# class TUnfold: public TObject

```
TUnfold solves the inverse problem

chi**2 = (y-Ax)# Vyy^-1 (y-Ax) + tau^2 (L(x-x0))# L(x-x0) + lambda sum_i(y_i -(Ax)_i)

where # means that the matrix is transposed

Monte Carlo input

y: vector of measured quantities  (dimension ny)
Vyy: covariance matrix for y (dimension ny x ny)
in many cases V is diagonal and calculated from the errors of y
A: migration matrix               (dimension ny x nx)
x: unknown underlying distribution (dimension nx)

Regularisation

tau: parameter, defining the regularisation strength
L: matrix of regularisation conditions (dimension nl x nx)
x0: bias distribution

Preservation of the area

lambda: lagrangian multiplier
y_i: one component of the vector y
(Ax)_i: one component of the vector Ax

and chi**2 is minimized
(a) not constrained: minimisation is performed a function of x for fixed lambda=0
or
(b) constrained: minimisation is performed a function of x and lambda

This applies to a very large number of problems, where the measured
distribution y is a linear superposition of several Monte Carlo shapes
and the sum of these shapes gives the output distribution x

The constraint can be useful to reduce biases on the result x
in cases where the vector y follows non-Gaussian probability densities
(example: Poisson statistics at counting experiments in particle physics)

Some random examples:

(1) measure a cross-section as a function of, say, E_T(detector)
and unfold it to obtain the underlying distribution E_T(generator)
(2) measure a lifetime distribution and unfold the contributions from
different flavours
(3) measure the transverse mass and decay angle
and unfold for the true mass distribution plus background

Documentation

Some technical documentation is available here:
http://www.desy.de/~sschmitt

References:

A nice overview of the method is given in:
The L-curve and Its Use in the Numerical Treatment of Inverse Problems
(2000) by P. C. Hansen, in Computational Inverse Problems in
Electrocardiology, ed. P. Johnston,
http://www.imm.dtu.dk/~pch/TR/Lcurve.ps
The relevant equations are (1), (2) for the unfolding
and (14) for the L-curve curvature definition

Related literature on unfolding:
The program package RUN and the web-page by V.Blobel
http://www.desy.de/~blobel/unfold.html
Talk by V. Blobel, Terascale Statistics school
https://indico.desy.de/contributionDisplay.py?contribId=23&confId=1149
References quoted in Blobel's talk:
Per Chistian Hansen, Rank-Deficient and Discrete Ill-posed Problems,
Siam (1998)
Jari Kaipio and Erkki Somersalo, Statistical and Computational
Inverse problems, Springer (2005)

Implementation

The result of the unfolding is calculated as follows:

Lsquared = L#L            regularisation conditions squared

epsilon_j = sum_i A_ij    vector of efficiencies

E^-1  = ((A# Vyy^-1 A)+tau^2 Lsquared)

x = E (A# Vyy^-1 y + tau^2 Lsquared x0 +lambda/2 * epsilon) is the result

The derivatives
dx_k/dy_i
dx_k/dA_ij
dx_k/d(tau^2)
are calculated for further usage.

The covariance matrix V_xx is calculated as:
Vxx_ij = sum_kl dx_i/dy_k Vyy_kl dx_j/dy_l

Warning:

The algorithm is based on "standard" matrix inversion, with the
known limitations in numerical accuracy and computing cost for
matrices with large dimensions.

Thus the algorithm should not used for large dimensions of x and y
nx should not be much larger than 200
ny should not be much larger than 1000

Example of using TUnfold:

imagine a 2-dimensional histogram is filled, named A
y-axis: generated quantity (e.g. 10 bins)
x-axis: reconstructed quantity (e.g. 20 bin)
The data are filled in a 1-dimensional histogram, named y
Note1: ALWAYS choose a higher number of bins on the reconstructed side
as compared to the generated size!
Note2: the events which are generated but not reconstructed
have to be added to the appropriate overflow bins of A
Note3: make sure all bins have sufficient statistics and their error is
non-zero. By default, bins with zero error are simply skipped;
however, this may cause problems if You try to unfold something
which depends on these input bins.

code fragment (with histograms A and y filled):

TUnfold unfold(A,TUnfold::kHistMapOutputHoriz);
Double_t tau=1.E-4;
Double_t biasScale=0.0;
unfold.DoUnfold(tau,y,biasScale);
TH1D *x=unfold.GetOutput("x","myVariable");
TH2D *rhoij=unfold.GetRhoIJ("correlation","myVariable");

will create histograms "x" and "correlation" from A and y.
if tau is very large, the output is biased to the generated distribution scaled by biasScale
if tau is very small, the output will show oscillations
and large entries in the correlation matrix

Proper choice of tau

One of the difficult questions is about the choice of tau. The most
common method is the L-curve method: a two-dimensional curve is plotted
x-axis: log10(chisquare)
y-axis: log10(regularisation condition)
In many cases this curve has an L-shape. The best choice of tau is in the
kink of the L

Within TUnfold a simple version of the L-curve analysis is included.
