TDecompQRH.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 TDecompQRH
    \ingroup Matrix
 
 QR Decomposition class
 
 Decompose  a general (m x n) matrix A into A = fQ' fR H   where
 
~~~
  fQ : (m x n) - internal Q' matrix (not orthoginal)
  fR : (n x n) - upper triangular matrix
  H  : HouseHolder matrix which is stored through
  fUp: (n) - vector with Householder up's
  fW : (n) - vector with Householder beta's
~~~
 
  If row/column index of A starts at (rowLwb,colLwb) then
  the decomposed matrices start from :
~~~
  fQ'  : (rowLwb,0)
  fR  : (0,colLwb)
  and the decomposed vectors start from :
  fUp : (0)
  fW  : (0)
~~~
 
   In order to get the QR dcomposition of A (i.e. A = QR )
   The orthoginal matrix Q needs to be computed from the internal Q' and
   the up's and beta's vector defining the Householder transformation
 
   The orthogonal Q matrix is returned to the user by calling the
   function TDecompQRH::GetOrthogonalMatrix()
 
 Errors arise from formation of reflectors i.e. singularity .
 Note it attempts to handle the cases where the nRow <= nCol .
*/
 
#include "TDecompQRH.h"
#include "TError.h" // For R__ASSERT
 
////////////////////////////////////////////////////////////////////////////////
/// Constructor for (nrows x ncols) matrix
 
TDecompQRH::TDecompQRH(Int_t nrows,Int_t ncols)
{
   if (nrows < ncols) {
      Error("TDecompQRH(Int_t,Int_t","matrix rows should be >= columns");
      return;
   }
 
   fQ.ResizeTo(nrows,ncols);
   fR.ResizeTo(ncols,ncols);
   if (nrows <= ncols) {
      fW.ResizeTo(nrows);
      fUp.ResizeTo(nrows);
   } else {
      fW.ResizeTo(ncols);
      fUp.ResizeTo(ncols);
   }
}
 
////////////////////////////////////////////////////////////////////////////////
/// Constructor for ([row_lwb..row_upb] x [col_lwb..col_upb]) matrix
 
TDecompQRH::TDecompQRH(Int_t row_lwb,Int_t row_upb,Int_t col_lwb,Int_t col_upb)
{
   const Int_t nrows = row_upb-row_lwb+1;
   const Int_t ncols = col_upb-col_lwb+1;
 
   if (nrows < ncols) {
      Error("TDecompQRH(Int_t,Int_t,Int_t,Int_t","matrix rows should be >= columns");
      return;
   }
 
   fRowLwb = row_lwb;
   fColLwb = col_lwb;
 
   fQ.ResizeTo(nrows,ncols);
   fR.ResizeTo(ncols,ncols);
   if (nrows <= ncols) {
      fW.ResizeTo(nrows);
      fUp.ResizeTo(nrows);
   } else {
      fW.ResizeTo(ncols);
      fUp.ResizeTo(ncols);
   }
}
 
////////////////////////////////////////////////////////////////////////////////
/// Constructor for general matrix A .
 
TDecompQRH::TDecompQRH(const TMatrixD &a,Double_t tol)
{
   R__ASSERT(a.IsValid());
   if (a.GetNrows() < a.GetNcols()) {
      Error("TDecompQRH(const TMatrixD &","matrix rows should be >= columns");
      return;
   }
 
   SetBit(kMatrixSet);
   fCondition = a.Norm1();
   fTol = a.GetTol();
   if (tol > 0.0)
      fTol = tol;
 
   fRowLwb = a.GetRowLwb();
   fColLwb = a.GetColLwb();
   const Int_t nRow = a.GetNrows();
   const Int_t nCol = a.GetNcols();
 
   fQ.ResizeTo(nRow,nCol);
   memcpy(fQ.GetMatrixArray(),a.GetMatrixArray(),nRow*nCol*sizeof(Double_t));
   fR.ResizeTo(nCol,nCol);
   if (nRow <= nCol) {
      fW.ResizeTo(nRow);
      fUp.ResizeTo(nRow);
   } else {
      fW.ResizeTo(nCol);
      fUp.ResizeTo(nCol);
   }
}
 
////////////////////////////////////////////////////////////////////////////////
/// Copy constructor
 
