    // @(#)root/matrix:$Name:  $:$Id: TMatrixFBase.cxx,v 1.14 2004/06/22 19:57:01 brun Exp $
    // 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.             *
     *************************************************************************/

    //////////////////////////////////////////////////////////////////////////
    //                                                                      //
    // Linear Algebra Package                                               //
    // ----------------------                                               //
    //                                                                      //
    // The present package implements all the basic algorithms dealing      //
    // with vectors, matrices, matrix columns, rows, diagonals, etc.        //
    // In addition eigen-Vector analysis and several matrix decomposition   //
    // have been added (LU,QRH,Cholesky and SVD) . The decompositions are   //
    // used in matrix inversion, equation solving.                          //
    //                                                                      //
    // Matrix elements are arranged in memory in a ROW-wise fashion         //
    // fashion . For (n x m) matrices where n*m < kSizeMax (=25 currently)  //
    // storage space is avaialble on the stack, thus avoiding expensive     //
    // allocation/deallocation of heap space . However, this introduces of  //
    // course kSizeMax overhead for each matrix object . If this is an      //
    // issue recompile with a new appropriate value (>=0) for kSizeMax      //
    //                                                                      //
    // Another way to assign and store matrix data is through Use           //
    // see for instance stressLinear.cxx file .                             //
    //                                                                      //
    // Unless otherwise specified, matrix and vector indices always start   //
    // with 0, spanning up to the specified limit-1. However, there are     //
    // constructors to which one can specify aribtrary lower and upper      //
    // bounds, e.g. TMatrixF m(1,10,1,5) defines a matrix that ranges,      //
    // nad that can be addresses, from 1..10, 1..5 (a(1,1)..a(10,5)).       //
    //                                                                      //
    // The present package provides all facilities to completely AVOID      //
    // returning matrices. Use "TMatrixF A(TMatrixF::kTransposed,B);"       //
    // and other fancy constructors as much as possible. If one really needs//
    // to return a matrix, return a TMatLazy object instead. The            //
    // conversion is completely transparent to the end user, e.g.           //
    // "TMatrixF m = THaarMatrixF(5);" and _is_ efficient.                  //
    //                                                                      //
    // Since TMatrixF et al. are fully integrated in ROOT they of course    //
    // can be stored in a ROOT database.                                    //
    //                                                                      //
    // For usage examples see $ROOTSYS/test/stressLinear.cxx                //
    //                                                                      //
    // Acknowledgements                                                     //
    // ----------------                                                     //
    // 1. Oleg E. Kiselyov                                                  //
    //  First implementations were based on the his code . We have diverged //
    //  quite a bit since then but the ideas/code for lazy matrix and       //
    //  "nested function" are 100% his .                                    //
    //  You can see him and his code in action at http://okmij.org/ftp      //
    // 2. Chris R. Birchenhall,                                             //
    //  We adapted his idea of the implementation for the decomposition     //
    //  classes instead of our messy installation of matrix inversion       //
    //  His installation of matrix condition number, using an iterative     //
    //  scheme using the Hage algorithm is worth looking at !               //
    //  Chris has a nice writeup (matdoc.ps) on his matrix classes at       //
    //   ftp://ftp.mcc.ac.uk/pub/matclass/                                  //
    // 3. Mark Fischler and Steven Haywood of CLHEP                         //
    //  They did the slave labor of spelling out all sub-determinants       //
    //   for Cramer inversion  of (4x4),(5x5) and (6x6) matrices            //
    //  The stack storage for small matrices was also taken from them       //
    // 4. Roldan Pozo of TNT (http://math.nist.gov/tnt/)                    //
    //  He converted the EISPACK routines for the eigen-vector analysis to  //
    //  C++ . We started with his implementation                            //
    // 5. Siegmund Brandt (http://siux00.physik.uni-siegen.de/~brandt/datan //
    //  We adapted his (very-well) documented SVD routines                  //
    //                                                                      //
    // How to efficiently use this package                                  //
    // -----------------------------------                                  //
    //                                                                      //
    // 1. Never return complex objects (matrices or vectors)                //
    //    Danger: For example, when the following snippet:                  //
    //        TMatrixF foo(int n)                                           //
    //        {                                                             //
    //           TMatrixF foom(n,n); fill_in(foom); return foom;            //
    //        }                                                             //
    //        TMatrixF m = foo(5);                                          //
    //    runs, it constructs matrix foo:foom, copies it onto stack as a    //
    //    return value and destroys foo:foom. Return value (a matrix)       //
    //    from foo() is then copied over to m (via a copy constructor),     //
    //    and the return value is destroyed. So, the matrix constructor is  //
    //    called 3 times and the destructor 2 times. For big matrices,      //
    //    the cost of multiple constructing/copying/destroying of objects   //
    //    may be very large. *Some* optimized compilers can cut down on 1   //
    //    copying/destroying, but still it leaves at least two calls to     //
    //    the constructor. Note, TMatLazy (see below) can construct         //
    //    TMatrixF m "inplace", with only a _single_ call to the            //
    //    constructor.                                                      //
    //                                                                      //
    // 2. Use "two-address instructions"                                    //