It tests a given number of points in a predefined tau-range and searches
for the maximum of the curvature in the L-curve (kink position).
if no tau range is given, the range of teh scan is determied automatically

Example: scan tau and produce the L-curve plot

Code fragment: assume A and y are filled

TUnfold unfold(A,TUnfold::kHistMapOutputHoriz);

unfold.SetInput(y);

Int_t nScan=30;
Int_t iBest;
TSpline *logTauX,*logTauY;
TGraph *lCurve;

iBest=unfold.ScanLcurve(nScan,0.0,0.0,&lCurve);

std::cout<<"tau="<<unfold.GetTau()<<"\n";

TH1D *x=unfold.GetOutput("x","myVariable");
TH2D *rhoij=unfold.GetRhoIJ("correlation","myVariable");

This creates
logTauX: the L-curve's x-coordinate as a function of log(tau)
logTauY: the L-curve's y-coordinate as a function of log(tau)
lCurve: a graph of the L-curve
x,rhoij: unfolding result for best choice of tau
iBest: the coordinate/spline knot number with the best choice of tau

Note: always check the L curve after unfolding. The algorithm is not
perfect

Bin averaging of the output

Sometimes it is useful to unfold for a fine binning in x and
calculate the final result with a smaller number of bins. The advantage
is a reduction in the correlation coefficients if bins are averaged.
For this type of averaging the full error matrix has to be used.
There are methods in TUnfold to support this type of calculation
Example:
The vector x has dimension 49, it consists of 7x7 bins
in two variables (Pt,Eta)
The unfolding result is to be presented as one-dimensional projections
in (Pt) and (Eta)
The bins of x are mapped as: bins 1..7 the first Eta bin
bins 2..14 the second Eta bin

bins 1,8,15,... the first Pt bin

code fragment:

TUnfold unfold(A,TUnfold::kHistMapOutputHoriz);
Double_t tau=1.E-4;
Double_t biasScale=0.0;
unfold.DoUnfold(tau,y,biasScale);
Int_t binMapEta[49+2];
Int_t binMapPt[49+2];
// overflow and underflow bins are not used
binMapEta[0]=-1;
binMapEta[49+1]=-1;
binMapPt[0]=-1;
binMapPt[49+1]=-1;
for(Int_t i=1;i<=49;i++) {
// all bins (i) with the same (i-1)/7 are added
binMapEta[i] = (i-1)/7 +1;
// all bins (i) with the same (i-1)%7 are added
binMapPt[i]  = (i-1)%7 +1;
}
TH1D *etaHist=new TH1D("eta(unfolded)",";eta",7,etamin,etamax);
TH1D *etaCorr=new TH2D("eta(unfolded)",";eta;eta",7,etamin,etamax,7,etamin,etamax);
TH1D *ptHist=new TH1D("pt(unfolded)",";pt",7,ptmin,ptmax);
TH1D *ptCorr=new TH2D("pt(unfolded)",";pt;pt",7,ptmin,ptmax,7,ptmin,ptmax);
unfold.GetOutput(etaHist,binMapEta);
unfold.GetRhoIJ(etaCorrt,binMapEta);
unfold.GetOutput(ptHist,binMapPt);
unfold.GetRhoIJ(ptCorrt,binMapPt);

Alternative Regularisation conditions

Regularisation is needed for most unfolding problems, in order to avoid
large oscillations and large correlations on the output bins.
It means that some extra conditions are applied on the output bins

Within TUnfold these conditions are posed on the difference (x-x0), where
x:  unfolding output
x0: the bias distribution, by default calculated from
the input matrix A. There is a method SetBias() to change the
bias distribution.
The 3rd argument to DoUnfold() is a scale factor applied to the bias
bias_default[j] = sum_i A[i][j]
x0[j] = scaleBias*bias[j]
The scale factor can be used to
(a) completely suppress the bias by setting it to zero
(b) compensate differences in the normalisation between data
and Monte Carlo

If the regularisation is strong, i.e. large parameter tau,
then the distribution x or its derivatives will look like the bias
distribution. If the parameter tau is small, the distribution x is
independent of the bias.

Three basic types of regularisation are implemented in TUnfold

condition            regularisation

kRegModeNone         none
kRegModeSize         minimize the size of (x-x0)
kRegModeDerivative   minimize the 1st derivative of (x-x0)
kRegModeCurvature    minimize the 2nd derivative of (x-x0)

kRegModeSize is the regularisation scheme which usually is found in
literature. In addition, the bias usually is not present
(bias scale factor is zero).

