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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()
#define b(i)
Definition RSha256.hxx:100
TMarker m
Definition textangle.C:8

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);
#define a(i)
Definition RSha256.hxx:99
Short_t Min(Short_t a, Short_t b)
Returns the smallest of a and b.
Definition TMathBase.h:198

For usage examples see \$ROOTSYS/test/stressLinear.cxx

Definition at line 33 of file TDecompBase.h.

## Public Member Functions

TDecompBase ()
Default constructor.

TDecompBase (const TDecompBase &another)
Copy constructor.

~TDecompBase () override

virtual Double_t Condition ()
Matrix condition number.

virtual Bool_t Decompose ()=0

virtual void Det (Double_t &d1, Double_t &d2)
Matrix determinant det = d1*TMath::Power(2.,d2)

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

TClassIsA () const override

virtual Bool_t MultiSolve (TMatrixD &B)
Solve set of equations with RHS in columns of B.

TDecompBaseoperator= (const TDecompBase &source)
Assignment operator.

void Print (Option_t *opt="") const override
Print class members.

Double_t SetTol (Double_t tol)

virtual TVectorD Solve (const TVectorD &b, Bool_t &ok)=0

virtual Bool_t Solve (TMatrixDColumn &b)=0

virtual Bool_t Solve (TVectorD &b)=0

void Streamer (TBuffer &) override
Stream an object of class TObject.

void StreamerNVirtual (TBuffer &ClassDef_StreamerNVirtual_b)

virtual TVectorD TransSolve (const TVectorD &b, Bool_t &ok)=0

virtual Bool_t TransSolve (TMatrixDColumn &b)=0

virtual Bool_t TransSolve (TVectorD &b)=0

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

TObject (const TObject &object)
TObject copy ctor.

virtual ~TObject ()
TObject destructor.

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

Append graphics object to current pad.

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

ULong_t CheckedHash ()
Check and record whether this class has a consistent Hash/RecursiveRemove setup (*) and then return the regular Hash value for this object.

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

virtual void Clear (Option_t *="")

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

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

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

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

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

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

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

virtual TObjectDrawClone (Option_t *option="") const
Draw a clone of this object in the current selected pad with: gROOT->SetSelectedPad(c1).

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

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

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

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

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

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

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

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

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

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

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

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

virtual Option_tGetOption () const

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

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

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

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

Bool_t HasInconsistentHash () const
Return true is the type of this object is known to have an inconsistent setup for Hash and RecursiveRemove (i.e.

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

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

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

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

void InvertBit (UInt_t f)

Bool_t IsDestructed () const
IsDestructed.

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

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

R__ALWAYS_INLINE Bool_t IsOnHeap () const

virtual Bool_t IsSortable () const

R__ALWAYS_INLINE Bool_t IsZombie () const

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

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).

virtual Bool_t Notify ()
This method must be overridden to handle object notification (the base implementation is no-op).

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

void operator delete (void *ptr)
Operator delete.

void operator delete (void *ptr, void *vp)
Only called by placement new when throwing an exception.

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

void operator delete[] (void *ptr, void *vp)
Only called by placement new[] when throwing an exception.

void * operator new (size_t sz)

void * operator new (size_t sz, void *vp)

void * operator new[] (size_t sz)

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

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

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

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

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

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

void ResetBit (UInt_t f)

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

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

void SetBit (UInt_t f)

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

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

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

void StreamerNVirtual (TBuffer &ClassDef_StreamerNVirtual_b)

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

R__ALWAYS_INLINE 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.

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

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

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

## Static Public Member Functions

static TClassClass ()

static const char * Class_Name ()

static constexpr Version_t Class_Version ()

static const char * DeclFileName ()

Static Public Member Functions inherited from TObject
static TClassClass ()

static const char * Class_Name ()

static constexpr Version_t Class_Version ()

static const char * DeclFileName ()

static Longptr_t GetDtorOnly ()
Return destructor only flag.

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

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

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

## Protected Types

enum  { kWorkMax = 100 }

enum  EMatrixDecompStat {
kInit = (1ULL << ( 14 )) , kPatternSet = (1ULL << ( 15 )) , kValuesSet = (1ULL << ( 16 )) , kMatrixSet = (1ULL << ( 17 )) ,
kDecomposed = (1ULL << ( 18 )) , kDetermined = (1ULL << ( 19 )) , kCondition = (1ULL << ( 20 )) , kSingular = (1ULL << ( 21 ))
}

Protected Types inherited from TObject
enum  { kOnlyPrepStep = (1ULL << ( 3 )) }

## 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).

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 , kInconsistent = 0x08000000 ,
}

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

enum  EDeprecatedStatusBits { kObjInCanvas = (1ULL << ( 3 )) }

enum  EStatusBits {
kCanDelete = (1ULL << ( 0 )) , kMustCleanup = (1ULL << ( 3 )) , kIsReferenced = (1ULL << ( 4 )) , kHasUUID = (1ULL << ( 5 )) ,
kCannotPick = (1ULL << ( 6 )) , kNoContextMenu = (1ULL << ( 8 )) , kInvalidObject = (1ULL << ( 13 ))
}

#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()

 TDecompBase::~TDecompBase ( )
inlineoverride

Definition at line 65 of file TDecompBase.h.

## ◆ Class()

 static TClass * TDecompBase::Class ( )
static
Returns
TClass describing this class

## ◆ Class_Name()

 static const char * TDecompBase::Class_Name ( )
static
Returns
Name of this class

## ◆ Class_Version()

 static constexpr Version_t TDecompBase::Class_Version ( )
inlinestaticconstexpr
Returns
Version of this class

Definition at line 93 of file TDecompBase.h.

## ◆ Condition()

 Double_t TDecompBase::Condition ( )
virtual

Matrix condition number.

Reimplemented in TDecompSVD.

Definition at line 257 of file TDecompBase.cxx.

## ◆ DeclFileName()

 static const char * TDecompBase::DeclFileName ( )
inlinestatic
Returns
Name of the file containing the class declaration

Definition at line 93 of file TDecompBase.h.

## ◆ 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 TDecompBK, TDecompSparse, TDecompChol, TDecompLU, TDecompQRH, and TDecompSVD.

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.

## ◆ IsA()

 TClass * TDecompBase::IsA ( ) const
inlineoverridevirtual
Returns
TClass describing current object

Reimplemented from TObject.

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

Definition at line 93 of file TDecompBase.h.

## ◆ 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
overridevirtual

Print class members.

Reimplemented from TObject.

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

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

## ◆ Solve() [2/3]

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

## ◆ Solve() [3/3]

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

## ◆ Streamer()

 void TDecompBase::Streamer ( TBuffer & R__b )
overridevirtual

Stream an object of class TObject.

Reimplemented from TObject.

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

## ◆ StreamerNVirtual()

 void TDecompBase::StreamerNVirtual ( TBuffer & ClassDef_StreamerNVirtual_b )
inline

Definition at line 93 of file TDecompBase.h.

## ◆ TransSolve() [1/3]

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

## ◆ TransSolve() [2/3]

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

## ◆ TransSolve() [3/3]

 virtual Bool_t TDecompBase::TransSolve ( TVectorD & 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.

Libraries for TDecompBase:

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