Public Types
enum	{ kC, kDelta, kXi, kSigma, kN, kA }

Public Member Functions
Double_t	NEstimate (Double_t x, Double_t delta, Double_t xi, Double_t sigma, Int_t n, Double_t *a)

Double_t	Evaluate (Double_t x) const

Double_t	Evaluate (Double_t x, Double_t &e) const

Double_t	EstimateNParticles (Double_t x, Short_t maxN=-1)

Double_t	EstimateNParticles (Double_t x, Double_t &e, Short_t maxN=-1)

Double_t	RandomEstimateNParticles (Double_t x, Short_t maxN=-1)

void	Draw (Option_t *option="")

Double_t	GetC () const

Double_t	GetDelta () const

Double_t	GetXi () const

Double_t	GetSigma () const

UShort_t	GetN () const

Double_t	GetA (Int_t i) const

Double_t *	GetAs () const

Double_t	GetEC () const

Double_t	GetEDelta () const

Double_t	GetEXi () const

Double_t	GetESigma () const

UShort_t	GetEN () const

Double_t	GetEA (Int_t i) const

Double_t *	GetEAs () const

Static Public Member Functions
static Double_t	Landau (Double_t x, Double_t delta, Double_t xi)

static Double_t	LandauGaus (Double_t x, Double_t delta, Double_t xi, Double_t sigma)

static Double_t	ILandauGaus (Int_t i, Double_t x, Double_t delta, Double_t xi, Double_t sigma)

static Double_t	IdLandauGausdPar (Int_t i, Double_t x, UShort_t ipar, Double_t dPar, Double_t delta, Double_t xi, Double_t sigma)

static Double_t	NLandauGaus (Double_t x, Double_t delta, Double_t xi, Double_t sigma, Int_t n, Double_t *a)

Public Attributes
TF1 *	fF

Static Public Attributes
static const Double_t	fgkMPShift = -0.22278298
	MP shift. More...

static const Double_t	fgkInvRoot2Pi = 1. / TMath::Sqrt(2*TMath::Pi())
	1 / sqrt(2 * pi) More...

static const Double_t	fgkConvNSigma = 5
	Number of sigmas to integrate over. More...

static const Double_t	fgkConvNSteps = 100
	Number of steps in integration. More...

Detailed Description

Definition at line 40 of file TestELossDist.C.

Member Enumeration Documentation

anonymous enum

Enumeration of parameters

Enumerator
kC	Constant.
kDelta	MPV of landau.
kXi	Width of landau.
kSigma	Sigma of gaussian.
kN	Number of particles.
kA	Weights.

Definition at line 46 of file TestELossDist.C.

Member Function Documentation

void Function::Draw ( Option_t * option = "" )

inline

Draw the function

Parameters

option Drawing option

Definition at line 689 of file TestELossDist.C.

Referenced by Generator::Draw().

Double_t Function::EstimateNParticles	(	Double_t	x,
		Short_t	maxN = `-1`
	)

inline

Evaluate

\[ E(x;\Delta,\xi,\sigma,\mathbf{a}) = \frac{\sum_{i=1}^{n} i a_i f_i(x;\Delta,\xi,\sigma)}{ f_N(x;\Delta,\xi,\sigma)} = \frac{\sum_{i=1}^{n} i a_i f(x;\Delta_i,\xi_i,\sigma_i)}{ \sum_{i=1}^{n} a_i f(x;\Delta_i,\xi_i,\sigma_i)} \]

If the denominator is less than $10^{-4}$, then 0 is returned.

Parameters

x	Where to evaluate $ E$
maxN	Maximum number of weights

Returns: $ E(x;\Delta,\xi,\sigma,\mathbf{a})$

Definition at line 576 of file TestELossDist.C.

Referenced by Generator::Draw().

Double_t Function::EstimateNParticles	(	Double_t	x,
		Double_t &	e,
		Short_t	maxN = `-1`
	)

inline

Evaluate

\[ E(x;\Delta,\xi,\sigma,\mathbf{a}) = \frac{\sum_{i=1}^{n} i a_i f_i(x;\Delta,\xi,\sigma)}{ f_N(x;\Delta,\xi,\sigma,\mathbf{a})} = \frac{\sum_{i=1}^{n} i a_i f(x;\Delta_i,\xi_i,\sigma_i)}{ \sum_{i=1}^{n} a_i f(x;\Delta_i,\xi_i,\sigma_i)} \]

If $f_N(x;\Delta,\xi,\sigma,\mathbf{a})<10^{-4}$ then 0 is returned.