The non-standard regularisation schemes kRegModeDerivative and
kRegModeCurvature have the nice feature that they create correlations
between x-bins, whereas the non-regularized unfolding tends to create
negative correlations between bins. For these regularisation schemes the
parameter tau could be tuned such that the correlations are smallest,
as an alternative to the L-curve method.

If kRegModeSize is chosen or if x is a smooth function through all bins,
the regularisation condition can be set on all bins together by giving
the appropriate argument in the constructor (see examples above).

If x is composed of independent groups of bins (for example,
signal and background binning in two variables), it may be necessary to
set regularisation conditions for the individual groups of bins.
In this case,  give  kRegModeNone  in the constructor and specify
the bin grouping with calls to
RegularizeBins()   specify a 1-dimensional group of bins
RegularizeBins2D() specify a 2-dimensional group of bins

For ultimate flexibility, the regularisation condition can be set on each
bin individually
-> give  kRegModeNone  in the constructor and use
RegularizeSize()        regularize one bin
RegularizeDerivative()  regularize the slope given by two bins
RegularizeCurvature()   regularize the curvature given by three bins
```

## Function Members (Methods)

public:
 TUnfold(const TUnfold&) TUnfold(const TH2* hist_A, TUnfold::EHistMap histmap, TUnfold::ERegMode regmode = kRegModeSize, TUnfold::EConstraint constraint = kEConstraintArea) virtual ~TUnfold() void TObject::AbstractMethod(const char* method) const virtual void TObject::AppendPad(Option_t* option = "") virtual void TObject::Browse(TBrowser* b) static TClass* Class() virtual const char* TObject::ClassName() const virtual void TObject::Clear(Option_t* = "") virtual TObject* TObject::Clone(const char* newname = "") const virtual Int_t TObject::Compare(const TObject* obj) const virtual void TObject::Copy(TObject& object) const virtual void TObject::Delete(Option_t* option = "")MENU virtual Int_t TObject::DistancetoPrimitive(Int_t px, Int_t py) virtual Double_t DoUnfold(Double_t tau) Double_t DoUnfold(Double_t tau, const TH1* hist_y, Double_t scaleBias = 0.0) virtual void TObject::Draw(Option_t* option = "") virtual void TObject::DrawClass() constMENU virtual TObject* TObject::DrawClone(Option_t* option = "") constMENU virtual void TObject::Dump() constMENU virtual void TObject::Error(const char* method, const char* msgfmt) const virtual void TObject::Execute(const char* method, const char* params, Int_t* error = 0) virtual void TObject::Execute(TMethod* method, TObjArray* params, Int_t* error = 0) virtual void TObject::ExecuteEvent(Int_t event, Int_t px, Int_t py) virtual void TObject::Fatal(const char* method, const char* msgfmt) const virtual TObject* TObject::FindObject(const char* name) const virtual TObject* TObject::FindObject(const TObject* obj) const TH1D* GetBias(const char* name, const char* title, Double_t x0 = 0.0, Double_t x1 = 0.0) const Double_t GetChi2A() const Double_t GetChi2L() const virtual Option_t* TObject::GetDrawOption() const static Long_t TObject::GetDtorOnly() void GetEmatrix(TH2* ematrix, const Int_t* binMap = 0) const TH2D* GetEmatrix(const char* name, const char* title, Double_t x0 = 0.0, Double_t x1 = 0.0) const TH1D* GetFoldedOutput(const char* name, const char* title, Double_t y0 = 0.0, Double_t y1 = 0.0) const virtual const char* TObject::GetIconName() const TH1D* GetInput(const char* name, const char* title, Double_t y0 = 0.0, Double_t y1 = 0.0) const virtual Double_t GetLcurveX() const virtual Double_t GetLcurveY() const TH2D* GetLsquared(const char* name, const char* title, Double_t x0 = 0.0, Double_t x1 = 0.0) const virtual const char* TObject::GetName() const Int_t GetNdf() const Int_t GetNpar() const virtual char* TObject::GetObjectInfo(Int_t px, Int_t py) const static Bool_t TObject::GetObjectStat() virtual Option_t* TObject::GetOption() const void GetOutput(TH1* output, const Int_t* binMap = 0) const TH1D* GetOutput(const char* name, const char* title, Double_t x0 = 0.0, Double_t x1 = 0.0) const Double_t GetRhoAvg() const Double_t GetRhoI(TH1* rhoi, TH2* ematrixinv = 0, const Int_t* binMap = 0) const TH1D* GetRhoI(const