF &source);
template TVectorF  operator*           <Float_t>(const TMatrixFSparse &a,      const TVectorF &source);
template TVectorF  operator*           <Float_t>(      Float_t         val,    const TVectorF &source);

template Float_t   Dot                 <Float_t>(const TVectorF       &v1,     const TVectorF &v2);
template TMatrixF  OuterProduct        <Float_t,Float_t>
                                                (const TVectorF       &v1,     const TVectorF &v2);
template TMatrixF &OuterProduct        <Float_t,Float_t,Float_t>
                                                (      TMatrixF       &target, const TVectorF &v1,     const TVectorF       &v2);
template Float_t   Mult                <Float_t,Float_t,Float_t>
                                                (const TVectorF       &v1,     const TMatrixF &m,      const TVectorF       &v2);

template TVectorF &Add                 <Float_t>(      TVectorF       &target,       Float_t   scalar, const TVectorF       &source);
template TVectorF &Add                 <Float_t>(      TVectorF       &target,       Float_t   scalar, const TMatrixF       &a,
                                                                               const TVectorF &source);
template TVectorF &Add                 <Float_t>(      TVectorF       &target,       Float_t   scalar, const TMatrixFSym    &a,
                                                                               const TVectorF &source);
template TVectorF &Add                 <Float_t>(      TVectorF       &target,       Float_t   scalar, const TMatrixFSparse &a,
                                                                               const TVectorF &source);
template TVectorF &AddElemMult         <Float_t>(      TVectorF       &target,       Float_t   scalar, const TVectorF       &source1,
                                                                               const TVectorF &source2);
template TVectorF &AddElemMult         <Float_t>(      TVectorF       &target,       Float_t   scalar, const TVectorF       &source1,
                                                                               const TVectorF &source2,const TVectorF       &select);
template TVectorF &AddElemDiv          <Float_t>(      TVectorF       &target,       Float_t   scalar, const TVectorF       &source1,
                                                                               const TVectorF &source2);
template TVectorF &AddElemDiv          <Float_t>(      TVectorF       &target,       Float_t   scalar, const TVectorF       &source1,
                                                                               const TVectorF &source2,const TVectorF       &select);
template TVectorF &ElementMult         <Float_t>(      TVectorF       &target, const TVectorF &source);
template TVectorF &ElementMult         <Float_t>(      TVectorF       &target, const TVectorF &source, const TVectorF       &select);
template TVectorF &ElementDiv          <Float_t>(      TVectorF       &target, const TVectorF &source);
template TVectorF &ElementDiv          <Float_t>(      TVectorF       &target, const TVectorF &source, const TVectorF       &select);

template Bool_t    AreCompatible       <Float_t,Float_t> (const TVectorF &v1,const TVectorF &v2,Int_t verbose);
template Bool_t    AreCompatible       <Float_t,Double_t>(const TVectorF &v1,const TVectorD &v2,Int_t verbose);
template Bool_t    AreCompatible       <Float_t,Float_t> (const TMatrixF &m, const TVectorF &v, Int_t verbose);
template Bool_t    AreCompatible       <Float_t,Float_t> (const TVectorF &v, const TMatrixF &m, Int_t verbose);

template void      Compare             <Float_t>         (const TVectorF &v1,const TVectorF &v2);
template Bool_t    VerifyVectorValue   <Float_t>         (const TVectorF &m,       Float_t   val,Int_t verbose,Float_t maxDevAllow);
template Bool_t    VerifyVectorIdentity<Float_t>         (const TVectorF &m1,const TVectorF &m2, Int_t verbose,Float_t maxDevAllow);