TDecompQRH::TDecompQRH(const TDecompQRH &another) : TDecompBase(another)
{
   *this = another;
}
 
////////////////////////////////////////////////////////////////////////////////
/// QR decomposition of matrix a by Householder transformations,
///  see Golub & Loan first edition p41 & Sec 6.2.
/// First fR is returned in upper triang of fQ and diagR. fQ returned in
/// 'u-form' in lower triang of fQ and fW, the latter containing the
///  "Householder betas".
/// If the decomposition succeeds, bit kDecomposed is set , otherwise kSingular
 
Bool_t TDecompQRH::Decompose()
{
   if (TestBit(kDecomposed)) return kTRUE;
 
   if ( !TestBit(kMatrixSet) ) {
      Error("Decompose()","Matrix has not been set");
      return kFALSE;
   }
 
   const Int_t nRow   = this->GetNrows();
   const Int_t nCol   = this->GetNcols();
   const Int_t rowLwb = this->GetRowLwb();
   const Int_t colLwb = this->GetColLwb();
 
   TVectorD diagR;
   Double_t work[kWorkMax];
   if (nCol > kWorkMax) diagR.ResizeTo(nCol);
   else                 diagR.Use(nCol,work);
 
   if (QRH(fQ,diagR,fUp,fW,fTol)) {
      for (Int_t i = 0; i < nRow; i++) {
         const Int_t ic = (i < nCol) ? i : nCol;
         for (Int_t j = ic ; j < nCol; j++)
            fR(i,j) = fQ(i,j);
      }
      TMatrixDDiag diag(fR); diag = diagR;
 
      fQ.Shift(rowLwb,0);
      fR.Shift(0,colLwb);
 
      SetBit(kDecomposed);
   }
 
   return kTRUE;
}
 
////////////////////////////////////////////////////////////////////////////////
/// Decomposition function .
 
Bool_t TDecompQRH::QRH(TMatrixD &q,TVectorD &diagR,TVectorD &up,TVectorD &w,Double_t tol)
{
   const Int_t nRow = q.GetNrows();
   const Int_t nCol = q.GetNcols();
 
   const Int_t n = (nRow <= nCol) ? nRow-1 : nCol;
 
   for (Int_t k = 0 ; k < n ; k++) {
      const TVectorD qc_k = TMatrixDColumn_const(q,k);
      if (!DefHouseHolder(qc_k,k,k+1,up(k),w(k),tol))
         return kFALSE;
      diagR(k) = qc_k(k)-up(k);
      if (k < nCol-1) {
         // Apply HouseHolder to sub-matrix
         for (Int_t j = k+1; j < nCol; j++) {
            TMatrixDColumn qc_j = TMatrixDColumn(q,j);
            ApplyHouseHolder(qc_k,up(k),w(k),k,k+1,qc_j);
         }
      }
   }
 
   if (nRow <= nCol) {
      diagR(nRow-1) = q(nRow-1,nRow-1);
      up(nRow-1) = 0;
      w(nRow-1)  = 0;
   }
 
   return kTRUE;
}
 
////////////////////////////////////////////////////////////////////////////////
/// Set matrix to be decomposed
 
void TDecompQRH::SetMatrix(const TMatrixD &a)
{
   R__ASSERT(a.IsValid());
 
   ResetStatus();
   if (a.GetNrows() < a.GetNcols()) {
      Error("TDecompQRH(const TMatrixD &","matrix rows should be >= columns");
      return;
   }
 
   SetBit(kMatrixSet);
   fCondition = a.Norm1();
 
   fRowLwb = a.GetRowLwb();
   fColLwb = a.GetColLwb();
   const Int_t nRow = a.GetNrows();
   const Int_t nCol = a.GetNcols();
 
   fQ.ResizeTo(nRow,nCol);
   memcpy(fQ.GetMatrixArray(),a.GetMatrixArray(),nRow*nCol*sizeof(Double_t));
   fR.ResizeTo(nCol,nCol);
   if (nRow <= nCol) {
      fW.ResizeTo(nRow);
      fUp.ResizeTo(nRow);
   } else {
      fW.ResizeTo(nCol);
      fUp.ResizeTo(nCol);
   }
}
 
////////////////////////////////////////////////////////////////////////////////
/// Solve Ax=b assuming the QR form of A is stored in fR,fQ and fW, but assume b
/// has *not* been transformed.  Solution returned in b.
 
