TMatrixTBase<double>

Class Description

                                                                      
 Linear Algebra Package                                               
 ----------------------                                               
                                                                      
 The present package implements all the basic algorithms dealing      
 with vectors, matrices, matrix columns, rows, diagonals, etc.        
 In addition eigen-Vector analysis and several matrix decomposition   
 have been added (LU,QRH,Cholesky,Bunch-Kaufman and SVD) .            
 The decompositions are used in matrix inversion, equation solving.   
                                                                      
 For a dense matrix, elements are arranged in memory in a ROW-wise    
 fashion . For (n x m) matrices where n*m <=kSizeMax (=25 currently)  
 storage space is available on the stack, thus avoiding expensive     
 allocation/deallocation of heap space . However, this introduces of  
 course kSizeMax overhead for each matrix object . If this is an      
 issue recompile with a new appropriate value (>=0) for kSizeMax      
                                                                      
 Sparse matrices are also stored in row-wise fashion but additional   
 row/column information is stored, see TMatrixTSparse source for      
 additional details .                                                 
                                                                      
 Another way to assign and store matrix data is through Use           
 see for instance stressLinear.cxx file .                             
                                                                      
 Unless otherwise specified, matrix and vector indices always start   
 with 0, spanning up to the specified limit-1. However, there are     
 constructors to which one can specify aribtrary lower and upper      
 bounds, e.g. TMatrixD m(1,10,1,5) defines a matrix that ranges       
 from 1..10, 1..5 (a(1,1)..a(10,5)).                                  
                                                                      
 The present package provides all facilities to completely AVOID      
 returning matrices. Use "TMatrixD A(TMatrixD::kTransposed,B);"       
 and other fancy constructors as much as possible. If one really needs
 to return a matrix, return a TMatrixTLazy object instead. The        
 conversion is completely transparent to the end user, e.g.           
 "TMatrixT m = THaarMatrixT(5);" and _is_ efficient.                  
                                                                      
 Since TMatrixT et al. are fully integrated in ROOT, they of course   
 can be stored in a ROOT database.                                    
                                                                      
 For usage examples see $ROOTSYS/test/stressLinear.cxx                
                                                                      
 Acknowledgements                                                     
 ----------------                                                     
 1. Oleg E. Kiselyov                                                  
  First implementations were based on the his code . We have diverged 
  quite a bit since then but the ideas/code for lazy matrix and       
  "nested function" are 100% his .                                    
  You can see him and his code in action at http://okmij.org/ftp      
 2. Chris R. Birchenhall,                                             
  We adapted his idea of the implementation for the decomposition     
  classes instead of our messy installation of matrix inversion       
  His installation of matrix condition number, using an iterative     
  scheme using the Hage algorithm is worth looking at !               
  Chris has a nice writeup (matdoc.ps) on his matrix classes at       
   ftp://ftp.mcc.ac.uk/pub/matclass/                                  
 3. Mark Fischler and Steven Haywood of CLHEP                         
  They did the slave labor of spelling out all sub-determinants       
   for Cramer inversion  of (4x4),(5x5) and (6x6) matrices            
  The stack storage for small matrices was also taken from them       
 4. Roldan Pozo of TNT (http://math.nist.gov/tnt/)                    
  He converted the EISPACK routines for the eigen-vector analysis to  
  C++ . We started with his implementation                            
 5. Siegmund Brandt (http://siux00.physik.uni-siegen.de/~brandt/datan 
  We adapted his (very-well) documented SVD routines                  
                                                                      
 How to efficiently use this package                                  
 -----------------------------------                                  
                                                                      
 1. Never return complex objects (matrices or vectors)                
    Danger: For example, when the following snippet:                  
        TMatrixD foo(int n)                                           
        {                                                             
           TMatrixD foom(n,n); fill_in(foom); return foom;            
        }                                                             
        TMatrixD m = foo(5);                                          
    runs, it constructs matrix foo:foom, copies it onto stack as a    
    return value and destroys foo:foom. Return value (a matrix)       
    from foo() is then copied over to m (via a copy constructor),     
    and the return value is destroyed. So, the matrix constructor is  
    called 3 times and the destructor 2 times. For big matrices,      
    the cost of multiple constructing/copying/destroying of objects   
    may be very large. *Some* optimized compilers can cut down on 1   
    copying/destroying, but still it leaves at least two calls to     
    the constructor. Note, TMatrixDLazy (see below) can construct     
    TMatrixD m "inplace", with only a _single_ call to the            
    constructor.                                                      
                                                                      
