ROOT 6.18/05 Reference Guide |
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:
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,
gives,
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 .
The determinant is d1*TMath::Power(2.,d2) Expressing the determinant this way makes under/over-flow very unlikely .
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 A x = b . x is supplied through the argument and replaced with the solution .
Solve A^T x = b . x is supplied through the argument and replaced with the solution .
Solve A X = B . where X and are now matrices . X is supplied through the argument and replaced with the solution .
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 .
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:
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 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 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 void | Print (Option_t *option="") const |
This method must be overridden when a class wants to print itself. 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 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>
|
protected |
Enumerator | |
---|---|
kWorkMax |
Definition at line 60 of file TDecompBase.h.
|
protected |
Enumerator | |
---|---|
kInit | |
kPatternSet | |
kValuesSet | |
kMatrixSet | |
kDecomposed | |
kDetermined | |
kCondition | |
kSingular |
Definition at line 49 of file TDecompBase.h.
TDecompBase::TDecompBase | ( | ) |
Default constructor.
Definition at line 130 of file TDecompBase.cxx.
TDecompBase::TDecompBase | ( | const TDecompBase & | another | ) |
Copy constructor.
Definition at line 143 of file TDecompBase.cxx.
|
inlinevirtual |
Definition at line 65 of file TDecompBase.h.
|
virtual |
Matrix condition number.
Reimplemented in TDecompSVD.
Definition at line 257 of file TDecompBase.cxx.
|
pure virtual |
Implemented in TDecompBK, TDecompChol, TDecompLU, TDecompQRH, TDecompSparse, and TDecompSVD.
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.
|
staticprotected |
Definition at line 209 of file TDecompBase.cxx.
|
inline |
Definition at line 74 of file TDecompBase.h.
|
inline |
Definition at line 70 of file TDecompBase.h.
|
protectedpure virtual |
Implemented in TDecompBK, TDecompChol, TDecompLU, TDecompQRH, TDecompSparse, and TDecompSVD.
|
inline |
Definition at line 68 of file TDecompBase.h.
|
inline |
Definition at line 69 of file TDecompBase.h.
|
pure virtual |
Implemented in TDecompBK, TDecompChol, TDecompLU, TDecompQRH, TDecompSparse, and TDecompSVD.
|
pure virtual |
Implemented in TDecompBK, TDecompChol, TDecompLU, TDecompQRH, TDecompSparse, and TDecompSVD.
|
inline |
Definition at line 73 of file TDecompBase.h.
|
inline |
Definition at line 67 of file TDecompBase.h.
Definition at line 150 of file TDecompBase.cxx.
Solve set of equations with RHS in columns of B.
Definition at line 280 of file TDecompBase.cxx.
TDecompBase & TDecompBase::operator= | ( | const TDecompBase & | source | ) |
Assignment operator.
Definition at line 337 of file TDecompBase.cxx.
Print class members.
Reimplemented from TObject.
Reimplemented in TDecompBK, TDecompChol, TDecompLU, TDecompQRH, TDecompSparse, and TDecompSVD.
Definition at line 324 of file TDecompBase.cxx.
|
inlineprotected |
Definition at line 43 of file TDecompBase.h.
Definition at line 96 of file TDecompBase.h.
Implemented in TDecompBK, TDecompChol, TDecompLU, TDecompQRH, TDecompSparse, and TDecompSVD.
|
pure virtual |
Implemented in TDecompSparse, TDecompBK, TDecompChol, TDecompLU, TDecompQRH, and TDecompSVD.
Implemented in TDecompBK, TDecompChol, TDecompLU, TDecompQRH, TDecompSparse, and TDecompSVD.
Implemented in TDecompBK, TDecompChol, TDecompLU, TDecompQRH, TDecompSparse, and TDecompSVD.
|
pure virtual |
Implemented in TDecompSparse, TDecompBK, TDecompChol, TDecompLU, TDecompQRH, and TDecompSVD.
Implemented in TDecompBK, TDecompChol, TDecompLU, TDecompQRH, TDecompSparse, and TDecompSVD.
|
protected |
Definition at line 41 of file TDecompBase.h.
|
protected |
Definition at line 39 of file TDecompBase.h.
|
protected |
Definition at line 37 of file TDecompBase.h.
|
protected |
Definition at line 38 of file TDecompBase.h.
|
protected |
Definition at line 40 of file TDecompBase.h.
|
protected |
Definition at line 36 of file TDecompBase.h.