TMatrixTSym<double> - source file

// @(#)root/matrix:$Id: TMatrixTSym.cxx 20882 2007-11-19 11:31:26Z rdm $
// 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.             *
 *************************************************************************/

//////////////////////////////////////////////////////////////////////////
//                                                                      //
// TMatrixTSym                                                          //
//                                                                      //
// Template class of a symmetric matrix in the linear algebra package   //
//                                                                      //
// Note that in this implementation both matrix element m[i][j] and     //
// m[j][i] are updated and stored in memory . However, when making the  //
// object persistent only the upper right triangle is stored .          //
//                                                                      //
//////////////////////////////////////////////////////////////////////////

#include "TMatrixTSym.h"
#include "TMatrixTLazy.h"
#include "TMatrixTSymCramerInv.h"
#include "TDecompLU.h"
#include "TMatrixDSymEigen.h"
#include "TClass.h"
#include "TMath.h"

#ifndef R__ALPHA
templateClassImp(TMatrixTSym)
#endif

//______________________________________________________________________________
template<class Element>
TMatrixTSym<Element>::TMatrixTSym(Int_t no_rows)
{
   Allocate(no_rows,no_rows,0,0,1);
}

//______________________________________________________________________________
template<class Element>
TMatrixTSym<Element>::TMatrixTSym(Int_t row_lwb,Int_t row_upb)
{
   const Int_t no_rows = row_upb-row_lwb+1;
   Allocate(no_rows,no_rows,row_lwb,row_lwb,1);
}

//______________________________________________________________________________
template<class Element>
TMatrixTSym<Element>::TMatrixTSym(Int_t no_rows,const Element *elements,Option_t *option)
{
  // option="F": array elements contains the matrix stored column-wise
  //             like in Fortran, so a[i,j] = elements[i+no_rows*j],
  // else        it is supposed that array elements are stored row-wise
  //             a[i,j] = elements[i*no_cols+j]
  //
  // array elements are copied

   Allocate(no_rows,no_rows);
   SetMatrixArray(elements,option);
   if (!this->IsSymmetric()) {
      Error("TMatrixTSym(Int_t,Element*,Option_t*)","matrix not symmetric");
   }
}

//______________________________________________________________________________
template<class Element>
TMatrixTSym<Element>::TMatrixTSym(Int_t row_lwb,Int_t row_upb,const Element *elements,Option_t *option)
{
  // array elements are copied

   const Int_t no_rows = row_upb-row_lwb+1;
   Allocate(no_rows,no_rows,row_lwb,row_lwb);
   SetMatrixArray(elements,option);
   if (!this->IsSymmetric()) {
      Error("TMatrixTSym(Int_t,Int_t,Element*,Option_t*)","matrix not symmetric");
   }
}

//______________________________________________________________________________
template<class Element>
TMatrixTSym<Element>::TMatrixTSym(const TMatrixTSym<Element> &another) : TMatrixTBase<Element>(another)
{
   R__ASSERT(another.IsValid());
   Allocate(another.GetNrows(),another.GetNcols(),another.GetRowLwb(),another.GetColLwb());
   *this = another;
}

//______________________________________________________________________________
template<class Element>
TMatrixTSym<Element>::TMatrixTSym(EMatrixCreatorsOp1 op,const TMatrixTSym<Element> &prototype)
{
  // Create a matrix applying a specific operation to the prototype.
  // Example: TMatrixTSym<Element> a(10,12); ...; TMatrixTSym<Element> b(TMatrixT::kTransposed, a);
  // Supported operations are: kZero, kUnit, kTransposed, kInverted and kAtA.

   R__ASSERT(prototype.IsValid());

   switch(op) {
      case kZero:
         Allocate(prototype.GetNrows(),prototype.GetNcols(),
                  prototype.GetRowLwb(),prototype.GetColLwb(),1);
         break;

      case kUnit:
         Allocate(prototype.GetNrows(),prototype.GetNcols(),
                  prototype.GetRowLwb(),prototype.GetColLwb(),1);
         this->UnitMatrix();
         break;

      case kTransposed:
         Allocate(prototype.GetNcols(), prototype.GetNrows(),
                  prototype.GetColLwb(),prototype.GetRowLwb());
         Transpose(prototype);
         break;

      case kInverted:
      {
         Allocate(prototype.GetNrows(),prototype.GetNcols(),
                  prototype.GetRowLwb(),prototype.GetColLwb(),1);
         *this = prototype;
         // Since the user can not control the tolerance of this newly created matrix
         // we put it to the smallest possible number
         const Element oldTol = this->SetTol(std::numeric_limits<Element>::min());
         this->Invert();
         this->SetTol(oldTol);
         break;
      }

      case kAtA:
         Allocate(prototype.GetNcols(),prototype.GetNcols(),prototype.GetColLwb(),prototype.GetColLwb(),1);
         TMult(prototype);
         break;

      default:
         Error("TMatrixTSym(EMatrixCreatorOp1,const TMatrixTSym)",
               "operation %d not yet implemented", op);
   }
}

//______________________________________________________________________________
template<class Element>
TMatrixTSym<Element>::TMatrixTSym(EMatrixCreatorsOp1 op,const TMatrixT<Element> &prototype)
{
   R__ASSERT(prototype.IsValid());

   switch(op) {
      case kAtA:
         Allocate(prototype.GetNcols(),prototype.GetNcols(),prototype.GetColLwb(),prototype.GetColLwb(),1);
         TMult(prototype);
         break;

      default:
         Error("TMatrixTSym(EMatrixCreatorOp1,const TMatrixT)",
                "operation %d not yet implemented", op);
   }
}

//______________________________________________________________________________
template<class Element>
TMatrixTSym<Element>::TMatrixTSym(const TMatrixTSym<Element> &a,EMatrixCreatorsOp2 op,const TMatrixTSym<Element> &b)
{
   R__ASSERT(a.IsValid());
   R__ASSERT(b.IsValid());

   switch(op) {
      case kPlus:
      {
         Plus(a,b);
         break;
      }

      case kMinus:
      {
         Minus(a,b);
         break;
      }

      default:
         Error("TMatrixTSym(EMatrixCreatorOp2)", "operation %d not yet implemented", op);
   }
}

//______________________________________________________________________________
template<class Element>
TMatrixTSym<Element>::TMatrixTSym(const TMatrixTSymLazy<Element> &lazy_constructor)
{
   Allocate(lazy_constructor.GetRowUpb()-lazy_constructor.GetRowLwb()+1,
            lazy_constructor.GetRowUpb()-lazy_constructor.GetRowLwb()+1,
            lazy_constructor.GetRowLwb(),lazy_constructor.GetRowLwb(),1);
   lazy_constructor.FillIn(*this);
   if (!this->IsSymmetric()) {
      Error("TMatrixTSym(TMatrixTSymLazy)","matrix not symmetric");
   }
}

//______________________________________________________________________________
template<class Element>
void TMatrixTSym<Element>::Delete_m(Int_t size,Element *&m)
{
  // delete data pointer m, if it was assigned on the heap

   if (m) {
      if (size > this->kSizeMax)
         delete [] m;
      m = 0;
   }
}

//______________________________________________________________________________
template<class Element>
Element* TMatrixTSym<Element>::New_m(Int_t size)
{
  // return data pointer . if requested size <= kSizeMax, assign pointer
  // to the stack space

   if (size == 0) return 0;
   else {
      if ( size <= this->kSizeMax )
         return fDataStack;
      else {
         Element *heap = new Element[size];
         return heap;
      }
   }
}

//______________________________________________________________________________
template<class Element>
Int_t TMatrixTSym<Element>::Memcpy_m(Element *newp,const Element *oldp,Int_t copySize,
                                     Int_t newSize,Int_t oldSize)
{
  // copy copySize doubles from *oldp to *newp . However take care of the
  // situation where both pointers are assigned to the same stack space

   if (copySize == 0 || oldp == newp)
      return 0;
   else {
      if ( newSize <= this->kSizeMax && oldSize <= this->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;
}

//______________________________________________________________________________
template<class Element>
void TMatrixTSym<Element>::Allocate(Int_t no_rows,Int_t no_cols,Int_t row_lwb,Int_t col_lwb,
                                    Int_t init,Int_t /*nr_nonzeros*/)
{
  // Allocate new matrix. Arguments are number of rows, columns, row
  // lowerbound (0 default) and column lowerbound (0 default).

   this->fIsOwner = kTRUE;
   this->fTol     = std::numeric_limits<Element>::epsilon();
   this->fNrows   = 0;
   this->fNcols   = 0;
   this->fRowLwb  = 0;
   this->fColLwb  = 0;
   this->fNelems  = 0;
   fElements      = 0;

   if (no_rows < 0 || no_cols < 0)
   {
      Error("Allocate","no_rows=%d no_cols=%d",no_rows,no_cols);
      this->Invalidate();
      return;
   }

   this->MakeValid();
   this->fNrows   = no_rows;
   this->fNcols   = no_cols;
   this->fRowLwb  = row_lwb;
   this->fColLwb  = col_lwb;
   this->fNelems  = this->fNrows*this->fNcols;

   if (this->fNelems > 0) {
      fElements = New_m(this->fNelems);
      if (init)
         memset(fElements,0,this->fNelems*sizeof(Element));
   } else
     fElements = 0;
}