This also calculatues the error $\delta E$ of the function value at $x$:

\[ \delta^2 E = \left(\frac{\partial E}{\partial\Delta_p}\right)^2\delta^2\Delta_p+ \left(\frac{\partial E}{\partial\xi }\right)^2\delta^2\xi+ \left(\frac{\partial E}{\partial\sigma }\right)^2\delta^2\sigma+ \sum_{i=2}^N\left(\frac{\partial E}{\partial a_i}\right)^2\delta^2a_i \]

The partial derivatives are evaluated numerically.

Parameters

x	Where to evaluate $ E$
e	On return, $\delta E\|_{x}$
maxN	Maximum number of weights

Returns: $ E(x;\Delta,\xi,\sigma,\mathbf{a})$

Definition at line 615 of file TestELossDist.C.

Double_t Function::Evaluate ( Double_t x ) const

inline

Calculate the partial derivative of the function $E(\Delta;\Delta_p,\xi,\sigma,\mathbf{a})$ with respect to one of the parameters $\Delta_{p}$, $\xi$, $\sigma$, or $a_i$. Note for that

\begin{eqnarray*} \frac{\partial E}{\partial C} & \equiv & 0\\ \frac{\partial E}{\partial a_1} & \equiv & 0\\ \frac{\partial E}{\partial N} & \equiv & 0\\ @f{eqnarray*} @param x Where to evaluate the derivative @param ipar Parameter to differentiate relative to @param dPar Variation in parameter to use in calculation @param delta @f$\Delta_{p}@f$ @param xi @f$\xi@f$ @param sigma @f$\sigma@f$ @param n @f$N@f$ @param a @f$\mathbf{a} = (a_2,...,a_N)@f$ @return @f$\partial E/\partial p|_{x}@f$ */ Double_t DEstimatePar(Double_t x, Int_t ipar, Double_t delta, Double_t xi, Double_t sigma, UShort_t n, Double_t* a, Double_t dPar=0.001) { Double_t dp = dPar; Double_t d2 = dPar / 2; Double_t y1 = 0; Double_t y2 = 0; Double_t y3 = 0; Double_t y4 = 0; switch (ipar) { case Function::kC: case Function::kN: return 0; case Function::kDelta: y1 = NEstimate(x, delta+dp, xi, sigma, n, a); y2 = NEstimate(x, delta+d2, xi, sigma, n, a); y3 = NEstimate(x, delta-d2, xi, sigma, n, a); y4 = NEstimate(x, delta-dp, xi, sigma, n, a); break; case Function::kXi: y1 = NEstimate(x, delta, xi+dp, sigma, n, a); y2 = NEstimate(x, delta, xi+d2, sigma, n, a); y3 = NEstimate(x, delta, xi-d2, sigma, n, a); y4 = NEstimate(x, delta, xi-dp, sigma, n, a); break; case Function::kSigma: y1 = NEstimate(x, delta, xi, sigma+dp, n, a); y2 = NEstimate(x, delta, xi, sigma+d2, n, a); y3 = NEstimate(x, delta, xi, sigma-d2, n, a); y4 = NEstimate(x, delta, xi, sigma-dp, n, a); break; default: { Int_t j = ipar-kA; if (j+1 > n) return 0; Double_t aa = a[j]; a[j] = aa+dp; y1 = NEstimate(x, delta, xi, sigma, n, a); a[j] = aa+d2; y2 = NEstimate(x, delta, xi, sigma, n, a); a[j] = aa-d2; y3 = NEstimate(x, delta, xi, sigma, n, a); a[j] = aa-dp; y4 = NEstimate(x, delta, xi, sigma, n, a); a[j] = aa; } break; } Double_t d0 = y1 - y4; Double_t d1 = 2 * (y2 - y3); Double_t g = 1/(2*dp) * (4*d1 - d0) / 3; return g; } /** Generate a TF1 object of @f$ f_I@f$ @param n @f$ n@f$ - the number of particles @param c Constant @param delta @f$ \Delta@f$ @param xi @f$ \xi_1@f$ @param sigma @f$ \sigma_1@f$ @param a Array of @f$a_i@f$ for @f$i > 1@f$ @param xmin Least value of range @param xmax Largest value of range @return Newly allocated TF1 object */ static TF1* MakeFunc(Int_t n, Double_t