ROOT   6.10/09 Reference Guide
TDecompBase Class Referenceabstract

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...

TDecompBaseoperator= (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

Public Member Functions inherited from TObject
TObject ()
TObject constructor. More...

TObject (const TObject &object)
TObject copy ctor. More...

virtual ~TObject ()
TObject destructor. More...

void AbstractMethod (const char *method) const
Use this method to implement an "abstract" method that you don't want to leave purely abstract. More...

Append graphics object to current pad. More...

virtual void Browse (TBrowser *b)
Browse object. May be overridden for another default action. More...

virtual const char * ClassName () const
Returns name of class to which the object belongs. More...

virtual void Clear (Option_t *="")

virtual TObjectClone (const char *newname="") const
Make a clone of an object using the Streamer facility. More...

virtual Int_t Compare (const TObject *obj) const
Compare abstract method. More...

virtual void Copy (TObject &object) const
Copy this to obj. More...

virtual void Delete (Option_t *option="")
Delete this object. More...

virtual Int_t DistancetoPrimitive (Int_t px, Int_t py)
Computes distance from point (px,py) to the object. More...

virtual void Draw (Option_t *option="")
Default Draw method for all objects. More...

virtual void DrawClass () const
Draw class inheritance tree of the class to which this object belongs. More...

virtual TObjectDrawClone (Option_t *option="") const
Draw a clone of this object in the current selected pad for instance with: gROOT->SetSelectedPad(gPad). More...

virtual void Dump () const
Dump contents of object on stdout. More...

virtual void Error (const char *method, const char *msgfmt,...) const
Issue error message. More...

virtual void Execute (const char *method, const char *params, Int_t *error=0)
Execute method on this object with the given parameter string, e.g. More...

virtual void Execute (TMethod *method, TObjArray *params, Int_t *error=0)
Execute method on this object with parameters stored in the TObjArray. More...

virtual void ExecuteEvent (Int_t event, Int_t px, Int_t py)
Execute action corresponding to an event at (px,py). More...

virtual void Fatal (const char *method, const char *msgfmt,...) const
Issue fatal error message. More...

virtual TObjectFindObject (const char *name) const
Must be redefined in derived classes. More...

virtual TObjectFindObject (const TObject *obj) const
Must be redefined in derived classes. More...

virtual Option_tGetDrawOption () const
Get option used by the graphics system to draw this object. More...

virtual const char * GetIconName () const
Returns mime type name of object. More...

virtual const char * GetName () const
Returns name of object. More...

virtual char * GetObjectInfo (Int_t px, Int_t py) const
Returns string containing info about the object at position (px,py). More...

virtual Option_tGetOption () const

virtual const char * GetTitle () const
Returns title of object. More...

virtual UInt_t GetUniqueID () const
Return the unique object id. More...

virtual Bool_t HandleTimer (TTimer *timer)
Execute action in response of a timer timing out. More...

virtual ULong_t Hash () const
Return hash value for this object. More...

virtual void Info (const char *method, const char *msgfmt,...) const
Issue info message. More...

virtual Bool_t InheritsFrom (const char *classname) const
Returns kTRUE if object inherits from class "classname". More...

virtual Bool_t InheritsFrom (const TClass *cl) const
Returns kTRUE if object inherits from TClass cl. More...

virtual void Inspect () const
Dump contents of this object in a graphics canvas. More...

void InvertBit (UInt_t f)

virtual Bool_t IsEqual (const TObject *obj) const
Default equal comparison (objects are equal if they have the same address in memory). More...

virtual Bool_t IsFolder () const
Returns kTRUE in case object contains browsable objects (like containers or lists of other objects). More...

Bool_t IsOnHeap () const

virtual Bool_t IsSortable () const

Bool_t IsZombie () const

virtual void ls (Option_t *option="") const
The ls function lists the contents of a class on stdout. More...

void MayNotUse (const char *method) const
Use this method to signal that a method (defined in a base class) may not be called in a derived class (in principle against good design since a child class should not provide less functionality than its parent, however, sometimes it is necessary). More...

virtual Bool_t Notify ()
This method must be overridden to handle object notification. More...

void Obsolete (const char *method, const char *asOfVers, const char *removedFromVers) const
Use this method to declare a method obsolete. More...

void operator delete (void *ptr)
Operator delete. More...