//______________________________________________________________________________
template<class Element>
void TMatrixTSym<Element>::Plus(const TMatrixTSym<Element> &a,const TMatrixTSym<Element> &b)
{
  // Symmetric matrix summation. Create a matrix C such that C = A + B.

   if (gMatrixCheck) {
      if (!AreCompatible(a,b)) {
         Error("Plus","matrices not compatible");
         return;
      }

      if (this->GetMatrixArray() == a.GetMatrixArray()) {
         Error("Plus","this->GetMatrixArray() == a.GetMatrixArray()");
         return;
      }

      if (this->GetMatrixArray() == b.GetMatrixArray()) {
         Error("Plus","this->GetMatrixArray() == b.GetMatrixArray()");
         return;
      }
   }

   const Element *       ap      = a.GetMatrixArray();
   const Element *       bp      = b.GetMatrixArray();
         Element *       cp      = this->GetMatrixArray();
   const Element * const cp_last = cp+this->fNelems;

   while (cp < cp_last) {
       *cp = *ap++ + *bp++;
       cp++;
   }
}

//______________________________________________________________________________
template<class Element>
void TMatrixTSym<Element>::Minus(const TMatrixTSym<Element> &a,const TMatrixTSym<Element> &b)
{
  // Symmetric matrix summation. Create a matrix C such that C = A + B.

   if (gMatrixCheck) {
      if (!AreCompatible(a,b)) {
         Error("Minus","matrices not compatible");
         return;
      }

      if (this->GetMatrixArray() == a.GetMatrixArray()) {
         Error("Minus","this->GetMatrixArray() == a.GetMatrixArray()");
         return;
      }

      if (this->GetMatrixArray() == b.GetMatrixArray()) {
         Error("Minus","this->GetMatrixArray() == b.GetMatrixArray()");
         return;
      }
   }

   const Element *       ap      = a.GetMatrixArray();
   const Element *       bp      = b.GetMatrixArray();
         Element *       cp      = this->GetMatrixArray();
   const Element * const cp_last = cp+this->fNelems;

   while (cp < cp_last) {
       *cp = *ap++ - *bp++;
       cp++;
   }
}

//______________________________________________________________________________
template<class Element>
void TMatrixTSym<Element>::TMult(const TMatrixT<Element> &a)
{
  // Create a matrix C such that C = A' * A. In other words,
  // c[i,j] = SUM{ a[k,i] * a[k,j] }.

   R__ASSERT(a.IsValid());

#ifdef CBLAS
   const Element *ap = a.GetMatrixArray();
         Element *cp = this->GetMatrixArray();
   if (typeid(Element) == typeid(Double_t))
      cblas_dgemm (CblasRowMajor,CblasTrans,CblasNoTrans,this->fNrows,this->fNcols,a.GetNrows(),
                   1.0,ap,a.GetNcols(),ap,a.GetNcols(),1.0,cp,this->fNcols);
   else if (typeid(Element) != typeid(Float_t))
      cblas_sgemm (CblasRowMajor,CblasTrans,CblasNoTrans,fNrows,fNcols,a.GetNrows(),
                   1.0,ap,a.GetNcols(),ap,a.GetNcols(),1.0,cp,fNcols);
  else
      Error("TMult","type %s not implemented in BLAS library",typeid(Element));
#else
   const Int_t nb     = a.GetNoElements();
   const Int_t ncolsa = a.GetNcols();
   const Int_t ncolsb = ncolsa;
  const Element * const ap = a.GetMatrixArray();
   const Element * const bp = ap;
         Element *       cp = this->GetMatrixArray();

   const Element *acp0 = ap;           // Pointer to  A[i,0];
   while (acp0 < ap+a.GetNcols()) {
      for (const Element *bcp = bp; bcp < bp+ncolsb; ) { // Pointer to the j-th column of A, Start bcp = A[0,0]
         const Element *acp = acp0;                       // Pointer to the i-th column of A, reset to A[0,i]
         Element cij = 0;
         while (bcp < bp+nb) {           // Scan the i-th column of A and
            cij += *acp * *bcp;           // the j-th col of A
            acp += ncolsa;
            bcp += ncolsb;
         }
         *cp++ = cij;
         bcp -= nb-1;                    // Set bcp to the (j+1)-th col
      }
      acp0++;                           // Set acp0 to the (i+1)-th col
   }

   R__ASSERT(cp == this->GetMatrixArray()+this->fNelems && acp0 == ap+ncolsa);
#endif
}

//______________________________________________________________________________
template<class Element>
void TMatrixTSym<Element>::TMult(const TMatrixTSym<Element> &a)
{
  // Matrix multiplication, with A symmetric
  // Create a matrix C such that C = A' * A = A * A = A * A'

   R__ASSERT(a.IsValid());

#ifdef CBLAS
   const Element *ap = a.GetMatrixArray();
         Element *cp = this->GetMatrixArray();
   if (typeid(Element) == typeid(Double_t))
      cblas_dsymm (CblasRowMajor,CblasLeft,CblasUpper,this->fNrows,this->fNcols,1.0,
                   ap,a.GetNcols(),ap,a.GetNcols(),0.0,cp,this->fNcols);
   else if (typeid(Element) != typeid(Float_t))
      cblas_ssymm (CblasRowMajor,CblasLeft,CblasUpper,fNrows,fNcols,1.0,
                   ap1,a.GetNcols(),ap,a.GetNcols(),0.0,cp,fNcols);
   else
      Error("TMult","type %s not implemented in BLAS library",typeid(Element));
#else
   const Int_t nb     = a.GetNoElements();
   const Int_t ncolsa = a.GetNcols();
   const Int_t ncolsb = ncolsa;
   const Element * const ap = a.GetMatrixArray();
   const Element * const bp = ap;
         Element *       cp = this->GetMatrixArray();

   const Element *acp0 = ap;           // Pointer to  A[i,0];
   while (acp0 < ap+a.GetNcols()) {
      for (const Element *bcp = bp; bcp < bp+ncolsb; ) { // Pointer to the j-th column of A, Start bcp = A[0,0]
         const Element *acp = acp0;                       // Pointer to the i-th column of A, reset to A[0,i]
         Element cij = 0;
         while (bcp < bp+nb) {           // Scan the i-th column of A and
            cij += *acp * *bcp;           // the j-th col of A
            acp += ncolsa;
            bcp += ncolsb;
         }
         *cp++ = cij;
         bcp -= nb-1;                    // Set bcp to the (j+1)-th col
      }
      acp0++;                           // Set acp0 to the (i+1)-th col
   }

   R__ASSERT(cp == this->GetMatrixArray()+this->fNelems && acp0 == ap+ncolsa);
#endif
}

//______________________________________________________________________________
template<class Element>
TMatrixTSym<Element> &TMatrixTSym<Element>::Use(Int_t row_lwb,Int_t row_upb,Element *data)
{
   if (gMatrixCheck && row_upb < row_lwb)
   {
      Error("Use","row_upb=%d < row_lwb=%d",row_upb,row_lwb);
      return *this;
   }

   this->Clear();
   this->fNrows    = row_upb-row_lwb+1;
   this->fNcols    = this->fNrows;
   this->fRowLwb   = row_lwb;
   this->fColLwb   = row_lwb;
   this->fNelems   = this->fNrows*this->fNcols;
   fElements = data;
   this->fIsOwner  = kFALSE;

   return *this;
}

//______________________________________________________________________________
template<class Element>
TMatrixTSym<Element> &TMatrixTSym<Element>::GetSub(Int_t row_lwb,Int_t row_upb,
                                                   TMatrixTSym<Element> &target,Option_t *option) const
{
  // Get submatrix [row_lwb..row_upb][row_lwb..row_upb]; The indexing range of the
  // returned matrix depends on the argument option:
  //
  // option == "S" : return [0..row_upb-row_lwb+1][0..row_upb-row_lwb+1] (default)
  // else          : return [row_lwb..row_upb][row_lwb..row_upb]

   if (gMatrixCheck) {
      R__ASSERT(this->IsValid());
      if (row_lwb < this->fRowLwb || row_lwb > this->fRowLwb+this->fNrows-1) {
         Error("GetSub","row_lwb out of bounds");
         return target;
      }
      if (row_upb < this->fRowLwb || row_upb > this->fRowLwb+this->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,row_lwb_sub,row_upb_sub);
   const Int_t nrows_sub = row_upb_sub-row_lwb_sub+1;