c, Double_t delta, Double_t xi, Double_t sigma, Double_t* a, Double_t xmin, Double_t xmax) { Int_t nPar = kN + n; TF1* f = new TF1(Form("landGaus%d",n), &landauGausN, xmin, xmax, nPar); f->SetNpx(500); f->SetParName(kC, "C"); f->SetParName(kDelta, "#Delta_{p}"); f->SetParName(kXi, "#xi"); f->SetParName(kSigma, "#sigma"); f->SetParName(kN, "N"); f->SetParameter(kC, c); f->SetParameter(kDelta, delta); f->SetParameter(kXi, xi); f->SetParameter(kSigma, sigma); f->FixParameter(kN, n); f->SetParLimits(kDelta, xmin, 4); f->SetParLimits(kXi, 0.00, 4); f->SetParLimits(kSigma, 0.01, 0.11); for (Int_t i = 2; i <= n; i++) { Int_t j = i-2; f->SetParName(kA+j, Form("a_{%d}", i)); f->SetParameter(kA+j, a[j]); f->SetParLimits(kA+j, 0, 1); } return f; } /** Make a ROOT TF1 function object corresponding to a single component for @f$ i@f$ particles @param i Number of particles @param c @f$C@f$ @param delta @f$ \Delta@f$ @param xi @f$ \xi_1@f$ @param sigma @f$ \sigma_1@f$ @param xmin Minimum of range of @f$\Delta@f$ @param xmax Maximum of range of @f$\Delta@f$ @return Pointer to newly allocated ROOT TF1 object */ static TF1* MakeIFunc(Int_t i, Double_t c, Double_t delta, Double_t xi, Double_t sigma, Double_t xmin, Double_t xmax) { Int_t nPar = 5; TF1* f = new TF1(Form("landGausI%d",i), &landauGausI, xmin, xmax, nPar); f->SetNpx(100); f->SetParName(kC, "C"); f->SetParName(kDelta, "#Delta_{p}"); f->SetParName(kXi, "#xi"); f->SetParName(kSigma, "#sigma"); f->SetParName(kN, "N"); f->SetParameter(kC, c); f->SetParameter(kDelta, delta); f->SetParameter(kXi, xi); f->SetParameter(kSigma, sigma); f->FixParameter(kN, i); return f; } // --- Object code ------------------------------------------------- /** Constructor @param n @param c @param delta @param xi @param sigma @param a @param xmin @param xmax @return */ Function(Int_t n, Double_t c, Double_t delta, Double_t xi, Double_t sigma, Double_t* a, Double_t xmin, Double_t xmax) : fF(MakeFunc(n, c, delta, xi, sigma, a, xmin, xmax)) {} Function(TF1* f) : fF(f) {} /** Get pointer to ROOT TF1 object @return Pointer to TF1 object */ TF1* GetF1() { return fF; } /** Evaluate the function at @a x @f[ f_N(x;\Delta,\xi,\sigma) = \sum_{i=1}^{n} a_i f(x;\Delta_i,\xi_i,\sigma_i) \]

Parameters

x	Where to evaluate

Returns

Definition at line 503 of file TestELossDist.C.

Referenced by Generator::Draw(), and RandomEstimateNParticles().

Double_t Function::Evaluate	(	Double_t	x,
		Double_t &	e
	)		const

inline

Evaluate the function at x

\[ f_N(x;\Delta,\xi,\sigma) = \sum_{i=1}^{n} a_i f(x;\Delta_i,\xi_i,\sigma_i) \]

This also calculates the error on the point like

\[ \delta^2 f_N = \left(\frac{\partial f_N}{\partial\Delta_p}\right)^2\delta^2\Delta_p+ \left(\frac{\partial f_N}{\partial\xi }\right)^2\delta^2\xi+ \left(\frac{\partial f_N}{\partial\sigma }\right)^2\delta^2\sigma+ \sum_{i=2}^N\left(\frac{\partial f_N}{\partial a_i}\right)^2\delta^2a_i \]

Parameters

x	Where to evaluate
e	On return, contains the error $\delta f_N$ on the function value

Returns: $ f_N(x;\Delta,\xi,\sigma)$

Definition at line 529 of file TestELossDist.C.

