TMatrixTBase.cxx

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// @(#)root/matrix:$Id: 2d00df45ce4c38c7ea0930d6b520cbf4cfb9152e $
// Authors: Fons Rademakers, Eddy Offermann   Nov 2003
 
/*************************************************************************
 * Copyright (C) 1995-2000, Rene Brun and Fons Rademakers.               *
 * All rights reserved.                                                  *
 *                                                                       *
 * For the licensing terms see $ROOTSYS/LICENSE.                         *
 * For the list of contributors see $ROOTSYS/README/CREDITS.             *
 *************************************************************************/
 
/** \class TMatrixTBase
    \ingroup Matrix
    \brief Linear Algebra Package
 
 ### 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,Bunch-Kaufman and SVD) .
 The decompositions are used in matrix inversion, equation solving.
 
 For a dense matrix, elements are arranged in memory in a ROW-wise
 fashion . For (n x m) matrices where n*m <=kSizeMax (=25 currently)
 storage space is available on the stack, thus avoiding expensive
 allocation/deallocation of heap space . However, this introduces of
 course kSizeMax overhead for each matrix object . If this is an
 issue recompile with a new appropriate value (>=0) for kSizeMax
 
 Sparse matrices are also stored in row-wise fashion but additional
 row/column information is stored, see TMatrixTSparse source for
 additional details .
 
 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. TMatrixD m(1,10,1,5) defines a matrix that ranges
 from 1..10, 1..5 (a(1,1)..a(10,5)).
 
 The present package provides all facilities to completely AVOID
 returning matrices. Use "TMatrixD A(TMatrixD::kTransposed,B);"
 and other fancy constructors as much as possible. If one really needs
 to return a matrix, return a TMatrixTLazy object instead. The
 conversion is completely transparent to the end user, e.g.
 "TMatrixT m = THaarMatrixT(5);" and _is_ efficient.
 
 Since TMatrixT 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:
~~~
   TMatrixD foo(int n)
   {
     TMatrixD foom(n,n); fill_in(foom); return foom;
   }
   TMatrixD 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, TMatrixDLazy (see below) can construct
   TMatrixD m "inplace", with only a _single_ call to the
   constructor.
 
#### 2. Use "two-address instructions"
~~~
   "void TMatrixD::operator += (const TMatrixD &B);"
~~~
    as much as possible.
    That is, to add two matrices, it's much more efficient to write
~~~
   A += B;
~~~
    than
~~~
   TMatrixD C = A + B;
~~~
   (if both operand should be preserved,
     TMatrixD C = A; C += B;
   is still better).
 
#### 3. Use glorified constructors when returning of an object seems inevitable:
~~~
   "TMatrixD A(TMatrixD::kTransposed,B);"
   "TMatrixD C(A,TMatrixD::kTransposeMult,B);"
~~~
 
  like in the following snippet (from $ROOTSYS/test/vmatrix.cxx)
  that verifies that for an orthogonal matrix T, T'T = TT' = E.
 
~~~
   TMatrixD haar = THaarMatrixD(5);
   TMatrixD unit(TMatrixD::kUnit,haar);
   TMatrixD haar_t(TMatrixD::kTransposed,haar);
   TMatrixD hth(haar,TMatrixD::kTransposeMult,haar);
   TMatrixD hht(haar,TMatrixD::kMult,haar_t);
   TMatrixD 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):
 
~~~
   TMatrixD m(n,n); TVectorD v(n); TMatrixDDiag(m) += 4;
   v = TMatrixDRow(m,0);
   TMatrixDColumn m1(m,1); m1(2) = 3; // the same as m(2,1)=3;
~~~
   Note, constructing of, say, TMatrixDDiag 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 TMatrixD &m)
   {
    TMatrixD m1(TMatrixD::kZero,m);
    struct MakeHilbert : public TElementPosActionD {
       void Operation(Double_t &element)
          { element = 1./(fI+fJ-1); }
    };
    m1.Apply(MakeHilbert());
   }
~~~
 
   of course, using a special method THilbertMatrixD() 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(TMatrixD &m,TMatrixD &m1)
   {
    typedef  double (*dfunc_t)(double);
    class ApplyFunction : public TElementActionD {
       dfunc_t fFunc;
       void Operation(Double_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:
~~~
   TMatrixD haar = THaarMatrixD(5);
~~~
   THaarMatrixD() 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. THaarMatrixD() constructs a
   TMatrixDLazy, an object of just a few bytes long. A special
   "TMatrixD(const TMatrixDLazy &recipe)" constructor follows the
   recipe and makes the matrix haar() right in place. No matrix
   element is moved whatsoever!
