TDecompBase.cxx

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// @(#)root/matrix:$Id$
// Authors: Fons Rademakers, Eddy Offermann  Dec 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 TDecompBase
    \ingroup Matrix

 Decomposition Base class

 This class forms the base for all the decompositions methods in the
 linear algebra package .
 It or its derived classes have installed the methods to solve
 equations,invert matrices and calculate determinants while monitoring
 the accuracy.

 Each derived class has always the following methods available:

 #### Condition() :
   In an iterative scheme the condition number for matrix inversion is
   calculated . This number is of interest for estimating the accuracy
   of x in the equation Ax=b
   For example:
     A is a (10x10) Hilbert matrix which looks deceivingly innocent
     and simple, A(i,j) = 1/(i+j+1)
     b(i) = Sum_j A(i,j), so a sum of a row in A

     the solution is x(i) = 1. i=0,.,9

   However,
~~~
     TMatrixD m....; TVectorD b.....
     TDecompLU lu(m); lu.SetTol(1.0e-12); lu.Solve(b); b.Print()
~~~
   gives,

~~~
   {1.000,1.000,1.000,1.000,0.998,1.000,0.993,1.001,0.996,1.000}
~~~

   Looking at the condition number, this is in line with expected the
   accuracy . The condition number is 3.957e+12 . As a simple rule of
   thumb, a condition number of 1.0e+n means that you lose up to n
   digits of accuracy in a solution . Since doubles are stored with 15
   digits, we can expect the accuracy to be as small as 3 digits .

 #### Det(Double_t &d1,Double_t &d2)
   The determinant is d1*TMath::Power(2.,d2)
   Expressing the determinant this way makes under/over-flow very
   unlikely .

 #### Decompose()
   Here the actually decomposition is performed . One can change the
   matrix A after the decomposition constructor has been called
   without effecting the decomposition result

 #### Solve(TVectorD &b)
  Solve A x = b . x is supplied through the argument and replaced with
  the solution .

 #### TransSolve(TVectorD &b)
  Solve A^T x = b . x is supplied through the argument and replaced
  with the solution .

 #### MultiSolve(TMatrixD    &B)
  Solve A X = B . where X and are now matrices . X is supplied through
  the argument and replaced with the solution .

 #### Invert(TMatrixD &inv)
  This is of course just a call to MultiSolve with as input argument
  the unit matrix . Note that for a matrix a(m,n) with m > n  a
  pseudo-inverse is calculated .

 ### Tolerances and Scaling

 The tolerance parameter (which is a member of this base class) plays
 a crucial role in all operations of the decomposition classes . It
 gives the user a powerful tool to monitor and steer the operations
 Its default value is sqrt(epsilon) where 1+epsilon = 1

 If you do not want to be bothered by the following considerations,
 like in most other linear algebra packages, just set the tolerance
 with SetTol to an arbitrary small number .

 The tolerance number is used by each decomposition method to decide
 whether the matrix is near singular, except of course SVD which can
 handle singular matrices .
 For each decomposition this will be checked in a different way; in LU
 the matrix is considered singular when, at some point in the
 decomposition, a diagonal element < fTol . Therefore, we had to set in
 the example above of the (10x10) Hilbert, which is near singular, the
 tolerance on 10e-12 . (The fact that we have to set the tolerance <
 sqrt(epsilon) is a clear indication that we are losing precision .)

 If the matrix is flagged as being singular, operations with the
 decomposition will fail and will return matrices/vectors that are
 invalid .

 The observant reader will notice that by scaling the complete matrix
 by some small number the decomposition will detect a singular matrix .
 In this case the user will have to reduce the tolerance number by this
 factor . (For CPU time saving we decided not to make this an automatic
 procedure) .

 Code for this could look as follows:
~~~
 const Double_t max_abs = Abs(a).Max();
 const Double_t scale = TMath::Min(max_abs,1.);
 a.SetTol(a.GetTol()*scale);
~~~

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

#include "TDecompBase.h"
#include "TMath.h"
#include "TError.h"

ClassImp(TDecompBase)

////////////////////////////////////////////////////////////////////////////////
/// Default constructor

TDecompBase::TDecompBase()
{
   fTol       = std::numeric_limits<double>::epsilon();
   fDet1      = 0;
   fDet2      = 0;
   fCondition = -1.0;
   fRowLwb    = 0;
   fColLwb    = 0;
}

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

TDecompBase::TDecompBase(const TDecompBase &another) : TObject(another)
{
   *this = another;
}

////////////////////////////////////////////////////////////////////////////////

Int_t TDecompBase::Hager(Double_t &est,Int_t iter)
{
// Estimates lower bound for norm1 of inverse of A. Returns norm
// estimate in est.  iter sets the maximum number of iterations to be used.
// The return value indicates the number of iterations remaining on exit from
// loop, hence if this is non-zero the processed "converged".
// This routine uses Hager's Convex Optimisation Algorithm.
// See Applied Numerical Linear Algebra, p139 & SIAM J Sci Stat Comp 1984 pp 311-16

   est = -1.0;

   const TMatrixDBase &m = GetDecompMatrix();
   if (!m.IsValid())
      return iter;

   const Int_t n = m.GetNrows();

   TVectorD b(n); TVectorD y(n); TVectorD z(n);
   b = Double_t(1.0/n);
   Double_t inv_norm1 = 0.0;
   Bool_t stop = kFALSE;
   do {
      y = b;
      if (!Solve(y))
         return iter;
      const Double_t ynorm1 = y.Norm1();
      if ( ynorm1 <= inv_norm1 ) {
         stop = kTRUE;
      } else {
         inv_norm1 = ynorm1;
         Int_t i;
         for (i = 0; i < n; i++)
            z(i) = ( y(i) >= 0.0 ? 1.0 : -1.0 );
         if (!TransSolve(z))
            return iter;
         Int_t imax = 0;
         Double_t maxz = TMath::Abs(z(0));
         for (i = 1; i < n; i++) {
            const Double_t absz = TMath::Abs(z(i));
            if ( absz > maxz ) {
               maxz = absz;
               imax = i;
            }
         }
         stop = (maxz <= b*z);
         if (!stop) {
            b = 0.0;
            b(imax) = 1.0;
         }
      }
      iter--;
   } while (!stop && iter);
   est = inv_norm1;

   return iter;
}

////////////////////////////////////////////////////////////////////////////////

void TDecompBase::DiagProd(const TVectorD &diag,Double_t tol,Double_t &d1,Double_t &d2)
{
// Returns product of matrix diagonal elements in d1 and d2. d1 is a mantissa and d2
// an exponent for powers of 2. If matrix is in diagonal or triangular-matrix form this
// will be the determinant.
// Based on Bowler, Martin, Peters and Wilkinson in HACLA

   const Double_t zero      = 0.0;
   const Double_t one       = 1.0;
   const Double_t four      = 4.0;
   const Double_t sixteen   = 16.0;
   const Double_t sixteenth = 0.0625;

   const Int_t n = diag.GetNrows();

   Double_t t1 = 1.0;
   Double_t t2 = 0.0;
   Int_t niter2 =0;
   Int_t niter3 =0;
   for (Int_t i = 0; (((i < n) && (t1 !=zero ))); i++) {
      if (TMath::Abs(diag(i)) > tol) {
         t1 *= (Double_t) diag(i);
         while ( TMath::Abs(t1) < one) {
            t1 *= sixteenth;
            t2 += four;
            niter2++;
            if ( niter2>100) break;
         }
         while ( TMath::Abs(t1) < sixteenth)  {
            t1 *= sixteen;
            t2 -= four;
            niter3++;
            if (niter3>100) break;
         }
      } else {
         t1 = zero;
         t2 = zero;
      }
   }
   d1 = t1;
   d2 = t2;

   return;
}

////////////////////////////////////////////////////////////////////////////////
/// Matrix condition number

Double_t TDecompBase::Condition()
{
   if ( !TestBit(kCondition) ) {
      fCondition = -1;
      if (TestBit(kSingular))
         return fCondition;
      if ( !TestBit(kDecomposed) ) {
         if (!Decompose())
            return fCondition;
      }
      Double_t invNorm;
      if (Hager(invNorm))
         fCondition *= invNorm;
      else // no convergence in Hager
         Error("Condition()","Hager procedure did NOT converge");
      SetBit(kCondition);
   }
   return fCondition;
}

////////////////////////////////////////////////////////////////////////////////
/// Solve set of equations with RHS in columns of B