Double_t Function::GetA ( Int_t i ) const

inline

Returns: $a_{i}$

Definition at line 701 of file TestELossDist.C.

Referenced by Generator::Draw(), Evaluate(), and RandomEstimateNParticles().

Double_t* Function::GetAs ( ) const

inline

Returns: $a_2,...,a_N$

Definition at line 703 of file TestELossDist.C.

Referenced by EstimateNParticles().

Double_t Function::GetC ( ) const

inline

Returns: $C$

Definition at line 691 of file TestELossDist.C.

Referenced by Generator::Draw().

Double_t Function::GetDelta ( ) const

inline

Returns: $\Delta_{mp}$

Definition at line 693 of file TestELossDist.C.

Referenced by Generator::Draw(), EstimateNParticles(), Evaluate(), and RandomEstimateNParticles().

Double_t Function::GetEA ( Int_t i ) const

inline

Returns: $\Delta a_{i}$

Definition at line 715 of file TestELossDist.C.

Referenced by EstimateNParticles(), and Evaluate().

Double_t* Function::GetEAs ( ) const

inline

Returns: $\Delta a_2,...,\Delta a_N$

Definition at line 717 of file TestELossDist.C.

Double_t Function::GetEC ( ) const

inline

Returns: $\Delta C$

Definition at line 705 of file TestELossDist.C.

Double_t Function::GetEDelta ( ) const

inline

Returns: $\Delta \Delta_{mp}$

Definition at line 707 of file TestELossDist.C.

Referenced by EstimateNParticles(), and Evaluate().

UShort_t Function::GetEN ( ) const

inline

Returns: $\Delta N$

Definition at line 713 of file TestELossDist.C.

Double_t Function::GetESigma ( ) const

inline

Returns: $\Delta \sigma$

Definition at line 711 of file TestELossDist.C.

Referenced by EstimateNParticles(), and Evaluate().

Double_t Function::GetEXi ( ) const

inline

Returns: $\Delta \xi$

Definition at line 709 of file TestELossDist.C.

Referenced by EstimateNParticles(), and Evaluate().

UShort_t Function::GetN ( ) const

inline

Returns: $N$

Definition at line 699 of file TestELossDist.C.

Referenced by EstimateNParticles(), Evaluate(), and RandomEstimateNParticles().

Double_t Function::GetSigma ( ) const

inline

Returns: $\sigma$

Definition at line 697 of file TestELossDist.C.

Referenced by Generator::Draw(), EstimateNParticles(), Evaluate(), and RandomEstimateNParticles().

Double_t Function::GetXi ( ) const

inline

Returns: $\xi$

Definition at line 695 of file TestELossDist.C.

Referenced by Generator::Draw(), EstimateNParticles(), Evaluate(), and RandomEstimateNParticles().

static Double_t Function::IdLandauGausdPar	(	Int_t	i,
		Double_t	x,
		UShort_t	ipar,
		Double_t	dPar,
		Double_t	delta,
		Double_t	xi,
		Double_t	sigma
	)

inlinestatic

Numerically evaluate

\[ \left.\frac{\partial f_i}{\partial p_i}\right|_{x} \]

where $ p_i$ is the $ i^{\mbox{th}}$ parameter. The mapping of the parameters is given by

0: $\Delta$
1: $\xi$
2: $\sigma$

This is the partial derivative with respect to the parameter of the response function corresponding to $ i$ particles i.e., with the substitutions

\[ \Delta \rightarrow \Delta_i = i(\Delta + \xi\log(i))\\ \xi \rightarrow \xi_i = i \xi\\ \sigma \rightarrow \sigma_i = \sqrt{i}\sigma\\ \]

Parameters

i	$ i$
x	Where to evaluate
ipar	Parameter number
dPar	$ \epsilon\delta p_i$ for some value of $\epsilon$
delta	$ \Delta$
xi	$ \xi$
sigma	$ \sigma$

Returns: $ f_i$ evaluated

Definition at line 206 of file TestELossDist.C.

