This is the base class for the ROOT Random number generators.

This class defines the ROOT Random number interface and it should not be instantiated directly but used via its derived classes The derived class are :

TRandom1 based on the RANLUX algorithm, has mathematically proven random proprieties and a period of about 10**171. It is however slower than the others.
TRandom2 is based on the Tausworthe generator of L'Ecuyer, and it has the advantage of being fast and using only 3 words (of 32 bits) for the state. The period is 10**26.
TRandom3 is based on the "Mersenne Twister generator", and is currently the recommended one, since it has quite good random proprieties (period of about 10**6000 ) and it is fast. However it fails some of the tests of the TestU)1 suite. In addition this generator provide only random number with 32 bits random.
Generator based on the template TRandomGen<Engine> class. Convenient typedef's, available also at the ROOT prompts, are defined for generator of the MIXMAX family (see the MIXMAX HEPFORGE Web page) and based on ROOT::Math::MixMaxEngine, and for the random engines from the standard library, based on ROOT::Math::StdEngine (see the C++ random documentation.). The typdefs are:
- TRandomMixMax : MIXMAX generator based on a state of N=240. This generator provides 61 bits random with a very large period (10**4839)
- TRandomMixMax17 : MIXMAX generator based on a state of N=17. This generator has a fast seeding time compared to N=240, but still a very large period (10**294)
- TRandomMixMax256 : MIXMAX generator based on a state of N=256. It is the generator described in the 2015 paper (based on a matrix with m=0 and special number s=-1), but with skipping to have higher randomness.
- TRandomMT64 : Generator based on a the Mersenne-Twister generator with 64 bits, using the implementation provided by the standard library ( std::mt19937_64 )
- TRandomRanlux48 : Generator based on a the RanLux generator with 48 bits, using the implementation provided by the standard library (std::ranlux48).

Note also that this class implements also a very simple generator (linear congruential) with periodicity = 10**9 which is known to have defects (the lower random bits are correlated) and therefore should NOT be used in any statistical study.

The following table shows some timings (in nanoseconds/call) for the random numbers obtained using a 2.6 GHz Intel Core i7 CPU:

TRandom 3 ns/call (but this is a very BAD Generator, not to be used)
TRandom1 82 ns/call
TRandom2 7 ns/call
TRandom3 5 ns/call
TRandomMixMax 6 ns/call
TRandomMixMax17 6 ns/call
TRandomMixMax256 10 ns/call
TRandomMT64 9 ns/call
TRandomRanlux48 270 ns/call

The following methods are provided to generate random numbers disctributed according to some basic distributions:

::Exp(tau)
::Integer(imax)
::Gaus(mean,sigma)
::Rndm()
::Uniform(x1)
::Landau(mpv,sigma)
::Poisson(mean)
::Binomial(ntot,prob)

Random numbers distributed according to 1-d, 2-d or 3-d distributions contained in TF1, TF2 or TF3 objects can also be generated. For example, to get a random number distributed following abs(sin(x)/x)*sqrt(x) you can do :

TF1 *f1 = new TF1("f1","abs(sin(x)/x)*sqrt(x)",0,10);

double r = f1->GetRandom();

or you can use the UNURAN package. You need in this case to initialize UNURAN to the function you would like to generate.

TUnuran u;
u.Init(TUnuranDistrCont(f1));
double r = u.Sample();

The techniques of using directly a TF1,2 or 3 function is powerful and can be used to generate numbers in the defined range of the function. Getting a number from a TF1,2,3 function is also quite fast. UNURAN is a powerful and flexible tool which containes various methods for generate random numbers for continuous distributions of one and multi-dimension. It requires some set-up (initialization) phase and can be very fast when the distribution parameters are not changed for every call.

The following table shows some timings (in nanosecond/call) for basic functions, TF1 functions and using UNURAN obtained running the tutorial math/testrandom.C Numbers have been obtained on an Intel Xeon Quad-core Harpertown (E5410) 2.33 GHz running Linux SLC4 64 bit and compiled with gcc 3.4

Distribution            nanoseconds/call
                    TRandom  TRandom1 TRandom2 TRandom3
Rndm..............    5.000  105.000    7.000   10.000
RndmArray.........    4.000  104.000    6.000    9.000
Gaus..............   36.000  180.000   40.000   48.000
Rannor............  118.000  220.000  120.000  124.000
Landau............   22.000  123.000   26.000   31.000
Exponential.......   93.000  198.000   98.000  104.000
Binomial(5,0.5)...   30.000  548.000   46.000   65.000
Binomial(15,0.5)..   75.000 1615.000  125.000  178.000
Poisson(3)........   96.000  494.000  109.000  125.000
Poisson(10).......  138.000 1236.000  165.000  203.000
Poisson(70).......  818.000 1195.000  835.000  844.000
Poisson(100)......  837.000 1218.000  849.000  864.000
GausTF1...........   83.000  180.000   87.000   88.000
LandauTF1.........   80.000  180.000   83.000   86.000
GausUNURAN........   40.000  139.000   41.000   44.000
PoissonUNURAN(10).   85.000  271.000   92.000  102.000
PoissonUNURAN(100)   62.000  256.000   69.000   78.000

Note that the time to generate a number from an arbitrary TF1 function using TF1::GetRandom or using TUnuran is independent of the complexity of the function.