Bool_t TDecompBase::MultiSolve(TMatrixD &B)
{
   const TMatrixDBase &m = GetDecompMatrix();
   R__ASSERT(m.IsValid() && B.IsValid());

   const Int_t colLwb = B.GetColLwb();
   const Int_t colUpb = B.GetColUpb();
   Bool_t status = kTRUE;
   for (Int_t icol = colLwb; icol <= colUpb && status; icol++) {
      TMatrixDColumn b(B,icol);
      status &= Solve(b);
   }

   return status;
}

////////////////////////////////////////////////////////////////////////////////
/// Matrix determinant det = d1*TMath::Power(2.,d2)

void TDecompBase::Det(Double_t &d1,Double_t &d2)
{
   if ( !TestBit(kDetermined) ) {
      if ( !TestBit(kDecomposed) )
         Decompose();
      if (TestBit(kSingular) ) {
         fDet1 = 0.0;
         fDet2 = 0.0;
      } else {
         const TMatrixDBase &m = GetDecompMatrix();
         R__ASSERT(m.IsValid());
         TVectorD diagv(m.GetNrows());
         for (Int_t i = 0; i < diagv.GetNrows(); i++)
            diagv(i) = m(i,i);
         DiagProd(diagv,fTol,fDet1,fDet2);
      }
      SetBit(kDetermined);
   }
   d1 = fDet1;
   d2 = fDet2;
}

////////////////////////////////////////////////////////////////////////////////
/// Print class members

void TDecompBase::Print(Option_t * /*opt*/) const
{
   printf("fTol       = %.4e\n",fTol);
   printf("fDet1      = %.4e\n",fDet1);
   printf("fDet2      = %.4e\n",fDet2);
   printf("fCondition = %.4e\n",fCondition);
   printf("fRowLwb    = %d\n",fRowLwb);
   printf("fColLwb    = %d\n",fColLwb);
}

////////////////////////////////////////////////////////////////////////////////
/// Assignment operator

TDecompBase &TDecompBase::operator=(const TDecompBase &source)
{
   if (this != &source) {
      TObject::operator=(source);
      fTol       = source.fTol;
      fDet1      = source.fDet1;
      fDet2      = source.fDet2;
      fCondition = source.fCondition;
      fRowLwb    = source.fRowLwb;
      fColLwb    = source.fColLwb;
   }
   return *this;
}

////////////////////////////////////////////////////////////////////////////////
/// Define a Householder-transformation through the parameters up and b .

Bool_t DefHouseHolder(const TVectorD &vc,Int_t lp,Int_t l,Double_t &up,Double_t &beta,
                      Double_t tol)
{
   const Int_t n = vc.GetNrows();
   const Double_t * const vp = vc.GetMatrixArray();

   Double_t c = TMath::Abs(vp[lp]);
   Int_t i;
   for (i = l; i < n; i++)
      c = TMath::Max(TMath::Abs(vp[i]),c);

   up   = 0.0;
   beta = 0.0;
   if (c <= tol) {
//     Warning("DefHouseHolder","max vector=%.4e < %.4e",c,tol);
      return kFALSE;
   }

   Double_t sd = vp[lp]/c; sd *= sd;
   for (i = l; i < n; i++) {
      const Double_t tmp = vp[i]/c;
      sd += tmp*tmp;
   }

   Double_t vpprim = c*TMath::Sqrt(sd);
   if (vp[lp] > 0.) vpprim = -vpprim;
   up = vp[lp]-vpprim;
   beta = 1./(vpprim*up);

   return kTRUE;
}

////////////////////////////////////////////////////////////////////////////////
/// Apply Householder-transformation.

void ApplyHouseHolder(const TVectorD &vc,Double_t up,Double_t beta,
                      Int_t lp,Int_t l,TMatrixDRow &cr)
{
   const Int_t nv = vc.GetNrows();
   const Int_t nc = (cr.GetMatrix())->GetNcols();

   if (nv > nc) {
      Error("ApplyHouseHolder(const TVectorD &,..,TMatrixDRow &)","matrix row too short");
      return;
   }

   const Int_t inc_c = cr.GetInc();
   const Double_t * const vp = vc.GetMatrixArray();
         Double_t *       cp = cr.GetPtr();

   Double_t s = cp[lp*inc_c]*up;
   Int_t i;
   for (i = l; i < nv; i++)
      s += cp[i*inc_c]*vp[i];

   s = s*beta;
   cp[lp*inc_c] += s*up;
   for (i = l; i < nv; i++)
      cp[i*inc_c] += s*vp[i];
}