 
*/
 
////////////////////////////////////////////////////////////////////////////////
///
/// TMatrixTBase
///
/// Template of base class in the linear algebra package
///
///  matrix properties are stored here, however the data storage is part
///  of the derived classes
 
#include "TMatrixTBase.h"
#include "TVectorT.h"
#include "TROOT.h"
#include "TClass.h"
#include "TMath.h"
#include <limits.h>
 
Int_t gMatrixCheck = 1;
 
templateClassImp(TMatrixTBase);
 
 
////////////////////////////////////////////////////////////////////////////////
/// Lexical sort on array data using indices first and second
 
template<class Element>
void TMatrixTBase<Element>::DoubleLexSort(Int_t n,Int_t *first,Int_t *second,Element *data)
{
   const int incs[] = {1,5,19,41,109,209,505,929,2161,3905,8929,16001,INT_MAX};
 
   Int_t kinc = 0;
   while (incs[kinc] <= n/2)
      kinc++;
   kinc -= 1;
 
   // incs[kinc] is the greatest value in the sequence that is also <= n/2.
   // If n == {0,1}, kinc == -1 and so no sort will take place.
 
   for( ; kinc >= 0; kinc--) {
      const Int_t inc = incs[kinc];
 
      for (Int_t k = inc; k < n; k++) {
         const Element tmp = data[k];
         const Int_t fi = first [k];
         const Int_t se = second[k];
         Int_t j;
         for (j = k; j >= inc; j -= inc) {
            if ( fi < first[j-inc] || (fi == first[j-inc] && se < second[j-inc]) ) {
               data  [j] = data  [j-inc];
               first [j] = first [j-inc];
               second[j] = second[j-inc];
            } else
               break;
         }
         data  [j] = tmp;
         first [j] = fi;
         second[j] = se;
      }
   }
}
 
////////////////////////////////////////////////////////////////////////////////
/// Lexical sort on array data using indices first and second
 
template<class Element>
void TMatrixTBase<Element>::IndexedLexSort(Int_t n,Int_t *first,Int_t swapFirst,
                                           Int_t *second,Int_t swapSecond,Int_t *index)
{
   const int incs[] = {1,5,19,41,109,209,505,929,2161,3905,8929,16001,INT_MAX};
 
   Int_t kinc = 0;
   while (incs[kinc] <= n/2)
      kinc++;
   kinc -= 1;
 
   // incs[kinc] is the greatest value in the sequence that is also less
   // than n/2.
 
   for( ; kinc >= 0; kinc--) {
      const Int_t inc = incs[kinc];
 
      if ( !swapFirst && !swapSecond ) {
         for (Int_t k = inc; k < n; k++) {
            // loop over all subarrays defined by the current increment
            const Int_t ktemp = index[k];
            const Int_t fi = first [ktemp];
            const Int_t se = second[ktemp];
            // Insert element k into the sorted subarray
            Int_t j;
            for (j = k; j >= inc; j -= inc) {
               // Loop over the elements in the current subarray
               if (fi < first[index[j-inc]] || (fi == first[index[j-inc]] && se < second[index[j-inc]])) {
                  // Swap elements j and j - inc, implicitly use the fact
                  // that ktemp hold element j to avoid having to assign to
                  // element j-inc
                    index[j] = index[j-inc];
               } else {
                  // There are no more elements in this sorted subarray which
                  // are less than element j
                  break;
               }
            } // End loop over the elements in the current subarray
            // Move index[j] out of temporary storage
            index[j] = ktemp;
            // The element has been inserted into the subarray.
