Matrix Inversion examples

#include "Riostream.h"
#include "TMatrixD.h"
#include "TVectorD.h"
#include "TDecompLU.h"
#include "TDecompSVD.h"

// This macro shows several ways to invert a matrix . Each  method
// is a trade-off between accuracy of the inversion and speed.
// Which method to chose depends on "how well-behaved" the matrix is.
// This is best checked through a call to Condition(), available in each
// decomposition class. A second possibilty (less preferred) would be to
// check the determinant
//
// You can run this script with different matrix sizes

void invertMatrix(Int_t msize=6)
{
  if (msize < 2 || msize > 10) {
    cout << "2 <= msize <= 10" <<endl;
    return;
  }
  cout << "--------------------------------------------------------" <<endl;
  cout << "Inversion results for a ("<<msize<<","<<msize<<") matrix" <<endl;
  cout << "For each inversion procedure we check the maxmimum size " <<endl;
  cout << "of the off-diagonal elements of Inv(A) * A              " <<endl;
  cout << "--------------------------------------------------------" <<endl;

  TMatrixD H_square = THilbertMatrixD(msize,msize);

//  1. InvertFast(Double_t *det=0)
//   This method is only available for the matrix class TMatrixD/F, because an
//   symmetric matrix is not necessarily symmetric after inversion .
//   It is identical to Invert() for sizes > 6 x 6 but for smaller sizes, the
//   inversion is performed according to Cramer's rule by explicitly calculating
//   all Jacobi's sub-determinants . For instance for a 6 x 6 matrix this means:
//    # of 5 x 5 determinant : 36 
//    # of 4 x 4 determinant : 75 
//    # of 3 x 3 determinant : 80 
//    # of 2 x 2 determinant : 45    (see TMatrixD/FCramerInv.cxx)
//
//    The only "quality" control in this process is to check whether the 6 x 6
//    determinant is unequal 0 . But speed gains are significant compared to Invert() ,
//    upto an order of magnitude for sizes <= 4 x 4
//
//    The inversion is done "in place", so the original matrix will be overwritten
//    If a pointer to a Double_t is supplied the determinant is calculated
//

  cout << "1. Use .InvertFast(&det)" <<endl;
  if (msize > 6)
    cout << " for ("<<msize<<","<<msize<<") this is identical to .Invert(&det)" <<endl;

  Double_t det1;
  TMatrixD H1 = H_square;
  H1.InvertFast(&det1);

  // Get the maximum off-diagonal matrix value . One way to do this is to set the
  // diagonal to zero .

  TMatrixD U1(H1,TMatrixD::kMult,H_square);
  TMatrixDDiag diag1(U1); diag1 = 0.0;
  const Double_t U1_max_offdiag = (U1.Abs()).Max();
  cout << "  Maximum off-diagonal = " << U1_max_offdiag << endl;
  cout << "  Determinant          = " << det1 <<endl;

// 2. Invert(Double_t *det=0)
//   Also only available for TMatrixD/F . Again the inversion is performed in place .
//   It consists out of a sequence of calls related to the LU decomposition:
//    - The matrix is decomposed using a scheme according to Crout which involves
//      "implicit partial pivoting", see for instance Num. Recip. (we have also available
//      a decomposition scheme that does not the scaling and is therefore even slightly
//      faster but less stable)
//      With each decomposition, a tolerance has to be specified . If this tolerance
//      requirement is not met, the matrix is regarded as being singular. The value
//      passed to this decomposition, is the data member fTol of the matrix . Its
//      default value is DBL_EPSILON, which is defined as the smallest nuber so that
//      1+DBL_EPSILON > 1
//    - The last step is a standard forward/backward substitution .
//
//   It is important to realize that both InvertFast() and Invert() are "one-shot" deals , speed
//   comes at a price . If something goes wrong because the matrix is (near) singular, you have
//   overwritten your original matrix and  no factorization is available anymore to get more
//   information like condition number or change the tolerance number .
//
//   All other calls in the matrix classes involving inversion like the ones with the "smart"
//   constructors (kInverted,kInvMult...) use this inversion method .
//

  cout << "2. Use .Invert(&det)" <<endl;
  cout << "  If the determinant is < 2^-52 (="<<TMath::Power(2,-52)<<"), a warning is generated" <<endl;

  Double_t det2;
  TMatrixD H2 = H_square;
  H2.Invert(&det2);

  TMatrixD U2(H2,TMatrixD::kMult,H_square);
  TMatrixDDiag diag2(U2); diag2 = 0.0;
  const Double_t U2_max_offdiag = (U2.Abs()).Max();
  cout << "  Maximum off-diagonal = " << U2_max_offdiag << endl;
  cout << "  Determinant          = " << det2 <<endl;

// 3. Inversion through LU decomposition
//   The (default) algorithms used are similar to 2. (Not identical because in 2, the whole
//   calculation is done "in-place". Here the orginal matrix is copied (so more memory
//   management => slower) and several operations can be performed without having to repeat
//   the decomposition step .
//   Inverting a matrix is nothing else than solving a set of equations where the rhs is given
//   by the unit matrix, so the steps to take are identical to those solving a linear equation :
//

  cout << "3. Use TDecompLU" <<endl;

  TMatrixD H3 = H_square;
  TDecompLU lu(H_square);

  // Any operation that requires a decomposition will trigger it . The class keeps
  // an internal state so that following operations will not perform the decomposition again
  // unless the matrix is changed through SetMatrix(..)
  // One might want to proceed more cautiously by invoking first Decompose() and check its
  // return value before proceeding....

  lu.Invert(H3);
  Double_t d1_lu; Double_t d2_lu;
  lu.Det(d1_lu,d2_lu);
  Double_t det3 = d1_lu*TMath::Power(2.,d2_lu);

  TMatrixD U3(H3,TMatrixD::kMult,H_square);
  TMatrixDDiag diag3(U3); diag3 = 0.0;
  const Double_t U3_max_offdiag = (U3.Abs()).Max();
  cout << "  Maximum off-diagonal = " << U3_max_offdiag << endl;
  cout << "  Determinant          = " << det3 <<endl;

// 4. Inversion through SVD decomposition
//   For SVD and QRH, the (n x m) matrix does only have to fulfill n >=m . In case n > m
//   a pseudo-inverse is calculated
  cout << "4. Use TDecompSVD on non-square matrix" <<endl;

  TMatrixD H_nsquare = THilbertMatrixD(msize,msize-1);

  TDecompSVD svd(H_nsquare);

  TMatrixD H4 = svd.Invert();
  Double_t d1_svd; Double_t d2_svd;
  svd.Det(d1_svd,d2_svd);
  Double_t det4 = d1_svd*TMath::Power(2.,d2_svd);

  TMatrixD U4(H4,TMatrixD::kMult,H_nsquare);
  TMatrixDDiag diag4(U4); diag4 = 0.0;
  const Double_t U4_max_offdiag = (U4.Abs()).Max();
  cout << "  Maximum off-diagonal = " << U4_max_offdiag << endl;
  cout << "  Determinant          = " << det4 <<endl;
}