    //        "void TMatrixF::operator += (const TMatrixF &B);"             //
    //    as much as possible.                                              //
    //    That is, to add two matrices, it's much more efficient to write   //
    //        A += B;                                                       //
    //    than                                                              //
    //        TMatrixF C = A + B;                                           //
    //    (if both operand should be preserved,                             //
    //        TMatrixF C = A; C += B;                                       //
    //    is still better).                                                 //
    //                                                                      //
    // 3. Use glorified constructors when returning of an object seems      //
    //    inevitable:                                                       //
    //        "TMatrixF A(TMatrixF::kTransposed,B);"                        //
    //        "TMatrixF C(A,TMatrixF::kTransposeMult,B);"                   //
    //                                                                      //
    //    like in the following snippet (from $ROOTSYS/test/vmatrix.cxx)    //
    //    that verifies that for an orthogonal matrix T, T'T = TT' = E.     //
    //                                                                      //
    //    TMatrixF haar = THaarMatrixF(5);                                  //
    //    TMatrixF unit(TMatrixF::kUnit,haar);                              //
    //    TMatrixF haar_t(TMatrixF::kTransposed,haar);                      //
    //    TMatrixF hth(haar,TMatrixF::kTransposeMult,haar);                 //
    //    TMatrixF hht(haar,TMatrixF::kMult,haar_t);                        //
    //    TMatrixF hht1 = haar; hht1 *= haar_t;                             //
    //    VerifyMatrixIdentity(unit,hth);                                   //
    //    VerifyMatrixIdentity(unit,hht);                                   //
    //    VerifyMatrixIdentity(unit,hht1);                                  //
    //                                                                      //
    // 4. Accessing row/col/diagonal of a matrix without much fuss          //
    //    (and without moving a lot of stuff around):                       //
    //                                                                      //
    //        TMatrixF m(n,n); TVectorF v(n); TMatrixFDiag(m) += 4;         //
    //        v = TMatrixFRow(m,0);                                         //
    //        TMatrixColumn m1(m,1); m1(2) = 3; // the same as m(2,1)=3;    //
    //    Note, constructing of, say, TMatrixFDiag does *not* involve any   //
    //    copying of any elements of the source matrix.                     //
    //                                                                      //
    // 5. It's possible (and encouraged) to use "nested" functions          //
    //    For example, creating of a Hilbert matrix can be done as follows: //
    //                                                                      //
    //    void foo(const TMatrixF &m)                                       //
    //    {                                                                 //
    //      TMatrixF m1(TMatrixF::kZero,m);                                 //
    //      struct MakeHilbert : public TElementPosActionF {                //
    //        void Operation(Float_t &element) { element = 1./(fI+fJ-1); }  //
    //      };                                                              //
    //      m1.Apply(MakeHilbert());                                        //
    //    }                                                                 //
    //                                                                      //
    //    of course, using a special method THilbertMatrixF() is            //
    //    still more optimal, but not by a whole lot. And that's right,     //
    //    class MakeHilbert is declared *within* a function and local to    //
    //    that function. It means one can define another MakeHilbert class  //
    //    (within another function or outside of any function, that is, in  //
    //    the global scope), and it still will be OK. Note, this currently  //
    //    is not yet supported by the interpreter CINT.                     //
    //                                                                      //
    //    Another example is applying of a simple function to each matrix   //
    //    element:                                                          //
    //                                                                      //
    //    void foo(TMatrixF &m,TMatrixF &m1)                                //
    //    {                                                                 //
    //      typedef  double (*dfunc_t)(double);                             //
    //      class ApplyFunction : public TElementActionF {                  //
    //        dfunc_t fFunc;                                                //
    //        void Operation(Float_t &element) { element=fFunc(element); }  //
    //      public:                                                         //
    //        ApplyFunction(dfunc_t func):fFunc(func) {}                    //
    //      };                                                              //
    //      ApplyFunction x(TMath::Sin);                                    //
    //      m.Apply(x);                                                     //
    //    }                                                                 //
    //                                                                      //
    //    Validation code $ROOTSYS/test/vmatrix.cxx and vvector.cxx contain //
    //    a few more examples of that kind.                                 //
    //                                                                      //
    // 6. Lazy matrices: instead of returning an object return a "recipe"   //
    //    how to make it. The full matrix would be rolled out only when     //
    //    and where it's needed:                                            //
    //       TMatrixF haar = THaarMatrixF(5);                               //
    //    THaarMatrixF() is a *class*, not a simple function. However       //
    //    similar this looks to a returning of an object (see note #1       //
    //    above), it's dramatically different. THaarMatrixF() constructs a  //
    //    TMatLazy, an object of just a few bytes long. A
    //    "TMatrixF(const TMatLazy &recipe)" constructor follows the        //
    //    recipe and makes the matrix haar() right in place. No matrix      //
    //    element is moved whatsoever!                                      //
    //                                                                      //
    //////////////////////////////////////////////////////////////////////////