         } // End loop over all subarrays defined by the current increment
      } else if ( swapSecond && !swapFirst ) {
         for (Int_t k = inc; k < n; k++) {
            const Int_t ktemp = index[k];
            const Int_t fi = first [ktemp];
            const Int_t se = second[k];
            Int_t j;
            for (j = k; j >= inc; j -= inc) {
               if (fi < first[index[j-inc]] || (fi == first[index[j-inc]] && se < second[j-inc])) {
                  index [j] = index[j-inc];
                  second[j] = second[j-inc];
               } else {
                  break;
               }
            }
            index[j]  = ktemp;
            second[j] = se;
         }
      } else if (swapFirst  && !swapSecond) {
         for (Int_t k = inc; k < n; k++ ) {
            const Int_t ktemp = index[k];
            const Int_t fi = first[k];
            const Int_t se = second[ktemp];
            Int_t j;
            for (j = k; j >= inc; j -= inc) {
               if ( fi < first[j-inc] || (fi == first[j-inc] && se < second[ index[j-inc]])) {
                  index[j] = index[j-inc];
                  first[j] = first[j-inc];
               } else {
                  break;
               }
            }
            index[j] = ktemp;
            first[j] = fi;
         }
      } else { // Swap both
         for (Int_t k = inc; k < n; k++ ) {
            const Int_t ktemp = index[k];
            const Int_t fi = first [k];
            const Int_t se = second[k];
            Int_t j;
            for (j = k; j >= inc; j -= inc) {
               if ( fi < first[j-inc] || (fi == first[j-inc] && se < second[j-inc])) {
                  index [j] = index [j-inc];
                  first [j] = first [j-inc];
                  second[j] = second[j-inc];
               } else {
                  break;
               }
            }
            index[j]  = ktemp;
            first[j]  = fi;
            second[j] = se;
         }
      }
   }
}
 
////////////////////////////////////////////////////////////////////////////////
/// 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)
 
template<class Element>
TMatrixTBase<Element> &TMatrixTBase<Element>::SetMatrixArray(const Element *data,Option_t *option)
{
   R__ASSERT(IsValid());
 
   TString opt = option;
   opt.ToUpper();
 
   Element *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(Element));
 
   return *this;
}
 
////////////////////////////////////////////////////////////////////////////////
/// Check whether matrix is symmetric
 
template<class Element>
Bool_t TMatrixTBase<Element>::IsSymmetric() const
{
  R__ASSERT(IsValid());
 
   if ((fNrows != fNcols) || (fRowLwb != fColLwb))
      return kFALSE;
 
   const Element * 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;
}
 
////////////////////////////////////////////////////////////////////////////////
/// 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)
 
template<class Element>
void TMatrixTBase<Element>::GetMatrix2Array(Element *data,Option_t *option) const
{
   R__ASSERT(IsValid());
 
   TString opt = option;
   opt.ToUpper();
 
   const Element * 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(Element));
}
 
////////////////////////////////////////////////////////////////////////////////
/// Copy n elements from array v to row rown starting at column coln
 
template<class Element>
TMatrixTBase<Element> &TMatrixTBase<Element>::InsertRow(Int_t rown,Int_t coln,const Element *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 (gMatrixCheck) {
      if (arown >= fNrows || arown < 0) {
         Error("InsertRow","row %d out of matrix range",rown);
         return *this;
      }
 
      if (acoln >= fNcols || acoln < 0) {
         Error("InsertRow","column %d out of matrix range",coln);
         return *this;
      }
 
      if (acoln+nr > fNcols || nr < 0) {
         Error("InsertRow","row length %d out of range",nr);
         return *this;
      }
   }
 
   const Int_t off = arown*fNcols+acoln;
   Element * const elem = GetMatrixArray()+off;
   memcpy(elem,v,nr*sizeof(Element));
 
   return *this;
}
 
////////////////////////////////////////////////////////////////////////////////
/// Store in array v, n matrix elements of row rown starting at column coln
 
template<class Element>
void TMatrixTBase<Element>::ExtractRow(Int_t rown,Int_t coln,Element *v,Int_t n) const
{
   const Int_t arown = rown-fRowLwb;
   const Int_t acoln = coln-fColLwb;
   const Int_t nr = (n > 0) ? n : fNcols;
 
   if (gMatrixCheck) {
      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 Element * const elem = GetMatrixArray()+off;
   memcpy(v,elem,nr*sizeof(Element));
}
 
////////////////////////////////////////////////////////////////////////////////
/// 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]
 
template<class Element>
TMatrixTBase<Element> &TMatrixTBase<Element>::Shift(Int_t row_shift,Int_t col_shift)
{
   fRowLwb += row_shift;
   fColLwb += col_shift;
 
   return *this;
}
 
////////////////////////////////////////////////////////////////////////////////
/// Set matrix elements to zero
 
template<class Element>
TMatrixTBase<Element> &TMatrixTBase<Element>::Zero()
{
   R__ASSERT(IsValid());
   memset(this->GetMatrixArray(),0,fNelems*sizeof(Element));
 
   return *this;
}
 
////////////////////////////////////////////////////////////////////////////////
/// Take an absolute value of a matrix, i.e. apply Abs() to each element.