////////////////////////////////////////////////////////////////////////////////
/// Apply Householder-transformation.

void ApplyHouseHolder(const TVectorD &vc,Double_t up,Double_t beta,
                      Int_t lp,Int_t l,TMatrixDColumn &cc)
{
   const Int_t nv = vc.GetNrows();
   const Int_t nc = (cc.GetMatrix())->GetNrows();

   if (nv > nc) {
      Error("ApplyHouseHolder(const TVectorD &,..,TMatrixDRow &)","matrix column too short");
      return;
   }

   const Int_t inc_c = cc.GetInc();
   const Double_t * const vp = vc.GetMatrixArray();
         Double_t *       cp = cc.GetPtr();

   Double_t s = cp[lp*inc_c]*up;
   Int_t i;
   for (i = l; i < nv; i++)
      s += cp[i*inc_c]*vp[i];

   s = s*beta;
   cp[lp*inc_c] += s*up;
   for (i = l; i < nv; i++)
      cp[i*inc_c] += s*vp[i];
}

////////////////////////////////////////////////////////////////////////////////
///  Apply Householder-transformation.

void ApplyHouseHolder(const TVectorD &vc,Double_t up,Double_t beta,
                      Int_t lp,Int_t l,TVectorD &cv)
{
   const Int_t nv = vc.GetNrows();
   const Int_t nc = cv.GetNrows();

   if (nv > nc) {
      Error("ApplyHouseHolder(const TVectorD &,..,TVectorD &)","vector too short");
      return;
   }

   const Double_t * const vp = vc.GetMatrixArray();
         Double_t *       cp = cv.GetMatrixArray();

   Double_t s = cp[lp]*up;
   Int_t i;
   for (i = l; i < nv; i++)
      s += cp[i]*vp[i];

   s = s*beta;
   cp[lp] += s*up;
   for (i = l; i < nv; i++)
      cp[i] += s*vp[i];
}

////////////////////////////////////////////////////////////////////////////////
/// Defines a Givens-rotation by calculating 2 rotation parameters c and s.
/// The rotation is defined with the vector components v1 and v2.

void DefGivens(Double_t v1,Double_t v2,Double_t &c,Double_t &s)
{
   const Double_t a1 = TMath::Abs(v1);
   const Double_t a2 = TMath::Abs(v2);
   if (a1 > a2) {
      const Double_t w = v2/v1;
      const Double_t q = TMath::Hypot(1.,w);
      c = 1./q;
      if (v1 < 0.) c = -c;
      s = c*w;
   } else {
      if (v2 != 0) {
         const Double_t w = v1/v2;
         const Double_t q = TMath::Hypot(1.,w);
         s = 1./q;
         if (v2 < 0.) s = -s;
         c = s*w;
      } else {
         c = 1.;
         s = 0.;
      }
   }
}

////////////////////////////////////////////////////////////////////////////////
/// Define and apply a Givens-rotation by calculating 2 rotation
/// parameters c and s. The rotation is defined with and applied to the vector
/// components v1 and v2.

void DefAplGivens(Double_t &v1,Double_t &v2,Double_t &c,Double_t &s)
{
   const Double_t a1 = TMath::Abs(v1);
   const Double_t a2 = TMath::Abs(v2);
   if (a1 > a2) {
      const Double_t w = v2/v1;
      const Double_t q = TMath::Hypot(1.,w);
      c = 1./q;
      if (v1 < 0.) c = -c;
      s  = c*w;
      v1 = a1*q;
      v2 = 0.;
   } else {
      if (v2 != 0) {
         const Double_t w = v1/v2;
         const Double_t q = TMath::Hypot(1.,w);
         s = 1./q;
         if (v2 < 0.) s = -s;
         c  = s*w;
         v1 = a2*q;
         v2 = 0.;
      } else {
         c = 1.;
         s = 0.;
      }
   }
}

////////////////////////////////////////////////////////////////////////////////
/// Apply a Givens transformation as defined by c and s to the vector components
/// v1 and v2 .

void ApplyGivens(Double_t &z1,Double_t &z2,Double_t c,Double_t s)
{
   const Double_t w = z1*c+z2*s;
   z2 = -z1*s+z2*c;
   z1 = w;
}