   if (target.GetRowIndexArray() && target.GetColIndexArray()) {
      for (Int_t irow = 0; irow < nrows_sub; irow++) {
         for (Int_t icol = 0; icol < nrows_sub; icol++) {
            target(irow+row_lwb_sub,icol+row_lwb_sub) = (*this)(row_lwb+irow,row_lwb+icol);
         }
      }
   } else {
      const Element *ap = this->GetMatrixArray()+(row_lwb-this->fRowLwb)*this->fNrows+(row_lwb-this->fRowLwb);
            Element *bp = target.GetMatrixArray();

      for (Int_t irow = 0; irow < nrows_sub; irow++) {
         const Element *ap_sub = ap;
         for (Int_t icol = 0; icol < nrows_sub; icol++) {
            *bp++ = *ap_sub++;
         }
         ap += this->fNrows;
      }
   }

   return target;
}

//______________________________________________________________________________
template<class Element>
TMatrixTBase<Element> &TMatrixTSym<Element>::GetSub(Int_t row_lwb,Int_t row_upb,Int_t col_lwb,Int_t col_upb,
                                                    TMatrixTBase<Element> &target,Option_t *option) const
{
  // Get submatrix [row_lwb..row_upb][col_lwb..col_upb]; The indexing range of the
  // returned matrix depends on the argument option:
  //
  // option == "S" : return [0..row_upb-row_lwb+1][0..col_upb-col_lwb+1] (default)
  // else          : return [row_lwb..row_upb][col_lwb..col_upb]

   if (gMatrixCheck) {
      R__ASSERT(this->IsValid());
      if (row_lwb < this->fRowLwb || row_lwb > this->fRowLwb+this->fNrows-1) {
         Error("GetSub","row_lwb out of bounds");
         return target;
      }
      if (col_lwb < this->fColLwb || col_lwb > this->fColLwb+this->fNcols-1) {
         Error("GetSub","col_lwb out of bounds");
         return target;
      }
      if (row_upb < this->fRowLwb || row_upb > this->fRowLwb+this->fNrows-1) {
         Error("GetSub","row_upb out of bounds");
         return target;
      }
      if (col_upb < this->fColLwb || col_upb > this->fColLwb+this->fNcols-1) {
         Error("GetSub","col_upb out of bounds");
         return target;
      }
      if (row_upb < row_lwb || col_upb < col_lwb) {
         Error("GetSub","row_upb < row_lwb || col_upb < col_lwb");
         return target;
      }
   }

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

   const Int_t row_lwb_sub = (shift) ? 0               : row_lwb;
   const Int_t row_upb_sub = (shift) ? row_upb-row_lwb : row_upb;
   const Int_t col_lwb_sub = (shift) ? 0               : col_lwb;
   const Int_t col_upb_sub = (shift) ? col_upb-col_lwb : col_upb;

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

   if (target.GetRowIndexArray() && target.GetColIndexArray()) {
      for (Int_t irow = 0; irow < nrows_sub; irow++) {
         for (Int_t icol = 0; icol < ncols_sub; icol++) {
            target(irow+row_lwb_sub,icol+col_lwb_sub) = (*this)(row_lwb+irow,col_lwb+icol);
         }
      }
   } else {
      const Element *ap = this->GetMatrixArray()+(row_lwb-this->fRowLwb)*this->fNcols+(col_lwb-this->fColLwb);
            Element *bp = target.GetMatrixArray();

      for (Int_t irow = 0; irow < nrows_sub; irow++) {
         const Element *ap_sub = ap;
         for (Int_t icol = 0; icol < ncols_sub; icol++) {
            *bp++ = *ap_sub++;
         }
         ap += this->fNcols;
      }
   }

   return target;
}

//______________________________________________________________________________
template<class Element>
TMatrixTSym<Element> &TMatrixTSym<Element>::SetSub(Int_t row_lwb,const TMatrixTBase<Element> &source)
{
  // Insert matrix source starting at [row_lwb][row_lwb], thereby overwriting the part
  // [row_lwb..row_lwb+nrows_source][row_lwb..row_lwb+nrows_source];

   if (gMatrixCheck) {
      R__ASSERT(this->IsValid());
      R__ASSERT(source.IsValid());

      if (!source.IsSymmetric()) {
         Error("SetSub","source matrix is not symmetric");
         return *this;
      }
      if (row_lwb < this->fRowLwb || row_lwb > this->fRowLwb+this->fNrows-1) {
         Error("SetSub","row_lwb outof bounds");
         return *this;
      }
      if (row_lwb+source.GetNrows() > this->fRowLwb+this->fNrows) {
         Error("SetSub","source matrix too large");
         return *this;
      }
   }

   const Int_t nRows_source = source.GetNrows();

   if (source.GetRowIndexArray() && source.GetColIndexArray()) {
      const Int_t rowlwb_s = source.GetRowLwb();
      for (Int_t irow = 0; irow < nRows_source; irow++) {
         for (Int_t icol = 0; icol < nRows_source; icol++) {
            (*this)(row_lwb+irow,row_lwb+icol) = source(rowlwb_s+irow,rowlwb_s+icol);
         }
      }
   } else {
      const Element *bp = source.GetMatrixArray();
            Element *ap = this->GetMatrixArray()+(row_lwb-this->fRowLwb)*this->fNrows+(row_lwb-this->fRowLwb);

      for (Int_t irow = 0; irow < nRows_source; irow++) {
         Element *ap_sub = ap;
         for (Int_t icol = 0; icol < nRows_source; icol++) {
            *ap_sub++ = *bp++;
         }
         ap += this->fNrows;
      }
   }

   return *this;
}

//______________________________________________________________________________
template<class Element>
TMatrixTBase<Element> &TMatrixTSym<Element>::SetSub(Int_t row_lwb,Int_t col_lwb,const TMatrixTBase<Element> &source)
{
  // Insert matrix source starting at [row_lwb][col_lwb] in a symmetric fashion, thereby overwriting the part
  // [row_lwb..row_lwb+nrows_source][row_lwb..row_lwb+nrows_source];

   if (gMatrixCheck) {
      R__ASSERT(this->IsValid());
      R__ASSERT(source.IsValid());

      if (row_lwb < this->fRowLwb || row_lwb > this->fRowLwb+this->fNrows-1) {
         Error("SetSub","row_lwb out of bounds");
         return *this;
      }
      if (col_lwb < this->fColLwb || col_lwb > this->fColLwb+this->fNcols-1) {
         Error("SetSub","col_lwb out of bounds");
         return *this;
      }

      if (row_lwb+source.GetNrows() > this->fRowLwb+this->fNrows || col_lwb+source.GetNcols() > this->fRowLwb+this->fNrows) {
         Error("SetSub","source matrix too large");
         return *this;
      }
      if (col_lwb+source.GetNcols() > this->fRowLwb+this->fNrows || row_lwb+source.GetNrows() > this->fRowLwb+this->fNrows) {
         Error("SetSub","source matrix too large");
         return *this;
      }
   }

   const Int_t nRows_source = source.GetNrows();
   const Int_t nCols_source = source.GetNcols();

   const Int_t rowlwb_s = source.GetRowLwb();
   const Int_t collwb_s = source.GetColLwb();
   if (row_lwb >= col_lwb) {
      // lower triangle
      Int_t irow;
      for (irow = 0; irow < nRows_source; irow++) {
         for (Int_t icol = 0; col_lwb+icol <= row_lwb+irow &&
                                icol < nCols_source; icol++) {
            (*this)(row_lwb+irow,col_lwb+icol) = source(irow+rowlwb_s,icol+collwb_s);
         }
      }

      // upper triangle
      for (irow = 0; irow < nCols_source; irow++) {
         for (Int_t icol = nRows_source-1; row_lwb+icol > irow+col_lwb &&
                                 icol >= 0; icol--) {
            (*this)(col_lwb+irow,row_lwb+icol) = source(icol+rowlwb_s,irow+collwb_s);
         }
      }
   } else {

   }

   return *this;
}

//______________________________________________________________________________
template<class Element>
TMatrixTBase<Element> &TMatrixTSym<Element>::SetMatrixArray(const Element *data,Option_t *option)
{
   TMatrixTBase<Element>::SetMatrixArray(data,option);
   if (!this->IsSymmetric()) {
      Error("SetMatrixArray","Matrix is not symmetric after Set");
   }

   return *this;
}

//______________________________________________________________________________
template<class Element>
TMatrixTBase<Element> &TMatrixTSym<Element>::Shift(Int_t row_shift,Int_t col_shift)
{
   if (row_shift != col_shift) {
      Error("Shift","row_shift != col_shift");
      return *this;
   }
   return TMatrixTBase<Element>::Shift(row_shift,col_shift);
}

//______________________________________________________________________________
template<class Element>
TMatrixTBase<Element> &TMatrixTSym<Element>::ResizeTo(Int_t nrows,Int_t ncols,Int_t /*nr_nonzeros*/)
{
  // Set size of the matrix to nrows x ncols
  // New dynamic elements are created, the overlapping part of the old ones are
  // copied to the new structures, then the old elements are deleted.