 
template<class Element>
TMatrixTBase<Element> &TMatrixTBase<Element>::Abs()
{
   R__ASSERT(IsValid());
 
         Element *ep = this->GetMatrixArray();
   const Element * const fp = ep+fNelems;
   while (ep < fp) {
      *ep = TMath::Abs(*ep);
      ep++;
   }
 
   return *this;
}
 
////////////////////////////////////////////////////////////////////////////////
/// Square each element of the matrix.
 
template<class Element>
TMatrixTBase<Element> &TMatrixTBase<Element>::Sqr()
{
   R__ASSERT(IsValid());
 
         Element *ep = this->GetMatrixArray();
   const Element * const fp = ep+fNelems;
   while (ep < fp) {
      *ep = (*ep) * (*ep);
      ep++;
   }
 
   return *this;
}
 
////////////////////////////////////////////////////////////////////////////////
/// Take square root of all elements.
 
template<class Element>
TMatrixTBase<Element> &TMatrixTBase<Element>::Sqrt()
{
   R__ASSERT(IsValid());
 
         Element *ep = this->GetMatrixArray();
   const Element * const fp = ep+fNelems;
   while (ep < fp) {
      *ep = TMath::Sqrt(*ep);
      ep++;
   }
 
   return *this;
}
 
////////////////////////////////////////////////////////////////////////////////
/// Make a unit matrix (matrix need not be a square one).
 
template<class Element>
TMatrixTBase<Element> &TMatrixTBase<Element>::UnitMatrix()
{
   R__ASSERT(IsValid());
 
   Element *ep = this->GetMatrixArray();
   memset(ep,0,fNelems*sizeof(Element));
   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;
}
 
////////////////////////////////////////////////////////////////////////////////
/// 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)
 
template<class Element>
TMatrixTBase<Element> &TMatrixTBase<Element>::NormByDiag(const TVectorT<Element> &v,Option_t *option)
{
   R__ASSERT(IsValid());
   R__ASSERT(v.IsValid());
 
   if (gMatrixCheck) {
      const Int_t nMax = TMath::Max(fNrows,fNcols);
      if (v.GetNoElements() < nMax) {
         Error("NormByDiag","vector shorter than matrix diagonal");
         return *this;
      }
   }
 
   TString opt(option);
   opt.ToUpper();
   const Int_t divide = (opt.Contains("D")) ? 1 : 0;
 
   const Element *pV = v.GetMatrixArray();
         Element *mp = this->GetMatrixArray();
 
   if (divide) {
      for (Int_t irow = 0; irow < fNrows; irow++) {
         if (pV[irow] != 0.0) {
            for (Int_t icol = 0; icol < fNcols; icol++) {
               if (pV[icol] != 0.0) {
                  const Element val = TMath::Sqrt(TMath::Abs(pV[irow]*pV[icol]));
                  *mp++ /= val;
               } else {
                  Error("NormbyDiag","vector element %d is zero",icol);
                  mp++;
               }
            }
         } else {
            Error("NormbyDiag","vector element %d is zero",irow);
            mp += fNcols;
         }
      }
   } else {
      for (Int_t irow = 0; irow < fNrows; irow++) {
         for (Int_t icol = 0; icol < fNcols; icol++) {
            const Element val = TMath::Sqrt(TMath::Abs(pV[irow]*pV[icol]));
            *mp++ *= val;
         }
      }
   }
 
   return *this;
}
 
////////////////////////////////////////////////////////////////////////////////
/// Row matrix norm, MAX{ SUM{ |M(i,j)|, over j}, over i}.
/// The norm is induced by the infinity vector norm.
 
template<class Element>
Element TMatrixTBase<Element>::RowNorm() const
{
   R__ASSERT(IsValid());
 
   const Element *       ep = GetMatrixArray();
   const Element * const fp = ep+fNelems;
         Element norm = 0;
 
   // Scan the matrix row-after-row
   while (ep < fp) {
      Element 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);
   }
 
   R__ASSERT(ep == fp);
 
   return norm;
}
 
////////////////////////////////////////////////////////////////////////////////
/// Column matrix norm, MAX{ SUM{ |M(i,j)|, over i}, over j}.
/// The norm is induced by the 1 vector norm.