 2. Use "two-address instructions"                                    
        "void TMatrixD::operator += (const TMatrixD &B);"             
    as much as possible.                                              
    That is, to add two matrices, it's much more efficient to write   
        A += B;                                                       
    than                                                              
        TMatrixD C = A + B;                                           
    (if both operand should be preserved,                             
        TMatrixD C = A; C += B;                                       
    is still better).                                                 
                                                                      
 3. Use glorified constructors when returning of an object seems      
    inevitable:                                                       
        "TMatrixD A(TMatrixD::kTransposed,B);"                        
        "TMatrixD C(A,TMatrixD::kTransposeMult,B);"                   
                                                                      
    like in the following snippet (from $ROOTSYS/test/vmatrix.cxx)    
    that verifies that for an orthogonal matrix T, T'T = TT' = E.     
                                                                      
    TMatrixD haar = THaarMatrixD(5);                                  
    TMatrixD unit(TMatrixD::kUnit,haar);                              
    TMatrixD haar_t(TMatrixD::kTransposed,haar);                      
    TMatrixD hth(haar,TMatrixD::kTransposeMult,haar);                 
    TMatrixD hht(haar,TMatrixD::kMult,haar_t);                        
    TMatrixD hht1 = haar; hht1 *= haar_t;                             
    VerifyMatrixIdentity(unit,hth);                                   
    VerifyMatrixIdentity(unit,hht);                                   
    VerifyMatrixIdentity(unit,hht1);                                  
                                                                      
 4. Accessing row/col/diagonal of a matrix without much fuss          
    (and without moving a lot of stuff around):                       
                                                                      
        TMatrixD m(n,n); TVectorD v(n); TMatrixDDiag(m) += 4;         
        v = TMatrixDRow(m,0);                                         
        TMatrixDColumn m1(m,1); m1(2) = 3; // the same as m(2,1)=3;   
    Note, constructing of, say, TMatrixDDiag does *not* involve any   
    copying of any elements of the source matrix.                     
                                                                      
 5. It's possible (and encouraged) to use "nested" functions          
    For example, creating of a Hilbert matrix can be done as follows: 
                                                                      
    void foo(const TMatrixD &m)                                       
    {                                                                 
       TMatrixD m1(TMatrixD::kZero,m);                                
       struct MakeHilbert : public TElementPosActionD {               
          void Operation(Double_t &element)                           
             { element = 1./(fI+fJ-1); }                              
       };                                                             
       m1.Apply(MakeHilbert());                                       
    }                                                                 
                                                                      
    of course, using a special method THilbertMatrixD() is            
    still more optimal, but not by a whole lot. And that's right,     
    class MakeHilbert is declared *within* a function and local to    
    that function. It means one can define another MakeHilbert class  
    (within another function or outside of any function, that is, in  
    the global scope), and it still will be OK. Note, this currently  
    is not yet supported by the interpreter CINT.                     
                                                                      
    Another example is applying of a simple function to each matrix   
    element:                                                          
                                                                      
    void foo(TMatrixD &m,TMatrixD &m1)                                
    {                                                                 
       typedef  double (*dfunc_t)(double);                            
       class ApplyFunction : public TElementActionD {                 
          dfunc_t fFunc;                                              
          void Operation(Double_t &element)                           
               { element=fFunc(element); }                            
        public:                                                       
          ApplyFunction(dfunc_t func):fFunc(func) {}                  
       };                                                             
       ApplyFunction x(TMath::Sin);                                   
       m.Apply(x);                                                    
    }                                                                 
                                                                      
    Validation code $ROOTSYS/test/vmatrix.cxx and vvector.cxx contain 
    a few more examples of that kind.                                 
                                                                      
 6. Lazy matrices: instead of returning an object return a "recipe"   
    how to make it. The full matrix would be rolled out only when     
    and where it's needed:                                            
       TMatrixD haar = THaarMatrixD(5);                               
    THaarMatrixD() is a *class*, not a simple function. However       
    similar this looks to a returning of an object (see note #1       
    above), it's dramatically different. THaarMatrixD() constructs a  
    TMatrixDLazy, an object of just a few bytes long. A special       
    "TMatrixD(const TMatrixDLazy &recipe)" constructor follows the    
    recipe and makes the matrix haar() right in place. No matrix      
    element is moved whatsoever!
TMatrixTBase<double>

class TMatrixTBase<double> : public TObject

Data Members

Class Description