   R__ASSERT(this->IsValid());
   if (!this->fIsOwner) {
      Error("ResizeTo(Int_t,Int_t)","Not owner of data array,cannot resize");
      return *this;
   }

   if (nrows != ncols) {
      Error("ResizeTo(Int_t,Int_t)","nrows != ncols");
      return *this;
   }

   if (this->fNelems > 0) {
      if (this->fNrows == nrows && this->fNcols == ncols)
         return *this;
      else if (nrows == 0 || ncols == 0) {
         this->fNrows = nrows; this->fNcols = ncols;
         this->Clear();
         return *this;
      }

      Element     *elements_old = GetMatrixArray();
      const Int_t  nelems_old   = this->fNelems;
      const Int_t  nrows_old    = this->fNrows;
      const Int_t  ncols_old    = this->fNcols;

      Allocate(nrows,ncols);
      R__ASSERT(this->IsValid());

      Element *elements_new = GetMatrixArray();
      // new memory should be initialized but be careful ot to wipe out the stack
      // storage. Initialize all when old or new storage was on the heap
      if (this->fNelems > this->kSizeMax || nelems_old > this->kSizeMax)
         memset(elements_new,0,this->fNelems*sizeof(Element));
      else if (this->fNelems > nelems_old)
         memset(elements_new+nelems_old,0,(this->fNelems-nelems_old)*sizeof(Element));

      // Copy overlap
      const Int_t ncols_copy = TMath::Min(this->fNcols,ncols_old);
      const Int_t nrows_copy = TMath::Min(this->fNrows,nrows_old);

      const Int_t nelems_new = this->fNelems;
      if (ncols_old < this->fNcols) {
         for (Int_t i = nrows_copy-1; i >= 0; i--)
           Memcpy_m(elements_new+i*this->fNcols,elements_old+i*ncols_old,ncols_copy,
                    nelems_new,nelems_old);
      } else {
         for (Int_t i = 0; i < nrows_copy; i++)
            Memcpy_m(elements_new+i*this->fNcols,elements_old+i*ncols_old,ncols_copy,
                     nelems_new,nelems_old);
      }

      Delete_m(nelems_old,elements_old);
   } else {
     Allocate(nrows,ncols,0,0,1);
   }

   return *this;
}

//______________________________________________________________________________
template<class Element>
TMatrixTBase<Element> &TMatrixTSym<Element>::ResizeTo(Int_t row_lwb,Int_t row_upb,Int_t col_lwb,Int_t col_upb,
                                                      Int_t /*nr_nonzeros*/)
{
  // Set size of the matrix to [row_lwb:row_upb] x [col_lwb:col_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.

   R__ASSERT(this->IsValid());
   if (!this->fIsOwner) {
      Error("ResizeTo(Int_t,Int_t,Int_t,Int_t)","Not owner of data array,cannot resize");
      return *this;
   }

   if (row_lwb != col_lwb) {
      Error("ResizeTo(Int_t,Int_t,Int_t,Int_t)","row_lwb != col_lwb");
      return *this;
   }
   if (row_upb != col_upb) {
      Error("ResizeTo(Int_t,Int_t,Int_t,Int_t)","row_upb != col_upb");
      return *this;
   }

   const Int_t new_nrows = row_upb-row_lwb+1;
   const Int_t new_ncols = col_upb-col_lwb+1;

   if (this->fNelems > 0) {

      if (this->fNrows  == new_nrows  && this->fNcols  == new_ncols &&
          this->fRowLwb == row_lwb    && this->fColLwb == col_lwb)
          return *this;
      else if (new_nrows == 0 || new_ncols == 0) {
         this->fNrows = new_nrows; this->fNcols = new_ncols;
         this->fRowLwb = row_lwb; this->fColLwb = col_lwb;
         this->Clear();
         return *this;
      }

      Element    *elements_old = GetMatrixArray();
      const Int_t  nelems_old   = this->fNelems;
      const Int_t  nrows_old    = this->fNrows;
      const Int_t  ncols_old    = this->fNcols;
      const Int_t  rowLwb_old   = this->fRowLwb;
      const Int_t  colLwb_old   = this->fColLwb;

      Allocate(new_nrows,new_ncols,row_lwb,col_lwb);
      R__ASSERT(this->IsValid());

      Element *elements_new = GetMatrixArray();
      // new memory should be initialized but be careful ot to wipe out the stack
      // storage. Initialize all when old or new storag ewas on the heap
      if (this->fNelems > this->kSizeMax || nelems_old > this->kSizeMax)
         memset(elements_new,0,this->fNelems*sizeof(Element));
      else if (this->fNelems > nelems_old)
         memset(elements_new+nelems_old,0,(this->fNelems-nelems_old)*sizeof(Element));

      // Copy overlap
      const Int_t rowLwb_copy = TMath::Max(this->fRowLwb,rowLwb_old);
      const Int_t colLwb_copy = TMath::Max(this->fColLwb,colLwb_old);
      const Int_t rowUpb_copy = TMath::Min(this->fRowLwb+this->fNrows-1,rowLwb_old+nrows_old-1);
      const Int_t colUpb_copy = TMath::Min(this->fColLwb+this->fNcols-1,colLwb_old+ncols_old-1);

      const Int_t nrows_copy = rowUpb_copy-rowLwb_copy+1;
      const Int_t ncols_copy = colUpb_copy-colLwb_copy+1;

      if (nrows_copy > 0 && ncols_copy > 0) {
         const Int_t colOldOff = colLwb_copy-colLwb_old;
         const Int_t colNewOff = colLwb_copy-this->fColLwb;
         if (ncols_old < this->fNcols) {
            for (Int_t i = nrows_copy-1; i >= 0; i--) {
               const Int_t iRowOld = rowLwb_copy+i-rowLwb_old;
               const Int_t iRowNew = rowLwb_copy+i-this->fRowLwb;
               Memcpy_m(elements_new+iRowNew*this->fNcols+colNewOff,
                        elements_old+iRowOld*ncols_old+colOldOff,ncols_copy,this->fNelems,nelems_old);
            }
         } else {
             for (Int_t i = 0; i < nrows_copy; i++) {
                const Int_t iRowOld = rowLwb_copy+i-rowLwb_old;
                const Int_t iRowNew = rowLwb_copy+i-this->fRowLwb;
                Memcpy_m(elements_new+iRowNew*this->fNcols+colNewOff,
                         elements_old+iRowOld*ncols_old+colOldOff,ncols_copy,this->fNelems,nelems_old);
            }
         }
      }

      Delete_m(nelems_old,elements_old);
   } else {
      Allocate(new_nrows,new_ncols,row_lwb,col_lwb,1);
   }

   return *this;
}

//______________________________________________________________________________
template<class Element>
Double_t TMatrixTSym<Element>::Determinant() const
{
   const TMatrixT<Element> &tmp = *this;
   TDecompLU lu(tmp,this->fTol);
   Double_t d1,d2;
   lu.Det(d1,d2);
   return d1*TMath::Power(2.0,d2);
}

//______________________________________________________________________________
template<class Element>
void TMatrixTSym<Element>::Determinant(Double_t &d1,Double_t &d2) const
{
   const TMatrixT<Element> &tmp = *this;
   TDecompLU lu(tmp,this->fTol);
   lu.Det(d1,d2);
}

//______________________________________________________________________________
template<class Element>
TMatrixTSym<Element> &TMatrixTSym<Element>::Invert(Double_t *det)
{
  // Invert the matrix and calculate its determinant
  // Notice that the LU decomposition is used instead of Bunch-Kaufman
  // Bunch-Kaufman guarantees a symmetric inverted matrix but is slower than LU .
  // The user can access Bunch-Kaufman through the TDecompBK class .