void operator delete[] (void *ptr)
Operator delete []. More...

voidoperator new (size_t sz)

voidoperator new (size_t sz, void *vp)

voidoperator new[] (size_t sz)

voidoperator new[] (size_t sz, void *vp)

TObjectoperator= (const TObject &rhs)
TObject assignment operator. More...

virtual void Paint (Option_t *option="")
This method must be overridden if a class wants to paint itself. More...

virtual void Pop ()
Pop on object drawn in a pad to the top of the display list. More...

virtual Int_t Read (const char *name)
Read contents of object with specified name from the current directory. More...

virtual void RecursiveRemove (TObject *obj)
Recursively remove this object from a list. More...

void ResetBit (UInt_t f)

virtual void SaveAs (const char *filename="", Option_t *option="") const
Save this object in the file specified by filename. More...

virtual void SavePrimitive (std::ostream &out, Option_t *option="")
Save a primitive as a C++ statement(s) on output stream "out". More...

void SetBit (UInt_t f, Bool_t set)
Set or unset the user status bits as specified in f. More...

void SetBit (UInt_t f)

virtual void SetDrawOption (Option_t *option="")
Set drawing option for object. More...

virtual void SetUniqueID (UInt_t uid)
Set the unique object id. More...

virtual void SysError (const char *method, const char *msgfmt,...) const
Issue system error message. More...

Bool_t TestBit (UInt_t f) const

Int_t TestBits (UInt_t f) const

virtual void UseCurrentStyle ()
Set current style settings in this object This function is called when either TCanvas::UseCurrentStyle or TROOT::ForceStyle have been invoked. More...

virtual void Warning (const char *method, const char *msgfmt,...) const
Issue warning message. More...

virtual Int_t Write (const char *name=0, Int_t option=0, Int_t bufsize=0)
Write this object to the current directory. More...

virtual Int_t Write (const char *name=0, Int_t option=0, Int_t bufsize=0) const
Write this object to the current directory. More...

## 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 TMatrixDBaseGetDecompMatrix () const =0

Int_t Hager (Double_t &est, Int_t iter=5)

void ResetStatus ()

Protected Member Functions inherited from TObject
virtual void DoError (int level, const char *location, const char *fmt, va_list va) const
Interface to ErrorHandler (protected). More...

void MakeZombie ()

## 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

Public Types inherited from TObject
enum  { kIsOnHeap = 0x01000000, kNotDeleted = 0x02000000, kZombie = 0x04000000, kBitMask = 0x00ffffff }

enum  { kSingleKey = BIT(0), kOverwrite = BIT(1), kWriteDelete = BIT(2) }

enum  EStatusBits {
kCanDelete = BIT(0), kMustCleanup = BIT(3), kObjInCanvas = BIT(3), kIsReferenced = BIT(4),
kHasUUID = BIT(5), kCannotPick = BIT(6), kNoContextMenu = BIT(8), kInvalidObject = BIT(13)
}

Static Public Member Functions inherited from TObject
static Long_t GetDtorOnly ()
Return destructor only flag. More...

static Bool_t GetObjectStat ()
Get status of object stat flag. More...

static void SetDtorOnly (void *obj)
Set destructor only flag. More...

static void SetObjectStat (Bool_t stat)
Turn on/off tracking of objects in the TObjectTable. More...

#include <TDecompBase.h>

Inheritance diagram for TDecompBase:
[legend]

## ◆ anonymous enum

 anonymous enum
protected
Enumerator
kWorkMax

Definition at line 60 of file TDecompBase.h.

## ◆ EMatrixDecompStat

 protected
Enumerator
kInit
kPatternSet
kValuesSet
kMatrixSet
kDecomposed
kDetermined
kCondition
kSingular

Definition at line 49 of file TDecompBase.h.

## ◆ 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.

## ◆ 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

## ◆ 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

## ◆ 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

## ◆ GetNrows()

 virtual Int_t TDecompBase::GetNrows ( ) const
pure virtual

## ◆ 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
virtual

Print class members.

Reimplemented from TObject.

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

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

## ◆ Solve() [2/3]

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

## ◆ Solve() [3/3]

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

## ◆ TransSolve() [1/3]

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

## ◆ TransSolve() [2/3]

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

## ◆ TransSolve() [3/3]

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

## ◆ 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.

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