TDecompSVD Class Reference

Single Value Decomposition class.

For an (m x n) matrix A with m >= n, the singular value decomposition is an (m x m) orthogonal matrix fU, an (m x n) diagonal matrix fS, and an (n x n) orthogonal matrix fV so that A = U*S*V'.

If the row/column index of A starts at (rowLwb,colLwb) then the decomposed matrices/vectors start at :

fU   : (rowLwb,colLwb)
fV   : (colLwb,colLwb)
fSig : (colLwb)

The diagonal matrix fS is stored in the singular values vector fSig . The singular values, fSig[k] = S[k][k], are ordered so that fSig[0] >= fSig[1] >= ... >= fSig[n-1].

The singular value decomposition always exists, so the decomposition will (as long as m >=n) never fail. If m < n, the user should add sufficient zero rows to A , so that m == n

Here fTol is used to set the threshold on the minimum allowed value of the singular values: min_singular = fTol*max(fSig[i])

Definition at line 23 of file TDecompSVD.h.

Public Types
enum	{ kWorkMax = 100 }

Public Member Functions
	TDecompSVD ()
	TDecompSVD (Int_t nrows, Int_t ncols)
	Constructor for (nrows x ncols) matrix. More...
	TDecompSVD (Int_t row_lwb, Int_t row_upb, Int_t col_lwb, Int_t col_upb)
	Constructor for ([row_lwb..row_upb] x [col_lwb..col_upb]) matrix. More...
	TDecompSVD (const TMatrixD &m, Double_t tol=0.0)
	Constructor for general matrix A . More...
	TDecompSVD (const TDecompSVD &another)
	Copy constructor. More...
virtual	~TDecompSVD ()
virtual Double_t	Condition ()
	Matrix condition number. More...
virtual Bool_t	Decompose ()
	SVD decomposition of matrix If the decomposition succeeds, bit kDecomposed is set , otherwise kSingular. More...
virtual void	Det (Double_t &d1, Double_t &d2)
	Matrix determinant det = d1*TMath::Power(2.,d2) More...
const TMatrixD	GetMatrix ()
	Reconstruct the original matrix using the decomposition parts. More...
virtual Int_t	GetNcols () const
virtual Int_t	GetNrows () const
const TVectorD &	GetSig ()
const TMatrixD &	GetU ()
const TMatrixD &	GetV ()
Bool_t	Invert (TMatrixD &inv)
	For a matrix A(m,n), its inverse A_inv is defined as A * A_inv = A_inv * A = unit The user should always supply a matrix of size (m x m) ! If m > n , only the (n x m) part of the returned (pseudo inverse) matrix should be used . More...
TMatrixD	Invert (Bool_t &status)
	For a matrix A(m,n), its inverse A_inv is defined as A * A_inv = A_inv * A = unit (n x m) Ainv is returned . More...
TMatrixD	Invert ()
TDecompSVD &	operator= (const TDecompSVD &source)
	Assignment operator. More...
void	Print (Option_t *opt="") const
	Print class members. More...
virtual void	SetMatrix (const TMatrixD &a)
	Set matrix to be decomposed. More...
virtual Bool_t	Solve (TVectorD &b)
	Solve Ax=b assuming the SVD form of A is stored . More...
virtual TVectorD	Solve (const TVectorD &b, Bool_t &ok)
virtual Bool_t	Solve (TMatrixDColumn &b)
	Solve Ax=b assuming the SVD form of A is stored . More...
virtual Bool_t	TransSolve (TVectorD &b)
	Solve A^T x=b assuming the SVD form of A is stored . Solution returned in b. More...
virtual TVectorD	TransSolve (const TVectorD &b, Bool_t &ok)
virtual Bool_t	TransSolve (TMatrixDColumn &b)
	Solve A^T x=b assuming the SVD form of A is stored . Solution returned in b. More...
Public Member Functions inherited from TDecompBase
	TDecompBase ()
	Default constructor. More...
	TDecompBase (const TDecompBase &another)
	Copy constructor. More...
virtual	~TDecompBase ()
Int_t	GetColLwb () const
Double_t	GetCondition () const
Double_t	GetDet1 () const
Double_t	GetDet2 () const
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)

Protected Member Functions
virtual const TMatrixDBase &	GetDecompMatrix () const
Protected Member Functions inherited from TDecompBase
Int_t	Hager (Double_t &est, Int_t iter=5)
void	ResetStatus ()