   R__ASSERT(this->IsValid());
   TMatrixD tmp(*this);
   if (TDecompLU::InvertLU(tmp,Double_t(this->fTol),det)) {
      const Double_t *p1 = tmp.GetMatrixArray();
            Element  *p2 = this->GetMatrixArray();
      for (Int_t i = 0; i < this->GetNoElements(); i++)
         p2[i] = p1[i];
   }

   return *this;
}

//______________________________________________________________________________
template<class Element>
TMatrixTSym<Element> &TMatrixTSym<Element>::InvertFast(Double_t *det)
{
  // Invert the matrix and calculate its determinant

   R__ASSERT(this->IsValid());

   const Char_t nRows = Char_t(this->GetNrows());
   switch (nRows) {
      case 1:
      {
         Element *pM = this->GetMatrixArray();
         if (*pM == 0.) {
            Error("InvertFast","matrix is singular");
            *det = 0;
         } else {
            *det = *pM;
            *pM = 1.0/(*pM);
         }
         return *this;
      }
      case 2:
      {
         TMatrixTSymCramerInv::Inv2x2<Element>(*this,det);
         return *this;
      }
      case 3:
      {
         TMatrixTSymCramerInv::Inv3x3<Element>(*this,det);
         return *this;
      }
      case 4:
      {
         TMatrixTSymCramerInv::Inv4x4<Element>(*this,det);
         return *this;
      }
      case 5:
      {
         TMatrixTSymCramerInv::Inv5x5<Element>(*this,det);
         return *this;
      }
      case 6:
      {
         TMatrixTSymCramerInv::Inv6x6<Element>(*this,det);
         return *this;
      }

      default:
      {
         TMatrixD tmp(*this);
         if (TDecompLU::InvertLU(tmp,Double_t(this->fTol),det)) {
            const Double_t *p1 = tmp.GetMatrixArray();
                  Element  *p2 = this->GetMatrixArray();
            for (Int_t i = 0; i < this->GetNoElements(); i++)
               p2[i] = p1[i];
         }
         return *this;
      }
   }
}

//______________________________________________________________________________
template<class Element>
TMatrixTSym<Element> &TMatrixTSym<Element>::Transpose(const TMatrixTSym<Element> &source)
{
  // Transpose a matrix.

   if (gMatrixCheck) {
      R__ASSERT(this->IsValid());
      R__ASSERT(source.IsValid());

      if (this->fNrows != source.GetNcols() || this->fRowLwb != source.GetColLwb())
      {
         Error("Transpose","matrix has wrong shape");
         return *this;
      }
   }

   *this = source;
   return *this;
}

//______________________________________________________________________________
template<class Element>
TMatrixTSym<Element> &TMatrixTSym<Element>::Rank1Update(const TVectorT<Element> &v,Element alpha)
{
  // Perform a rank 1 operation on the matrix:
  //     A += alpha * v * v^T

   if (gMatrixCheck) {
      R__ASSERT(this->IsValid());
      R__ASSERT(v.IsValid());
      if (v.GetNoElements() < this->fNrows) {
         Error("Rank1Update","vector too short");
         return *this;
      }
   }

   const Element * const pv = v.GetMatrixArray();
         Element *trp = this->GetMatrixArray(); // pointer to UR part and diagonal, traverse row-wise
         Element *tcp = trp;                    // pointer to LL part,              traverse col-wise
   for (Int_t i = 0; i < this->fNrows; i++) {
      trp += i;               // point to [i,i]
      tcp += i*this->fNcols;  // point to [i,i]
      const Element tmp = alpha*pv[i];
      for (Int_t j = i; j < this->fNcols; j++) {
         if (j > i) *tcp += tmp*pv[j];
         *trp++ += tmp*pv[j];
         tcp += this->fNcols;
      }
      tcp -= this->fNelems-1; // point to [0,i]
   }

   return *this;
}

//______________________________________________________________________________
template<class Element>
TMatrixTSym<Element> &TMatrixTSym<Element>::Similarity(const TMatrixT<Element> &b)
{
// Calculate B * (*this) * B^T , final matrix will be (nrowsb x nrowsb)
// This is a similarity transform when B is orthogonal . It is more
// efficient than applying the actual multiplication because this
// routine realizes that  the final matrix is symmetric .

   if (gMatrixCheck) {
      R__ASSERT(this->IsValid());
      R__ASSERT(b.IsValid());
      if (this->fNcols != b.GetNcols() || this->fColLwb != b.GetColLwb()) {
         Error("Similarity(const TMatrixT &)","matrices incompatible");
         return *this;
      }
   }

   const Int_t ncolsa  = this->fNcols;
   const Int_t nb      = b.GetNoElements();
   const Int_t nrowsb  = b.GetNrows();
   const Int_t ncolsb  = b.GetNcols();

   const Element * const bp = b.GetMatrixArray();

   Element work[kWorkMax];
   Bool_t isAllocated = kFALSE;
   Element *bap = work;
   if (nrowsb*ncolsa > kWorkMax) {
      isAllocated = kTRUE;
      bap = new Element[nrowsb*ncolsa];
   }

   AMultB(bp,nb,ncolsb,this->fElements,this->fNelems,this->fNcols,bap);

   if (nrowsb != this->fNrows)
      this->ResizeTo(nrowsb,nrowsb);

#ifdef CBLAS
         Element *cp = this->GetMatrixArray();
   if (typeid(Element) == typeid(Double_t))
      cblas_dgemm (CblasRowMajor,CblasNoTrans,CblasTrans,this->fNrows,this->fNcols,ba.GetNcols(),
                   1.0,bap,ba.GetNcols(),bp,b.GetNcols(),1.0,cp,this->fNcols);
   else if (typeid(Element) != typeid(Float_t))
      cblas_sgemm (CblasRowMajor,CblasNoTrans,CblasTrans,this->fNrows,this->fNcols,ba.GetNcols(),
                   1.0,bap,ba.GetNcols(),bp,b.GetNcols(),1.0,cp,this->fNcols);
   else
      Error("Similarity","type %s not implemented in BLAS library",typeid(Element));
#else
   const Int_t nba     = nrowsb*ncolsa;
   const Int_t ncolsba = ncolsa;
   const Element *       bi1p = bp;
         Element *       cp   = this->GetMatrixArray();
         Element * const cp0  = cp;

   Int_t ishift = 0;
   const Element *barp0 = bap;
   while (barp0 < bap+nba) {
      const Element *brp0 = bi1p;
      while (brp0 < bp+nb) {
         const Element *barp = barp0;
         const Element *brp  = brp0;
         Element cij = 0;
         while (brp < brp0+ncolsb)
            cij += *barp++ * *brp++;
         *cp++ = cij;
         brp0 += ncolsb;
      }
      barp0 += ncolsba;
      bi1p += ncolsb;
      cp += ++ishift;
   }

   R__ASSERT(cp == cp0+this->fNelems+ishift && barp0 == bap+nba);

   cp = cp0;
   for (Int_t irow = 0; irow < this->fNrows; irow++) {
      const Int_t rowOff1 = irow*this->fNrows;
      for (Int_t icol = 0; icol < irow; icol++) {
         const Int_t rowOff2 = icol*this->fNrows;
         cp[rowOff1+icol] = cp[rowOff2+irow];
      }
   }
#endif

   if (isAllocated)
      delete [] bap;

   return *this;
}

//______________________________________________________________________________
template<class Element>
TMatrixTSym<Element> &TMatrixTSym<Element>::Similarity(const TMatrixTSym<Element> &b)
{
// Calculate B * (*this) * B^T , final matrix will be (nrowsb x nrowsb)
// This is a similarity transform when B is orthogonal . It is more
// efficient than applying the actual multiplication because this
// routine realizes that  the final matrix is symmetric .

   if (gMatrixCheck) {
      R__ASSERT(this->IsValid());
      R__ASSERT(b.IsValid());
      if (this->fNcols != b.GetNcols() || this->fColLwb != b.GetColLwb()) {
         Error("Similarity(const TMatrixTSym &)","matrices incompatible");
         return *this;
      }
   }

#ifdef CBLAS
   const Int_t nrowsb = b.GetNrows();
   const Int_t ncolsa = this->GetNcols();

   Element work[kWorkMax];
   Bool_t isAllocated = kFALSE;
   Element *abtp = work;
   if (this->fNcols > kWorkMax) {
      isAllocated = kTRUE;
      abtp = new Element[this->fNcols];
   }

   const TMatrixT<Element> abt(*this,TMatrixT<Element>::kMultTranspose,b);

   const Element *bp = b.GetMatrixArray();
         Element *cp = this->GetMatrixArray();
   if (typeid(Element) == typeid(Double_t))
      cblas_dsymm (CblasRowMajor,CblasLeft,CblasUpper,this->fNrows,this->fNcols,1.0,
                   bp,b.GetNcols(),abtp,abt.GetNcols(),0.0,cp,this->fNcols);
   else if (typeid(Element) != typeid(Float_t))
      cblas_ssymm (CblasRowMajor,CblasLeft,CblasUpper,this->fNrows,this->fNcols,1.0,
                   bp,b.GetNcols(),abtp,abt.GetNcols(),0.0,cp,this->fNcols);
   else
      Error("Similarity","type %s not implemented in BLAS library",typeid(Element));

   if (isAllocated)
      delete [] abtp;
#else
   const Int_t ncolsa = this->GetNcols();
   const Int_t nb     = b.GetNoElements();
   const Int_t nrowsb = b.GetNrows();
   const Int_t ncolsb = b.GetNcols();

   const Element * const bp = b.GetMatrixArray();

   Element work[kWorkMax];
   Bool_t isAllocated = kFALSE;
   Element *bap = work;
   if (nrowsb*ncolsa > kWorkMax) {
      isAllocated = kTRUE;
      bap = new Element[nrowsb*ncolsa];
   }