Bool_t TDecompQRH::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 (fQ.GetNrows() != b.GetNrows() || fQ.GetRowLwb() != b.GetLwb()) {
      Error("Solve(TVectorD &","vector and matrix incompatible");
      return kFALSE;
   }
 
   const Int_t nQRow = fQ.GetNrows();
   const Int_t nQCol = fQ.GetNcols();
 
   // Calculate  Q^T.b
   const Int_t nQ = (nQRow <= nQCol) ? nQRow-1 : nQCol;
   for (Int_t k = 0; k < nQ; k++) {
      const TVectorD qc_k = TMatrixDColumn_const(fQ,k);
      ApplyHouseHolder(qc_k,fUp(k),fW(k),k,k+1,b);
   }
 
   const Int_t nRCol = fR.GetNcols();
 
   const Double_t *pR = fR.GetMatrixArray();
         Double_t *pb = b.GetMatrixArray();
 
   // Backward substitution
   for (Int_t i = nRCol-1; i >= 0; i--) {
      const Int_t off_i = i*nRCol;
      Double_t r = pb[i];
      for (Int_t j = i+1; j < nRCol; j++)
         r -= pR[off_i+j]*pb[j];
      if (TMath::Abs(pR[off_i+i]) < fTol)
      {
         Error("Solve(TVectorD &)","R[%d,%d]=%.4e < %.4e",i,i,pR[off_i+i],fTol);
         return kFALSE;
      }
      pb[i] = r/pR[off_i+i];
   }
 
   return kTRUE;
}
 
////////////////////////////////////////////////////////////////////////////////
/// Solve Ax=b assuming the QR form of A is stored in fR,fQ and fW, but assume b
/// has *not* been transformed.  Solution returned in b.
 
Bool_t TDecompQRH::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 (fQ.GetNrows() != b->GetNrows() || fQ.GetRowLwb() != b->GetRowLwb())
   {
      Error("Solve(TMatrixDColumn &","vector and matrix incompatible");
      return kFALSE;
   }
 
   const Int_t nQRow = fQ.GetNrows();
   const Int_t nQCol = fQ.GetNcols();
 
   // Calculate  Q^T.b
   const Int_t nQ = (nQRow <= nQCol) ? nQRow-1 : nQCol;
   for (Int_t k = 0; k < nQ; k++) {
      const TVectorD qc_k = TMatrixDColumn_const(fQ,k);
      ApplyHouseHolder(qc_k,fUp(k),fW(k),k,k+1,cb);
   }
 
   const Int_t nRCol = fR.GetNcols();
 
   const Double_t *pR  = fR.GetMatrixArray();
         Double_t *pcb = cb.GetPtr();
   const Int_t     inc = cb.GetInc();
 
   // Backward substitution
   for (Int_t i = nRCol-1; i >= 0; i--) {
      const Int_t off_i  = i*nRCol;
      const Int_t off_i2 = i*inc;
      Double_t r = pcb[off_i2];
      for (Int_t j = i+1; j < nRCol; j++)
         r -= pR[off_i+j]*pcb[j*inc];
      if (TMath::Abs(pR[off_i+i]) < fTol)
      {
         Error("Solve(TMatrixDColumn &)","R[%d,%d]=%.4e < %.4e",i,i,pR[off_i+i],fTol);
         return kFALSE;
      }
      pcb[off_i2] = r/pR[off_i+i];
   }
 
   return kTRUE;
}
 
////////////////////////////////////////////////////////////////////////////////
/// Solve A^T x=b assuming the QR form of A is stored in fR,fQ and fW, but assume b
/// has *not* been transformed.  Solution returned in b.
 
Bool_t TDecompQRH::TransSolve(TVectorD &b)
{
   R__ASSERT(b.IsValid());
   if (TestBit(kSingular)) {
      Error("TransSolve()","Matrix is singular");
      return kFALSE;
   }
   if ( !TestBit(kDecomposed) ) {
      if (!Decompose()) {
         Error("TransSolve()","Decomposition failed");
         return kFALSE;
      }
   }
 
   if (fQ.GetNrows() != fQ.GetNcols() || fQ.GetRowLwb() != fQ.GetColLwb()) {
      Error("TransSolve(TVectorD &","matrix should be square");
      return kFALSE;
   }
 
   if (fR.GetNrows() != b.GetNrows() || fR.GetRowLwb() != b.GetLwb()) {
      Error("TransSolve(TVectorD &","vector and matrix incompatible");
      return kFALSE;
   }
 
   const Double_t *pR = fR.GetMatrixArray();
         Double_t *pb = b.GetMatrixArray();
 
   const Int_t nRCol = fR.GetNcols();
 
   // Backward substitution
   for (Int_t i = 0; i < nRCol; i++) {
      const Int_t off_i = i*nRCol;
      Double_t r = pb[i];
      for (Int_t j = 0; j < i; j++) {
         const Int_t off_j = j*nRCol;
         r -= pR[off_j+i]*pb[j];
      }
      if (TMath::Abs(pR[off_i+i]) < fTol)
      {
         Error("TransSolve(TVectorD &)","R[%d,%d]=%.4e < %.4e",i,i,pR[off_i+i],fTol);
         return kFALSE;
      }
      pb[i] = r/pR[off_i+i];
   }
 
   const Int_t nQRow = fQ.GetNrows();
 
   // Calculate  Q.b; it was checked nQRow == nQCol
   for (Int_t k = nQRow-1; k >= 0; k--) {
      const TVectorD qc_k = TMatrixDColumn_const(fQ,k);
      ApplyHouseHolder(qc_k,fUp(k),fW(k),k,k+1,b);
   }
 
   return kTRUE;
}
 
////////////////////////////////////////////////////////////////////////////////
/// Solve A^T x=b assuming the QR form of A is stored in fR,fQ and fW, but assume b
/// has *not* been transformed.  Solution returned in b.
 
Bool_t TDecompQRH::TransSolve(TMatrixDColumn &cb)
{
   TMatrixDBase *b = const_cast<TMatrixDBase *>(cb.GetMatrix());
   R__ASSERT(b->IsValid());
   if (TestBit(kSingular)) {
      Error("TransSolve()","Matrix is singular");
      return kFALSE;
   }
   if ( !TestBit(kDecomposed) ) {
      if (!Decompose()) {
         Error("TransSolve()","Decomposition failed");
         return kFALSE;
      }
   }
 
   if (fQ.GetNrows() != fQ.GetNcols() || fQ.GetRowLwb() != fQ.GetColLwb()) {
      Error("TransSolve(TMatrixDColumn &","matrix should be square");
      return kFALSE;
   }
 
   if (fR.GetNrows() != b->GetNrows() || fR.GetRowLwb() != b->GetRowLwb()) {
      Error("TransSolve(TMatrixDColumn &","vector and matrix incompatible");
      return kFALSE;
   }
 
   const Double_t *pR  = fR.GetMatrixArray();
         Double_t *pcb = cb.GetPtr();
   const Int_t     inc = cb.GetInc();
 
   const Int_t nRCol = fR.GetNcols();
 
   // Backward substitution
   for (Int_t i = 0; i < nRCol; i++) {
      const Int_t off_i  = i*nRCol;
      const Int_t off_i2 = i*inc;
      Double_t r = pcb[off_i2];
      for (Int_t j = 0; j < i; j++) {
         const Int_t off_j = j*nRCol;
         r -= pR[off_j+i]*pcb[j*inc];
      }
      if (TMath::Abs(pR[off_i+i]) < fTol)
      {
         Error("TransSolve(TMatrixDColumn &)","R[%d,%d]=%.4e < %.4e",i,i,pR[off_i+i],fTol);
         return kFALSE;
      }
      pcb[off_i2] = r/pR[off_i+i];
   }
 
   const Int_t nQRow = fQ.GetNrows();
 
   // Calculate  Q.b; it was checked nQRow == nQCol
   for (Int_t k = nQRow-1; k >= 0; k--) {
      const TVectorD qc_k = TMatrixDColumn_const(fQ,k);
      ApplyHouseHolder(qc_k,fUp(k),fW(k),k,k+1,cb);
   }
 
   return kTRUE;
}
 
////////////////////////////////////////////////////////////////////////////////
/// This routine calculates the absolute (!) value of the determinant
/// |det| = d1*TMath::Power(2.,d2)
 
void TDecompQRH::Det(Double_t &d1,Double_t &d2)
{
   if ( !TestBit(kDetermined) ) {
      if ( !TestBit(kDecomposed) )
        Decompose();
      if (TestBit(kSingular)) {
         fDet1 = 0.0;
         fDet2 = 0.0;
      } else
         TDecompBase::Det(d1,d2);
      SetBit(kDetermined);
   }
   d1 = fDet1;
   d2 = fDet2;
}
 
////////////////////////////////////////////////////////////////////////////////
/// For a matrix A(m,n), return the OtrhogonalMatrix Q such as
///    A = Q * R
///
///  Note that this Q is not th einternal fQ matrix obtained in the QRH decomposition, but can be computed
///  from the fQ and the up and beta vector's defining the Householder transformation
 
TMatrixD TDecompQRH::GetOrthogonalMatrix() const
{
   // apply HouseHolder transformation starting from the identity
   // Calculate  Q.b; it was checked nQRow == nQCol
 
   const Int_t nRow = this->GetNrows();
   const Int_t nCol = this->GetNcols();
   // remmber nCol <= nRow
   TMatrixD orthogQ(nRow, nCol);
   // start from identity matrix
   for (int i = 0; i < nCol; ++i)
      orthogQ(i, i) = 1;
 
 
   // apply the HouseHolder transformations for each column of Q
   for (int j = 0; j < nCol; ++j) {
      TMatrixDColumn b = TMatrixDColumn(orthogQ, j);
      int nQRow = fQ.GetNrows();
      for (Int_t k = nQRow - 1; k >= 0; k--) {
         const TVectorD qc_k = TMatrixDColumn_const(fQ, k);
         ApplyHouseHolder(qc_k, fUp(k), fW(k), k, k + 1, b);
      }
   }
   return orthogQ;
}
////////////////////////////////////////////////////////////////////////////////
/// For a matrix A(m,n), its inverse A_inv is defined as A * A_inv = A_inv * A = unit
/// The user should always supply a matrix of size (m x m) !
/// If m > n , only the (n x m) part of the returned (pseudo inverse) matrix
/// should be used .
 
Bool_t TDecompQRH::Invert(TMatrixD &inv)
{
   if (inv.GetNrows()  != GetNrows()  || inv.GetNcols()  != GetNrows() ||
       inv.GetRowLwb() != GetRowLwb() || inv.GetColLwb() != GetColLwb()) {
      Error("Invert(TMatrixD &","Input matrix has wrong shape");
      return kFALSE;
   }
 
   inv.UnitMatrix();
   const Bool_t status = MultiSolve(inv);
 
   return status;
}
 
////////////////////////////////////////////////////////////////////////////////
/// For a matrix A(m,n), its inverse A_inv is defined as A * A_inv = A_inv * A = unit
/// (n x m) Ainv is returned .
 
TMatrixD TDecompQRH::Invert(Bool_t &status)
{
   const Int_t rowLwb = GetRowLwb();
   const Int_t colLwb = GetColLwb();
   const Int_t rowUpb = rowLwb+GetNrows()-1;
   TMatrixD inv(rowLwb,rowUpb,colLwb,colLwb+GetNrows()-1);
   inv.UnitMatrix();
   status = MultiSolve(inv);
   inv.ResizeTo(rowLwb,rowLwb+GetNcols()-1,colLwb,colLwb+GetNrows()-1);
 
   return inv;
}
 
////////////////////////////////////////////////////////////////////////////////
/// Print the class members
 
void TDecompQRH::Print(Option_t *opt) const
{
   TDecompBase::Print(opt);
   fQ.Print("fQ");
   fR.Print("fR");
   fUp.Print("fUp");
   fW.Print("fW");
}
 
////////////////////////////////////////////////////////////////////////////////
/// Assignment operator
 
TDecompQRH &TDecompQRH::operator=(const TDecompQRH &source)
{
   if (this != &source) {
      TDecompBase::operator=(source);
      fQ.ResizeTo(source.fQ);
      fR.ResizeTo(source.fR);
      fUp.ResizeTo(source.fUp);
      fW.ResizeTo(source.fW);
      fQ  = source.fQ;
      fR  = source.fR;
      fUp = source.fUp;
      fW  = source.fW;
   }
   return *this;
}