 
template<class Element>
Element TMatrixTBase<Element>::ColNorm() const
{
   R__ASSERT(IsValid());
 
   const Element *       ep = GetMatrixArray();
   const Element * const fp = ep+fNcols;
         Element norm = 0;
 
   // Scan the matrix col-after-col
   while (ep < fp) {
      Element 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);
   }
 
   R__ASSERT(ep == fp);
 
   return norm;
}
 
////////////////////////////////////////////////////////////////////////////////
/// Square of the Euclidian norm, SUM{ m(i,j)^2 }.
 
template<class Element>
Element TMatrixTBase<Element>::E2Norm() const
{
   R__ASSERT(IsValid());
 
   const Element *       ep = GetMatrixArray();
   const Element * const fp = ep+fNelems;
         Element sum = 0;
 
   for ( ; ep < fp; ep++)
      sum += (*ep) * (*ep);
 
   return sum;
}
 
////////////////////////////////////////////////////////////////////////////////
/// Compute the number of elements != 0.0
 
template<class Element>
Int_t TMatrixTBase<Element>::NonZeros() const
{
   R__ASSERT(IsValid());
 
   Int_t nr_nonzeros = 0;
   const Element *ep = this->GetMatrixArray();
   const Element * const fp = ep+fNelems;
   while (ep < fp)
      if (*ep++ != 0.0) nr_nonzeros++;
 
   return nr_nonzeros;
}
 
////////////////////////////////////////////////////////////////////////////////
/// Compute sum of elements
 
template<class Element>
Element TMatrixTBase<Element>::Sum() const
{
   R__ASSERT(IsValid());
 
   Element sum = 0.0;
   const Element *ep = this->GetMatrixArray();
   const Element * const fp = ep+fNelems;
   while (ep < fp)
      sum += *ep++;
 
   return sum;
}
 
////////////////////////////////////////////////////////////////////////////////
/// return minimum matrix element value
 
template<class Element>
Element TMatrixTBase<Element>::Min() const
{
   R__ASSERT(IsValid());
 
   const Element * const ep = this->GetMatrixArray();
   const Int_t index = TMath::LocMin(fNelems,ep);
   return ep[index];
}
 
////////////////////////////////////////////////////////////////////////////////
/// return maximum vector element value
 
template<class Element>
Element TMatrixTBase<Element>::Max() const
{
   R__ASSERT(IsValid());
 
   const Element * const ep = this->GetMatrixArray();
   const Int_t index = TMath::LocMax(fNelems,ep);
   return ep[index];
}
 
////////////////////////////////////////////////////////////////////////////////
/// Draw this matrix
/// The histogram is named "TMatrixT" by default and no title
 
template<class Element>
void TMatrixTBase<Element>::Draw(Option_t *option)
{
   gROOT->ProcessLine(Form("THistPainter::PaintSpecialObjects((TObject*)0x%lx,\"%s\");",
                           (ULong_t)this, option));
}
 
////////////////////////////////////////////////////////////////////////////////
/// Print the matrix as a table of elements.
/// By default the format "%11.4g" is used to print one element.
/// One can specify an alternative format with eg
///  option ="f=  %6.2f  "
 
template<class Element>
void TMatrixTBase<Element>::Print(Option_t *option) const
{
   if (!IsValid()) {
      Error("Print","Matrix is invalid");
      return;
   }
 
   //build format
   const char *format = "%11.4g ";
   if (option) {
      const char *f = strstr(option,"f=");
      if (f) format = f+2;
   }
   char topbar[100];
   snprintf(topbar,100,format,123.456789);
   Int_t nch = strlen(topbar)+1;
   if (nch > 18) nch = 18;
   char ftopbar[20];
   for (Int_t i = 0; i < nch; i++) ftopbar[i] = ' ';
   Int_t nk = 1 + Int_t(TMath::Log10(fNcols));
   snprintf(ftopbar+nch/2,20-nch/2,"%s%dd","%",nk);
   Int_t nch2 = strlen(ftopbar);
   for (Int_t i = nch2; i < nch; i++) ftopbar[i] = ' ';
   ftopbar[nch] = '|';
   ftopbar[nch+1] = 0;
 
   printf("\n%dx%d matrix is as follows",fNrows,fNcols);
 
   Int_t cols_per_sheet = 5;
   if (nch <= 8) cols_per_sheet =10;
   const Int_t ncols  = fNcols;
   const Int_t nrows  = fNrows;
   const Int_t collwb = fColLwb;
   const Int_t rowlwb = fRowLwb;
   nk = 5+nch*TMath::Min(cols_per_sheet,fNcols);
   for (Int_t i = 0; i < nk; i++) topbar[i] = '-';
   topbar[nk] = 0;
   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(ftopbar,j+collwb-1);
      printf("\n%s\n",topbar);
      if (fNelems <= 0) continue;
      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(format,(*this)(i+rowlwb-1,j+collwb-1));
         printf("\n");
      }
   }
   printf("\n");
}
 
////////////////////////////////////////////////////////////////////////////////
/// Are all matrix elements equal to val?
 
template<class Element>
Bool_t TMatrixTBase<Element>::operator==(Element val) const
{
   R__ASSERT(IsValid());
 
   if (val == 0. && fNelems == 0)
      return kTRUE;
 
   const Element *       ep = GetMatrixArray();
   const Element * const fp = ep+fNelems;
   for (; ep < fp; ep++)
      if (!(*ep == val))
         return kFALSE;
 
   return kTRUE;
}
 
////////////////////////////////////////////////////////////////////////////////
/// Are all matrix elements not equal to val?
 
template<class Element>
Bool_t TMatrixTBase<Element>::operator!=(Element val) const
{
   R__ASSERT(IsValid());
 
   if (val == 0. && fNelems == 0)
      return kFALSE;
 
   const Element *       ep = GetMatrixArray();
   const Element * const fp = ep+fNelems;
   for (; ep < fp; ep++)
      if (!(*ep != val))
         return kFALSE;
 
   return kTRUE;
}
 
////////////////////////////////////////////////////////////////////////////////
/// Are all matrix elements < val?
 
template<class Element>
Bool_t TMatrixTBase<Element>::operator<(Element val) const
{
   R__ASSERT(IsValid());
 
   const Element *       ep = GetMatrixArray();
   const Element * const fp = ep+fNelems;
   for (; ep < fp; ep++)
      if (!(*ep < val))
         return kFALSE;
 
   return kTRUE;
}
 
////////////////////////////////////////////////////////////////////////////////
/// Are all matrix elements <= val?
 
template<class Element>
Bool_t TMatrixTBase<Element>::operator<=(Element val) const
{
   R__ASSERT(IsValid());
 
   const Element *       ep = GetMatrixArray();
   const Element * const fp = ep+fNelems;
   for (; ep < fp; ep++)
      if (!(*ep <= val))
         return kFALSE;
 
   return kTRUE;
}
 
////////////////////////////////////////////////////////////////////////////////
/// Are all matrix elements > val?
 
template<class Element>
Bool_t TMatrixTBase<Element>::operator>(Element val) const
{
   R__ASSERT(IsValid());
 
   const Element *       ep = GetMatrixArray();
   const Element * const fp = ep+fNelems;
   for (; ep < fp; ep++)
      if (!(*ep > val))
         return kFALSE;
 
   return kTRUE;
}
 
////////////////////////////////////////////////////////////////////////////////
/// Are all matrix elements >= val?
 
template<class Element>
Bool_t TMatrixTBase<Element>::operator>=(Element val) const
{
   R__ASSERT(IsValid());
 
   const Element *       ep = GetMatrixArray();
   const Element * const fp = ep+fNelems;
   for (; ep < fp; ep++)
      if (!(*ep >= val))
         return kFALSE;
 
   return kTRUE;
}
 
////////////////////////////////////////////////////////////////////////////////
/// Apply action to each matrix element
 
template<class Element>
TMatrixTBase<Element> &TMatrixTBase<Element>::Apply(const TElementActionT<Element> &action)
{
   R__ASSERT(IsValid());
 
   Element *ep = this->GetMatrixArray();
   const Element * const ep_last = ep+fNelems;
   while (ep < ep_last)
      action.Operation(*ep++);
 
   return *this;
}
 
////////////////////////////////////////////////////////////////////////////////
/// Apply action to each element of the matrix. To action the location
/// of the current element is passed.
 
template<class Element>
TMatrixTBase<Element> &TMatrixTBase<Element>::Apply(const TElementPosActionT<Element> &action)
{
   R__ASSERT(IsValid());
 
   Element *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++);
 
   R__ASSERT(ep == this->GetMatrixArray()+fNelems);
 
   return *this;
}
 
////////////////////////////////////////////////////////////////////////////////
/// Randomize matrix element values
 
template<class Element>
TMatrixTBase<Element> &TMatrixTBase<Element>::Randomize(Element alpha,Element beta,Double_t &seed)
{
   R__ASSERT(IsValid());
 
   const Element scale = beta-alpha;
   const Element shift = alpha/scale;
 
         Element *       ep = GetMatrixArray();
   const Element * const fp = ep+fNelems;
   while (ep < fp)
      *ep++ = scale*(Drand(seed)+shift);
 
   return *this;
}
 
////////////////////////////////////////////////////////////////////////////////
/// Check to see if two matrices are identical.