   AMultB(bp,nb,ncolsb,this->fElements,this->fNelems,this->fNcols,bap);

   const Int_t nba     = nrowsb*ncolsa;
   const Int_t ncolsba = ncolsa;
   const Element *       bi1p = bp;
         Element *       cp   = this->GetMatrixArray();
         Element * const cp0  = cp;

   Int_t ishift = 0;
   const Element *barp0 = bap;
   while (barp0 < bap+nba) {
      const Element *brp0 = bi1p;
      while (brp0 < bp+nb) {
         const Element *barp = barp0;
         const Element *brp  = brp0;
         Element cij = 0;
         while (brp < brp0+ncolsb)
            cij += *barp++ * *brp++;
         *cp++ = cij;
         brp0 += ncolsb;
      }
      barp0 += ncolsba;
      bi1p += ncolsb;
      cp += ++ishift;
   }

   R__ASSERT(cp == cp0+this->fNelems+ishift && barp0 == bap+nba);

   cp = cp0;
   for (Int_t irow = 0; irow < this->fNrows; irow++) {
      const Int_t rowOff1 = irow*this->fNrows;
      for (Int_t icol = 0; icol < irow; icol++) {
         const Int_t rowOff2 = icol*this->fNrows;
         cp[rowOff1+icol] = cp[rowOff2+irow];
      }
   }

   if (isAllocated)
      delete [] bap;
#endif

   return *this;
}

//______________________________________________________________________________
template<class Element>
Element TMatrixTSym<Element>::Similarity(const TVectorT<Element> &v) const
{
// Calculate scalar v * (*this) * v^T

   if (gMatrixCheck) {
      R__ASSERT(this->IsValid());
      R__ASSERT(v.IsValid());
      if (this->fNcols != v.GetNrows() || this->fColLwb != v.GetLwb()) {
         Error("Similarity(const TVectorT &)","vector and matrix incompatible");
         return -1.;
      }
   }

   const Element *mp = this->GetMatrixArray(); // Matrix row ptr
   const Element *vp = v.GetMatrixArray();     // vector ptr

   Element sum1 = 0;
   const Element * const vp_first = vp;
   const Element * const vp_last  = vp+v.GetNrows();
   while (vp < vp_last) {
      Element sum2 = 0;
      for (const Element *sp = vp_first; sp < vp_last; )
         sum2 += *mp++ * *sp++;
      sum1 += sum2 * *vp++;
   }

   R__ASSERT(mp == this->GetMatrixArray()+this->GetNoElements());

   return sum1;
}

//______________________________________________________________________________
template<class Element>
TMatrixTSym<Element> &TMatrixTSym<Element>::SimilarityT(const TMatrixT<Element> &b)
{
// Calculate B^T * (*this) * B , final matrix will be (ncolsb x ncolsb)
// It is more efficient than applying the actual multiplication because this
// routine realizes that  the final matrix is symmetric .

  if (gMatrixCheck) {
     R__ASSERT(this->IsValid());
     R__ASSERT(b.IsValid());
     if (this->fNrows != b.GetNrows() || this->fRowLwb != b.GetRowLwb()) {
        Error("SimilarityT(const TMatrixT &)","matrices incompatible");
        return *this;
     }
  }

   const Int_t ncolsb = b.GetNcols();
   const Int_t ncolsa = this->GetNcols();

   Element work[kWorkMax];
   Bool_t isAllocated = kFALSE;
   Element *btap = work;
   if (ncolsb*ncolsa > kWorkMax) {
      isAllocated = kTRUE;
      btap = new Element[ncolsb*ncolsa];
   }

   TMatrixT<Element> bta; bta.Use(ncolsb,ncolsa,btap);
   bta.TMult(b,*this);

   if (ncolsb != this->fNcols)
      this->ResizeTo(ncolsb,ncolsb);

#ifdef CBLAS
   const Element *bp = b.GetMatrixArray();
         Element *cp = this->GetMatrixArray();
   if (typeid(Element) == typeid(Double_t))
      cblas_dgemm (CblasRowMajor,CblasNoTrans,CblasNoTrans,this->fNrows,this->fNcols,bta.GetNcols(),
                   1.0,btap,bta.GetNcols(),bp,b.GetNcols(),1.0,cp,this->fNcols);
   else if (typeid(Element) != typeid(Float_t))
      cblas_sgemm (CblasRowMajor,CblasNoTrans,CblasNoTrans,this->fNrows,this->fNcols,bta.GetNcols(),
                   1.0,btap,bta.GetNcols(),bp,b.GetNcols(),1.0,cp,this->fNcols);
   else
      Error("similarityT","type %s not implemented in BLAS library",typeid(Element));
#else
   const Int_t nbta     = bta.GetNoElements();
   const Int_t nb       = b.GetNoElements();
   const Int_t ncolsbta = bta.GetNcols();
   const Element * const bp   = b.GetMatrixArray();
         Element *       cp   = this->GetMatrixArray();
         Element * const cp0  = cp;

   Int_t ishift = 0;
   const Element *btarp0 = btap;                     // Pointer to  A[i,0];
   const Element *bcp0   = bp;
   while (btarp0 < btap+nbta) {
      for (const Element *bcp = bcp0; bcp < bp+ncolsb; ) { // Pointer to the j-th column of B, Start bcp = B[0,0]
         const Element *btarp = btarp0;                   // Pointer to the i-th row of A, reset to A[i,0]
         Element cij = 0;
         while (bcp < bp+nb) {                         // Scan the i-th row of A and
            cij += *btarp++ * *bcp;                     // the j-th col of B
            bcp += ncolsb;
         }
         *cp++ = cij;
         bcp -= nb-1;                                  // Set bcp to the (j+1)-th col
      }
      btarp0 += ncolsbta;                             // Set ap to the (i+1)-th row
      bcp0++;
      cp += ++ishift;
   }

   R__ASSERT(cp == cp0+this->fNelems+ishift && btarp0 == btap+nbta);

   cp = cp0;
   for (Int_t irow = 0; irow < this->fNrows; irow++) {
      const Int_t rowOff1 = irow*this->fNrows;
      for (Int_t icol = 0; icol < irow; icol++) {
         const Int_t rowOff2 = icol*this->fNrows;
         cp[rowOff1+icol] = cp[rowOff2+irow];
      }
   }
#endif

   if (isAllocated)
      delete [] btap;

   return *this;
}

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

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

//______________________________________________________________________________
template<class Element>
TMatrixTSym<Element> &TMatrixTSym<Element>::operator=(const TMatrixTSymLazy<Element> &lazy_constructor)
{
   R__ASSERT(this->IsValid());

   if (lazy_constructor.fRowUpb != this->GetRowUpb() ||
       lazy_constructor.fRowLwb != this->GetRowLwb()) {
       Error("operator=(const TMatrixTSymLazy&)", "matrix is incompatible with "
             "the assigned Lazy matrix");
      return *this;
   }

   lazy_constructor.FillIn(*this);
   return *this;
}

//______________________________________________________________________________
template<class Element>
TMatrixTSym<Element> &TMatrixTSym<Element>::operator=(Element val)
{
  // Assign val to every element of the matrix.

   R__ASSERT(this->IsValid());

   Element *ep = fElements;
   const Element * const ep_last = ep+this->fNelems;
   while (ep < ep_last)
      *ep++ = val;

   return *this;
}

//______________________________________________________________________________
template<class Element>
TMatrixTSym<Element> &TMatrixTSym<Element>::operator+=(Element val)
{
  // Add val to every element of the matrix.

   R__ASSERT(this->IsValid());

   Element *ep = fElements;
   const Element * const ep_last = ep+this->fNelems;
   while (ep < ep_last)
      *ep++ += val;

   return *this;
}

//______________________________________________________________________________
template<class Element>
TMatrixTSym<Element> &TMatrixTSym<Element>::operator-=(Element val)
{
  // Subtract val from every element of the matrix.

   R__ASSERT(this->IsValid());

   Element *ep = fElements;
   const Element * const ep_last = ep+this->fNelems;
   while (ep < ep_last)
      *ep++ -= val;

   return *this;
}

//______________________________________________________________________________
template<class Element>
TMatrixTSym<Element> &TMatrixTSym<Element>::operator*=(Element val)
{
  // Multiply every element of the matrix with val.

