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

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...

virtual void	AppendPad (Option_t *option="")
	Append graphics object to current pad. More...

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

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

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

virtual void	Clear (Option_t *="")

virtual TObject *	Clone (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 TObject *	DrawClone (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 TObject *	FindObject (const char *name) const
	Must be redefined in derived classes. More...

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

virtual Option_t *	GetDrawOption () 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_t *	GetOption () 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...

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. 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...

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. 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...

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)

TObject &	operator= (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...

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. 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 TMatrixDBase &	GetDecompMatrix () 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

Additional Inherited Members
Public Types inherited from TObject
enum	{ kIsOnHeap = 0x01000000, kNotDeleted = 0x02000000, kZombie = 0x04000000, kInconsistent = 0x08000000, kBitMask = 0x00ffffff }

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

enum	EDeprecatedStatusBits { kObjInCanvas = BIT(3) }

enum	EStatusBits { kCanDelete = BIT(0), kMustCleanup = 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]

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

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

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:

[legend]

Condition() :

Det(Double_t &d1,Double_t &d2)

Decompose()

Solve(TVectorD &b)

TransSolve(TVectorD &b)

MultiSolve(TMatrixD &B)

Invert(TMatrixD &inv)

Tolerances and Scaling

Public Member Functions

Protected Types

Protected Member Functions

Static Protected Member Functions

Protected Attributes

Additional Inherited Members

Member Enumeration Documentation

◆ anonymous enum

◆ EMatrixDecompStat

Constructor & Destructor Documentation

◆ TDecompBase() [1/2]

◆ TDecompBase() [2/2]

◆ ~TDecompBase()

Member Function Documentation

◆ Condition()

◆ Decompose()

◆ Det()

◆ DiagProd()

◆ GetColLwb()

◆ GetCondition()

◆ GetDecompMatrix()

◆ GetDet1()

◆ GetDet2()

◆ GetNcols()

◆ GetNrows()

◆ GetRowLwb()

◆ GetTol()

◆ Hager()

◆ MultiSolve()

◆ operator=()

◆ Print()

◆ ResetStatus()

◆ SetTol()

◆ Solve() [1/3]

◆ Solve() [2/3]

◆ Solve() [3/3]

◆ TransSolve() [1/3]

◆ TransSolve() [2/3]

◆ TransSolve() [3/3]

Member Data Documentation

◆ fColLwb

◆ fCondition

◆ fDet1

◆ fDet2

◆ fRowLwb

◆ fTol