TDecompChol.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 TDecompChol
    \ingroup Matrix

 Cholesky Decomposition class

 Decompose a symmetric, positive definite matrix A = U^T * U

 where U is a upper triangular matrix

 The decomposition fails if a diagonal element of fU is <= 0, the
 matrix is not positive negative . The matrix fU is made invalid .

 fU has the same index range as A .
*/

#include "TDecompChol.h"
#include "TMath.h"

ClassImp(TDecompChol);

////////////////////////////////////////////////////////////////////////////////
/// Constructor for (nrows x nrows) matrix

TDecompChol::TDecompChol(Int_t nrows)
{
   fU.ResizeTo(nrows,nrows);
}

////////////////////////////////////////////////////////////////////////////////
/// Constructor for ([row_lwb..row_upb] x [row_lwb..row_upb]) matrix

TDecompChol::TDecompChol(Int_t row_lwb,Int_t row_upb)
{
   const Int_t nrows = row_upb-row_lwb+1;
   fRowLwb = row_lwb;
   fColLwb = row_lwb;
   fU.ResizeTo(row_lwb,row_lwb+nrows-1,row_lwb,row_lwb+nrows-1);
}

////////////////////////////////////////////////////////////////////////////////
/// Constructor for symmetric matrix A . Matrix should be positive definite

TDecompChol::TDecompChol(const TMatrixDSym &a,Double_t tol)
{
   R__ASSERT(a.IsValid());

   SetBit(kMatrixSet);
   fCondition = a.Norm1();
   fTol = a.GetTol();
   if (tol > 0)
      fTol = tol;

   fRowLwb = a.GetRowLwb();
   fColLwb = a.GetColLwb();
   fU.ResizeTo(a);
   fU = a;
}

////////////////////////////////////////////////////////////////////////////////
/// Constructor for general matrix A . Matrix should be symmetric positive definite

TDecompChol::TDecompChol(const TMatrixD &a,Double_t tol)
{
   R__ASSERT(a.IsValid());

   if (a.GetNrows() != a.GetNcols() || a.GetRowLwb() != a.GetColLwb()) {
      Error("TDecompChol(const TMatrixD &","matrix should be square");
      return;
   }

   SetBit(kMatrixSet);
   fCondition = a.Norm1();
   fTol = a.GetTol();
   if (tol > 0)
      fTol = tol;

   fRowLwb = a.GetRowLwb();
   fColLwb = a.GetColLwb();
   fU.ResizeTo(a);
   fU = a;
}

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

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

////////////////////////////////////////////////////////////////////////////////
/// Matrix A is decomposed in component U so that A = U^T * U
/// If the decomposition succeeds, bit kDecomposed is set , otherwise kSingular

Bool_t TDecompChol::Decompose()
{
   if (TestBit(kDecomposed)) return kTRUE;

   if ( !TestBit(kMatrixSet) ) {
      Error("Decompose()","Matrix has not been set");
      return kFALSE;
   }

   Int_t i,j,icol,irow;
   const Int_t     n  = fU.GetNrows();
         Double_t *pU = fU.GetMatrixArray();
   for (icol = 0; icol < n; icol++) {
      const Int_t rowOff = icol*n;

      //Compute fU(j,j) and test for non-positive-definiteness.
      Double_t ujj = pU[rowOff+icol];
      for (irow = 0; irow < icol; irow++) {
         const Int_t pos_ij = irow*n+icol;
         ujj -= pU[pos_ij]*pU[pos_ij];
      }
      if (ujj <= 0) {
         Error("Decompose()","matrix not positive definite");
         return kFALSE;
      }
      ujj = TMath::Sqrt(ujj);
      pU[rowOff+icol] = ujj;

      if (icol < n-1) {
         for (j = icol+1; j < n; j++) {
            for (i = 0; i < icol; i++) {
               const Int_t rowOff2 = i*n;
               pU[rowOff+j] -= pU[rowOff2+j]*pU[rowOff2+icol];
            }
         }
         for (j = icol+1; j < n; j++)
            pU[rowOff+j] /= ujj;
      }
   }

   for (irow = 0; irow < n; irow++) {
      const Int_t rowOff = irow*n;
      for (icol = 0; icol < irow; icol++)
         pU[rowOff+icol] = 0.;
   }

   SetBit(kDecomposed);

   return kTRUE;
}

////////////////////////////////////////////////////////////////////////////////
/// Reconstruct the original matrix using the decomposition parts

const TMatrixDSym TDecompChol::GetMatrix()
{
   if (TestBit(kSingular)) {
      Error("GetMatrix()","Matrix is singular");
      return TMatrixDSym();
   }
   if ( !TestBit(kDecomposed) ) {
      if (!Decompose()) {
         Error("GetMatrix()","Decomposition failed");
         return TMatrixDSym();
      }
   }

   return TMatrixDSym(TMatrixDSym::kAtA,fU);
}

////////////////////////////////////////////////////////////////////////////////
/// Set the matrix to be decomposed, decomposition status is reset.