Referenced by Evaluate().

static Double_t Function::ILandauGaus	(	Int_t	i,
		Double_t	x,
		Double_t	delta,
		Double_t	xi,
		Double_t	sigma
	)

inlinestatic

Evaluate

\[ f_i(x;\Delta,\xi,\sigma) = f(x;\Delta_i,\xi_i,\sigma_i) \]

corresponding to $ i$ particles i.e., with the substitutions

\begin{eqnarray*} \Delta \rightarrow \Delta_i &=& i(\Delta + \xi\log(i))\\ \xi \rightarrow \xi_i &=& i \xi\\ \sigma \rightarrow \sigma_i &=& \sqrt{i}\sigma\\ \end{eqnarray*}

Parameters

x	Where to evaluate
delta	$ \Delta$
xi	$ \xi$
sigma	$ \sigma$
i	$ i $

Returns: $ f_i $ evaluated *

Definition at line 160 of file TestELossDist.C.

Referenced by Evaluate(), IdLandauGausdPar(), landauGausI(), NEstimate(), NLandauGaus(), and RandomEstimateNParticles().

static Double_t Function::Landau	(	Double_t	x,
		Double_t	delta,
		Double_t	xi
	)

inlinestatic

Evaluate shifted landay

\[ f'_{L}(x;\Delta,\xi) = f_L(x;\Delta+0.22278298\xi) \]

where $ f_{L}$ is the ROOT implementation of the Landau distribution (known to have $ \Delta_{p}=-0.22278298$ for $\Delta=0,\xi=1$.

Parameters

x	Where to evaluate $ f'_{L}$
delta	Most probable value
xi	The 'width' of the distribution

Returns: $ f'_{L}(x;\Delta,\xi) $

Definition at line 84 of file TestELossDist.C.

Referenced by ILandauGaus().

static Double_t Function::LandauGaus	(	Double_t	x,
		Double_t	delta,
		Double_t	xi,
		Double_t	sigma
	)

inlinestatic

Calculate the value of a Landau convolved with a Gaussian

\[ f(x;\Delta,\xi,\sigma') = \frac{1}{\sigma' \sqrt{2 \pi}} \int_{-\infty}^{+\infty} d\Delta' f'_{L}(x;\Delta',\xi) \exp{-\frac{(\Delta-\Delta')^2}{2\sigma^2}} \]

where $ f'_{L}$ is the Landau distribution, $ \Delta$ the energy loss, $ \xi$ the width of the Landau, and $\sigma$ is the variance of the Gaussian.

Note that this function uses the constants fgkConvNSteps and fgkConvNSigma

References:

Parameters

x	where to evaluate $ f$
delta	$ \Delta$ of $ f(x;\Delta,\xi,\sigma')$
xi	$ \xi$ of $ f(x;\Delta,\xi,\sigma')$
sigma	$ \sigma$ of Gaussian

Returns: $ f$ evaluated at $ x$.

Definition at line 118 of file TestELossDist.C.

Referenced by ILandauGaus(), landauGaus1(), NEstimate(), and NLandauGaus().

Double_t Function::NEstimate	(	Double_t	x,
		Double_t	delta,
		Double_t	xi,
		Double_t	sigma,
		Int_t	n,
		Double_t *	a
	)

inline

Calculate the the estimate of number of particle contribtutions at energy loss x

\[ E(\Delta;\Delta_p,\xi,\sigma,\mathbf{a}) = \frac{\sum_{i=1}^N i a_i f_i(x;\Delta_p,\xi,\sigma)}{ \sum_{i=1}^N a_i f_i(x;\Delta_p,\xi,\sigma)} \]

where $a_1=1$

Parameters

x	Energy loss $\Delta$
delta	$\Delta_{p}$
xi	$\xi$
sigma	$\sigma$
n	Maximum number of particles $N$
a	Weights $a_i$ for $i=2,...,N$

Returns: $E(\Delta;\Delta_p,\xi,\sigma,\mathbf{a})$

Definition at line 304 of file TestELossDist.C.

Referenced by EstimateNParticles().

static Double_t Function::NLandauGaus	(	Double_t	x,
		Double_t	delta,
		Double_t	xi,
		Double_t	sigma,
		Int_t	n,
		Double_t *	a
	)

inlinestatic

Evaluate

\[ f_N(x;\Delta,\xi,\sigma') = \sum_{i=1}^N a_i f_i(x;\Delta,\xi,\sigma'a) \]

where $ f(x;\Delta,\xi,\sigma')$ is the convolution of a Landau with a Gaussian (see LandauGaus). Note that $ a_1 = 1$, $\Delta_i = i(\Delta_1 + \xi\log(i))$, $\xi_i=i\xi_1$, and $\sigma_i'^2 = \sigma_n^2 + i\sigma_1^2$.