#include "TMatrixDfwd.h"
#include "TMatrixDSymfwd.h"
#include "TMatrixDSparsefwd.h"

template class TVectorT<Double_t>;

template Bool_t    operator==          <Double_t>(const TVectorD       &source1,const TVectorD &source2);
template TVectorD  operator+           <Double_t>(const TVectorD       &source1,const TVectorD &source2);
template TVectorD  operator-           <Double_t>(const TVectorD       &source1,const TVectorD &source2);
template Double_t  operator*           <Double_t>(const TVectorD       &source1,const TVectorD &source2);
template TVectorD  operator*           <Double_t>(const TMatrixD       &a,      const TVectorD &source);
template TVectorD  operator*           <Double_t>(const TMatrixDSym    &a,      const TVectorD &source);
template TVectorD  operator*           <Double_t>(const TMatrixDSparse &a,      const TVectorD &source);
template TVectorD  operator*           <Double_t>(      Double_t        val,    const TVectorD &source);

template Double_t  Dot                 <Double_t>(const TVectorD       &v1,     const TVectorD &v2);
template TMatrixD  OuterProduct        <Double_t,Double_t>
                                                 (const TVectorD       &v1,     const TVectorD &v2);
template TMatrixD &OuterProduct        <Double_t,Double_t,Double_t>
                                                 (      TMatrixD       &target, const TVectorD &v1,     const TVectorD       &v2);
template Double_t  Mult                <Double_t,Double_t,Double_t>
                                                 (const TVectorD       &v1,     const TMatrixD &m,      const TVectorD       &v2);

template TVectorD &Add                 <Double_t>(      TVectorD       &target,       Double_t  scalar, const TVectorD       &source);
template TVectorD &Add                 <Double_t>(      TVectorD       &target,       Double_t  scalar, const TMatrixD       &a,
                                                                                const TVectorD &source);
template TVectorD &Add                 <Double_t>(      TVectorD       &target,       Double_t  scalar, const TMatrixDSym    &a
                                                                                ,      const TVectorD &source);
template TVectorD &Add                 <Double_t>(      TVectorD       &target,       Double_t  scalar, const TMatrixDSparse &a
                                                                                ,      const TVectorD &source);
template TVectorD &AddElemMult         <Double_t>(      TVectorD       &target,       Double_t  scalar, const TVectorD       &source1,
                                                                                const TVectorD &source2);
template TVectorD &AddElemMult         <Double_t>(      TVectorD       &target,       Double_t  scalar, const TVectorD       &source1,
                                                                                const TVectorD &source2,const TVectorD       &select);
template TVectorD &AddElemDiv          <Double_t>(      TVectorD       &target,       Double_t  scalar, const TVectorD       &source1,
                                                                                const TVectorD &source2);
template TVectorD &AddElemDiv          <Double_t>(      TVectorD       &target,       Double_t  scalar, const TVectorD       &source1,
                                                                                const TVectorD &source2,const TVectorD       &select);
template TVectorD &ElementMult         <Double_t>(      TVectorD       &target, const TVectorD &source);
template TVectorD &ElementMult         <Double_t>(      TVectorD       &target, const TVectorD &source, const TVectorD       &select);
template TVectorD &ElementDiv          <Double_t>(      TVectorD       &target, const TVectorD &source);
template TVectorD &ElementDiv          <Double_t>(      TVectorD       &target, const TVectorD &source, const TVectorD       &select);

template Bool_t    AreCompatible       <Double_t,Double_t>(const TVectorD &v1,const TVectorD &v2,Int_t verbose);
template Bool_t    AreCompatible       <Double_t,Float_t> (const TVectorD &v1,const TVectorF &v2,Int_t verbose);
template Bool_t    AreCompatible       <Double_t,Double_t>(const TMatrixD &m, const TVectorD &v, Int_t verbose);
template Bool_t    AreCompatible       <Double_t,Double_t>(const TVectorD &v, const TMatrixD &m, Int_t verbose);

template void      Compare             <Double_t>         (const TVectorD &v1,const TVectorD &v2);
template Bool_t    VerifyVectorValue   <Double_t>         (const TVectorD &m,       Double_t  val,Int_t verbose,Double_t maxDevAllow);
template Bool_t    VerifyVectorIdentity<Double_t>         (const TVectorD &m1,const TVectorD &m2, Int_t verbose,Double_t maxDevAllow);