void TDecompChol::SetMatrix(const TMatrixDSym &a)
{
   R__ASSERT(a.IsValid());

   ResetStatus();
   if (a.GetNrows() != a.GetNcols() || a.GetRowLwb() != a.GetColLwb()) {
      Error("SetMatrix(const TMatrixDSym &","matrix should be square");
      return;
   }

   SetBit(kMatrixSet);
   fCondition = -1.0;

   fRowLwb = a.GetRowLwb();
   fColLwb = a.GetColLwb();
   fU.ResizeTo(a);
   fU = a;
}

////////////////////////////////////////////////////////////////////////////////
/// Solve equations Ax=b assuming A has been factored by Cholesky. The factor U is
/// assumed to be in upper triang of fU. fTol is used to determine if diagonal
/// element is zero. The solution is returned in b.

Bool_t TDecompChol::Solve(TVectorD &b)
{
   R__ASSERT(b.IsValid());
   if (TestBit(kSingular)) {
      Error("Solve()","Matrix is singular");
      return kFALSE;
   }
   if ( !TestBit(kDecomposed) ) {
      if (!Decompose()) {
         Error("Solve()","Decomposition failed");
         return kFALSE;
      }
   }

   if (fU.GetNrows() != b.GetNrows() || fU.GetRowLwb() != b.GetLwb()) {
      Error("Solve(TVectorD &","vector and matrix incompatible");
      return kFALSE;
   }

   const Int_t n = fU.GetNrows();

   const Double_t *pU = fU.GetMatrixArray();
         Double_t *pb = b.GetMatrixArray();

   Int_t i;
   // step 1: Forward substitution on U^T
   for (i = 0; i < n; i++) {
      const Int_t off_i = i*n;
      if (pU[off_i+i] < fTol)
      {
         Error("Solve(TVectorD &b)","u[%d,%d]=%.4e < %.4e",i,i,pU[off_i+i],fTol);
         return kFALSE;
      }
      Double_t r = pb[i];
      for (Int_t j = 0; j < i; j++) {
         const Int_t off_j = j*n;
         r -= pU[off_j+i]*pb[j];
      }
      pb[i] = r/pU[off_i+i];
   }

   // step 2: Backward substitution on U
   for (i = n-1; i >= 0; i--) {
      const Int_t off_i = i*n;
      Double_t r = pb[i];
      for (Int_t j = i+1; j < n; j++)
         r -= pU[off_i+j]*pb[j];
      pb[i] = r/pU[off_i+i];
   }

   return kTRUE;
}

////////////////////////////////////////////////////////////////////////////////
/// Solve equations Ax=b assuming A has been factored by Cholesky. The factor U is
/// assumed to be in upper triang of fU. fTol is used to determine if diagonal
/// element is zero. The solution is returned in b.

Bool_t TDecompChol::Solve(TMatrixDColumn &cb)
{
   TMatrixDBase *b = const_cast<TMatrixDBase *>(cb.GetMatrix());
   R__ASSERT(b->IsValid());
   if (TestBit(kSingular)) {
      Error("Solve()","Matrix is singular");
      return kFALSE;
   }
   if ( !TestBit(kDecomposed) ) {
      if (!Decompose()) {
         Error("Solve()","Decomposition failed");
         return kFALSE;
      }
   }

   if (fU.GetNrows() != b->GetNrows() || fU.GetRowLwb() != b->GetRowLwb())
   {
      Error("Solve(TMatrixDColumn &cb","vector and matrix incompatible");
      return kFALSE;
   }

   const Int_t n = fU.GetNrows();

   const Double_t *pU  = fU.GetMatrixArray();
         Double_t *pcb = cb.GetPtr();
   const Int_t     inc = cb.GetInc();

   Int_t i;
   // step 1: Forward substitution U^T
   for (i = 0; i < n; i++) {
      const Int_t off_i  = i*n;
      const Int_t off_i2 = i*inc;
      if (pU[off_i+i] < fTol)
      {
         Error("Solve(TMatrixDColumn &cb)","u[%d,%d]=%.4e < %.4e",i,i,pU[off_i+i],fTol);
         return kFALSE;
      }
      Double_t r = pcb[off_i2];
      for (Int_t j = 0; j < i; j++) {
         const Int_t off_j = j*n;
         r -= pU[off_j+i]*pcb[j*inc];
      }
      pcb[off_i2] = r/pU[off_i+i];
   }

   // step 2: Backward substitution U
   for (i = n-1; i >= 0; i--) {
      const Int_t off_i  = i*n;
      const Int_t off_i2 = i*inc;
      Double_t r = pcb[off_i2];
      for (Int_t j = i+1; j < n; j++)
         r -= pU[off_i+j]*pcb[j*inc];
      pcb[off_i2] = r/pU[off_i+i];
   }

   return kTRUE;
}

////////////////////////////////////////////////////////////////////////////////
/// Matrix determinant det = d1*TMath::Power(2.,d2) is square of diagProd
/// of cholesky factor

void TDecompChol::Det(Double_t &d1,Double_t &d2)
{
   if ( !TestBit(kDetermined) ) {
      if ( !TestBit(kDecomposed) )
         Decompose();
      TDecompBase::Det(d1,d2);
      // square det as calculated by above
      fDet1 *= fDet1;
      fDet2 += fDet2;
      SetBit(kDetermined);
   }
   d1 = fDet1;
   d2 = fDet2;
}

////////////////////////////////////////////////////////////////////////////////
/// For a symmetric matrix A(m,m), its inverse A_inv(m,m) is returned .