Static Protected Member Functions
static Bool_t	Bidiagonalize (TMatrixD &v, TMatrixD &u, TVectorD &sDiag, TVectorD &oDiag)
	Bidiagonalize the (m x n) - matrix a (stored in v) through a series of Householder transformations applied to the left (Q^T) and to the right (H) of a , so that A = Q . More...
static void	Diag_1 (TMatrixD &v, TVectorD &sDiag, TVectorD &oDiag, Int_t k)
	Step 1 in the matrix diagonalization. More...
static void	Diag_2 (TVectorD &sDiag, TVectorD &oDiag, Int_t k, Int_t l)
	Step 2 in the matrix diagonalization. More...
static void	Diag_3 (TMatrixD &v, TMatrixD &u, TVectorD &sDiag, TVectorD &oDiag, Int_t k, Int_t l)
	Step 3 in the matrix diagonalization. More...
static Bool_t	Diagonalize (TMatrixD &v, TMatrixD &u, TVectorD &sDiag, TVectorD &oDiag)
	Diagonalizes in an iterative fashion the bidiagonal matrix C as described through sDiag and oDiag, so that S' = U'^T . More...
static void	SortSingular (TMatrixD &v, TMatrixD &u, TVectorD &sDiag)
	Perform a permutation transformation on the diagonal matrix S', so that matrix S'' = U''^T . More...
Static Protected Member Functions inherited from TDecompBase
static void	DiagProd (const TVectorD &diag, Double_t tol, Double_t &d1, Double_t &d2)

Protected Attributes
TVectorD	fSig
TMatrixD	fU
TMatrixD	fV
Protected Attributes inherited from TDecompBase
Int_t	fColLwb
Double_t	fCondition
Double_t	fDet1
Double_t	fDet2
Int_t	fRowLwb
Double_t	fTol

Additional Inherited Members
Protected Types inherited from TDecompBase
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) }

#include <TDecompSVD.h>

Inheritance diagram for TDecompSVD:

Member Enumeration Documentation

anonymous enum

anonymous enum

Enumerator
kWorkMax

Definition at line 43 of file TDecompSVD.h.

Constructor & Destructor Documentation

TDecompSVD() [1/5]

TDecompSVD::TDecompSVD ( )

inline

Definition at line 45 of file TDecompSVD.h.

TDecompSVD() [2/5]

TDecompSVD::TDecompSVD	(	Int_t	nrows,
		Int_t	ncols
	)

Constructor for (nrows x ncols) matrix.

Definition at line 52 of file TDecompSVD.cxx.

TDecompSVD() [3/5]

TDecompSVD::TDecompSVD	(	Int_t	row_lwb,
		Int_t	row_upb,
		Int_t	col_lwb,
		Int_t	col_upb
	)

Constructor for ([row_lwb..row_upb] x [col_lwb..col_upb]) matrix.

Definition at line 66 of file TDecompSVD.cxx.

TDecompSVD() [4/5]

TDecompSVD::TDecompSVD	(	const TMatrixD &	m,
		Double_t	tol = 0.0
	)

Constructor for general matrix A .

Definition at line 85 of file TDecompSVD.cxx.

TDecompSVD() [5/5]

TDecompSVD::TDecompSVD

(

const TDecompSVD &

another

)

Copy constructor.

Definition at line 114 of file TDecompSVD.cxx.

~TDecompSVD()

virtual TDecompSVD::~TDecompSVD ( )

inlinevirtual

Definition at line 50 of file TDecompSVD.h.

Member Function Documentation

Bidiagonalize()

Bool_t TDecompSVD::Bidiagonalize ( TMatrixD &  v, TMatrixD &  u, TVectorD &  sDiag, TVectorD &  oDiag  )

staticprotected

Bidiagonalize the (m x n) - matrix a (stored in v) through a series of Householder transformations applied to the left (Q^T) and to the right (H) of a , so that A = Q .

C . H^T with matrix C bidiagonal. Q and H are orthogonal matrices .

Output: v - (n x n) - matrix H in the (n x n) part of v u - (m x m) - matrix Q^T sDiag - diagonal of the (m x n) C oDiag - off-diagonal elements of matrix C