    //////////////////////////////////////////////////////////////////////////
    //                                                                      //
    // Matrix Base class                                                    //
    //                                                                      //
    //  matrix properties are stored here, however the data storage is part //
    //  of the derived classes                                              //
    //                                                                      //
    //////////////////////////////////////////////////////////////////////////

    #include "TMatrixFBase.h"

    ClassImp(TMatrixFBase)

    //______________________________________________________________________________
     void TMatrixFBase::Delete_m(Int_t size,Float_t *&m)
    {
      // delete data pointer m, if it was assigned on the heap

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

    //______________________________________________________________________________
     Float_t* TMatrixFBase::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 <= kSizeMax )
          return fDataStack;
        else {
          Float_t *heap = new Float_t[size];
          return heap;
        }
      }
    }

    //______________________________________________________________________________
     Int_t TMatrixFBase::Memcpy_m(Float_t *newp,const Float_t *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 <= 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(Float_t));
      }
      return 0;
    }

    //______________________________________________________________________________
     TMatrixFBase &TMatrixFBase::SetMatrixArray(const Float_t *data,Option_t *option)
    {
      // Copy array data to matrix . It is assumed that array is of size >= fNelems
      // (=)))) fNrows*fNcols
      // option indicates how the data is stored in the array:
      // option =
      //          'F'   : column major (Fortran) m[i][j] = array[i+j*fNrows]
      //          else  : row major    (C)       m[i][j] = array[i*fNcols+j] (default)

      Assert(IsValid());

      TString opt = option;
      opt.ToUpper();

      Float_t *elem = GetMatrixArray();
      if (opt.Contains("F")) {
        for (Int_t irow = 0; irow < fNrows; irow++) {
          const Int_t off1 = irow*fNcols;
          Int_t off2 = 0;
          for (Int_t icol = 0; icol < fNcols; icol++) {
            elem[off1+icol] = data[off2+irow];
            off2 += fNrows;
          }
        }
      }
      else
        memcpy(elem,data,fNelems*sizeof(Float_t));

      return *this;
    }

    //______________________________________________________________________________
     Bool_t TMatrixFBase::IsSymmetric() const
    {
      Assert(IsValid());

      if ((fNrows != fNcols) || (fRowLwb != fColLwb))
        return kFALSE;

      const Float_t * const elem = GetMatrixArray();
      for (Int_t irow = 0; irow < fNrows; irow++) {
        const Int_t rowOff = irow*fNcols;
        Int_t colOff = 0;
        for (Int_t icol = 0; icol < irow; icol++) {
          if (elem[rowOff+icol] != elem[colOff+irow])
            return kFALSE;
          colOff += fNrows;
        }
      }
      return kTRUE;
    }

    //______________________________________________________________________________
     void TMatrixFBase::GetMatrix2Array(Float_t *data,Option_t *option) const
    {
      // Copy matrix data to array . It is assumed that array is of size >= fNelems
      // (=)))) fNrows*fNcols
      // option indicates how the data is stored in the array:
      // option =
      //          'F'   : column major (Fortran) array[i+j*fNrows] = m[i][j]
      //          else  : row major    (C)       array[i*fNcols+j] = m[i][j] (default)

      Assert(IsValid());

      TString opt = option;
      opt.ToUpper();

      const Float_t * const elem = GetMatrixArray();
      if (opt.Contains("F")) {
        for (Int_t irow = 0; irow < fNrows; irow++) {
          const Int_t off1 = irow*fNcols;
          Int_t off2 = 0;
          for (Int_t icol = 0; icol < fNcols; icol++)
            data[off2+irow] = elem[off1+icol];
            off2 += fNrows;
        }
      }
      else
        memcpy(data,elem,fNelems*sizeof(Float_t));
    }