char* name, const char* title, Double_t x0 = 0.0, Double_t x1 = 0.0) const void GetRhoIJ(TH2* rhoij, const Int_t* binMap = 0) const TH2D* GetRhoIJ(const char* name, const char* title, Double_t x0 = 0.0, Double_t x1 = 0.0) const Double_t GetRhoMax() const Double_t GetTau() const virtual const char* TObject::GetTitle() const static const char* GetTUnfoldVersion() virtual UInt_t TObject::GetUniqueID() const virtual Bool_t TObject::HandleTimer(TTimer* timer) virtual ULong_t TObject::Hash() const virtual void TObject::Info(const char* method, const char* msgfmt) const virtual Bool_t TObject::InheritsFrom(const char* classname) const virtual Bool_t TObject::InheritsFrom(const TClass* cl) const virtual void TObject::Inspect() constMENU void TObject::InvertBit(UInt_t f) virtual TClass* IsA() const virtual Bool_t TObject::IsEqual(const TObject* obj) const virtual Bool_t TObject::IsFolder() const Bool_t TObject::IsOnHeap() const virtual Bool_t TObject::IsSortable() const Bool_t TObject::IsZombie() const virtual void TObject::ls(Option_t* option = "") const void TObject::MayNotUse(const char* method) const virtual Bool_t TObject::Notify() void TObject::Obsolete(const char* method, const char* asOfVers, const char* removedFromVers) const static void TObject::operator delete(void* ptr) static void TObject::operator delete(void* ptr, void* vp) static void TObject::operator delete[](void* ptr) static void TObject::operator delete[](void* ptr, void* vp) void* TObject::operator new(size_t sz) void* TObject::operator new(size_t sz, void* vp) void* TObject::operator new[](size_t sz) void* TObject::operator new[](size_t sz, void* vp) TUnfold& operator=(const TUnfold&) virtual void TObject::Paint(Option_t* option = "") virtual void TObject::Pop() virtual void TObject::Print(Option_t* option = "") const virtual Int_t TObject::Read(const char* name) virtual void TObject::RecursiveRemove(TObject* obj) Int_t RegularizeBins(int start, int step, int nbin, TUnfold::ERegMode regmode) Int_t RegularizeBins2D(int start_bin, int step1, int nbin1, int step2, int nbin2, TUnfold::ERegMode regmode) Int_t RegularizeCurvature(int left_bin, int center_bin, int right_bin, Double_t scale_left = 1.0, Double_t scale_right = 1.0) Int_t RegularizeDerivative(int left_bin, int right_bin, Double_t scale = 1.0) Int_t RegularizeSize(int bin, Double_t scale = 1.0) void TObject::ResetBit(UInt_t f) virtual void TObject::SaveAs(const char* filename = "", Option_t* option = "") constMENU virtual void TObject::SavePrimitive(ostream& out, Option_t* option = "") virtual Int_t ScanLcurve(Int_t nPoint, Double_t tauMin, Double_t tauMax, TGraph** lCurve, TSpline** logTauX = 0, TSpline** logTauY = 0) void SetBias(const TH1* bias) void TObject::SetBit(UInt_t f) void TObject::SetBit(UInt_t f, Bool_t set) void SetConstraint(TUnfold::EConstraint constraint) virtual void TObject::SetDrawOption(Option_t* option = "")MENU static void TObject::SetDtorOnly(void* obj) virtual Int_t SetInput(const TH1* hist_y, Double_t scaleBias = 0.0, Double_t oneOverZeroError = 0.0) static void TObject::SetObjectStat(Bool_t stat) virtual void TObject::SetUniqueID(UInt_t uid) virtual void ShowMembers(TMemberInspector& insp) virtual void Streamer(TBuffer& b) void StreamerNVirtual(TBuffer& b) virtual void TObject::SysError(const char* method, const char* msgfmt) const Bool_t TObject::TestBit(UInt_t f) const Int_t TObject::TestBits(UInt_t f) const virtual void TObject::UseCurrentStyle() virtual void TObject::Warning(const char* method, const char* msgfmt) const virtual Int_t TObject::Write(const char* name = 0, Int_t option = 0, Int_t bufsize = 0) virtual Int_t TObject::Write(const char* name = 0, Int_t option = 0, Int_t bufsize = 0) const
protected:
 TUnfold() void AddMSparse(TMatrixDSparse* dest, Double_t f, const TMatrixDSparse* src) virtual void ClearResults() TMatrixDSparse* CreateSparseMatrix(Int_t nrow, Int_t ncol, Int_t nele, Int_t* row, Int_t* col, Double_t* data) const static void DeleteMatrix(TMatrixD** m) static void DeleteMatrix(TMatrixDSparse** m) virtual void TObject::DoError(int level, const char* location, const char* fmt, va_list va) const virtual Double_t DoUnfold() void ErrorMatrixToHist(TH2* ematrix, const TMatrixDSparse* emat, const Int_t* binMap, Bool_t doClear) const const TMatrixDSparse* GetAx() const const TMatrixDSparse* GetDXDAM(int i) const const TMatrixDSparse* GetDXDAZ(int i) const const TMatrixDSparse* GetDXDtauSquared() const const TMatrixDSparse* GetDXDY() const const TMatrixDSparse* GetE() const const TMatrixDSparse* GetEinv() const Int_t GetNx() const Int_t GetNy() const const TMatrixDSparse* GetVxx() const const TMatrixDSparse* GetVxxInv() const const TMatrixD* GetX() const static Bool_t InvertMConditioned(TMatrixD* A) TMatrixD* InvertMSparse(const TMatrixDSparse* A) const void TObject::MakeZombie() TMatrixDSparse* MultiplyMSparseM(const TMatrixDSparse* a, const TMatrixD* b) const TMatrixDSparse* MultiplyMSparseMSparse(const TMatrixDSparse* a, const TMatrixDSparse* b) const TMatrixDSparse* MultiplyMSparseMSparseTranspVector(const TMatrixDSparse* m1, const TMatrixDSparse* m2, const TMatrixTBase* v) const TMatrixDSparse* MultiplyMSparseTranspMSparse(const TMatrixDSparse* a, const TMatrixDSparse* b) const
private:
 void InitTUnfold()

## Data Members

public:
 enum EConstraint { kEConstraintNone kEConstraintArea }; enum ERegMode { kRegModeNone kRegModeSize kRegModeDerivative kRegModeCurvature kRegModeMixed }; enum EHistMap { kHistMapOutputHoriz kHistMapOutputVert }; enum TObject::EStatusBits { kCanDelete kMustCleanup kObjInCanvas kIsReferenced kHasUUID kCannotPick kNoContextMenu kInvalidObject }; enum TObject::[unnamed] { kIsOnHeap kNotDeleted kZombie kBitMask kSingleKey kOverwrite kWriteDelete };
protected:
 TMatrixDSparse* fA Input: matrix Double_t fBiasScale Input: scale factor for the bias TUnfold::EConstraint fConstraint Input: type of constraint to use TArrayI fHistToX Input: histogram bins -> matrix indices TMatrixDSparse* fLsquared Input: regularisation conditions squared TUnfold::ERegMode fRegMode Input: type of regularisation TArrayD fSumOverY Input: sum of all columns Double_t fTauSquared Input: regularisation parameter TMatrixDSparse* fVyy Input: covariance matrix for y TMatrixD* fX0 Input: x0 TArrayI fXToHist Input: matrix indices -> histogram bins TMatrixD* fY Input: y
private:
 TMatrixDSparse* fAx Result: Ax Double_t fChi2A Result: chi**2 contribution from (y-Ax)V(y-Ax) TMatrixDSparse* fDXDAM[2] Result: part of derivative dx_k/dA_ij TMatrixDSparse* fDXDAZ[2] Result: part of derivative dx_k/dA_ij TMatrixDSparse* fDXDY Result: derivative dx/dy TMatrixDSparse* fDXDtauSquared Result: derivative dx/dtau TMatrixDSparse* fE Result: matrix E TMatrixDSparse* fEinv Result: matrix E^(-1) Double_t fLXsquared Result: chi**2 contribution from (x-s*x0)Lsquared(x-s*x0) Int_t fNdf Result: number of degrees of freedom Double_t fRhoAvg Result: average global correlation Double_t fRhoMax Result: maximum global correlation TMatrixDSparse* fVxx Result: covariance matrix on x TMatrixDSparse* fVxxInv Result: inverse of covariance matrix on x TMatrixD* fX Result: x

## Function documentation

const char * GetTUnfoldVersion(void)
void InitTUnfold(void)
``` reset all data members
```
void DeleteMatrix(TMatrixD** m)
void DeleteMatrix(TMatrixDSparse** m)
void ClearResults(void)
``` delete old results (if any)
this function is virtual, so derived classes may flag their results
```
TUnfold(const TUnfold& )
``` set all matrix pointers to zero
```

``` main unfolding algorithm. Declared virtual, because other algorithms
could be implemented

Purpose: unfold y -> x
Data members required:
fA:  matrix to relate x and y
fY:  measured data points
fX0: bias on x
fBiasScale: scale factor for fX0
fVyy:  covariance matrix for y
fLsquared: regularisation conditions
fTauSquared: regularisation strength
fConstraint: whether the constraint is applied
Data members modified:
fEinv: inverse of the covariance matrix of x
fE:    covariance matrix of x
fX:    unfolded data points
fDXDY: derivative of x wrt y (for error propagation)
fVxx:  error matrix (covariance matrix) on x
fAx:   estimate of distribution y from unfolded data
fChi2A:  contribution to chi**2 from y-Ax
fChi2L:  contribution to chi**2 from L*(x-x0)
fDXDtauSquared: derivative of x wrt tau
fDXDAM[0,1]: matrix parts of derivative x wrt A
fDXDAZ[0,1]: vector parts of derivative x wrt A
fRhoMax: maximum global correlation coefficient
fRhoAvg: average global correlation coefficient
return code:
fRhoMax   if(fRhoMax>=1.0) then the unfolding has failed!