Test code for the output:

const Int_t nRow = v.GetNrows();
const Int_t nCol = v.GetNcols();
TMatrixD H(v); H.ResizeTo(nCol,nCol);
TMatrixD E1(nCol,nCol); E1.UnitMatrix();
TMatrixD Ht(TMatrixDBase::kTransposed,H);
Bool_t ok = kTRUE;
ok &= VerifyMatrixIdentity(Ht * H,E1,kTRUE,1.0e-13);
ok &= VerifyMatrixIdentity(H * Ht,E1,kTRUE,1.0e-13);
TMatrixD E2(nRow,nRow); E2.UnitMatrix();
TMatrixD Qt(u);
TMatrixD Q(TMatrixDBase::kTransposed,Qt);
ok &= VerifyMatrixIdentity(Q * Qt,E2,kTRUE,1.0e-13);
TMatrixD C(nRow,nCol);
TMatrixDDiag(C) = sDiag;
for (Int_t i = 0; i < nCol-1; i++)
  C(i,i+1) = oDiag(i+1);
TMatrixD A = Q*C*Ht;
ok &= VerifyMatrixIdentity(A,a,kTRUE,1.0e-13);

Definition at line 192 of file TDecompSVD.cxx.

Condition()

Double_t TDecompSVD::Condition ( )

virtual

Matrix condition number.

Reimplemented from TDecompBase.

Definition at line 823 of file TDecompSVD.cxx.

Decompose()

Bool_t TDecompSVD::Decompose ( )

virtual

SVD decomposition of matrix If the decomposition succeeds, bit kDecomposed is set , otherwise kSingular.

Implements TDecompBase.

Definition at line 123 of file TDecompSVD.cxx.

Det()

void TDecompSVD::Det ( Double_t &  d1, Double_t &  d2  )

virtual

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

Reimplemented from TDecompBase.

Definition at line 846 of file TDecompSVD.cxx.

Diag_1()

void TDecompSVD::Diag_1 ( TMatrixD &  v, TVectorD &  sDiag, TVectorD &  oDiag, Int_t  k  )

staticprotected

Step 1 in the matrix diagonalization.

Definition at line 374 of file TDecompSVD.cxx.

Diag_2()

void TDecompSVD::Diag_2 ( TVectorD &  sDiag, TVectorD &  oDiag, Int_t  k, Int_t  l  )

staticprotected

Step 2 in the matrix diagonalization.

Definition at line 398 of file TDecompSVD.cxx.

Diag_3()

void TDecompSVD::Diag_3 ( TMatrixD &  v, TMatrixD &  u, TVectorD &  sDiag, TVectorD &  oDiag, Int_t  k, Int_t  l  )

staticprotected

Step 3 in the matrix diagonalization.

Definition at line 416 of file TDecompSVD.cxx.

Diagonalize()

Bool_t TDecompSVD::Diagonalize ( TMatrixD &  v, TMatrixD &  u, TVectorD &  sDiag, TVectorD &  oDiag  )

staticprotected

Diagonalizes in an iterative fashion the bidiagonal matrix C as described through sDiag and oDiag, so that S' = U'^T .

C . V' is diagonal. U' and V' are orthogonal matrices .

Output:

• v - (n x n) - matrix H . V' in the (n x n) part of v
• u - (m x m) - matrix U'^T . Q^T

• sDiag - diagonal of the (m x n) S'

  return convergence flag: 0 -> no convergence 1 -> convergence

Test code for the output:

const Int_t nRow = v.GetNrows();
const Int_t nCol = v.GetNcols();
TMatrixD tmp = v; tmp.ResizeTo(nCol,nCol);
TMatrixD Vprime  = Ht*tmp;
TMatrixD Vprimet(TMatrixDBase::kTransposed,Vprime);
TMatrixD Uprimet = u*Q;
TMatrixD Uprime(TMatrixDBase::kTransposed,Uprimet);
TMatrixD Sprime(nRow,nCol);
TMatrixDDiag(Sprime) = sDiag;
ok &= VerifyMatrixIdentity(Uprimet * C * Vprime,Sprime,kTRUE,1.0e-13);
ok &= VerifyMatrixIdentity(Q*Uprime * Sprime * Vprimet * Ht,a,kTRUE,1.0e-13);

Definition at line 307 of file TDecompSVD.cxx.

GetDecompMatrix()

virtual const TMatrixDBase& TDecompSVD::GetDecompMatrix ( ) const

inlineprotectedvirtual

Implements TDecompBase.

Definition at line 39 of file TDecompSVD.h.

GetMatrix()

const TMatrixD TDecompSVD::GetMatrix

(

)

Reconstruct the original matrix using the decomposition parts.

Definition at line 557 of file TDecompSVD.cxx.

GetNcols()

Int_t TDecompSVD::GetNcols ( ) const

virtual

Implements TDecompBase.

Definition at line 870 of file TDecompSVD.cxx.

GetNrows()

Int_t TDecompSVD::GetNrows ( ) const

virtual

Implements TDecompBase.

Definition at line 865 of file TDecompSVD.cxx.