    //______________________________________________________________________________
     TMatrixFBase &TMatrixFBase::InsertRow(Int_t rown,Int_t coln,const Float_t *v,Int_t n)
    {
      const Int_t arown = rown-fRowLwb;
      const Int_t acoln = coln-fColLwb;
      const Int_t nr = (n > 0) ? n : fNcols;

      if (arown >= fNrows || arown < 0) {
        Error("InsertRow","row %d out of matrix range",rown);
        Invalidate();
        return *this;
      }

      if (acoln >= fNcols || acoln < 0) {
        Error("InsertRow","column %d out of matrix range",coln);
        Invalidate();
        return *this;
      }

      if (acoln+nr >= fNcols || nr < 0) {
        Error("InsertRow","row length %d out of range",nr);
        Invalidate();
        return *this;
      }

      const Int_t off = arown*fNcols+acoln;
      Float_t * const elem = GetMatrixArray()+off;
      memcpy(elem,v,nr*sizeof(Float_t));

      return *this;
    }

    //______________________________________________________________________________
     void TMatrixFBase::ExtractRow(Int_t rown,Int_t coln,Float_t *v,Int_t n) const
    {
      // Store in array v, n matrix elements of row rown starting at column coln

      const Int_t arown = rown-fRowLwb;
      const Int_t acoln = coln-fColLwb;
      const Int_t nr = (n > 0) ? n : fNcols;

      if (arown >= fNrows || arown < 0) {
        Error("ExtractRow","row %d out of matrix range",rown);
        return;
      }

      if (acoln >= fNcols || acoln < 0) {
        Error("ExtractRow","column %d out of matrix range",coln);
        return;
      }

      if (acoln+n >= fNcols || nr < 0) {
        Error("ExtractRow","row length %d out of range",nr);
        return;
      }

      const Int_t off = arown*fNcols+acoln;
      const Float_t * const elem = GetMatrixArray()+off;
      memcpy(v,elem,nr*sizeof(Double_t));
    }

    //______________________________________________________________________________
     TMatrixFBase &TMatrixFBase::Shift(Int_t row_shift,Int_t col_shift)
    {
      // Shift the row index by adding row_shift and the column index by adding
      // col_shift, respectively. So [rowLwb..rowUpb][colLwb..colUpb] becomes
      // [rowLwb+row_shift..rowUpb+row_shift][colLwb+col_shift..colUpb+col_shift]

      fRowLwb += row_shift;
      fColLwb += col_shift;

      return *this;
    }

    //______________________________________________________________________________
     TMatrixFBase &TMatrixFBase::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.

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

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

        Float_t     *elements_old = GetMatrixArray();
        const Int_t  nelems_old   = fNelems;
        const Int_t  nrows_old    = fNrows;
        const Int_t  ncols_old    = fNcols;

        Allocate(nrows,ncols);
        Assert(IsValid());

        Float_t  *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 (fNelems > kSizeMax || nelems_old > kSizeMax)
          memset(elements_new,0,fNelems*sizeof(Float_t));
        else if (fNelems > nelems_old)
          memset(elements_new+nelems_old,0,(fNelems-nelems_old)*sizeof(Float_t));

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

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

    //______________________________________________________________________________
     TMatrixFBase &TMatrixFBase::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.

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

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

      if (fNelems > 0) {

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

        Float_t     *elements_old = GetMatrixArray();
        const Int_t  nelems_old   = fNelems;
        const Int_t  nrows_old    = fNrows;
        const Int_t  ncols_old    = fNcols;
        const Int_t  rowLwb_old   = fRowLwb;
        const Int_t  colLwb_old   = fColLwb;

        Allocate(new_nrows,new_ncols,row_lwb,col_lwb);
        Assert(IsValid());

        Float_t *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 (fNelems > kSizeMax || nelems_old > kSizeMax)
          memset(elements_new,0,fNelems*sizeof(Float_t));
        else if (fNelems > nelems_old)
          memset(elements_new+nelems_old,0,(fNelems-nelems_old)*sizeof(Float_t));

        // Copy overlap
        const Int_t rowLwb_copy = TMath::Max(fRowLwb,rowLwb_old);
        const Int_t colLwb_copy = TMath::Max(fColLwb,colLwb_old);
        const Int_t rowUpb_copy = TMath::Min(fRowLwb+fNrows-1,rowLwb_old+nrows_old-1);
        const Int_t colUpb_copy = TMath::Min(fColLwb+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-fColLwb;
          if (ncols_old < 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-fRowLwb;
              Memcpy_m(elements_new+iRowNew*fNcols+colNewOff,
                       elements_old+iRowOld*ncols_old+colOldOff,ncols_copy,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-fRowLwb;
              Memcpy_m(elements_new+iRowNew*fNcols+colNewOff,
                       elements_old+iRowOld*ncols_old+colOldOff,ncols_copy,fNelems,nelems_old);
            }
          }
        }

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

      return *this;
    }

    //______________________________________________________________________________
     TMatrixFBase &TMatrixFBase::Zero()
    {
      Assert(IsValid());
      memset(this->GetMatrixArray(),0,fNelems*sizeof(Float_t));

      return *this;
    }

    //______________________________________________________________________________
     TMatrixFBase &TMatrixFBase::Abs()
    {
      // Take an absolute value of a matrix, i.e. apply Abs() to each element.