```
TMatrixDSparse * MultiplyMSparseMSparse(const TMatrixDSparse* a, const TMatrixDSparse* b) const
``` calculate the product of two sparse matrices
a,b: pointers to sparse matrices, where a->GetNcols()==b->GetNrows()
this is a replacement for the call
new TMatrixDSparse(*a,TMatrixDSparse::kMult,*b);
```
TMatrixDSparse * MultiplyMSparseTranspMSparse(const TMatrixDSparse* a, const TMatrixDSparse* b) const
``` multiply a transposed Sparse matrix with another Sparse matrix
a:  pointer to sparse matrix (to be transposed)
b:  pointer to sparse matrix
this is a replacement for the call
new TMatrixDSparse(TMatrixDSparse(TMatrixDSparse::kTransposed,*a),
TMatrixDSparse::kMult,*b)
```
TMatrixDSparse * MultiplyMSparseM(const TMatrixDSparse* a, const TMatrixD* b) const
``` multiply a Sparse matrix with a non-sparse matrix
a:  pointer to sparse matrix
b:  pointer to non-sparse matrix
this is a replacement for the call
new TMatrixDSparse(*a,TMatrixDSparse::kMult,*b);
```
void AddMSparse(TMatrixDSparse* dest, Double_t f, const TMatrixDSparse* src)
``` a replacement for
(*dest) += f*(*src)
```
TMatrixD * InvertMSparse(const TMatrixDSparse* A) const
``` get the inverse of a sparse matrix
A: the original matrix
this is a replacement of the call
new TMatrixD(TMatrixD::kInverted, a);
the matrix inversion is optimized for the case
where a large submatrix of A is diagonal
```

``` invert the matrix A
the inversion is done with pre-conditioning
all rows and columns are normalized to sqrt(abs(a_ii*a_jj))
such that the diagonals are equal to 1.0
This type of preconditioning improves the numerival results
for the symmetric, positive definite matrices which are
treated here in the context of unfolding
```
TUnfold(const TH2* hist_A, TUnfold::EHistMap histmap, TUnfold::ERegMode regmode = kRegModeSize, TUnfold::EConstraint constraint = kEConstraintArea)
``` set up unfolding matrix and initial regularisation scheme
hist_A:  matrix that describes the migrations
histmap: mapping of the histogram axes to the unfolding output
regmode: global regularisation mode
constraint: type of constraint to use
data members initialized to something different from zero:
fA: filled from hist_A
fDA: filled from hist_A
fX0: filled from hist_A
fLsquared: filled depending on the regularisation scheme
Treatment of overflow bins
Bins where the unfolding input (Detector level) is in overflow
are used for the efficiency correction. They have to be filled
properly!
Bins where the unfolding output (Generator level) is in overflow
are treated as a part of the generator level distribution.
I.e. the unfolding output could have non-zero overflow bins if the
input matrix does have such bins.
```
~TUnfold(void)
``` delete all data members
```
void SetBias(const TH1* bias)
``` initialize alternative bias from histogram
modifies data member fX0
```
Int_t RegularizeSize(int bin, Double_t scale = 1.0)
``` add regularisation on the size of bin i
bin: bin number
scale: size of the regularisation
return value: number of conditions which have been skipped
modifies data member fLsquared
```
Int_t RegularizeDerivative(int left_bin, int right_bin, Double_t scale = 1.0)
``` add regularisation on the difference of two bins
left_bin: 1st bin
right_bin: 2nd bin
scale: size of the regularisation
return value: number of conditions which have been skipped
modifies data member fLsquared
```
Int_t RegularizeCurvature(int left_bin, int center_bin, int right_bin, Double_t scale_left = 1.0, Double_t scale_right = 1.0)
``` add regularisation on the curvature through 3 bins (2nd derivative)
left_bin: 1st bin
center_bin: 2nd bin
right_bin: 3rd bin
scale_left: scale factor on center-left difference
scale_right: scale factor on right-center difference
return value: number of conditions which have been skipped
modifies data member fLsquared
```
Int_t RegularizeBins(int start, int step, int nbin, TUnfold::ERegMode regmode)
``` set regulatisation on a 1-dimensional curve
start: first bin
step:  distance between neighbouring bins
nbin:  total number of bins
regmode:  regularisation mode
return value:
number of errors (i.e. conditions which have been skipped)
modifies data member fLsquared
```
Int_t RegularizeBins2D(int start_bin, int step1, int nbin1, int step2, int nbin2, TUnfold::ERegMode regmode)
``` set regularisation on a 2-dimensional grid of bins
start: first bin
step1: distance between bins in 1st direction
nbin1: number of bins in 1st direction
step2: distance between bins in 2nd direction
nbin2: number of bins in 2nd direction
return value:
number of errors (i.e. conditions which have been skipped)
modifies data member fLsquared
```
Double_t DoUnfold(Double_t tau, const TH1* hist_y, Double_t scaleBias = 0.0)
``` Do unfolding of an input histogram
tau_reg: regularisation parameter
input:   input distribution with errors
scaleBias:  scale factor applied to the bias
Data members required:
fA, fX0, fLsquared
Data members modified:
those documented in SetInput()
and those documented in DoUnfold(Double_t)
Return value:
maximum global correlation coefficient
NOTE!!! return value >=1.0 means error, and the result is junk

Overflow bins of the input distribution are ignored!