   R__ASSERT(this->IsValid());

   Element *ep = fElements;
   const Element * const ep_last = ep+this->fNelems;
   while (ep < ep_last)
      *ep++ *= val;

   return *this;
}

//______________________________________________________________________________
template<class Element>
TMatrixTSym<Element> &TMatrixTSym<Element>::operator+=(const TMatrixTSym<Element> &source)
{
  // Add the source matrix.

   if (gMatrixCheck && !AreCompatible(*this,source)) {
      Error("operator+=","matrices not compatible");
      return *this;
   }

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

   return *this;
}

//______________________________________________________________________________
template<class Element>
TMatrixTSym<Element> &TMatrixTSym<Element>::operator-=(const TMatrixTSym<Element> &source)
{
  // Subtract the source matrix.

   if (gMatrixCheck && !AreCompatible(*this,source)) {
      Error("operator-=","matrices not compatible");
      return *this;
   }

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

   return *this;
}

//______________________________________________________________________________
template<class Element>
TMatrixTBase<Element> &TMatrixTSym<Element>::Apply(const TElementActionT<Element> &action)
{
   R__ASSERT(this->IsValid());

   Element val = 0;
   Element *trp = this->GetMatrixArray(); // pointer to UR part and diagonal, traverse row-wise
   Element *tcp = trp;                    // pointer to LL part,              traverse col-wise
   for (Int_t i = 0; i < this->fNrows; i++) {
      trp += i;               // point to [i,i]
      tcp += i*this->fNcols;  // point to [i,i]
      for (Int_t j = i; j < this->fNcols; j++) {
         action.Operation(val);
         if (j > i) *tcp = val;
         *trp++ = val;
         tcp += this->fNcols;
      }
      tcp -= this->fNelems-1; // point to [0,i]
   }

   return *this;
}

//______________________________________________________________________________
template<class Element>
TMatrixTBase<Element> &TMatrixTSym<Element>::Apply(const TElementPosActionT<Element> &action)
{
  // Apply action to each element of the matrix. To action the location
  // of the current element is passed.

   R__ASSERT(this->IsValid());

   Element val = 0;
   Element *trp = this->GetMatrixArray(); // pointer to UR part and diagonal, traverse row-wise
   Element *tcp = trp;                    // pointer to LL part,              traverse col-wise
   for (Int_t i = 0; i < this->fNrows; i++) {
      action.fI = i+this->fRowLwb;
      trp += i;               // point to [i,i]
      tcp += i*this->fNcols;  // point to [i,i]
      for (Int_t j = i; j < this->fNcols; j++) {
         action.fJ = j+this->fColLwb;
         action.Operation(val);
         if (j > i) *tcp = val;
         *trp++ = val;
         tcp += this->fNcols;
      }
      tcp -= this->fNelems-1; // point to [0,i]
   }

   return *this;
}

//______________________________________________________________________________
template<class Element>
TMatrixTBase<Element> &TMatrixTSym<Element>::Randomize(Element alpha,Element beta,Double_t &seed)
{
  // randomize matrix element values but keep matrix symmetric

   if (gMatrixCheck) {
      R__ASSERT(this->IsValid());
      if (this->fNrows != this->fNcols || this->fRowLwb != this->fColLwb) {
         Error("Randomize(Element,Element,Element &","matrix should be square");
         return *this;
      }
   }

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

   Element *ep = GetMatrixArray();
   for (Int_t i = 0; i < this->fNrows; i++) {
      const Int_t off = i*this->fNcols;
      for (Int_t j = 0; j <= i; j++) {
         ep[off+j] = scale*(Drand(seed)+shift);
         if (i != j) {
            ep[j*this->fNcols+i] = ep[off+j];
         }
      }
   }

   return *this;
}

//______________________________________________________________________________
template<class Element>
TMatrixTSym<Element> &TMatrixTSym<Element>::RandomizePD(Element alpha,Element beta,Double_t &seed)
{
  // randomize matrix element values but keep matrix symmetric positive definite

   if (gMatrixCheck) {
      R__ASSERT(this->IsValid());
      if (this->fNrows != this->fNcols || this->fRowLwb != this->fColLwb) {
         Error("RandomizeSym(Element,Element,Element &","matrix should be square");
         return *this;
      }
   }

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

   Element *ep = GetMatrixArray();
   Int_t i;
   for (i = 0; i < this->fNrows; i++) {
      const Int_t off = i*this->fNcols;
      for (Int_t j = 0; j <= i; j++)
         ep[off+j] = scale*(Drand(seed)+shift);
   }

   for (i = this->fNrows-1; i >= 0; i--) {
      const Int_t off1 = i*this->fNcols;
      for (Int_t j = i; j >= 0; j--) {
         const Int_t off2 = j*this->fNcols;
         ep[off1+j] *= ep[off2+j];
         for (Int_t k = j-1; k >= 0; k--) {
            ep[off1+j] += ep[off1+k]*ep[off2+k];
         }
         if (i != j)
            ep[off2+i] = ep[off1+j];
      }
   }

   return *this;
}

//______________________________________________________________________________
template<class Element>
const TMatrixT<Element> TMatrixTSym<Element>::EigenVectors(TVectorT<Element> &eigenValues) const
{
  // Return a matrix containing the eigen-vectors ordered by descending eigen-values.
  // For full functionality use TMatrixDSymEigen .

   TMatrixDSym tmp = *this;
   TMatrixDSymEigen eigen(tmp);
   eigenValues.ResizeTo(this->fNrows);
   eigenValues = eigen.GetEigenValues();
   return eigen.GetEigenVectors();
}

//______________________________________________________________________________
template<class Element>
Bool_t operator==(const TMatrixTSym<Element> &m1,const TMatrixTSym<Element> &m2)
{
  // Check to see if two matrices are identical.

   if (!AreCompatible(m1,m2)) return kFALSE;
   return (memcmp(m1.GetMatrixArray(),m2.GetMatrixArray(),
                  m1.GetNoElements()*sizeof(Element)) == 0);
}

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

//______________________________________________________________________________
template<class Element>
TMatrixTSym<Element> operator+(const TMatrixTSym<Element> &source1,Element val)
{
   TMatrixTSym<Element> target(source1);
   target += val;
   return target;
}

//______________________________________________________________________________
template<class Element>
TMatrixTSym<Element> operator+(Element val,const TMatrixTSym<Element> &source1)
{
   return operator+(source1,val);
}

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

//______________________________________________________________________________
template<class Element>
TMatrixTSym<Element> operator-(const TMatrixTSym<Element> &source1,Element val)
{
   TMatrixTSym<Element> target(source1);
   target -= val;
   return target;
}

//______________________________________________________________________________
template<class Element>
TMatrixTSym<Element> operator-(Element val,const TMatrixTSym<Element> &source1)
{
   return Element(-1.0)*operator-(source1,val);
}

//______________________________________________________________________________
template<class Element>
TMatrixTSym<Element> operator*(const TMatrixTSym<Element> &source1,Element val)
{
   TMatrixTSym<Element> target(source1);
   target *= val;
   return target;
}

//______________________________________________________________________________
template<class Element>
TMatrixTSym<Element> operator*(Element val,const TMatrixTSym<Element> &source1)
{
  return operator*(source1,val);
}

//______________________________________________________________________________
template<class Element>
TMatrixTSym<Element> operator&&(const TMatrixTSym<Element> &source1,const TMatrixTSym<Element> &source2)
{
  // Logical AND

   TMatrixTSym<Element> target;

   if (gMatrixCheck && !AreCompatible(source1,source2)) {
      Error("operator&&(const TMatrixTSym&,const TMatrixTSym&)","matrices not compatible");
      return target;
   }

   target.ResizeTo(source1);

   const Element *sp1 = source1.GetMatrixArray();
   const Element *sp2 = source2.GetMatrixArray();
         Element *tp  = target.GetMatrixArray();
   const Element * const tp_last = tp+target.GetNoElements();
   while (tp < tp_last)
      *tp++ = (*sp1++ != 0.0 && *sp2++ != 0.0);

   return target;
}

//______________________________________________________________________________
template<class Element>
TMatrixTSym<Element> operator||(const TMatrixTSym<Element> &source1,const TMatrixTSym<Element> &source2)
{
  // Logical Or

   TMatrixTSym<Element> target;

   if (gMatrixCheck && !AreCompatible(source1,source2)) {
      Error("operator||(const TMatrixTSym&,const TMatrixTSym&)","matrices not compatible");
      return target;
   }

   target.ResizeTo(source1);

   const Element *sp1 = source1.GetMatrixArray();
   const Element *sp2 = source2.GetMatrixArray();
         Element *tp  = target.GetMatrixArray();
   const Element * const tp_last = tp+target.GetNoElements();
   while (tp < tp_last)
      *tp++ = (*sp1++ != 0.0 || *sp2++ != 0.0);

   return target;
}

//______________________________________________________________________________
template<class Element>
TMatrixTSym<Element> operator>(const TMatrixTSym<Element> &source1,const TMatrixTSym<Element> &source2)
{
  // source1 > source2

  TMatrixTSym<Element> target;

   if (gMatrixCheck && !AreCompatible(source1,source2)) {
      Error("operator>(const TMatrixTSym&,const TMatrixTSym&)","matrices not compatible");
      return target;
   }

   target.ResizeTo(source1);

   const Element *sp1 = source1.GetMatrixArray();
   const Element *sp2 = source2.GetMatrixArray();
         Element *tp  = target.GetMatrixArray();
   const Element * const tp_last = tp+target.GetNoElements();
   while (tp < tp_last) {
      *tp++ = (*sp1) > (*sp2); sp1++; sp2++;
   }

   return target;
}

//______________________________________________________________________________
template<class Element>
TMatrixTSym<Element> operator>=(const TMatrixTSym<Element> &source1,const TMatrixTSym<Element> &source2)
{
  // source1 >= source2