Bool_t TDecompChol::Invert(TMatrixDSym &inv)
{
   if (inv.GetNrows() != GetNrows() || inv.GetRowLwb() != GetRowLwb()) {
      Error("Invert(TMatrixDSym &","Input matrix has wrong shape");
      return kFALSE;
   }

   inv.UnitMatrix();

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

   return status;
}

////////////////////////////////////////////////////////////////////////////////
/// For a symmetric matrix A(m,m), its inverse A_inv(m,m) is returned .

TMatrixDSym TDecompChol::Invert(Bool_t &status)
{
   const Int_t rowLwb = GetRowLwb();
   const Int_t rowUpb = rowLwb+GetNrows()-1;

   TMatrixDSym inv(rowLwb,rowUpb);
   inv.UnitMatrix();
   status = Invert(inv);

   return inv;
}

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

void TDecompChol::Print(Option_t *opt) const
{
   TDecompBase::Print(opt);
   fU.Print("fU");
}

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

TDecompChol &TDecompChol::operator=(const TDecompChol &source)
{
   if (this != &source) {
      TDecompBase::operator=(source);
      fU.ResizeTo(source.fU);
      fU = source.fU;
   }
   return *this;
}

////////////////////////////////////////////////////////////////////////////////
/// Solve min {(A . x - b)^T (A . x - b)} for vector x where
///   A : (m x n) matrix, m >= n
///   b : (m)     vector
///   x : (n)     vector

TVectorD NormalEqn(const TMatrixD &A,const TVectorD &b)
{
   TDecompChol ch(TMatrixDSym(TMatrixDSym::kAtA,A));
   Bool_t ok;
   return ch.Solve(TMatrixD(TMatrixD::kTransposed,A)*b,ok);
}

////////////////////////////////////////////////////////////////////////////////
/// Solve min {(A . x - b)^T W (A . x - b)} for vector x where
///   A : (m x n) matrix, m >= n
///   b : (m)     vector
///   x : (n)     vector
///   W : (m x m) weight matrix with W(i,j) = 1/std(i)^2  for i == j
///                                         = 0           for i != j

TVectorD NormalEqn(const TMatrixD &A,const TVectorD &b,const TVectorD &std)
{
   if (!AreCompatible(b,std)) {
      ::Error("NormalEqn","vectors b and std are not compatible");
      return TVectorD();
   }

   TMatrixD mAw = A;
   TVectorD mBw = b;
   for (Int_t irow = 0; irow < A.GetNrows(); irow++) {
      TMatrixDRow(mAw,irow) *= 1/std(irow);
      mBw(irow) /= std(irow);
   }
   TDecompChol ch(TMatrixDSym(TMatrixDSym::kAtA,mAw));
   Bool_t ok;
   return ch.Solve(TMatrixD(TMatrixD::kTransposed,mAw)*mBw,ok);
}

////////////////////////////////////////////////////////////////////////////////
/// Solve min {(A . X_j - B_j)^T (A . X_j - B_j)} for each column j in
/// B and X
///   A : (m x n ) matrix, m >= n
///   B : (m x nb) matrix, nb >= 1
///  mX : (n x nb) matrix

TMatrixD NormalEqn(const TMatrixD &A,const TMatrixD &B)
{
   TDecompChol ch(TMatrixDSym(TMatrixDSym::kAtA,A));
   TMatrixD mX(A,TMatrixD::kTransposeMult,B);
   ch.MultiSolve(mX);
   return mX;
}

////////////////////////////////////////////////////////////////////////////////
/// Solve min {(A . X_j - B_j)^T W (A . X_j - B_j)} for each column j in
/// B and X
/// ~~~
///   A : (m x n ) matrix, m >= n
///   B : (m x nb) matrix, nb >= 1
///  mX : (n x nb) matrix
///   W : (m x m) weight matrix with W(i,j) = 1/std(i)^2  for i == j
///                                         = 0           for i != j
/// ~~~

TMatrixD NormalEqn(const TMatrixD &A,const TMatrixD &B,const TVectorD &std)
{
   TMatrixD mAw = A;
   TMatrixD mBw = B;
   for (Int_t irow = 0; irow < A.GetNrows(); irow++) {
      TMatrixDRow(mAw,irow) *= 1/std(irow);
      TMatrixDRow(mBw,irow) *= 1/std(irow);
   }

   TDecompChol ch(TMatrixDSym(TMatrixDSym::kAtA,mAw));
   TMatrixD mX(mAw,TMatrixD::kTransposeMult,mBw);
   ch.MultiSolve(mX);
   return mX;
}