TH1::FillRandom(TH1 *) or TH1::FillRandom(const char *tf1name) can be used to fill an histogram (1-d, 2-d, 3-d from an existing histogram or from an existing function.

Note this interesting feature when working with objects. You can use several TRandom objects, each with their "independent" random sequence. For example, one can imagine

TRandom *eventGenerator = new TRandom();

TRandom *tracking = new TRandom();

eventGenerator can be used to generate the event kinematics. tracking can be used to track the generated particles with random numbers independent from eventGenerator. This very interesting feature gives the possibility to work with simple and very fast random number generators without worrying about random number periodicity as it was the case with Fortran. One can use TRandom::SetSeed to modify the seed of one generator.

A TRandom object may be written to a Root file

as part of another object
or with its own key (example: gRandom->Write("Random") ) ;

Definition at line 27 of file TRandom.h.

Public Member Functions
	TRandom (UInt_t seed=65539)
	Default constructor. For seed see SetSeed(). More...

virtual	~TRandom ()
	Default destructor. More...

virtual Int_t	Binomial (Int_t ntot, Double_t prob)
	Generates a random integer N according to the binomial law. More...

virtual Double_t	BreitWigner (Double_t mean=0, Double_t gamma=1)
	Return a number distributed following a BreitWigner function with mean and gamma. More...

virtual void	Circle (Double_t &x, Double_t &y, Double_t r)
	Generates random vectors, uniformly distributed over a circle of given radius. More...

virtual Double_t	Exp (Double_t tau)
	Returns an exponential deviate. More...

virtual Double_t	Gaus (Double_t mean=0, Double_t sigma=1)
	Samples a random number from the standard Normal (Gaussian) Distribution with the given mean and sigma. More...

virtual UInt_t	GetSeed () const

virtual UInt_t	Integer (UInt_t imax)
	Returns a random integer on [ 0, imax-1 ]. More...

virtual Double_t	Landau (Double_t mean=0, Double_t sigma=1)
	Generate a random number following a Landau distribution with location parameter mu and scale parameter sigma: Landau( (x-mu)/sigma ) Note that mu is not the mpv(most probable value) of the Landa distribution and sigma is not the standard deviation of the distribution which is not defined. More...

virtual Int_t	Poisson (Double_t mean)
	Generates a random integer N according to a Poisson law. More...

virtual Double_t	PoissonD (Double_t mean)
	Generates a random number according to a Poisson law. More...

virtual void	Rannor (Float_t &a, Float_t &b)
	Return 2 numbers distributed following a gaussian with mean=0 and sigma=1. More...

virtual void	Rannor (Double_t &a, Double_t &b)
	Return 2 numbers distributed following a gaussian with mean=0 and sigma=1. More...

virtual void	ReadRandom (const char *filename)
	Reads saved random generator status from filename. More...

virtual Double_t	Rndm ()
	Machine independent random number generator. More...

virtual Double_t	Rndm (Int_t)

virtual void	RndmArray (Int_t n, Float_t *array)
	Return an array of n random numbers uniformly distributed in ]0,1]. More...

virtual void	RndmArray (Int_t n, Double_t *array)
	Return an array of n random numbers uniformly distributed in ]0,1]. More...

virtual void	SetSeed (ULong_t seed=0)
	Set the random generator seed. More...

virtual void	Sphere (Double_t &x, Double_t &y, Double_t &z, Double_t r)
	Generates random vectors, uniformly distributed over the surface of a sphere of given radius. More...

virtual Double_t	Uniform (Double_t x1=1)
	Returns a uniform deviate on the interval (0, x1). More...

virtual Double_t	Uniform (Double_t x1, Double_t x2)
	Returns a uniform deviate on the interval (x1, x2). More...

virtual void	WriteRandom (const char *filename) const
	Writes random generator status to filename. More...

Public Member Functions inherited from ROOT::Math::TRandomEngine
virtual	~TRandomEngine ()

Protected Attributes
UInt_t	fSeed

#include <TRandom.h>

Inheritance diagram for TRandom:

[legend]

Constructor & Destructor Documentation

◆ TRandom()

TRandom::TRandom ( UInt_t seed = 65539 )

Default constructor. For seed see SetSeed().