References:

Nucl.Instrum.Meth.B1:16
Phys.Rev.A28:615
Parameters

x Where to evaluate $ f_N$

delta $ \Delta_1$

xi $ \xi_1$

sigma $ \sigma_1$

n $ N$ in the sum above.

a Array of size $ N-1$ of the weights $ a_i$ for $ i > 1$

Returns
$ f_N(x;\Delta,\xi,\sigma)$

Definition at line 277 of file TestELossDist.C.

Referenced by landauGausN().

Double_t Function::RandomEstimateNParticles	(	Double_t	x,
		Short_t	maxN = `-1`
	)

inline

Estimate the number of particles by calculating the weights

\[ w_i = a_i f_i(x;\Delta_p,\xi,\sigma) \]

and then draw a random number $r$ in the range $[0,\sum_i^N w_i]$ and return the $i$ for which $w_{i-1} < @r \leq w_{i}$ (notem $w_{-1}=0$).

Parameters

x	Where to evaluate the component functions
maxN	Maximum weight to use

Returns: Estimate of the number of particles using the procedure outlined above.

Definition at line 654 of file TestELossDist.C.

Referenced by Generator::Draw().

Member Data Documentation

TF1* Function::fF

The ROOT object

Definition at line 719 of file TestELossDist.C.

Referenced by Draw(), Evaluate(), GetA(), GetAs(), GetC(), GetDelta(), GetEA(), GetEAs(), GetEC(), GetEDelta(), GetESigma(), GetEXi(), GetN(), GetSigma(), and GetXi().

const Double_t Function::fgkConvNSigma = 5

static

Number of sigmas to integrate over.

Definition at line 65 of file TestELossDist.C.

Referenced by LandauGaus().

const Double_t Function::fgkConvNSteps = 100

static

Number of steps in integration.

Definition at line 67 of file TestELossDist.C.

Referenced by LandauGaus().

const Double_t Function::fgkInvRoot2Pi = 1. / TMath::Sqrt(2*TMath::Pi())

static

1 / sqrt(2 * pi)

Definition at line 63 of file TestELossDist.C.

Referenced by LandauGaus().

const Double_t Function::fgkMPShift = -0.22278298

static

MP shift.

Definition at line 61 of file TestELossDist.C.

Referenced by Landau(), and LandauGaus().

The documentation for this struct was generated from the following file:

PWGLF/FORWARD/analysis2/tests/TestELossDist.C

x	where to evaluate \( f\)
delta	\( \Delta\) of \( f(x;\Delta,\xi,\sigma')\)
xi	\( \xi\) of \( f(x;\Delta,\xi,\sigma')\)
sigma	\( \sigma\) of Gaussian

x	Energy loss \(\Delta\)
delta	\(\Delta_{p}\)
xi	\(\xi\)
sigma	\(\sigma\)
n	Maximum number of particles \(N\)
a	Weights \(a_i\) for \(i=2,...,N\)

x	Where to evaluate \( f_N\)
delta	\( \Delta_1\)
xi	\( \xi_1\)
sigma	\( \sigma_1\)
n	\( N\) in the sum above.
a	Array of size \( N-1\) of the weights \( a_i\) for \( i > 1\)

x	Where to evaluate \( E\)
e	On return, \(\delta E\|_{x}\)
maxN	Maximum number of weights

x	Where to evaluate
e	On return, contains the error \(\delta f_N\) on the function value

i	\( i\)
x	Where to evaluate
ipar	Parameter number
dPar	\( \epsilon\delta p_i\) for some value of \(\epsilon\)
delta	\( \Delta\)
xi	\( \xi\)
sigma	\( \sigma\)

x	Where to evaluate \( f'_{L}\)
delta	Most probable value
xi	The 'width' of the distribution

Public Types

Public Member Functions

Static Public Member Functions

Public Attributes

Static Public Attributes

Detailed Description

Member Enumeration Documentation

Member Function Documentation

Member Data Documentation