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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 TVectorD Solve (const TVectorD &b, Bool_t &ok)=0
 
virtual Bool_t Solve (TMatrixDColumn &b)=0
 
virtual Bool_t Solve (TVectorD &b)=0
 
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. 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 ()
 Check and record whether 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 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...
 
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...
 
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)
 
void SetBit (UInt_t f, Bool_t set)
 Set or unset the user status bits as specified in f. More...
 
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 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
 

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 66 of file TDecompBase.h.

◆ EMatrixDecompStat

Enumerator
kInit 
kPatternSet 
kValuesSet 
kMatrixSet 
kDecomposed 
kDetermined 
kCondition 
kSingular 

Definition at line 55 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 71 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

◆ Det()

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

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

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

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 80 of file TDecompBase.h.

◆ GetCondition()

Double_t TDecompBase::GetCondition ( ) const
inline

Definition at line 76 of file TDecompBase.h.

◆ GetDecompMatrix()

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

◆ GetDet1()

Double_t TDecompBase::GetDet1 ( ) const
inline

Definition at line 74 of file TDecompBase.h.

◆ GetDet2()

Double_t TDecompBase::GetDet2 ( ) const
inline

Definition at line 75 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 79 of file TDecompBase.h.

◆ GetTol()

Double_t TDecompBase::GetTol ( ) const
inline

Definition at line 73 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 49 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

◆ 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

Member Data Documentation

◆ fColLwb

Int_t TDecompBase::fColLwb
protected

Definition at line 47 of file TDecompBase.h.

◆ fCondition

Double_t TDecompBase::fCondition
protected

Definition at line 45 of file TDecompBase.h.

◆ fDet1

Double_t TDecompBase::fDet1
protected

Definition at line 43 of file TDecompBase.h.

◆ fDet2

Double_t TDecompBase::fDet2
protected

Definition at line 44 of file TDecompBase.h.

◆ fRowLwb

Int_t TDecompBase::fRowLwb
protected

Definition at line 46 of file TDecompBase.h.

◆ fTol

Double_t TDecompBase::fTol
protected

Definition at line 42 of file TDecompBase.h.

Libraries for TDecompBase:
[legend]

The documentation for this class was generated from the following files:
m
auto * m
Definition: textangle.C:8
e
#define e(i)
Definition: RSha256.hxx:121
TMath::Abs
Short_t Abs(Short_t d)
Definition: TMathBase.h:120
TMatrixT
Definition: TMatrixDfwd.h:22
b
#define b(i)
Definition: RSha256.hxx:118
a
auto * a
Definition: textangle.C:12
TMath::Min
Short_t Min(Short_t a, Short_t b)
Definition: TMathBase.h:180
TVectorT
Definition: TMatrixTBase.h:78
Double_t
double Double_t
Definition: RtypesCore.h:59