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TDecompBase Class Referenceabstract
Math » Matrix Linear Algebra

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

Definition at line 33 of file TDecompBase.h.

Public Member Functions

 TDecompBase ()
 Default constructor. More...
 
 TDecompBase (const TDecompBase &another)
 Copy constructor. More...
 
virtual ~TDecompBase ()
 
virtual Double_t Condition ()
 Matrix condition number. More...
 
virtual Bool_t Decompose ()=0
 
virtual void Det (Double_t &d1, Double_t &d2)
 Matrix determinant det = d1*TMath::Power(2.,d2) More...
 
Int_t GetColLwb () const
 
Double_t GetCondition () const
 
Double_t GetDet1 () const
 
Double_t GetDet2 () const
 
virtual Int_t GetNcols () const =0
 
virtual Int_t GetNrows () const =0
 
Int_t GetRowLwb () const
 
Double_t GetTol () const
 
virtual Bool_t MultiSolve (TMatrixD &B)
 Solve set of equations with RHS in columns of B. More...
 
TDecompBase & operator= (const TDecompBase &source)
 Assignment operator. More...
 
void Print (Option_t *opt="") const
 Print class members. More...
 
Double_t SetTol (Double_t tol)
 
virtual Bool_t Solve (TVectorD &b)=0
 
virtual TVectorD Solve (const TVectorD &b, Bool_t &ok)=0
 
virtual Bool_t Solve (TMatrixDColumn &b)=0
 
virtual Bool_t TransSolve (TVectorD &b)=0
 
virtual TVectorD TransSolve (const TVectorD &b, Bool_t &ok)=0
 
virtual Bool_t TransSolve (TMatrixDColumn &b)=0
 

Protected Types

enum  { kWorkMax = 100 }
 
enum  EMatrixDecompStat {
  kInit = BIT(14), kPatternSet = BIT(15), kValuesSet = BIT(16), kMatrixSet = BIT(17),
  kDecomposed = BIT(18), kDetermined = BIT(19), kCondition = BIT(20), kSingular = BIT(21)
}
 

Protected Member Functions

virtual const TMatrixDBase & GetDecompMatrix () const =0
 
Int_t Hager (Double_t &est, Int_t iter=5)
 
void ResetStatus ()
 

Static Protected Member Functions

static void DiagProd (const TVectorD &diag, Double_t tol, Double_t &d1, Double_t &d2)
 

Protected Attributes

Int_t fColLwb
 
Double_t fCondition
 
Double_t fDet1
 
Double_t fDet2
 
Int_t fRowLwb
 
Double_t fTol
 

#include <TDecompBase.h>

Inheritance diagram for TDecompBase:
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Member Enumeration Documentation

◆ anonymous enum

anonymous enum
protected
Enumerator
kWorkMax 

Definition at line 60 of file TDecompBase.h.

◆ EMatrixDecompStat

enum TDecompBase::EMatrixDecompStat
protected
Enumerator
kInit 
kPatternSet 
kValuesSet 
kMatrixSet 
kDecomposed 
kDetermined 
kCondition 
kSingular 

Definition at line 49 of file TDecompBase.h.

Constructor & Destructor Documentation

◆ TDecompBase() [1/2]

TDecompBase::TDecompBase ( )

Default constructor.

Definition at line 130 of file TDecompBase.cxx.

◆ TDecompBase() [2/2]

TDecompBase::TDecompBase ( const TDecompBase &  another)

Copy constructor.

Definition at line 143 of file TDecompBase.cxx.

◆ ~TDecompBase()

virtual TDecompBase::~TDecompBase ( )
inlinevirtual

Definition at line 65 of file TDecompBase.h.

Member Function Documentation

◆ Condition()

Double_t TDecompBase::Condition ( )
virtual

Matrix condition number.

Reimplemented in TDecompSVD.

Definition at line 257 of file TDecompBase.cxx.

◆ Decompose()

virtual Bool_t TDecompBase::Decompose ( )
pure virtual

Implemented in TDecompSparse, TDecompSVD, TDecompQRH, TDecompLU, TDecompBK, and TDecompChol.

◆ Det()

void TDecompBase::Det ( Double_t &  d1,
Double_t &  d2 
)
virtual

Matrix determinant det = d1*TMath::Power(2.,d2)

Reimplemented in TDecompSparse, TDecompSVD, TDecompQRH, TDecompLU, TDecompBK, and TDecompChol.

Definition at line 299 of file TDecompBase.cxx.

◆ DiagProd()

void TDecompBase::DiagProd ( const TVectorD &  diag,
Double_t  tol,
Double_t &  d1,
Double_t &  d2 
)
staticprotected

Definition at line 209 of file TDecompBase.cxx.

◆ GetColLwb()

Int_t TDecompBase::GetColLwb ( ) const
inline

Definition at line 74 of file TDecompBase.h.