 
template<class Element>
Bool_t operator==(const TMatrixTBase<Element> &m1,const TMatrixTBase<Element> &m2)
{
   if (!AreCompatible(m1,m2)) return kFALSE;
   return (memcmp(m1.GetMatrixArray(),m2.GetMatrixArray(),
                   m1.GetNoElements()*sizeof(Element)) == 0);
}
 
////////////////////////////////////////////////////////////////////////////////
/// Square of the Euclidian norm of the difference between two matrices.
 
template<class Element>
Element E2Norm(const TMatrixTBase<Element> &m1,const TMatrixTBase<Element> &m2)
{
   if (gMatrixCheck && !AreCompatible(m1,m2)) {
      ::Error("E2Norm","matrices not compatible");
      return -1.0;
   }
 
   const Element *        mp1 = m1.GetMatrixArray();
   const Element *        mp2 = m2.GetMatrixArray();
   const Element * const fmp1 = mp1+m1.GetNoElements();
 
   Element sum = 0.0;
   for (; mp1 < fmp1; mp1++, mp2++)
      sum += (*mp1 - *mp2)*(*mp1 - *mp2);
 
   return sum;
}
 
////////////////////////////////////////////////////////////////////////////////
/// Check that matrice sm1 and m2 areboth valid and have identical shapes .
 
template<class Element1,class Element2>
Bool_t AreCompatible(const TMatrixTBase<Element1> &m1,const TMatrixTBase<Element2> &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;
}
 
////////////////////////////////////////////////////////////////////////////////
/// Compare two matrices and print out the result of the comparison.
 
template<class Element>
void Compare(const TMatrixTBase<Element> &m1,const TMatrixTBase<Element> &m2)
{
   if (!AreCompatible(m1,m2)) {
      Error("Compare(const TMatrixTBase<Element> &,const TMatrixTBase<Element> &)","matrices are incompatible");
      return;
   }
 
   printf("\n\nComparison of two TMatrices:\n");
 
   Element norm1  = 0;      // Norm of the Matrices
   Element norm2  = 0;
   Element ndiff  = 0;      // Norm of the difference
   Int_t   imax   = 0;      // For the elements that differ most
   Int_t   jmax   = 0;
   Element difmax = -1;
 
   for (Int_t i = m1.GetRowLwb(); i <= m1.GetRowUpb(); i++) {
      for (Int_t j = m1.GetColLwb(); j < m1.GetColUpb(); j++) {
         const Element mv1 = m1(i,j);
         const Element mv2 = m2(i,j);
         const Element 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 Element mv1 = m1(imax,jmax);
   const Element 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,(Element)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));
}
 
////////////////////////////////////////////////////////////////////////////////
/// Validate that all elements of matrix have value val within maxDevAllow.
 
template<class Element>
Bool_t VerifyMatrixValue(const TMatrixTBase<Element> &m,Element val,Int_t verbose,Element maxDevAllow)
{
   R__ASSERT(m.IsValid());
 
   if (m == 0)
      return kTRUE;
 
   Int_t   imax      = 0;
   Int_t   jmax      = 0;
   Element maxDevObs = 0;
 
   if (TMath::Abs(maxDevAllow) <= 0.0)
      maxDevAllow = std::numeric_limits<Element>::epsilon();
 
   for (Int_t i = m.GetRowLwb(); i <= m.GetRowUpb(); i++) {
      for (Int_t j = m.GetColLwb(); j <= m.GetColUpb(); j++) {
         const Element 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;
}
 
////////////////////////////////////////////////////////////////////////////////
/// Verify that elements of the two matrices are equal within MaxDevAllow .