Definition at line 168 of file TRandom.cxx.

◆ ~TRandom()

TRandom::~TRandom ( )

virtual

Default destructor.

Can reset gRandom to 0 if gRandom points to this generator.

Definition at line 177 of file TRandom.cxx.

Member Function Documentation

◆ Binomial()

Int_t TRandom::Binomial	(	Int_t	ntot,
		Double_t	prob
	)

virtual

Generates a random integer N according to the binomial law.

Coded from Los Alamos report LA-5061-MS.

N is binomially distributed between 0 and ntot inclusive with mean prob*ntot and prob is between 0 and 1.

Note: This function should not be used when ntot is large (say >100). The normal approximation is then recommended instead (with mean =*ntot+0.5 and standard deviation sqrt(ntot*prob*(1-prob)).

Definition at line 193 of file TRandom.cxx.

◆ BreitWigner()

Double_t TRandom::BreitWigner	(	Double_t	mean = `0`,
		Double_t	gamma = `1`
	)

virtual

Return a number distributed following a BreitWigner function with mean and gamma.

Definition at line 207 of file TRandom.cxx.

◆ Circle()

void TRandom::Circle	(	Double_t &	x,
		Double_t &	y,
		Double_t	r
	)

virtual

Generates random vectors, uniformly distributed over a circle of given radius.

Input : r = circle radius Output: x,y a random 2-d vector of length r

Definition at line 221 of file TRandom.cxx.

◆ Exp()

Double_t TRandom::Exp ( Double_t tau )

virtual

Returns an exponential deviate.

exp( -t/tau )

Definition at line 233 of file TRandom.cxx.

◆ Gaus()

Double_t TRandom::Gaus	(	Double_t	mean = `0`,
		Double_t	sigma = `1`
	)

virtual

Samples a random number from the standard Normal (Gaussian) Distribution with the given mean and sigma.

Uses the Acceptance-complement ratio from W. Hoermann and G. Derflinger This is one of the fastest existing method for generating normal random variables. It is a factor 2/3 faster than the polar (Box-Muller) method used in the previous version of TRandom::Gaus. The speed is comparable to the Ziggurat method (from Marsaglia) implemented for example in GSL and available in the MathMore library.

REFERENCE: - W. Hoermann and G. Derflinger (1990): The ACR Method for generating normal random variables, OR Spektrum 12 (1990), 181-185.

Definition at line 256 of file TRandom.cxx.

◆ GetSeed()

virtual UInt_t TRandom::GetSeed ( ) const

inlinevirtual

Reimplemented in TRandom1, and TRandom3.

Definition at line 40 of file TRandom.h.

◆ Integer()

UInt_t TRandom::Integer ( UInt_t imax )

virtual

Returns a random integer on [ 0, imax-1 ].

Definition at line 341 of file TRandom.cxx.

◆ Landau()

Double_t TRandom::Landau	(	Double_t	mu = `0`,
		Double_t	sigma = `1`
	)

virtual

Generate a random number following a Landau distribution with location parameter mu and scale parameter sigma: Landau( (x-mu)/sigma ) Note that mu is not the mpv(most probable value) of the Landa distribution and sigma is not the standard deviation of the distribution which is not defined.

For mu =0 and sigma=1, the mpv = -0.22278

The Landau random number generation is implemented using the function landau_quantile(x,sigma), which provides the inverse of the landau cumulative distribution. landau_quantile has been converted from CERNLIB ranlan(G110).

Definition at line 361 of file TRandom.cxx.

◆ Poisson()

Int_t TRandom::Poisson ( Double_t mean )

virtual

Generates a random integer N according to a Poisson law.

Prob(N) = exp(-mean)*mean^N/Factorial(N)

Use a different procedure according to the mean value. The algorithm is the same used by CLHEP. For lower value (mean < 25) use the rejection method based on the exponential. For higher values use a rejection method comparing with a Lorentzian distribution, as suggested by several authors. This routine since is returning 32 bits integer will not work for values larger than 2*10**9. One should then use the Trandom::PoissonD for such large values.

Definition at line 383 of file TRandom.cxx.

◆ PoissonD()

Double_t TRandom::PoissonD ( Double_t mean )

virtual

Generates a random number according to a Poisson law.

Prob(N) = exp(-mean)*mean^N/Factorial(N)

This function is a variant of TRandom::Poisson returning a double instead of an integer.

Definition at line 435 of file TRandom.cxx.

◆ Rannor() [1/2]

void TRandom::Rannor	(	Float_t &	a,
		Float_t &	b
	)

virtual

Return 2 numbers distributed following a gaussian with mean=0 and sigma=1.

Definition at line 481 of file TRandom.cxx.