      Assert(IsValid());

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

      return *this;
    }
    //______________________________________________________________________________
     TMatrixFBase &TMatrixFBase::Sqr()
    {
      // Square each element of the matrix.

      Assert(IsValid());

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

      return *this;
    }

    //______________________________________________________________________________
     TMatrixFBase &TMatrixFBase::Sqrt()
    {
      // Take square root of all elements.

      Assert(IsValid());

            Float_t *ep = this->GetMatrixArray();
      const Float_t * const fp = ep+fNelems;
      while (ep < fp) {
        *ep = TMath::Sqrt(*ep);
        ep++;
      }

      return *this;
    }

    //______________________________________________________________________________
     TMatrixFBase &TMatrixFBase::UnitMatrix()
    {
      // Make a unit matrix (matrix need not be a square one).

      Assert(IsValid());

      Float_t *ep = this->GetMatrixArray();
      memset(ep,0,fNelems*sizeof(Float_t));
      for (Int_t i = fRowLwb; i <= fRowLwb+fNrows-1; i++)
        for (Int_t j = fColLwb; j <= fColLwb+fNcols-1; j++)
          *ep++ = (i==j ? 1.0 : 0.0);

      return *this;
    }

    //______________________________________________________________________________
     TMatrixFBase &TMatrixFBase::NormByDiag(const TVectorF &v,Option_t *option)
    {
      // option:
      // "D"   :  b(i,j) = a(i,j)/sqrt(abs*(v(i)*v(j)))  (default)
      // else  :  b(i,j) = a(i,j)*sqrt(abs*(v(i)*v(j)))  (default)

      Assert(IsValid());
      Assert(v.IsValid());

      const Int_t nMax = TMath::Max(fNrows,fNcols);
      if (v.GetNoElements() < nMax) {
        Error("NormByDiag","vector shorter than matrix diagonal");
        Invalidate();
        return *this;
      }

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

      const Float_t *pV = v.GetMatrixArray();
            Float_t *mp = this->GetMatrixArray();

      if (divide) {
        for (Int_t irow = 0; irow < fNrows; irow++) {
          for (Int_t icol = 0; icol < fNcols; icol++) {
            const Float_t val = TMath::Sqrt(TMath::Abs(pV[irow]*pV[icol]));
            Assert(val != 0.0);
            *mp++ /= val;
          }
        }
      } else {
        for (Int_t irow = 0; irow < fNrows; irow++) {
          for (Int_t icol = 0; icol < fNcols; icol++) {
            const Float_t val = TMath::Sqrt(TMath::Abs(pV[irow]*pV[icol]));
            *mp++ *= val;
          }
        }
      }

      return *this;
    }

    //______________________________________________________________________________
     Float_t TMatrixFBase::RowNorm() const
    {
      // Row matrix norm, MAX{ SUM{ |M(i,j)|, over j}, over i}.
      // The norm is induced by the infinity vector norm.

      Assert(IsValid());

      const Float_t *       ep = GetMatrixArray();
      const Float_t * const fp = ep+fNelems;
            Float_t norm = 0;

      // Scan the matrix row-after-row
      while (ep < fp) {
        Float_t sum = 0;
        // Scan a row to compute the sum
        for (Int_t j = 0; j < fNcols; j++)
          sum += TMath::Abs(*ep++);
        norm = TMath::Max(norm,sum);
      }

      Assert(ep == fp);

      return norm;
    }

    //______________________________________________________________________________
     Float_t TMatrixFBase::ColNorm() const
    {
      // Column matrix norm, MAX{ SUM{ |M(i,j)|, over i}, over j}.
      // The norm is induced by the 1 vector norm.

      Assert(IsValid());

      const Float_t *       ep = GetMatrixArray();
      const Float_t * const fp = ep+fNcols;
            Float_t norm = 0;

      // Scan the matrix col-after-col
      while (ep < fp) {
        Float_t sum = 0;
        // Scan a col to compute the sum
        for (Int_t i = 0; i < fNrows; i++,ep += fNcols)
          sum += TMath::Abs(*ep);
        ep -= fNelems-1;         // Point ep to the beginning of the next col
        norm = TMath::Max(norm,sum);
      }

      Assert(ep == fp);

      return norm;
    }

    //______________________________________________________________________________
     Float_t TMatrixFBase::E2Norm() const
    {
      // Square of the Euclidian norm, SUM{ m(i,j)^2 }.