 
template<class Element>
Bool_t VerifyMatrixIdentity(const TMatrixTBase<Element> &m1,const TMatrixTBase<Element> &m2,Int_t verbose,
                            Element maxDevAllow)
{
   if (!AreCompatible(m1,m2,verbose))
      return kFALSE;
 
   if (m1 == 0 && m2 == 0)
      return kTRUE;
 
   Int_t   imax      = 0;
   Int_t   jmax      = 0;
   Element maxDevObs = 0;
 
   if (TMath::Abs(maxDevAllow) <= 0.0)
      maxDevAllow = std::numeric_limits<Element>::epsilon();
 
   for (Int_t i = m1.GetRowLwb(); i <= m1.GetRowUpb(); i++) {
      for (Int_t j = m1.GetColLwb(); j <= m1.GetColUpb(); j++) {
         const Element 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;
}
 
////////////////////////////////////////////////////////////////////////////////
/// Stream an object of class TMatrixTBase<Element>.
 
template<class Element>
void TMatrixTBase<Element>::Streamer(TBuffer &R__b)
{
   if (R__b.IsReading()) {
      UInt_t R__s, R__c;
      Version_t R__v = R__b.ReadVersion(&R__s, &R__c);
      if (R__v > 1) {
         R__b.ReadClassBuffer(TMatrixTBase<Element>::Class(),this,R__v,R__s,R__c);
      } else {
         Error("TMatrixTBase<Element>::Streamer","Unknown version number: %d",R__v);
         R__ASSERT(0);
      }
      if (R__v < 4) MakeValid();
   } else {
      R__b.WriteClassBuffer(TMatrixTBase<Element>::Class(),this);
   }
}
 
// trick to return a reference to nan in operator(i,j_ when i,j are outside of range
template<class Element>
struct nan_value_t {
   static Element gNanValue;
};
template<>
Double_t nan_value_t<Double_t>::gNanValue = std::numeric_limits<Double_t>::quiet_NaN();
template<>
Float_t nan_value_t<Float_t>::gNanValue = std::numeric_limits<Float_t>::quiet_NaN();
 
template<class Element>
Element & TMatrixTBase<Element>::NaNValue()
{
   return nan_value_t<Element>::gNanValue;
}
 
 
template class TMatrixTBase<Float_t>;
 
template Bool_t   operator==          <Float_t>(const TMatrixFBase &m1,const TMatrixFBase &m2);
template Float_t  E2Norm              <Float_t>(const TMatrixFBase &m1,const TMatrixFBase &m2);
template Bool_t   AreCompatible<Float_t,Float_t>
                                               (const TMatrixFBase &m1,const TMatrixFBase &m2,Int_t verbose);
template Bool_t   AreCompatible<Float_t,Double_t>
                                               (const TMatrixFBase &m1,const TMatrixDBase &m2,Int_t verbose);
template void     Compare             <Float_t>(const TMatrixFBase &m1,const TMatrixFBase &m2);
template Bool_t   VerifyMatrixValue   <Float_t>(const TMatrixFBase &m,Float_t val,Int_t verbose,Float_t maxDevAllow);
template Bool_t   VerifyMatrixValue   <Float_t>(const TMatrixFBase &m,Float_t val);
template Bool_t   VerifyMatrixIdentity<Float_t>(const TMatrixFBase &m1,const TMatrixFBase &m2,
                                                Int_t verbose,Float_t maxDevAllowN);
template Bool_t   VerifyMatrixIdentity<Float_t>(const TMatrixFBase &m1,const TMatrixFBase &m2);
 
template class TMatrixTBase<Double_t>;
 
template Bool_t   operator==          <Double_t>(const TMatrixDBase &m1,const TMatrixDBase &m2);
template Double_t E2Norm              <Double_t>(const TMatrixDBase &m1,const TMatrixDBase &m2);
template Bool_t   AreCompatible<Double_t,Double_t>
                                               (const TMatrixDBase &m1,const TMatrixDBase &m2,Int_t verbose);
template Bool_t   AreCompatible<Double_t,Float_t>
                                               (const TMatrixDBase &m1,const TMatrixFBase &m2,Int_t verbose);
template void     Compare             <Double_t>(const TMatrixDBase &m1,const TMatrixDBase &m2);
template Bool_t   VerifyMatrixValue   <Double_t>(const TMatrixDBase &m,Double_t val,Int_t verbose,Double_t maxDevAllow);
template Bool_t   VerifyMatrixValue   <Double_t>(const TMatrixDBase &m,Double_t val);
template Bool_t   VerifyMatrixIdentity<Double_t>(const TMatrixDBase &m1,const TMatrixDBase &m2,
                                                 Int_t verbose,Double_t maxDevAllow);
template Bool_t   VerifyMatrixIdentity<Double_t>(const TMatrixDBase &m1,const TMatrixDBase &m2);