TDecompBase

Class Description

                                                                       
 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