      Assert(IsValid());

      const Float_t *       ep = GetMatrixArray();
      const Float_t * const fp = ep+fNelems;
            Float_t sum = 0;

      for ( ; ep < fp; ep++)
        sum += (*ep) * (*ep);

      return sum;
    }

    //______________________________________________________________________________
     Int_t TMatrixFBase::NonZeros() const
    {
      // Compute the number of elements != 0.0

      Assert(IsValid());

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

      return nr_nonzeros;
    }

    //______________________________________________________________________________
     Float_t TMatrixFBase::Sum() const
    {
      // Compute sum of elements

      Assert(IsValid());

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

      return sum;
    }

    //______________________________________________________________________________
     Float_t TMatrixFBase::Min() const
    {
      // return minimum matrix element value

      Assert(IsValid());

      const Float_t * const ep = this->GetMatrixArray();
      const Int_t index = TMath::LocMin(fNelems,ep);
      return ep[index];
    }

    //______________________________________________________________________________
     Float_t TMatrixFBase::Max() const
    {
      // return maximum vector element value

      Assert(IsValid());

      const Float_t * const ep = this->GetMatrixArray();
      const Int_t index = TMath::LocMax(fNelems,ep);
      return ep[index];
    }

    //______________________________________________________________________________
     void TMatrixFBase::Draw(Option_t *option)
    {
      // Draw this matrix using an intermediate histogram
      // The histogram is named "TMatrixF" by default and no title

      //create the hist utility manager (a plugin)
      TVirtualUtilHist *util = (TVirtualUtilHist*)gROOT->GetListOfSpecials()->FindObject("R__TVirtualUtilHist");
      if (!util) {
        TPluginHandler *h;
        if ((h = gROOT->GetPluginManager()->FindHandler("TVirtualUtilHist"))) {
          if (h->LoadPlugin() == -1)
            return;
          h->ExecPlugin(0);
          util = (TVirtualUtilHist*)gROOT->GetListOfSpecials()->FindObject("R__TVirtualUtilHist");
        }
      }
     util->PaintMatrix(*this,option);
    }

    //______________________________________________________________________________
     void TMatrixFBase::Print(Option_t *name) const
    {
      // Print the matrix as a table of elements .

      if (!IsValid()) {
         Error("Print","Matrix is invalid");
         return;
      }

      printf("\n%dx%d matrix %s is as follows",fNrows,fNcols,name);

      const Int_t cols_per_sheet = 5;

      const Int_t ncols  = fNcols;
      const Int_t nrows  = fNrows;
      const Int_t collwb = fColLwb;
      const Int_t rowlwb = fRowLwb;
      for (Int_t sheet_counter = 1; sheet_counter <= ncols; sheet_counter += cols_per_sheet) {
        printf("\n\n     |");
        for (Int_t j = sheet_counter; j < sheet_counter+cols_per_sheet && j <= ncols; j++)
          printf("   %6d  |",j+collwb-1);
        printf("\n%s\n","------------------------------------------------------------------");
        for (Int_t i = 1; i <= nrows; i++) {
          printf("%4d |",i+rowlwb-1);
          for (Int_t j = sheet_counter; j < sheet_counter+cols_per_sheet && j <= ncols; j++)
            printf("%11.4g ",(*this)(i+rowlwb-1,j+collwb-1));
            printf("\n");
        }
      }
      printf("\n");
    }

    //______________________________________________________________________________
    Bool_t TMatrixFBase::operator==(Float_t val) const
    {
      // Are all matrix elements equal to val?

      Assert(IsValid());

      const Float_t *       ep = GetMatrixArray();
      const Float_t * const fp = ep+fNelems;
      for (; ep < fp; ep++)
        if (!(*ep == val))
          return kFALSE;

      return kTRUE;
    }

    //______________________________________________________________________________
    Bool_t TMatrixFBase::operator!=(Float_t val) const
    {
      // Are all matrix elements not equal to val?

      Assert(IsValid());

      const Float_t *       ep = GetMatrixArray();
      const Float_t * const fp = ep+fNelems;
      for (; ep < fp; ep++)
        if (!(*ep != val))
          return kFALSE;

      return kTRUE;
    }

    //______________________________________________________________________________
    Bool_t TMatrixFBase::operator<(Float_t val) const
    {
      // Are all matrix elements < val?

      Assert(IsValid());

      const Float_t *       ep = GetMatrixArray();
      const Float_t * const fp = ep+fNelems;
      for (; ep < fp; ep++)
        if (!(*ep < val))
          return kFALSE;

      return kTRUE;
    }

    //______________________________________________________________________________
    Bool_t TMatrixFBase::operator<=(Float_t val) const
    {
      // Are all matrix elements <= val?

      Assert(IsValid());

      const Float_t *       ep = GetMatrixArray();
      const Float_t * const fp = ep+fNelems;
      for (; ep < fp; ep++)
        if (!(*ep <= val))
          return kFALSE;

      return kTRUE;
    }

    //______________________________________________________________________________
    Bool_t TMatrixFBase::operator>(Float_t val) const
    {
      // Are all matrix elements > val?

      Assert(IsValid());

      const Float_t *       ep = GetMatrixArray();
      const Float_t * const fp = ep+fNelems;
      for (; ep < fp; ep++)
        if (!(*ep > val))
          return kFALSE;

      return kTRUE;
    }

    //______________________________________________________________________________
    Bool_t TMatrixFBase::operator>=(Float_t val) const
    {
      // Are all matrix elements >= val?

      Assert(IsValid());

      const Float_t *       ep = GetMatrixArray();
      const Float_t * const fp = ep+fNelems;
      for (; ep < fp; ep++)
        if (!(*ep >= val))
          return kFALSE;

      return kTRUE;
    }

    //______________________________________________________________________________
     TMatrixFBase &TMatrixFBase::Apply(const TElementActionF &action)
    {
      Assert(IsValid());

      Float_t *ep = this->GetMatrixArray();
      const Float_t * const ep_last = ep+fNelems;
      while (ep < ep_last)
        action.Operation(*ep++);

      return *this;
    }

    //______________________________________________________________________________
     TMatrixFBase &TMatrixFBase::Apply(const TElementPosActionF &action)
    {
      // Apply action to each element of the matrix. To action the location
      // of the current element is passed.

      Assert(IsValid());

      Float_t *ep = this->GetMatrixArray();
      for (action.fI = fRowLwb; action.fI < fRowLwb+fNrows; action.fI++)
        for (action.fJ = fColLwb; action.fJ < fColLwb+fNcols; action.fJ++)
          action.Operation(*ep++);

      Assert(ep == this->GetMatrixArray()+fNelems);

      return *this;
    }

    //______________________________________________________________________________
     TMatrixFBase &TMatrixFBase::Randomize(Float_t alpha,Float_t beta,Double_t &seed)
    {
      // randomize matrix element values

      Assert(IsValid());

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

            Float_t *       ep = GetMatrixArray();
      const Float_t * const fp = ep+fNelems;
      while (ep < fp)
        *ep++ = scale*(Drand(seed)+shift);

      return *this;
    }

    //______________________________________________________________________________
    Bool_t operator==(const TMatrixFBase &m1,const TMatrixFBase &m2)
    {
      // Check to see if two matrices are identical.

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

    //______________________________________________________________________________
    Float_t E2Norm(const TMatrixFBase &m1,const TMatrixFBase &m2)
    {
      // Square of the Euclidian norm of the difference between two matrices.

      if (!AreCompatible(m1,m2)) {
        ::Error("E2Norm","matrices not compatible");
        return -1.0;
      }

      const Float_t *        mp1 = m1.GetMatrixArray();
      const Float_t *        mp2 = m2.GetMatrixArray();
      const Float_t * const fmp1 = mp1+m1.GetNoElements();

      Float_t sum = 0.0;
      for (; mp1 < fmp1; mp1++, mp2++)
        sum += (*mp1 - *mp2)*(*mp1 - *mp2);

      return sum;
    }

    //______________________________________________________________________________
    Bool_t AreCompatible(const TMatrixFBase &m1,const TMatrixFBase &m2,Int_t verbose)
    {
      if (!m1.IsValid()) {
        if (verbose)
          ::Error("AreCompatible", "matrix 1 not valid");
        return kFALSE;
      }
      if (!m2.IsValid()) {
        if (verbose)
          ::Error("AreCompatible", "matrix 2 not valid");
        return kFALSE;
      }

      if (m1.GetNrows()  != m2.GetNrows()  || m1.GetNcols()  != m2.GetNcols() ||
          m1.GetRowLwb() != m2.GetRowLwb() || m1.GetColLwb() != m2.GetColLwb()) {
        if (verbose)
          ::Error("AreCompatible", "matrices 1 and 2 not compatible");
        return kFALSE;
      }

      return kTRUE;
    }

    //______________________________________________________________________________
    Bool_t AreCompatible(const TMatrixFBase &m1,const TMatrixDBase &m2,Int_t verbose)
    {
      if (!m1.IsValid()) {
        if (verbose)
          ::Error("AreCompatible", "matrix 1 not valid");
        return kFALSE;
      }
      if (!m2.IsValid()) {
        if (verbose)
          ::Error("AreCompatible", "matrix 2 not valid");
        return kFALSE;
      }

      if (m1.GetNrows()  != m2.GetNrows()  || m1.GetNcols()  != m2.GetNcols() ||
          m1.GetRowLwb() != m2.GetRowLwb() || m1.GetColLwb() != m2.GetColLwb()) {
        if (verbose)
          ::Error("AreCompatible", "matrices 1 and 2 not compatible");
        return kFALSE;
      }

      return kTRUE;
    }

    //______________________________________________________________________________
    void Compare(const TMatrixFBase &m1,const TMatrixFBase &m2)
    {
      // Compare two matrices and print out the result of the comparison.

      if (!AreCompatible(m1,m2)) {
        Error("Compare(const TMatrixFBase &,const TMatrixFBase &)","matrices are incompatible");
        return;
      }

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

      Float_t  norm1  = 0;      // Norm of the Matrices
      Float_t  norm2  = 0;
      Float_t  ndiff  = 0;      // Norm of the difference
      Int_t    imax   = 0;      // For the elements that differ most
      Int_t    jmax   = 0;
      Float_t  difmax = -1;

      for (Int_t i = m1.GetRowLwb(); i <= m1.GetRowUpb(); i++) {
        for (Int_t j = m1.GetColLwb(); j < m1.GetColUpb(); j++) {
          const Float_t mv1 = m1(i,j);
          const Float_t mv2 = m2(i,j);
          const Float_t diff = TMath::Abs(mv1-mv2);

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

      printf("\nMaximal discrepancy    \t\t%g", difmax);
      printf("\n   occured at the point\t\t(%d,%d)",imax,jmax);
      const Float_t mv1 = m1(imax,jmax);
      const Float_t mv2 = m2(imax,jmax);
      printf("\n Matrix 1 element is    \t\t%g", mv1);
      printf("\n Matrix 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,(Float_t)1e-7));

      printf("\n||Matrix 1||   \t\t\t%g", norm1);
      printf("\n||Matrix 2||   \t\t\t%g", norm2);
      printf("\n||Matrix1-Matrix2||\t\t\t\t%g", ndiff);
      printf("\n||Matrix1-Matrix2||/sqrt(||Matrix1|| ||Matrix2||)\t%g\n\n",
             ndiff/TMath::Max(TMath::Sqrt(norm1*norm2),1e-7));
    }

    //______________________________________________________________________________
    Bool_t VerifyMatrixValue(const TMatrixFBase &m,Float_t val,Int_t verbose,Float_t maxDevAllow)
    {
      // Validate that all elements of matrix have value val within maxDevAllow.

      Assert(m.IsValid());

      Int_t    imax      = 0;
      Int_t    jmax      = 0;
      Float_t  maxDevObs = 0;

      for (Int_t i = m.GetRowLwb(); i <= m.GetRowUpb(); i++) {
        for (Int_t j = m.GetColLwb(); j <= m.GetColUpb(); j++) {
          const Float_t dev = TMath::Abs(m(i,j)-val);
          if (dev > maxDevObs) {
            imax    = i;
            jmax    = j;
            maxDevObs = dev;
          }
        }
      }

      if (maxDevObs == 0)
        return kTRUE;

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

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

    //______________________________________________________________________________
    Bool_t VerifyMatrixIdentity(const TMatrixFBase &m1,const TMatrixFBase &m2,Int_t verbose,
                                Float_t maxDevAllow)
    {
       // Verify that elements of the two matrices are equal within MaxDevAllow .

      if (!AreCompatible(m1,m2))
        return kFALSE;

      Int_t    imax      = 0;
      Int_t    jmax      = 0;
      Float_t  maxDevObs = 0;

      for (Int_t i = m1.GetRowLwb(); i <= m1.GetRowUpb(); i++) {
        for (Int_t j = m1.GetColLwb(); j <= m1.GetColUpb(); j++) {
          const Float_t dev = TMath::Abs(m1(i,j)-m2(i,j));
          if (dev > maxDevObs) {
            imax = i;
            jmax = j;
            maxDevObs = dev;
          }
        }
      }

      if (maxDevObs == 0)
        return kTRUE;

      if (verbose) {
        printf("Largest dev for (%d,%d); dev = |%g - %g| = %g\n",
                imax,jmax,m1(imax,jmax),m2(imax,jmax),maxDevObs);
        if(maxDevObs > maxDevAllow)
          Error("VerifyMatrixValue","Deviation > %g\n",maxDevAllow);
      }

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

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

      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) {
          TMatrixFBase::Class()->ReadBuffer(R__b,this,R__v,R__s,R__c);
        } else {
          Error("TMatrixFBase::Streamer","Unknown version number: %d",R__v);
          Assert(0);
        }
        if (R__v < 4)
          MakeValid();
      } else {
        TMatrixFBase::Class()->WriteBuffer(R__b,this);
      }
    }