◆ Rannor() [2/2]

void TRandom::Rannor	(	Double_t &	a,
		Double_t &	b
	)

virtual

Return 2 numbers distributed following a gaussian with mean=0 and sigma=1.

Definition at line 496 of file TRandom.cxx.

◆ ReadRandom()

void TRandom::ReadRandom ( const char * filename )

virtual

Reads saved random generator status from filename.

Definition at line 511 of file TRandom.cxx.

◆ Rndm() [1/2]

Double_t TRandom::Rndm ( )

virtual

Machine independent random number generator.

Based on the BSD Unix (Rand) Linear congrential generator. Produces uniformly-distributed floating points between 0 and 1. Identical sequence on all machines of >= 32 bits. Periodicity = 2**31, generates a number in (0,1). Note that this is a generator which is known to have defects (the lower random bits are correlated) and therefore should NOT be used in any statistical study).

Implements ROOT::Math::TRandomEngine.

Reimplemented in TRandomGen< Engine >, TRandom1, TRandom3, and TRandom2.

Definition at line 533 of file TRandom.cxx.

◆ Rndm() [2/2]

virtual Double_t TRandom::Rndm ( Int_t )

inlinevirtual

Definition at line 51 of file TRandom.h.

◆ RndmArray() [1/2]

void TRandom::RndmArray	(	Int_t	n,
		Float_t *	array
	)

virtual

Return an array of n random numbers uniformly distributed in ]0,1].

Reimplemented in TRandomGen< Engine >, TRandom1, TRandom3, and TRandom2.

Definition at line 569 of file TRandom.cxx.

◆ RndmArray() [2/2]

void TRandom::RndmArray	(	Int_t	n,
		Double_t *	array
	)

virtual

Return an array of n random numbers uniformly distributed in ]0,1].

Reimplemented in TRandomGen< Engine >, TRandom1, TRandom3, and TRandom2.

Definition at line 556 of file TRandom.cxx.

◆ SetSeed()

void TRandom::SetSeed ( ULong_t seed = 0 )

virtual

Set the random generator seed.

Note that default value is zero, which is different than the default value used when constructing the class. If the seed is zero the seed is set to a random value which in case of TRandom depends on the lowest 4 bytes of TUUID The UUID will be identical if SetSeed(0) is called with time smaller than 100 ns Instead if a different generator implementation is used (TRandom1, 2 or 3) the seed is generated using a 128 bit UUID. This results in different seeds and then random sequence for every SetSeed(0) call.

Reimplemented in TRandomGen< Engine >, TRandom1, TRandom3, and TRandom2.

Definition at line 589 of file TRandom.cxx.

◆ Sphere()

void TRandom::Sphere	(	Double_t &	x,
		Double_t &	y,
		Double_t &	z,
		Double_t	r
	)

virtual

Generates random vectors, uniformly distributed over the surface of a sphere of given radius.

Input : r = sphere radius Output: x,y,z a random 3-d vector of length r Method: (based on algorithm suggested by Knuth and attributed to Robert E Knop) which uses less random numbers than the CERNLIB RN23DIM algorithm

Definition at line 609 of file TRandom.cxx.

◆ Uniform() [1/2]

Double_t TRandom::Uniform ( Double_t x1 = 1 )

virtual

Returns a uniform deviate on the interval (0, x1).

Definition at line 627 of file TRandom.cxx.

◆ Uniform() [2/2]

Double_t TRandom::Uniform	(	Double_t	x1,
		Double_t	x2
	)

virtual

Returns a uniform deviate on the interval (x1, x2).

Definition at line 636 of file TRandom.cxx.

◆ WriteRandom()

void TRandom::WriteRandom ( const char * filename ) const

virtual

Writes random generator status to filename.

Definition at line 645 of file TRandom.cxx.

Member Data Documentation

◆ fSeed

UInt_t TRandom::fSeed

protected

Definition at line 30 of file TRandom.h.

Libraries for TRandom:

[legend]

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

math/mathcore/inc/TRandom.h
math/mathcore/src/TRandom.cxx

Public Member Functions

Protected Attributes

Constructor & Destructor Documentation

◆ TRandom()

◆ ~TRandom()

Member Function Documentation

◆ Binomial()

◆ BreitWigner()

◆ Circle()

◆ Exp()

◆ Gaus()

◆ GetSeed()

◆ Integer()

◆ Landau()

◆ Poisson()

◆ PoissonD()

◆ Rannor() [1/2]

◆ Rannor() [2/2]

◆ ReadRandom()

◆ Rndm() [1/2]

◆ Rndm() [2/2]

◆ RndmArray() [1/2]

◆ RndmArray() [2/2]

◆ SetSeed()

◆ Sphere()

◆ Uniform() [1/2]

◆ Uniform() [2/2]

◆ WriteRandom()

Member Data Documentation

◆ fSeed