```
Int_t SetInput(const TH1* hist_y, Double_t scaleBias = 0.0, Double_t oneOverZeroError = 0.0)
``` Define the input data for subsequent calls to DoUnfold(Double_t)
input:   input distribution with errors
scaleBias:  scale factor applied to the bias
oneOverZeroError: for bins with zero error, this number defines 1/error.
Return value: number of bins with bad error
+10000*number of unconstrained output bins
Note: return values>=10000 are fatal errors,
for the given input, the unfolding can not be done!
Data members modified:
fY, fVyy, fVyyinv, fBiasScale, fNdf
Data members cleared
see ClearResults
```

``` Unfold with given value of regularisation parameter tau
tau: new tau parameter
required data members:
fA:  matrix to relate x and y
fY:  measured data points
fX0: bias on x
fBiasScale: scale factor for fX0
fV:  inverse of covariance matrix for y
fLsquared: regularisation conditions
modified data members:
fTauSquared and those documented in DoUnfold(void)
```
Int_t ScanLcurve(Int_t nPoint, Double_t tauMin, Double_t tauMax, TGraph** lCurve, TSpline** logTauX = 0, TSpline** logTauY = 0)
``` scan the L curve
nPoint: number of points on the resulting curve
tauMin: smallest tau value to study
tauMax: largest tau value to study
lCurve: the L curve as graph
logTauX: output spline of x-coordinates vs tau for the L curve
logTauY: output spline of y-coordinates vs tau for the L curve
return value: the coordinate number (0..nPoint-1) with the "best" choice
of tau
```
TH1D * GetOutput(const char* name, const char* title, Double_t x0 = 0.0, Double_t x1 = 0.0) const
``` retreive unfolding result as histogram
name:  name of the histogram
title: title of the histogram
x0,x1: lower/upper edge of histogram.
if (x0>=x1) then x0=0 and x1=nbin are used
```
TH1D * GetBias(const char* name, const char* title, Double_t x0 = 0.0, Double_t x1 = 0.0) const
``` retreive bias as histogram
name:  name of the histogram
title: title of the histogram
x0,x1: lower/upper edge of histogram.
if (x0>=x1) then x0=0 and x1=nbin are used
```
TH1D * GetFoldedOutput(const char* name, const char* title, Double_t y0 = 0.0, Double_t y1 = 0.0) const
``` retreive unfolding result folded back by the matrix
name:  name of the histogram
title: title of the histogram
y0,y1: lower/upper edge of histogram.
if (y0>=y1) then y0=0 and y1=nbin are used
```
TH1D * GetInput(const char* name, const char* title, Double_t y0 = 0.0, Double_t y1 = 0.0) const
``` retreive input distribution
name:  name of the histogram
title: title of the histogram
y0,y1: lower/upper edge of histogram.
if (y0>=y1) then y0=0 and y1=nbin are used
```
TH2D * GetRhoIJ(const char* name, const char* title, Double_t x0 = 0.0, Double_t x1 = 0.0) const
``` retreive full matrix of correlation coefficients
name:  name of the histogram
title: title of the histogram
x0,x1: lower/upper edge of histogram.
if (x0>=x1) then x0=0 and x1=nbin are used
```
TH2D * GetEmatrix(const char* name, const char* title, Double_t x0 = 0.0, Double_t x1 = 0.0) const
``` retreive full error matrix
name:  name of the histogram
title: title of the histogram
x0,x1: lower/upper edge of histogram.