GetSig()

const TVectorD& TDecompSVD::GetSig ( )

inline

Definition at line 59 of file TDecompSVD.h.

GetU()

const TMatrixD& TDecompSVD::GetU ( )

inline

Definition at line 55 of file TDecompSVD.h.

GetV()

const TMatrixD& TDecompSVD::GetV ( )

inline

Definition at line 57 of file TDecompSVD.h.

Invert() [1/3]

Bool_t TDecompSVD::Invert

(

TMatrixD &

inv

)

For a matrix A(m,n), its inverse A_inv is defined as A * A_inv = A_inv * A = unit The user should always supply a matrix of size (m x m) ! If m > n , only the (n x m) part of the returned (pseudo inverse) matrix should be used .

Definition at line 881 of file TDecompSVD.cxx.

Invert() [2/3]

TMatrixD TDecompSVD::Invert

(

Bool_t &

status

)

For a matrix A(m,n), its inverse A_inv is defined as A * A_inv = A_inv * A = unit (n x m) Ainv is returned .

Definition at line 903 of file TDecompSVD.cxx.

Invert() [3/3]

TMatrixD TDecompSVD::Invert ( )

inline

Definition at line 82 of file TDecompSVD.h.

operator=()

TDecompSVD & TDecompSVD::operator=

(

const TDecompSVD &

source

)

Assignment operator.

Definition at line 930 of file TDecompSVD.cxx.

Print()

void TDecompSVD::Print

(

Option_t *

opt = ""

)

const

Print class members.

Definition at line 919 of file TDecompSVD.cxx.

SetMatrix()

void TDecompSVD::SetMatrix ( const TMatrixD &  a )

virtual

Set matrix to be decomposed.

Definition at line 582 of file TDecompSVD.cxx.

Solve() [1/3]

Bool_t TDecompSVD::Solve ( TVectorD &  b )

virtual

Solve Ax=b assuming the SVD form of A is stored .

Solution returned in b. If A is of size (m x n), input vector b should be of size (m), however, the solution, returned in b, will be in the first (n) elements .

For m > n , x is the least-squares solution of min(A . x - b)

Implements TDecompBase.

Definition at line 615 of file TDecompSVD.cxx.

Solve() [2/3]

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

inlinevirtual

Implements TDecompBase.

Definition at line 66 of file TDecompSVD.h.

Solve() [3/3]

Bool_t TDecompSVD::Solve ( TMatrixDColumn &  cb )

virtual

Solve Ax=b assuming the SVD form of A is stored .

Solution returned in the matrix column cb b. If A is of size (m x n), input vector b should be of size (m), however, the solution, returned in b, will be in the first (n) elements .

For m > n , x is the least-squares solution of min(A . x - b)

Implements TDecompBase.

Definition at line 670 of file TDecompSVD.cxx.

SortSingular()

void TDecompSVD::SortSingular ( TMatrixD &  v, TMatrixD &  u, TVectorD &  sDiag  )

staticprotected

Perform a permutation transformation on the diagonal matrix S', so that matrix S'' = U''^T .

S' . V'' has diagonal elements ordered such that they do not increase.

Output:

• v - (n x n) - matrix H . V' . V'' in the (n x n) part of v
• u - (m x m) - matrix U''^T . U'^T . Q^T
• sDiag - diagonal of the (m x n) S''

Definition at line 497 of file TDecompSVD.cxx.

TransSolve() [1/3]

Bool_t TDecompSVD::TransSolve ( TVectorD &  b )

virtual

Solve A^T x=b assuming the SVD form of A is stored . Solution returned in b.

Implements TDecompBase.

Definition at line 723 of file TDecompSVD.cxx.

TransSolve() [2/3]

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

inlinevirtual

Implements TDecompBase.

Definition at line 72 of file TDecompSVD.h.

TransSolve() [3/3]

Bool_t TDecompSVD::TransSolve ( TMatrixDColumn &  b )

virtual

Solve A^T x=b assuming the SVD form of A is stored . Solution returned in b.

Implements TDecompBase.

Definition at line 772 of file TDecompSVD.cxx.

Member Data Documentation

fSig

TVectorD TDecompSVD::fSig

protected

Definition at line 30 of file TDecompSVD.h.

fU

TMatrixD TDecompSVD::fU

protected

Definition at line 28 of file TDecompSVD.h.

fV

TMatrixD TDecompSVD::fV

protected

Definition at line 29 of file TDecompSVD.h.

Libraries for TDecompSVD:

---

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

• math/matrix/inc/TDecompSVD.h
• math/matrix/src/TDecompSVD.cxx