   TMatrixTSym<Element> target;

   if (gMatrixCheck && !AreCompatible(source1,source2)) {
      Error("operator>=(const TMatrixTSym&,const TMatrixTSym&)","matrices not compatible");
      return target;
   }

   target.ResizeTo(source1);

   const Element *sp1 = source1.GetMatrixArray();
   const Element *sp2 = source2.GetMatrixArray();
         Element *tp  = target.GetMatrixArray();
   const Element * const tp_last = tp+target.GetNoElements();
   while (tp < tp_last) {
      *tp++ = (*sp1) >= (*sp2); sp1++; sp2++;
   }

   return target;
}

//______________________________________________________________________________
template<class Element>
TMatrixTSym<Element> operator<=(const TMatrixTSym<Element> &source1,const TMatrixTSym<Element> &source2)
{
  // source1 <= source2

   TMatrixTSym<Element> target;

   if (gMatrixCheck && !AreCompatible(source1,source2)) {
      Error("operator<=(const TMatrixTSym&,const TMatrixTSym&)","matrices not compatible");
      return target;
   }

   target.ResizeTo(source1);

   const Element *sp1 = source1.GetMatrixArray();
   const Element *sp2 = source2.GetMatrixArray();
         Element *tp  = target.GetMatrixArray();
   const Element * const tp_last = tp+target.GetNoElements();
   while (tp < tp_last) {
      *tp++ = (*sp1) <= (*sp2); sp1++; sp2++;
   }

   return target;
}

//______________________________________________________________________________
template<class Element>
TMatrixTSym<Element> operator<(const TMatrixTSym<Element> &source1,const TMatrixTSym<Element> &source2)
{
  // source1 < source2

  TMatrixTSym<Element> target;

   if (gMatrixCheck && !AreCompatible(source1,source2)) {
      Error("operator<(const TMatrixTSym&,const TMatrixTSym&)","matrices not compatible");
      return target;
   }

   target.ResizeTo(source1);

   const Element *sp1 = source1.GetMatrixArray();
   const Element *sp2 = source2.GetMatrixArray();
         Element *tp  = target.GetMatrixArray();
   const Element * const tp_last = tp+target.GetNoElements();
   while (tp < tp_last) {
      *tp++ = (*sp1) < (*sp2); sp1++; sp2++;
   }

   return target;
}

//______________________________________________________________________________
template<class Element>
TMatrixTSym<Element> &Add(TMatrixTSym<Element> &target,Element scalar,const TMatrixTSym<Element> &source)
{
   // Modify addition: target += scalar * source.

   if (gMatrixCheck && !AreCompatible(target,source)) {
      ::Error("Add","matrices not compatible");
      return target;
   }

   const Int_t nrows  = target.GetNrows();
   const Int_t ncols  = target.GetNcols();
   const Int_t nelems = target.GetNoElements();
   const Element *sp  = source.GetMatrixArray();
         Element *trp = target.GetMatrixArray(); // pointer to UR part and diagonal, traverse row-wise
         Element *tcp = target.GetMatrixArray(); // pointer to LL part,              traverse col-wise
   for (Int_t i = 0; i < nrows; i++) {
      sp  += i;
      trp += i;        // point to [i,i]
      tcp += i*ncols;  // point to [i,i]
      for (Int_t j = i; j < ncols; j++) {
         const Element tmp = scalar * *sp++;
         if (j > i) *tcp += tmp;
         *trp++ += tmp;
         tcp += ncols;
      }
      tcp -= nelems-1; // point to [0,i]
   }

   return target;
}

//______________________________________________________________________________
template<class Element>
TMatrixTSym<Element> &ElementMult(TMatrixTSym<Element> &target,const TMatrixTSym<Element> &source)
{
  // Multiply target by the source, element-by-element.

   if (gMatrixCheck && !AreCompatible(target,source)) {
      ::Error("ElementMult","matrices not compatible");
      return target;
   }

   const Int_t nrows  = target.GetNrows();
   const Int_t ncols  = target.GetNcols();
   const Int_t nelems = target.GetNoElements();
   const Element *sp  = source.GetMatrixArray();
         Element *trp = target.GetMatrixArray(); // pointer to UR part and diagonal, traverse row-wise
         Element *tcp = target.GetMatrixArray(); // pointer to LL part,              traverse col-wise
   for (Int_t i = 0; i < nrows; i++) {
      sp  += i;
      trp += i;        // point to [i,i]
      tcp += i*ncols;  // point to [i,i]
      for (Int_t j = i; j < ncols; j++) {
         if (j > i) *tcp *= *sp;
         *trp++ *= *sp++;
         tcp += ncols;
      }
      tcp -= nelems-1; // point to [0,i]
   }

   return target;
}

//______________________________________________________________________________
template<class Element>
TMatrixTSym<Element> &ElementDiv(TMatrixTSym<Element> &target,const TMatrixTSym<Element> &source)
{
  // Multiply target by the source, element-by-element.

   if (gMatrixCheck && !AreCompatible(target,source)) {
      ::Error("ElementDiv","matrices not compatible");
      return target;
   }

   const Int_t nrows  = target.GetNrows();
   const Int_t ncols  = target.GetNcols();
   const Int_t nelems = target.GetNoElements();
   const Element *sp  = source.GetMatrixArray();
         Element *trp = target.GetMatrixArray(); // pointer to UR part and diagonal, traverse row-wise
         Element *tcp = target.GetMatrixArray(); // pointer to LL part,              traverse col-wise
   for (Int_t i = 0; i < nrows; i++) {
      sp  += i;
      trp += i;        // point to [i,i]
      tcp += i*ncols;  // point to [i,i]
      for (Int_t j = i; j < ncols; j++) {
         if (*sp != 0.0) {
            if (j > i) *tcp /= *sp;
            *trp++ /= *sp++;
         } else {
            const Int_t irow = (sp-source.GetMatrixArray())/source.GetNcols();
            const Int_t icol = (sp-source.GetMatrixArray())%source.GetNcols();
            Error("ElementDiv","source (%d,%d) is zero",irow,icol);
            trp++;
         }
         tcp += ncols;
      }
      tcp -= nelems-1; // point to [0,i]
   }

   return target;
}

//______________________________________________________________________________
template<class Element>
void TMatrixTSym<Element>::Streamer(TBuffer &R__b)
{
  // Stream an object of class TMatrixTSym.

   if (R__b.IsReading()) {
      UInt_t R__s, R__c;
      Version_t R__v = R__b.ReadVersion(&R__s, &R__c);
      Clear();
      R__b.ReadClassBuffer(TMatrixTBase<Element>::Class(),this,R__v,R__s,R__c);
      fElements = new Element[this->fNelems];
      Int_t i;
      for (i = 0; i < this->fNrows; i++) {
         R__b.ReadFastArray(fElements+i*this->fNcols+i,this->fNcols-i);
      }
      // copy to Lower left triangle
      for (i = 0; i < this->fNrows; i++) {
         for (Int_t j = 0; j < i; j++) {
            fElements[i*this->fNcols+j] = fElements[j*this->fNrows+i];
        }
      }
      if (this->fNelems <= this->kSizeMax) {
         memcpy(fDataStack,fElements,this->fNelems*sizeof(Element));
         delete [] fElements;
         fElements = fDataStack;
      }
   } else {
      R__b.WriteClassBuffer(TMatrixTBase<Element>::Class(),this);
      // Only write the Upper right triangle
      for (Int_t i = 0; i < this->fNrows; i++) {
         R__b.WriteFastArray(fElements+i*this->fNcols+i,this->fNcols-i);
      }
   }
}

#ifndef ROOT_TMatrixFSymfwd
#include "TMatrixFSymfwd.h"
#endif

template class TMatrixTSym<Float_t>;

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

template TMatrixFSym &Add        <Float_t>(TMatrixFSym &target,      Float_t      scalar,const TMatrixFSym &source);
template TMatrixFSym &ElementMult<Float_t>(TMatrixFSym &target,const TMatrixFSym &source);
template TMatrixFSym &ElementDiv <Float_t>(TMatrixFSym &target,const TMatrixFSym &source);

#ifndef ROOT_TMatrixDSymfwd
#include "TMatrixDSymfwd.h"
#endif

template class TMatrixTSym<Double_t>;

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

template TMatrixDSym &Add        <Double_t>(TMatrixDSym &target,      Double_t     scalar,const TMatrixDSym &source);
template TMatrixDSym &ElementMult<Double_t>(TMatrixDSym &target,const TMatrixDSym &source);
template TMatrixDSym &ElementDiv <Double_t>(TMatrixDSym &target,const TMatrixDSym &source);
Last update: Thu Jan 17 09:00:04 2008
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