if (x0>=x1) then x0=0 and x1=nbin are used
```
TH1D * GetRhoI(const char* name, const char* title, Double_t x0 = 0.0, Double_t x1 = 0.0) const
``` retreive matrix of global correlation coefficients
name:  name of the histogram
title: title of the histogram
x0,x1: lower/upper edge of histogram.
if (x0>=x1) then x0=0 and x1=nbin are used
```
TH2D * GetLsquared(const char* name, const char* title, Double_t x0 = 0.0, Double_t x1 = 0.0) const
``` retreive ix of regularisation conditions squared
name:  name of the histogram
title: title of the histogram
x0,x1: lower/upper edge of histogram.
if (x0>=x1) then x0=0 and x1=nbin are used
```
void SetConstraint(TUnfold::EConstraint constraint)
``` set type of constraint for the next unfolding
```
Double_t GetTau(void)
``` return regularisation parameter
```
Double_t GetChi2L(void)
``` return chi**2 contribution from regularisation conditions
```
Int_t GetNpar(void)
``` return number of parameters
```
Double_t GetLcurveX(void)
``` return value on x axis of L curve
```
Double_t GetLcurveY(void)
``` return value on y axis of L curve
```
void GetOutput(TH1* output, const Int_t* binMap = 0) const
``` get output distribution, cumulated over several bins
output: output histogram
binMap: for each bin of the original output distribution
specify the destination bin. A value of -1 means that the bin
is discarded. 0 means underflow bin, 1 first bin, ...
binMap[0] : destination of underflow bin
binMap[1] : destination of first bin

```
void ErrorMatrixToHist(TH2* ematrix, const TMatrixDSparse* emat, const Int_t* binMap, Bool_t doClear) const
``` get an error matrix, cumulated over several bins
ematrix: output error matrix histogram
emat: error matrix
binMap: for each bin of the original output distribution
specify the destination bin. A value of -1 means that the bin
is discarded. 0 means underflow bin, 1 first bin, ...
binMap[0] : destination of underflow bin
binMap[1] : destination of first bin

```
void GetEmatrix(TH2* ematrix, const Int_t* binMap = 0) const
``` get output error matrix, cumulated over several bins
ematrix: output error matrix histogram
binMap: for each bin of the original output distribution
specify the destination bin. A value of -1 means that the bin
is discarded. 0 means underflow bin, 1 first bin, ...
binMap[0] : destination of underflow bin
binMap[1] : destination of first bin

```
Double_t GetRhoI(TH1* rhoi, TH2* ematrixinv = 0, const Int_t* binMap = 0) const
``` get global correlation coefficients and inverted error matrix,
cumulated over several bins
rhoi: global correlation histogram
ematrixinv: inverse of error matrix (if pointer==0 it is not returned)
binMap: for each bin of the original output distribution
specify the destination bin. A value of -1 means that the bin
is discarded. 0 means underflow bin, 1 first bin, ...
binMap[0] : destination of underflow bin
binMap[1] : destination of first bin

return value: average global correlation
```
void GetRhoIJ(TH2* rhoij, const Int_t* binMap = 0) const
``` get correlation coefficient matrix, cumulated over several bins
rhoij:  correlation coefficient matrix histogram
binMap: for each bin of the original output distribution
specify the destination bin. A value of -1 means that the bin
is discarded. 0 means underflow bin, 1 first bin, ...
binMap[0] : destination of underflow bin
binMap[1] : destination of first bin

```
TUnfold(const TUnfold& )
TMatrixDSparse * CreateSparseMatrix(Int_t nrow, Int_t ncol, Int_t nele, Int_t* row, Int_t* col, Double_t* data) const
Int_t GetNx(void)
Int_t GetNy(void)
const TMatrixDSparse * GetDXDY(void)
`{ return fDXDY; }`
const TMatrixDSparse * GetDXDAM(int i) const
`{ return fDXDAM[i]; }`
const TMatrixDSparse * GetDXDAZ(int i) const
`{ return fDXDAZ[i]; }`
const TMatrixDSparse * GetDXDtauSquared(void)
`{ return fDXDtauSquared; }`
const TMatrixDSparse * GetAx(void)
`{ return fAx; }`
const TMatrixDSparse * GetEinv(void)
`{ return fEinv; }`
const TMatrixDSparse * GetE(void)
`{ return fE; }`
const TMatrixDSparse * GetVxx(void)
`{ return fVxx; }`
const TMatrixDSparse * GetVxxInv(void)
`{ return fVxxInv; }`
const TMatrixD * GetX(void)
`{ return fX; }`
Double_t GetRhoMax(void)
`{ return fRhoMax; }`
Double_t GetRhoAvg(void)
`{ return fRhoAvg; }`
Double_t GetChi2A(void)
`{ return fChi2A; }`
Int_t GetNdf(void)
`{ return fNdf; }`