◆ GetCondition()

Double_t TDecompBase::GetCondition ( ) const
inline

Definition at line 70 of file TDecompBase.h.

◆ GetDecompMatrix()

virtual const TMatrixDBase& TDecompBase::GetDecompMatrix ( ) const
protectedpure virtual

Implemented in TDecompSparse, TDecompSVD, TDecompLU, TDecompQRH, TDecompBK, and TDecompChol.

◆ GetDet1()

Double_t TDecompBase::GetDet1 ( ) const
inline

Definition at line 68 of file TDecompBase.h.

◆ GetDet2()

Double_t TDecompBase::GetDet2 ( ) const
inline

Definition at line 69 of file TDecompBase.h.

◆ GetNcols()

virtual Int_t TDecompBase::GetNcols ( ) const
pure virtual

Implemented in TDecompSparse, TDecompSVD, TDecompLU, TDecompQRH, TDecompBK, and TDecompChol.

◆ GetNrows()

virtual Int_t TDecompBase::GetNrows ( ) const
pure virtual

Implemented in TDecompSparse, TDecompSVD, TDecompLU, TDecompQRH, TDecompBK, and TDecompChol.

◆ GetRowLwb()

Int_t TDecompBase::GetRowLwb ( ) const
inline

Definition at line 73 of file TDecompBase.h.

◆ GetTol()

Double_t TDecompBase::GetTol ( ) const
inline

Definition at line 67 of file TDecompBase.h.

◆ Hager()

Int_t TDecompBase::Hager ( Double_t &  est,
Int_t  iter = 5 
)
protected

Definition at line 150 of file TDecompBase.cxx.

◆ MultiSolve()

Bool_t TDecompBase::MultiSolve ( TMatrixD &  B)
virtual

Solve set of equations with RHS in columns of B.

Definition at line 280 of file TDecompBase.cxx.

◆ operator=()

TDecompBase & TDecompBase::operator= ( const TDecompBase &  source)

Assignment operator.

Definition at line 337 of file TDecompBase.cxx.

◆ Print()

void TDecompBase::Print ( Option_t *  opt = "") const

Print class members.

Definition at line 324 of file TDecompBase.cxx.

◆ ResetStatus()

void TDecompBase::ResetStatus ( )
inlineprotected

Definition at line 43 of file TDecompBase.h.

◆ SetTol()

Double_t TDecompBase::SetTol ( Double_t  tol)
inline

Definition at line 96 of file TDecompBase.h.

◆ Solve() [1/3]

virtual Bool_t TDecompBase::Solve ( TVectorD &  b)
pure virtual

Implemented in TDecompSparse, TDecompSVD, TDecompQRH, TDecompLU, TDecompBK, and TDecompChol.

◆ Solve() [2/3]

virtual TVectorD TDecompBase::Solve ( const TVectorD &  b,
Bool_t &  ok 
)
pure virtual

Implemented in TDecompSparse, TDecompSVD, TDecompQRH, TDecompLU, TDecompBK, and TDecompChol.

◆ Solve() [3/3]

virtual Bool_t TDecompBase::Solve ( TMatrixDColumn &  b)
pure virtual

Implemented in TDecompSparse, TDecompSVD, TDecompQRH, TDecompLU, TDecompBK, and TDecompChol.

◆ TransSolve() [1/3]

virtual Bool_t TDecompBase::TransSolve ( TVectorD &  b)
pure virtual

Implemented in TDecompSparse, TDecompSVD, TDecompQRH, TDecompLU, TDecompBK, and TDecompChol.

◆ TransSolve() [2/3]

virtual TVectorD TDecompBase::TransSolve ( const TVectorD &  b,
Bool_t &  ok 
)
pure virtual

Implemented in TDecompSparse, TDecompSVD, TDecompQRH, TDecompLU, TDecompBK, and TDecompChol.

◆ TransSolve() [3/3]

virtual Bool_t TDecompBase::TransSolve ( TMatrixDColumn &  b)
pure virtual

Implemented in TDecompSparse, TDecompSVD, TDecompQRH, TDecompLU, TDecompBK, and TDecompChol.

Member Data Documentation

◆ fColLwb

Int_t TDecompBase::fColLwb
protected

Definition at line 41 of file TDecompBase.h.

◆ fCondition

Double_t TDecompBase::fCondition
protected

Definition at line 39 of file TDecompBase.h.

◆ fDet1

Double_t TDecompBase::fDet1
protected

Definition at line 37 of file TDecompBase.h.

◆ fDet2

Double_t TDecompBase::fDet2
protected

Definition at line 38 of file TDecompBase.h.

◆ fRowLwb

Int_t TDecompBase::fRowLwb
protected

Definition at line 40 of file TDecompBase.h.

◆ fTol

Double_t TDecompBase::fTol
protected

Definition at line 36 of file TDecompBase.h.

Libraries for TDecompBase:
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The documentation for this class was generated from the following files: