namespace TMath

```
TMath

Encapsulate math routines.

```

Function Members (Methods)

public:
 Short_t Abs(Short_t d) Int_t Abs(Int_t d) Long_t Abs(Long_t d) Long64_t Abs(Long64_t d) Float_t Abs(Float_t d) Double_t Abs(Double_t d) LongDouble_t Abs(LongDouble_t d) Double_t ACos(Double_t x) Double_t ACosH(Double_t) Bool_t AreEqualAbs(Double_t af, Double_t bf, Double_t epsilon) Bool_t AreEqualRel(Double_t af, Double_t bf, Double_t relPrec) Double_t ASin(Double_t x) Double_t ASinH(Double_t) Double_t ATan(Double_t x) Double_t ATan2(Double_t y, Double_t x) Double_t ATanH(Double_t) Double_t BesselI(Int_t n, Double_t x) Double_t BesselI0(Double_t x) Double_t BesselI1(Double_t x) Double_t BesselJ0(Double_t x) Double_t BesselJ1(Double_t x) Double_t BesselK(Int_t n, Double_t x) Double_t BesselK0(Double_t x) Double_t BesselK1(Double_t x) Double_t BesselY0(Double_t x) Double_t BesselY1(Double_t x) Double_t Beta(Double_t p, Double_t q) Double_t BetaCf(Double_t x, Double_t a, Double_t b) Double_t BetaDist(Double_t x, Double_t p, Double_t q) Double_t BetaDistI(Double_t x, Double_t p, Double_t q) Double_t BetaIncomplete(Double_t x, Double_t a, Double_t b) Long64_t BinarySearch(Long64_t n, const short* array, short value) Long64_t BinarySearch(Long64_t n, const short** array, short value) Long64_t BinarySearch(Long64_t n, const int* array, int value) Long64_t BinarySearch(Long64_t n, const int** array, int value) Long64_t BinarySearch(Long64_t n, const float* array, float value) Long64_t BinarySearch(Long64_t n, const float** array, float value) Long64_t BinarySearch(Long64_t n, const double* array, double value) Long64_t BinarySearch(Long64_t n, const double** array, double value) Long64_t BinarySearch(Long64_t n, const long* array, long value) Long64_t BinarySearch(Long64_t n, const long** array, long value) Long64_t BinarySearch(Long64_t n, const long long* array, long long value) Long64_t BinarySearch(Long64_t n, const long long** array, long long value) Double_t Binomial(Int_t n, Int_t k) Double_t BinomialI(Double_t p, Int_t n, Int_t k) Double_t BreitWigner(Double_t x, Double_t mean = 0, Double_t gamma = 1) void BubbleHigh(Int_t Narr, Double_t* arr1, Int_t* arr2) void BubbleLow(Int_t Narr, Double_t* arr1, Int_t* arr2) Double_t C() Double_t CauchyDist(Double_t x, Double_t t = 0, Double_t s = 1) Double_t Ccgs() Double_t Ceil(Double_t x) Int_t CeilNint(Double_t x) Double_t ChisquareQuantile(Double_t p, Double_t ndf) Double_t Cos(Double_t x) Double_t CosH(Double_t x) float* Cross(const float* v1, const float* v2, float* out) double* Cross(const double* v1, const double* v2, double* out) Double_t CUncertainty() Double_t DegToRad() Double_t DiLog(Double_t x) Double_t E() Double_t Erf(Double_t x) Double_t Erfc(Double_t x) Double_t ErfcInverse(Double_t x) Double_t ErfInverse(Double_t x) Double_t EulerGamma() Bool_t Even(Long_t a) Double_t Exp(Double_t x) Double_t Factorial(Int_t i) Double_t FDist(Double_t F, Double_t N, Double_t M) Double_t FDistI(Double_t F, Double_t N, Double_t M) Int_t Finite(Double_t x) Double_t Floor(Double_t x) Int_t FloorNint(Double_t x) Double_t Freq(Double_t x) Double_t G() Double_t Gamma(Double_t z) Double_t Gamma(Double_t a, Double_t x) Double_t GammaDist(Double_t x, Double_t gamma, Double_t mu = 0, Double_t beta = 1) Double_t Gaus(Double_t x, Double_t mean = 0, Double_t sigma = 1, Bool_t norm = kFALSE) Double_t Gcgs() Double_t GeomMean(Long64_t n, const short* a) Double_t GeomMean(Long64_t n, const int* a) Double_t GeomMean(Long64_t n, const float* a) Double_t GeomMean(Long64_t n, const double* a) Double_t GeomMean(Long64_t n, const long* a) Double_t GeomMean(Long64_t n, const long long* a) Double_t GhbarC() Double_t GhbarCUncertainty() Double_t Gn() Double_t GnUncertainty() Double_t GUncertainty() Double_t H() ULong_t Hash(const char* str) ULong_t Hash(const void* txt, Int_t ntxt) Double_t Hbar() Double_t Hbarcgs() Double_t HbarUncertainty() Double_t HC() Double_t HCcgs() Double_t Hcgs() Double_t HUncertainty() Double_t Hypot(Double_t x, Double_t y) Long_t Hypot(Long_t x, Long_t y) Double_t Infinity() Double_t InvPi() Bool_t IsInside(float xp, float yp, Int_t np, float* x, float* y) Bool_t IsInside(int xp, int yp, Int_t np, int* x, int* y) Int_t IsNaN(Double_t x) Double_t K() Double_t Kcgs() Double_t KolmogorovProb(Double_t z) Double_t KolmogorovTest(Int_t na, const Double_t* a, Int_t nb, const Double_t* b, Option_t* option) short KOrdStat(long long n, const short* a, long long k, long long* work = 0) int KOrdStat(long long n, const int* a, long long k, long long* work = 0) float KOrdStat(long long n, const float* a, long long k, long long* work = 0) double KOrdStat(long long n, const double* a, long long k, long long* work = 0) long KOrdStat(long long n, const long* a, long long k, long long* work = 0) long long KOrdStat(long long n, const long long* a, long long k, long long* work = 0) Double_t KUncertainty() Double_t Landau(Double_t x, Double_t mpv = 0, Double_t sigma = 1, Bool_t norm = kFALSE) Double_t LandauI(Double_t x) Double_t LaplaceDist(Double_t x, Double_t alpha = 0, Double_t beta = 1) Double_t LaplaceDistI(Double_t x, Double_t alpha = 0, Double_t beta = 1) Double_t Ldexp(Double_t x, Int_t exp) Double_t Ln10() Double_t LnGamma(Double_t z) Long64_t LocMax(Long64_t n, const short* a) Long64_t LocMax(Long64_t n, const int* a) Long64_t LocMax(Long64_t n, const float* a) Long64_t LocMax(Long64_t n, const double* a) Long64_t LocMax(Long64_t n, const long* a) Long64_t LocMax(Long64_t n, const long long* a) Long64_t LocMin(Long64_t n, const short* a) Long64_t LocMin(Long64_t n, const int* a) Long64_t LocMin(Long64_t n, const float* a) Long64_t LocMin(Long64_t n, const double* a) Long64_t LocMin(Long64_t n, const long* a) Long64_t LocMin(Long64_t n, const long long* a) Double_t Log(Double_t x) Double_t Log10(Double_t x) Double_t Log2(Double_t x) Double_t LogE() Double_t LogNormal(Double_t x, Double_t sigma, Double_t theta = 0, Double_t m = 1) Short_t Max(Short_t a, Short_t b) UShort_t Max(UShort_t a, UShort_t b) Int_t Max(Int_t a, Int_t b) UInt_t Max(UInt_t a, UInt_t b) Long_t Max(Long_t a, Long_t b) ULong_t Max(ULong_t a, ULong_t b) Long64_t Max(Long64_t a, Long64_t b) ULong64_t Max(ULong64_t a, ULong64_t b) Float_t Max(Float_t a, Float_t b) Double_t Max(Double_t a, Double_t b) short MaxElement(Long64_t n, const short* a) int MaxElement(Long64_t n, const int* a) float MaxElement(Long64_t n, const float* a) double MaxElement(Long64_t n, const double* a) long MaxElement(Long64_t n, const long* a) long long MaxElement(Long64_t n, const long long* a) Double_t Mean(Long64_t n, const short* a, const Double_t* w = 0) Double_t Mean(Long64_t n, const int* a, const Double_t* w = 0) Double_t Mean(Long64_t n, const float* a, const Double_t* w = 0) Double_t Mean(Long64_t n, const double* a, const Double_t* w = 0) Double_t Mean(Long64_t n, const long* a, const Double_t* w = 0) Double_t Mean(Long64_t n, const long long* a, const Double_t* w = 0) Double_t Median(Long64_t n, const short* a, const Double_t* w = 0, Long64_t* work = 0) Double_t Median(Long64_t n, const int* a, const Double_t* w = 0, Long64_t* work = 0) Double_t Median(Long64_t n, const float* a, const Double_t* w = 0, Long64_t* work = 0) Double_t Median(Long64_t n, const double* a, const Double_t* w = 0, Long64_t* work = 0) Double_t Median(Long64_t n, const long* a, const Double_t* w = 0, Long64_t* work = 0) Double_t Median(Long64_t n, const long long* a, const Double_t* w = 0, Long64_t* work = 0) Short_t Min(Short_t a, Short_t b) UShort_t Min(UShort_t a, UShort_t b) Int_t Min(Int_t a, Int_t b) UInt_t Min(UInt_t a, UInt_t b) Long_t Min(Long_t a, Long_t b) ULong_t Min(ULong_t a, ULong_t b) Long64_t Min(Long64_t a, Long64_t b) ULong64_t Min(ULong64_t a, ULong64_t b) Float_t Min(Float_t a, Float_t b) Double_t Min(Double_t a, Double_t b) short MinElement(Long64_t n, const short* a) int MinElement(Long64_t n, const int* a) float MinElement(Long64_t n, const float* a) double MinElement(Long64_t n, const double* a) long MinElement(Long64_t n, const long* a) long long MinElement(Long64_t n, const long long* a) Double_t MWair() Double_t Na() Double_t NaUncertainty() Long_t NextPrime(Long_t x) Int_t Nint(float x) Int_t Nint(double x) Int_t Nint(double x) Int_t Nint(float x) float* Normal2Plane(const float* p1, const float* p2, const float* p3, float* normal) double* Normal2Plane(const double* p1, const double* p2, const double* p3, double* normal) Float_t Normalize(Float_t* v) Double_t Normalize(Double_t* v) float NormCross(const float* v1, const float* v2, float* out) double NormCross(const double* v1, const double* v2, double* out) Double_t NormQuantile(Double_t p) Bool_t Odd(Long_t a) Bool_t Permute(Int_t n, Int_t* a) Double_t Pi() Double_t PiOver2() Double_t PiOver4() Double_t Poisson(Double_t x, Double_t par) Double_t PoissonI(Double_t x, Double_t par) LongDouble_t Power(LongDouble_t x, LongDouble_t y) LongDouble_t Power(LongDouble_t x, Long64_t y) LongDouble_t Power(Long64_t x, Long64_t y) Double_t Power(Double_t x, Double_t y) Double_t Power(Double_t x, Int_t y) Double_t Prob(Double_t chi2, Int_t ndf) Double_t Qe() Double_t QeUncertainty() void Quantiles(Int_t n, Int_t nprob, Double_t* x, Double_t* quantiles, Double_t* prob, Bool_t isSorted = kTRUE, Int_t* index = 0, Int_t type = 7) Double_t QuietNaN() Double_t R() Double_t RadToDeg() Short_t Range(Short_t lb, Short_t ub, Short_t x) Int_t Range(Int_t lb, Int_t ub, Int_t x) Long_t Range(Long_t lb, Long_t ub, Long_t x) ULong_t Range(ULong_t lb, ULong_t ub, ULong_t x) Double_t Range(Double_t lb, Double_t ub, Double_t x) Double_t Rgair() Double_t RMS(Long64_t n, const short* a, const Double_t* w = 0) Double_t RMS(Long64_t n, const int* a, const Double_t* w = 0) Double_t RMS(Long64_t n, const float* a, const Double_t* w = 0) Double_t RMS(Long64_t n, const double* a, const Double_t* w = 0) Double_t RMS(Long64_t n, const long* a, const Double_t* w = 0) Double_t RMS(Long64_t n, const long long* a, const Double_t* w = 0) Bool_t RootsCubic(const Double_t* coef, Double_t& a, Double_t& b, Double_t& c) Double_t RUncertainty() Double_t Sigma() Double_t SigmaUncertainty() Short_t Sign(Short_t a, Short_t b) Int_t Sign(Int_t a, Int_t b) Long_t Sign(Long_t a, Long_t b) Long64_t Sign(Long64_t a, Long64_t b) Float_t Sign(Float_t a, Float_t b) Double_t Sign(Double_t a, Double_t b) Double_t SignalingNaN() Double_t Sin(Double_t x) Double_t SinH(Double_t x) void Sort(long long n, const short* a, long long* index, Bool_t down = kTRUE) void Sort(long long n, const int* a, long long* index, Bool_t down = kTRUE) void Sort(long long n, const float* a, long long* index, Bool_t down = kTRUE) void Sort(long long n, const double* a, long long* index, Bool_t down = kTRUE) void Sort(long long n, const long* a, long long* index, Bool_t down = kTRUE) void Sort(long long n, const long long* a, long long* index, Bool_t down = kTRUE) void Sort(int n, const short* a, int* index, Bool_t down = kTRUE) void Sort(int n, const int* a, int* index, Bool_t down = kTRUE) void Sort(int n, const float* a, int* index, Bool_t down = kTRUE) void Sort(int n, const double* a, int* index, Bool_t down = kTRUE) void Sort(int n, const long* a, int* index, Bool_t down = kTRUE) void Sort(int n, const long long* a, int* index, Bool_t down = kTRUE) Double_t Sqrt(Double_t x) Double_t Sqrt2() Double_t StruveH0(Double_t x) Double_t StruveH1(Double_t x) Double_t StruveL0(Double_t x) Double_t StruveL1(Double_t x) Double_t Student(Double_t T, Double_t ndf) Double_t StudentI(Double_t T, Double_t ndf) Double_t StudentQuantile(Double_t p, Double_t ndf, Bool_t lower_tail = kTRUE) Double_t Tan(Double_t x) Double_t TanH(Double_t x) Double_t TwoPi() Double_t Vavilov(Double_t x, Double_t kappa, Double_t beta2) Double_t VavilovI(Double_t x, Double_t kappa, Double_t beta2) Double_t Voigt(Double_t x, Double_t sigma, Double_t lg, Int_t r = 4)

Function documentation

``` The DiLogarithm function
Code translated by R.Brun from CERNLIB DILOG function C332
```

``` Computation of the error function erf(x).
Erf(x) = (2/sqrt(pi)) Integral(exp(-t^2))dt between 0 and x
```

``` Compute the complementary error function erfc(x).
Erfc(x) = (2/sqrt(pi)) Integral(exp(-t^2))dt between x and infinity

```

``` returns  the inverse error function
x must be  <-1<x<1
```

``` returns  the inverse of the complementary error function
x must be  0<x<2
implement using  the quantile of the normal distribution
instead of ErfInverse for better numerical precision for large x
```

``` Compute factorial(n).
```

``` Computation of the normal frequency function freq(x).
Freq(x) = (1/sqrt(2pi)) Integral(exp(-t^2/2))dt between -infinity and x.

Translated from CERNLIB C300 by Rene Brun.
```

``` Computation of gamma(z) for all z.

C.Lanczos, SIAM Journal of Numerical Analysis B1 (1964), 86.

```

``` Computation of the normalized lower incomplete gamma function P(a,x) as defined in the
Handbook of Mathematical Functions by Abramowitz and Stegun, formula 6.5.1 on page 260 .
Its normalization is such that TMath::Gamma(a,+infinity) = 1 .

$P(a, x) = #frac{1}{#Gamma(a) } #int_{0}^{x} t^{a-1} e^{-t} dt$

--- Nve 14-nov-1998 UU-SAP Utrecht
```
Double_t BreitWigner(Double_t x, Double_t mean = 0, Double_t gamma = 1)
``` Calculate a Breit Wigner function with mean and gamma.
```
Double_t Gaus(Double_t x, Double_t mean = 0, Double_t sigma = 1, Bool_t norm = kFALSE)
``` Calculate a gaussian function with mean and sigma.
If norm=kTRUE (default is kFALSE) the result is divided
by sqrt(2*Pi)*sigma.
```
Double_t Landau(Double_t x, Double_t mpv = 0, Double_t sigma = 1, Bool_t norm = kFALSE)
``` The LANDAU function.
mu is a location parameter and correspond approximatly to the most probable value
and sigma is a scale parameter (not the sigma of the full distribution which is not defined)
Note that for mu=0 and sigma=1 (default values) the exact location of the maximum of the distribution
(most proble value) is at x = -0.22278
This function has been adapted from the CERNLIB routine G110 denlan.
If norm=kTRUE (default is kFALSE) the result is divided by sigma
```

``` Computation of ln[gamma(z)] for all z.

C.Lanczos, SIAM Journal of Numerical Analysis B1 (1964), 86.

The accuracy of the result is better than 2e-10.

--- Nve 14-nov-1998 UU-SAP Utrecht
```
Float_t Normalize(Float_t v[3])
``` Normalize a vector v in place.
Returns the norm of the original vector.
```

``` Normalize a vector v in place.
Returns the norm of the original vector.
This implementation (thanks Kevin Lynch <krlynch@bu.edu>) is protected
against possible overflows.
```

``` compute the Poisson distribution function for (x,par)
The Poisson PDF is implemented by means of Euler's Gamma-function
(for the factorial), so for any x integer argument it is correct.
BUT for non-integer x values, it IS NOT equal to the Poisson distribution.
see TMath::PoissonI to get a non-smooth function.
Note that for large values of par, it is better to call
TMath::Gaus(x,par,sqrt(par),kTRUE)

```
/* */

``` compute the Poisson distribution function for (x,par)
This is a non-smooth function.
This function is equivalent to ROOT::Math::poisson_pdf

```
/* */
Double_t Prob(Double_t chi2, Int_t ndf)
``` Computation of the probability for a certain Chi-squared (chi2)
and number of degrees of freedom (ndf).

Calculations are based on the incomplete gamma function P(a,x),
where a=ndf/2 and x=chi2/2.

P(a,x) represents the probability that the observed Chi-squared
for a correct model should be less than the value chi2.

The returned probability corresponds to 1-P(a,x),
which denotes the probability that an observed Chi-squared exceeds
the value chi2 by chance, even for a correct model.

--- NvE 14-nov-1998 UU-SAP Utrecht
```

``` Calculates the Kolmogorov distribution function,

```
/* */
``` which gives the probability that Kolmogorov's test statistic will exceed
the value z assuming the null hypothesis. This gives a very powerful
test for comparing two one-dimensional distributions.
see, for example, Eadie et al, "statistocal Methods in Experimental
Physics', pp 269-270).

This function returns the confidence level for the null hypothesis, where:
z = dn*sqrt(n), and
dn  is the maximum deviation between a hypothetical distribution
function and an experimental distribution with
n    events

NOTE: To compare two experimental distributions with m and n events,
use z = sqrt(m*n/(m+n))*dn

Accuracy: The function is far too accurate for any imaginable application.
Probabilities less than 10^-15 are returned as zero.
However, remember that the formula is only valid for "large" n.
Theta function inversion formula is used for z <= 1

This function was translated by Rene Brun from PROBKL in CERNLIB.
```
Double_t KolmogorovTest(Int_t na, const Double_t* a, Int_t nb, const Double_t* b, Option_t* option)
```  Statistical test whether two one-dimensional sets of points are compatible
with coming from the same parent distribution, using the Kolmogorov test.
That is, it is used to compare two experimental distributions of unbinned data.

Input:
a,b: One-dimensional arrays of length na, nb, respectively.
The elements of a and b must be given in ascending order.
option is a character string to specify options
"D" Put out a line of "Debug" printout
"M" Return the Maximum Kolmogorov distance instead of prob

Output:
The returned value prob is a calculated confidence level which gives a
statistical test for compatibility of a and b.
Values of prob close to zero are taken as indicating a small probability
of compatibility. For two point sets drawn randomly from the same parent
distribution, the value of prob should be uniformly distributed between
zero and one.
in case of error the function return -1
If the 2 sets have a different number of points, the minimum of
the two sets is used.

Method:
The Kolmogorov test is used. The test statistic is the maximum deviation
between the two integrated distribution functions, multiplied by the
normalizing factor (rdmax*sqrt(na*nb/(na+nb)).

Code adapted by Rene Brun from CERNLIB routine TKOLMO (Fred James)
(W.T. Eadie, D. Drijard, F.E. James, M. Roos and B. Sadoulet,
Statistical Methods in Experimental Physics, (North-Holland,
Amsterdam 1971) 269-271)

Method Improvement by Jason A Detwiler (JADetwiler@lbl.gov)

The nuts-and-bolts of the TMath::KolmogorovTest() algorithm is a for-loop
over the two sorted arrays a and b representing empirical distribution
functions. The for-loop handles 3 cases: when the next points to be
evaluated satisfy a>b, a<b, or a=b:

for (Int_t i=0;i<na+nb;i++) {
if (a[ia-1] < b[ib-1]) {
rdiff -= sa;
ia++;
if (ia > na) {ok = kTRUE; break;}
} else if (a[ia-1] > b[ib-1]) {
rdiff += sb;
ib++;
if (ib > nb) {ok = kTRUE; break;}
} else {
rdiff += sb - sa;
ia++;
ib++;
if (ia > na) {ok = kTRUE; break;}
if (ib > nb) {ok = kTRUE; break;}
}
rdmax = TMath::Max(rdmax,TMath::Abs(rdiff));
}

For the last case, a=b, the algorithm advances each array by one index in an
attempt to move through the equality. However, this is incorrect when one or
the other of a or b (or both) have a repeated value, call it x. For the KS
statistic to be computed properly, rdiff needs to be calculated after all of
the a and b at x have been tallied (this is due to the definition of the
empirical distribution function; another way to convince yourself that the
old CERNLIB method is wrong is that it implies that the function defined as the
difference between a and b is multi-valued at x -- besides being ugly, this
would invalidate Kolmogorov's theorem).

The solution is to just add while-loops into the equality-case handling to
perform the tally:

} else {
double x = a[ia-1];
while(a[ia-1] == x && ia <= na) {
rdiff -= sa;
ia++;
}
while(b[ib-1] == x && ib <= nb) {
rdiff += sb;
ib++;
}
if (ia > na) {ok = kTRUE; break;}
if (ib > nb) {ok = kTRUE; break;}
}

NOTE1
A good description of the Kolmogorov test can be seen at:
http://www.itl.nist.gov/div898/handbook/eda/section3/eda35g.htm
```
Double_t Voigt(Double_t x, Double_t sigma, Double_t lg, Int_t r = 4)
``` Computation of Voigt function (normalised).
Voigt is a convolution of
gauss(xx) = 1/(sqrt(2*pi)*sigma) * exp(xx*xx/(2*sigma*sigma)
and
lorentz(xx) = (1/pi) * (lg/2) / (xx*xx + lg*lg/4)
functions.

The Voigt function is known to be the real part of Faddeeva function also
called complex error function [2].

The algoritm was developed by J. Humlicek [1].
This code is based on fortran code presented by R. J. Wells [2].
Translated and adapted by Miha D. Puc

To calculate the Faddeeva function with relative error less than 10^(-r).
r can be set by the the user subject to the constraints 2 <= r <= 5.

[1] J. Humlicek, JQSRT, 21, 437 (1982).
[2] R.J. Wells "Rapid Approximation to the Voigt/Faddeeva Function and its
Derivatives" JQSRT 62 (1999), pp 29-48.
http://www-atm.physics.ox.ac.uk/user/wells/voigt.html
```
Bool_t RootsCubic(const Double_t* coef, Double_t& a, Double_t& b, Double_t& c)
``` Calculates roots of polynomial of 3rd order a*x^3 + b*x^2 + c*x + d, where
a == coef[3], b == coef[2], c == coef[1], d == coef[0]
coef[3] must be different from 0
If the boolean returned by the method is false:
==> there are 3 real roots a,b,c
If the boolean returned by the method is true:
==> there is one real root a and 2 complex conjugates roots (b+i*c,b-i*c)
Author: Francois-Xavier Gentit
```
void Quantiles(Int_t n, Int_t nprob, Double_t* x, Double_t* quantiles, Double_t* prob, Bool_t isSorted = kTRUE, Int_t* index = 0, Int_t type = 7)
```Computes sample quantiles, corresponding to the given probabilities
Parameters:
x -the data sample
n - its size
quantiles - computed quantiles are returned in there
prob - probabilities where to compute quantiles
nprob - size of prob array
isSorted - is the input array x sorted?
NOTE, that when the input is not sorted, an array of integers of size n needs
to be allocated. It can be passed by the user in parameter index,
or, if not passed, it will be allocated inside the function

type - method to compute (from 1 to 9). Following types are provided:
Discontinuous:
type=1 - inverse of the empirical distribution function
type=2 - like type 1, but with averaging at discontinuities
type=3 - SAS definition: nearest even order statistic
Piecwise linear continuous:
In this case, sample quantiles can be obtained by linear interpolation
between the k-th order statistic and p(k).
type=4 - linear interpolation of empirical cdf, p(k)=k/n;
type=5 - a very popular definition, p(k) = (k-0.5)/n;
type=6 - used by Minitab and SPSS, p(k) = k/(n+1);
type=7 - used by S-Plus and R, p(k) = (k-1)/(n-1);
type=8 - resulting sample quantiles are approximately median unbiased
regardless of the distribution of x. p(k) = (k-1/3)/(n+1/3);
type=9 - resulting sample quantiles are approximately unbiased, when
the sample comes from Normal distribution. p(k)=(k-3/8)/(n+1/4);

default type = 7

References:
1) Hyndman, R.J and Fan, Y, (1996) "Sample quantiles in statistical packages"
American Statistician, 50, 361-365
2) R Project documentation for the function quantile of package {stats}
```
void BubbleHigh(Int_t Narr, Double_t* arr1, Int_t* arr2)
``` Bubble sort variant to obtain the order of an array's elements into
an index in order to do more useful things than the standard built
in functions.
*arr1 is unchanged;
*arr2 is the array of indicies corresponding to the decending value
of arr1 with arr2[0] corresponding to the largest arr1 value and
arr2[Narr] the smallest.

Author: Adrian Bevan (bevan@slac.stanford.edu)
```
void BubbleLow(Int_t Narr, Double_t* arr1, Int_t* arr2)
``` Opposite ordering of the array arr2[] to that of BubbleHigh.

Author: Adrian Bevan (bevan@slac.stanford.edu)
```
ULong_t Hash(const void* txt, Int_t ntxt)
``` Calculates hash index from any char string.
Based on precalculated table of 256 specially selected numbers.
These numbers are selected in such a way, that for string
length == 4 (integer number) the hash is unambigous, i.e.
from hash value we can recalculate input (no degeneration).

The quality of hash method is good enough, that
"random" numbers made as R = Hash(1), Hash(2), ...Hash(N)
tested by <R>, <R*R>, <Ri*Ri+1> gives the same result
as for libc rand().

For string:  i = TMath::Hash(string,nstring);
For int:     i = TMath::Hash(&intword,sizeof(int));
For pointer: i = TMath::Hash(&pointer,sizeof(void*));

V.Perev
This function is kept for back compatibility. The code previously in this function
has been moved to the static function TString::Hash
```
ULong_t Hash(const char* str)

``` Compute the modified Bessel function I_0(x) for any real x.

--- NvE 12-mar-2000 UU-SAP Utrecht
```

``` Compute the modified Bessel function K_0(x) for positive real x.

M.Abramowitz and I.A.Stegun, Handbook of Mathematical Functions,
Applied Mathematics Series vol. 55 (1964), Washington.

--- NvE 12-mar-2000 UU-SAP Utrecht
```

``` Compute the modified Bessel function I_1(x) for any real x.

M.Abramowitz and I.A.Stegun, Handbook of Mathematical Functions,
Applied Mathematics Series vol. 55 (1964), Washington.

--- NvE 12-mar-2000 UU-SAP Utrecht
```

``` Compute the modified Bessel function K_1(x) for positive real x.

M.Abramowitz and I.A.Stegun, Handbook of Mathematical Functions,
Applied Mathematics Series vol. 55 (1964), Washington.

--- NvE 12-mar-2000 UU-SAP Utrecht
```

``` Compute the Integer Order Modified Bessel function K_n(x)
for n=0,1,2,... and positive real x.

--- NvE 12-mar-2000 UU-SAP Utrecht
```

``` Compute the Integer Order Modified Bessel function I_n(x)
for n=0,1,2,... and any real x.

--- NvE 12-mar-2000 UU-SAP Utrecht
```

``` Returns the Bessel function J0(x) for any real x.
```

``` Returns the Bessel function J1(x) for any real x.
```

``` Returns the Bessel function Y0(x) for positive x.
```

``` Returns the Bessel function Y1(x) for positive x.
```

``` Struve Functions of Order 0

Converted from CERNLIB M342 by Rene Brun.
```

``` Struve Functions of Order 1

Converted from CERNLIB M342 by Rene Brun.
```

``` Modified Struve Function of Order 0.
By Kirill Filimonov.
```

``` Modified Struve Function of Order 1.
By Kirill Filimonov.
```

``` Calculates Beta-function Gamma(p)*Gamma(q)/Gamma(p+q).
```

``` Continued fraction evaluation by modified Lentz's method
used in calculation of incomplete Beta function.
```

``` Computes the probability density function of the Beta distribution
(the distribution function is computed in BetaDistI).
The first argument is the point, where the function will be
computed, second and third are the function parameters.
Since the Beta distribution is bounded on both sides, it's often
used to represent processes with natural lower and upper limits.
```

``` Computes the distribution function of the Beta distribution.
The first argument is the point, where the function will be
computed, second and third are the function parameters.
Since the Beta distribution is bounded on both sides, it's often
used to represent processes with natural lower and upper limits.
```

``` Calculates the incomplete Beta-function.
```

``` Calculate the binomial coefficient n over k.
```

``` Suppose an event occurs with probability _p_ per trial
Then the probability P of its occuring _k_ or more times
in _n_ trials is termed a cumulative binomial probability
the formula is P = sum_from_j=k_to_n(TMath::Binomial(n, j)*
*TMath::Power(p, j)*TMath::Power(1-p, n-j)
For _n_ larger than 12 BetaIncomplete is a much better way
to evaluate the sum than would be the straightforward sum calculation
for _n_ smaller than 12 either method is acceptable
("Numerical Recipes")
--implementation by Anna Kreshuk
```
Double_t CauchyDist(Double_t x, Double_t t = 0, Double_t s = 1)
``` Computes the density of Cauchy distribution at point x
by default, standard Cauchy distribution is used (t=0, s=1)
t is the location parameter
s is the scale parameter
The Cauchy distribution, also called Lorentzian distribution,
is a continuous distribution describing resonance behavior
The mean and standard deviation of the Cauchy distribution are undefined.
The practical meaning of this is that collecting 1,000 data points gives
no more accurate an estimate of the mean and standard deviation than
does a single point.
The formula was taken from "Engineering Statistics Handbook" on site
http://www.itl.nist.gov/div898/handbook/eda/section3/eda3663.htm
Implementation by Anna Kreshuk.
Example:
TF1* fc = new TF1("fc", "TMath::CauchyDist(x, [0], [1])", -5, 5);
fc->SetParameters(0, 1);
fc->Draw();
```

``` Evaluate the quantiles of the chi-squared probability distribution function.
Algorithm AS 91   Appl. Statist. (1975) Vol.24, P.35
implemented by Anna Kreshuk.
Incorporates the suggested changes in AS R85 (vol.40(1), pp.233-5, 1991)
Parameters:
p   - the probability value, at which the quantile is computed
ndf - number of degrees of freedom
```

``` Computes the density function of F-distribution
(probability function, integral of density, is computed in FDistI).

Parameters N and M stand for degrees of freedom of chi-squares
mentioned above parameter F is the actual variable x of the
density function p(x) and the point at which the density function
is calculated.

F-distribution arises in testing whether two random samples
have the same variance. It is the ratio of two chi-square
distributions, with N and M degrees of freedom respectively,
where each chi-square is first divided by it's number of degrees
of freedom.
Implementation by Anna Kreshuk.
```

``` Calculates the cumulative distribution function of F-distribution,
this function occurs in the statistical test of whether two observed
samples have the same variance. For this test a certain statistic F,
the ratio of observed dispersion of the first sample to that of the
second sample, is calculated. N and M stand for numbers of degrees
of freedom in the samples 1-FDistI() is the significance level at
which the hypothesis "1 has smaller variance than 2" can be rejected.
A small numerical value of 1 - FDistI() implies a very significant
rejection, in turn implying high confidence in the hypothesis
"1 has variance greater than 2".
Implementation by Anna Kreshuk.
```
Double_t GammaDist(Double_t x, Double_t gamma, Double_t mu = 0, Double_t beta = 1)
``` Computes the density function of Gamma distribution at point x.
gamma - shape parameter
mu    - location parameter
beta  - scale parameter

The definition can be found in "Engineering Statistics Handbook" on site
http://www.itl.nist.gov/div898/handbook/eda/section3/eda366b.htm
use now implementation in ROOT::Math::gamma_pdf

```
/* */
Double_t LaplaceDist(Double_t x, Double_t alpha = 0, Double_t beta = 1)
``` Computes the probability density function of Laplace distribution
at point x, with location parameter alpha and shape parameter beta.
By default, alpha=0, beta=1
This distribution is known under different names, most common is
double exponential distribution, but it also appears as
the two-tailed exponential or the bilateral exponential distribution
```
Double_t LaplaceDistI(Double_t x, Double_t alpha = 0, Double_t beta = 1)
``` Computes the distribution function of Laplace distribution
at point x, with location parameter alpha and shape parameter beta.
By default, alpha=0, beta=1
This distribution is known under different names, most common is
double exponential distribution, but it also appears as
the two-tailed exponential or the bilateral exponential distribution
```
Double_t LogNormal(Double_t x, Double_t sigma, Double_t theta = 0, Double_t m = 1)
``` Computes the density of LogNormal distribution at point x.
Variable X has lognormal distribution if Y=Ln(X) has normal distribution
sigma is the shape parameter
theta is the location parameter
m is the scale parameter
The formula was taken from "Engineering Statistics Handbook" on site
http://www.itl.nist.gov/div898/handbook/eda/section3/eda3669.htm
Implementation using ROOT::Math::lognormal_pdf

```
/* */

``` Computes quantiles for standard normal distribution N(0, 1)
at probability p
ALGORITHM AS241  APPL. STATIST. (1988) VOL. 37, NO. 3, 477-484.
```
Bool_t Permute(Int_t n, Int_t* a)
``` Simple recursive algorithm to find the permutations of
n natural numbers, not necessarily all distinct
adapted from CERNLIB routine PERMU.
The input array has to be initialised with a non descending
sequence. The method returns kFALSE when
all combinations are exhausted.
```

``` Computes density function for Student's t- distribution
(the probability function (integral of density) is computed in StudentI).

First parameter stands for x - the actual variable of the
density function p(x) and the point at which the density is calculated.
Second parameter stands for number of degrees of freedom.

Student's t-distribution is used for many significance tests, for example,
for the Student's t-tests for the statistical significance of difference
between two sample means and for confidence intervals for the difference
between two population means.

Example: suppose we have a random sample of size n drawn from normal
distribution with mean Mu and st.deviation Sigma. Then the variable

t = (sample_mean - Mu)/(sample_deviation / sqrt(n))

has Student's t-distribution with n-1 degrees of freedom.

NOTE that this function's second argument is number of degrees of freedom,
not the sample size.

As the number of degrees of freedom grows, t-distribution approaches
Normal(0,1) distribution.
Implementation by Anna Kreshuk.
```

``` Calculates the cumulative distribution function of Student's
t-distribution second parameter stands for number of degrees of freedom,
not for the number of samples
if x has Student's t-distribution, the function returns the probability of
x being less than T.
Implementation by Anna Kreshuk.
```
Double_t StudentQuantile(Double_t p, Double_t ndf, Bool_t lower_tail = kTRUE)
``` Computes quantiles of the Student's t-distribution
1st argument is the probability, at which the quantile is computed
2nd argument - the number of degrees of freedom of the
Student distribution
When the 3rd argument lower_tail is kTRUE (default)-
the algorithm returns such x0, that
P(x < x0)=p
upper tail (lower_tail is kFALSE)- the algorithm returns such x0, that
P(x > x0)=p
the algorithm was taken from
G.W.Hill, "Algorithm 396, Student's t-quantiles"
"Communications of the ACM", 13(10), October 1970
```
Double_t Vavilov(Double_t x, Double_t kappa, Double_t beta2)
```Returns the value of the Vavilov density function
Parameters: 1st - the point were the density function is evaluated
2nd - value of kappa (distribution parameter)
3rd - value of beta2 (distribution parameter)
The algorithm was taken from the CernLib function vavden(G115)
Reference: A.Rotondi and P.Montagna, Fast Calculation of Vavilov distribution
Nucl.Instr. and Meth. B47(1990), 215-224
Accuracy: quote from the reference above:
"The resuls of our code have been compared with the values of the Vavilov
density function computed numerically in an accurate way: our approximation
shows a difference of less than 3% around the peak of the density function, slowly
increasing going towards the extreme tails to the right and to the left"

```
/* */
Double_t VavilovI(Double_t x, Double_t kappa, Double_t beta2)
```Returns the value of the Vavilov distribution function
Parameters: 1st - the point were the density function is evaluated
2nd - value of kappa (distribution parameter)
3rd - value of beta2 (distribution parameter)
The algorithm was taken from the CernLib function vavden(G115)
Reference: A.Rotondi and P.Montagna, Fast Calculation of Vavilov distribution
Nucl.Instr. and Meth. B47(1990), 215-224
Accuracy: quote from the reference above:
"The resuls of our code have been compared with the values of the Vavilov
density function computed numerically in an accurate way: our approximation
shows a difference of less than 3% around the peak of the density function, slowly
increasing going towards the extreme tails to the right and to the left"
```

```Returns the value of the Landau distribution function at point x.
The algorithm was taken from the Cernlib function dislan(G110)
Reference: K.S.Kolbig and B.Schorr, "A program package for the Landau
distribution", Computer Phys.Comm., 31(1984), 97-111
```

`{ return sin(x); }`

`{ return cos(x); }`

`{ return tan(x); }`

`{ return sinh(x); }`

`{ return cosh(x); }`

`{ return tanh(x); }`

`{ return atan(x); }`

`{ return sqrt(x); }`

`{ return ceil(x); }`

`{ return TMath::Nint(ceil(x)); }`

`{ return floor(x); }`

`{ return TMath::Nint(floor(x)); }`
Int_t Nint(T x)
``` Round to nearest integer. Rounds half integers to the nearest
even integer.
```

`{ return exp(x); }`
Double_t Ldexp(Double_t x, Int_t exp)
`{ return ldexp(x, exp); }`

`{ return std::pow(x,y); }`

`{ return std::pow(x,(LongDouble_t)y); }`

`{ return std::pow(x,y); }`

`{ return pow(x, y); }`

`{ return log(x); }`

`{ return log10(x); }`

`{ return isfinite(x); }`

``` from math.h
```
`{ return ::isnan(x); }`

```--------wrapper to numeric_limits

```

``` returns an infinity as defined by the IEEE standard
```
template <typename T> inline T NormCross(const T v1[3],const T v2[3],T out[3])
``` Calculate the Normalized Cross Product of two vectors
```
T MinElement(Long64_t n, const T *a)
``` Return minimum of array a of length n.
```
T MaxElement(Long64_t n, const T *a)
``` Return maximum of array a of length n.
```
Long64_t LocMin(Long64_t n, const T *a)
``` Return index of array with the minimum element.
If more than one element is minimum returns first found.
```
Iterator LocMin(Iterator first, Iterator last)
``` Return index of array with the minimum element.
If more than one element is minimum returns first found.
```
Long64_t LocMax(Long64_t n, const T *a)
``` Return index of array with the maximum element.
If more than one element is maximum returns first found.
```
Iterator LocMax(Iterator first, Iterator last)
``` Return index of array with the maximum element.
If more than one element is maximum returns first found.
```
Double_t Mean(Iterator first, Iterator last)
``` Return the weighted mean of an array defined by the iterators.
```
Double_t Mean(Iterator first, Iterator last, WeightIterator w)
``` Return the weighted mean of an array defined by the first and
last iterators. The w iterator should point to the first element
of a vector of weights of the same size as the main array.
```
Double_t Mean(Long64_t n, const T *a, const Double_t *w)
``` Return the weighted mean of an array a with length n.
```
Double_t GeomMean(Iterator first, Iterator last)
``` Return the geometric mean of an array defined by the iterators.
geometric_mean = (Prod_i=0,n-1 |a[i]|)^1/n
```
Double_t GeomMean(Long64_t n, const T *a)
``` Return the geometric mean of an array a of size n.
geometric_mean = (Prod_i=0,n-1 |a[i]|)^1/n
```
Double_t RMS(Iterator first, Iterator last)
``` Return the Standard Deviation of an array defined by the iterators.
Note that this function returns the sigma(standard deviation) and
not the root mean square of the array.
```
Double_t RMS(Iterator first, Iterator last, WeightIterator w)
``` Return the weighted Standard Deviation of an array defined by the iterators.
Note that this function returns the sigma(standard deviation) and
not the root mean square of the array.
```
Double_t RMS(Long64_t n, const T *a, const Double_t * w)
``` Return the Standard Deviation of an array a with length n.
Note that this function returns the sigma(standard deviation) and
not the root mean square of the array.
```
Iterator BinarySearch(Iterator first, Iterator last, Element value)
``` Binary search in an array defined by its iterators.

The values in the iterators range are supposed to be sorted
prior to this call.  If match is found, function returns
position of element.  If no match found, function gives nearest
element smaller than value.
```
template <typename T> Long64_t BinarySearch(Long64_t n, const T *array, T value)
``` Binary search in an array of n values to locate value.

Array is supposed  to be sorted prior to this call.
If match is found, function returns position of element.
If no match found, function gives nearest element smaller than value.
```
template <typename T> Long64_t BinarySearch(Long64_t n, const T **array, T value)
``` Binary search in an array of n values to locate value.

Array is supposed  to be sorted prior to this call.
If match is found, function returns position of element.
If no match found, function gives nearest element smaller than value.
```
template <typename Element, typename Index> void Sort(Index n, const Element* a, Index* index, Bool_t down)
``` Sort the n elements of the  array a of generic templated type Element.
In output the array index of type Index contains the indices of the sorted array.
If down is false sort in increasing order (default is decreasing order).
```
template <typename T> T * Cross(const T v1[3],const T v2[3], T out[3])
``` Calculate the Cross Product of two vectors:
out = [v1 x v2]
```
template <typename T> T * Normal2Plane(const T p1[3],const T p2[3],const T p3[3], T normal[3])
``` Calculate a normal vector of a plane.

Input:
Float_t *p1,*p2,*p3  -  3 3D points belonged the plane to define it.

Return:
Pointer to 3D normal vector (normalized)
```
template <typename T> Bool_t IsInside(T xp, T yp, Int_t np, T *x, T *y)
``` Function which returns kTRUE if point xp,yp lies inside the
polygon defined by the np points in arrays x and y, kFALSE otherwise.
Note that the polygon may be open or closed.
```
template <typename T> Double_t Median(Long64_t n, const T *a, const Double_t *w, Long64_t *work)
``` Return the median of the array a where each entry i has weight w[i] .
Both arrays have a length of at least n . The median is a number obtained
from the sorted array a through

median = (a[jl]+a[jh])/2.  where (using also the sorted index on the array w)

sum_i=0,jl w[i] <= sumTot/2
sum_i=0,jh w[i] >= sumTot/2
sumTot = sum_i=0,n w[i]

If w=0, the algorithm defaults to the median definition where it is
a number that divides the sorted sequence into 2 halves.
When n is odd or n > 1000, the median is kth element k = (n + 1) / 2.
when n is even and n < 1000the median is a mean of the elements k = n/2 and k = n/2 + 1.

If the weights are supplied (w not 0) all weights must be >= 0

If work is supplied, it is used to store the sorting index and assumed to be
>= n . If work=0, local storage is used, either on the stack if n < kWorkMax
or on the heap for n >= kWorkMax .
```
Element KOrdStat(Size n, const Element *a, Size k, Size *work)
``` Returns k_th order statistic of the array a of size n
(k_th smallest element out of n elements).

C-convention is used for array indexing, so if you want
the second smallest element, call KOrdStat(n, a, 1).

If work is supplied, it is used to store the sorting index and
assumed to be >= n. If work=0, local storage is used, either on
the stack if n < kWorkMax or on the heap for n >= kWorkMax.
Note that the work index array will not contain the sorted indices but
all indeces of the smaller element in arbitrary order in work[0,...,k-1] and
all indeces of the larger element in arbitrary order in work[k+1,..,n-1]
work[k] will contain instead the index of the returned element.

Taken from "Numerical Recipes in C++" without the index array
implemented by Anna Khreshuk.

See also the declarations at the top of this file
```

```Fundamental constants

```
`{ return 3.14159265358979323846; }`

`{ return 2.0 * Pi(); }`

`{ return Pi() / 2.0; }`

`{ return Pi() / 4.0; }`

`{ return 1.0 / Pi(); }`

`{ return 180.0 / Pi(); }`

`{ return Pi() / 180.0; }`

`{ return 1.4142135623730950488016887242097; }`

``` e (base of natural log)
```
`{ return 2.71828182845904523536; }`

``` natural log of 10 (to convert log to ln)
```
`{ return 2.30258509299404568402; }`

``` base-10 log of e  (to convert ln to log)
```
`{ return 0.43429448190325182765; }`

``` velocity of light
```
`{ return 2.99792458e8; }`

`{ return 100.0 * C(); }`

`{ return 0.0; }`

``` gravitational constant
```
`{ return 6.673e-11; }`

`{ return G() / 1000.0; }`

`{ return 0.010e-11; }`

``` G over h-bar C
```
`{ return 6.707e-39; }`

`{ return 0.010e-39; }`

``` standard acceleration of gravity
```
`{ return 9.80665; }`

`{ return 0.0; }`

``` Planck's constant
```
`{ return 6.62606876e-34; }`

`{ return 1.0e7 * H(); }`

`{ return 0.00000052e-34; }`

``` h-bar (h over 2 pi)
```
`{ return 1.054571596e-34; }`

`{ return 1.0e7 * Hbar(); }`

`{ return 0.000000082e-34; }`

``` hc (h * c)
```
`{ return H() * C(); }`

`{ return Hcgs() * Ccgs(); }`

``` Boltzmann's constant
```
`{ return 1.3806503e-23; }`

`{ return 1.0e7 * K(); }`

`{ return 0.0000024e-23; }`

``` Stefan-Boltzmann constant
```
`{ return 5.6704e-8; }`

`{ return 0.000040e-8; }`

``` Avogadro constant (Avogadro's Number)
```
`{ return 6.02214199e+23; }`

`{ return 0.00000047e+23; }`

``` universal gas constant (Na * K)
http://scienceworld.wolfram.com/physics/UniversalGasConstant.html
```
`{ return K() * Na(); }`

`{ return R()*((KUncertainty()/K()) + (NaUncertainty()/Na())); }`

``` Molecular weight of dry air
1976 US Standard Atmosphere,
also see http://atmos.nmsu.edu/jsdap/encyclopediawork.html
```
`{ return 28.9644; }`

``` Dry Air Gas Constant (R / MWair)
http://atmos.nmsu.edu/education_and_outreach/encyclopedia/gas_constant.htm
```
`{ return (1000.0 * R()) / MWair(); }`

``` Euler-Mascheroni Constant
```
`{ return 0.577215664901532860606512090082402431042; }`

``` Elementary charge
```
`{ return 1.602176462e-19; }`

`{ return 0.000000063e-19; }`
Int_t Nint(T x)
T Min()
T Max()
Bool_t AreEqualAbs(Double_t af, Double_t bf, Double_t epsilon)
``` Comparing floating points
```
return Abs(af-bf)
```return kTRUE if absolute difference between af and bf is less than epsilon
```
Bool_t AreEqualRel(Double_t af, Double_t bf, Double_t relPrec)
```return kTRUE if relative difference between af and bf is less than relPrec
```
return Abs(af-bf)
template <typename T> T MinElement(Long64_t n, const T *a)
```Array Algorithms

Min, Max of an array
```
template <typename T> T MaxElement(Long64_t n, const T *a)
template <typename T> Long64_t LocMin(Long64_t n, const T *a)
``` Locate Min, Max element number in an array
```
template <typename Iterator> Iterator LocMin(Iterator first, Iterator last)
template <typename T> Long64_t LocMax(Long64_t n, const T *a)
template <typename Iterator> Iterator LocMax(Iterator first, Iterator last)
template <typename T> Long64_t BinarySearch(Long64_t n, const T *array, T value)
``` Binary search
```
template <typename T> Long64_t BinarySearch(Long64_t n, const T **array, T value)
template <typename Iterator, typename Element> Iterator BinarySearch(Iterator first, Iterator last, Element value)
void Sort(Index n, const Element* a, Index* index, Bool_t down=kTRUE)
``` Sorting
```
template <typename T> Bool_t IsInside(T xp, T yp, Int_t np, T *x, T *y)
``` IsInside
```
template <typename T> T * Cross(const T v1[3],const T v2[3], T out[3])
``` Calculate the Cross Product of two vectors
```
template <typename T> inline T NormCross(const T v1[3],const T v2[3],T out[3])
```Calculate the Normalized Cross Product of two vectors
```
template <typename T> T * Normal2Plane(const T v1[3],const T v2[3],const T v3[3], T normal[3])
``` Calculate a normal vector of a plane
```
template <typename T> Double_t Mean(Long64_t n, const T *a, const Double_t *w=0)
```Statistics over arrays

Mean, Geometric Mean, Median, RMS(sigma)
```
template <typename Iterator> Double_t Mean(Iterator first, Iterator last)
template <typename Iterator, typename WeightIterator> Double_t Mean(Iterator first, Iterator last, WeightIterator wfirst)
template <typename T> Double_t GeomMean(Long64_t n, const T *a)
template <typename Iterator> Double_t GeomMean(Iterator first, Iterator last)
template <typename T> Double_t RMS(Long64_t n, const T *a, const Double_t *w=0)
template <typename Iterator> Double_t RMS(Iterator first, Iterator last)
template <typename Iterator, typename WeightIterator> Double_t RMS(Iterator first, Iterator last, WeightIterator wfirst)
template <typename T> Double_t Median(Long64_t n, const T *a, const Double_t *w=0, Long64_t *work=0)
template <class Element, typename Size> Element KOrdStat(Size n, const Element *a, Size k, Size *work = 0)
```k-th order statistic
```
T TMath::Limits<T> Min()
``` returns maximum representation for type T
```
T TMath::Limits<T> Max()
``` returns minimum double representation
```
T MinElement(Long64_t n, const T *a)
``` Return minimum of array a of length n.
```
T MaxElement(Long64_t n, const T *a)
``` Return maximum of array a of length n.
```
Long64_t LocMin(Long64_t n, const T *a)
``` Return index of array with the minimum element.
If more than one element is minimum returns first found.
```
Iterator LocMin(Iterator first, Iterator last)
``` Return index of array with the minimum element.
If more than one element is minimum returns first found.
```
Long64_t LocMax(Long64_t n, const T *a)
``` Return index of array with the maximum element.
If more than one element is maximum returns first found.
```
Iterator LocMax(Iterator first, Iterator last)
``` Return index of array with the maximum element.
If more than one element is maximum returns first found.
```
Double_t GeomMean(Iterator first, Iterator last)
``` Return the geometric mean of an array defined by the iterators.
geometric_mean = (Prod_i=0,n-1 |a[i]|)^1/n
```
Double_t GeomMean(Long64_t n, const T *a)
``` Return the geometric mean of an array a of size n.
geometric_mean = (Prod_i=0,n-1 |a[i]|)^1/n
```
Iterator BinarySearch(Iterator first, Iterator last, Element value)
``` Binary search in an array defined by its iterators.

The values in the iterators range are supposed to be sorted
prior to this call.  If match is found, function returns
position of element.  If no match found, function gives nearest
element smaller than value.
```
template <typename T> Long64_t BinarySearch(Long64_t n, const T *array, T value)
``` Binary search in an array of n values to locate value.

Array is supposed  to be sorted prior to this call.
If match is found, function returns position of element.
If no match found, function gives nearest element smaller than value.
```
template <typename T> Long64_t BinarySearch(Long64_t n, const T **array, T value)
``` Binary search in an array of n values to locate value.

Array is supposed  to be sorted prior to this call.
If match is found, function returns position of element.
If no match found, function gives nearest element smaller than value.
```
template <typename Element, typename Index> void Sort(Index n, const Element* a, Index* index, Bool_t down)
``` Sort the n elements of the  array a of generic templated type Element.
In output the array index of type Index contains the indices of the sorted array.
If down is false sort in increasing order (default is decreasing order).
```
template <typename T> Double_t Median(Long64_t n, const T *a, const Double_t *w, Long64_t *work)
``` Return the median of the array a where each entry i has weight w[i] .
Both arrays have a length of at least n . The median is a number obtained
from the sorted array a through

median = (a[jl]+a[jh])/2.  where (using also the sorted index on the array w)

sum_i=0,jl w[i] <= sumTot/2
sum_i=0,jh w[i] >= sumTot/2
sumTot = sum_i=0,n w[i]

If w=0, the algorithm defaults to the median definition where it is
a number that divides the sorted sequence into 2 halves.
When n is odd or n > 1000, the median is kth element k = (n + 1) / 2.
when n is even and n < 1000the median is a mean of the elements k = n/2 and k = n/2 + 1.

If the weights are supplied (w not 0) all weights must be >= 0

If work is supplied, it is used to store the sorting index and assumed to be
>= n . If work=0, local storage is used, either on the stack if n < kWorkMax
or on the heap for n >= kWorkMax .
```
Sort(n, a, ind, kFALSE)
Element KOrdStat(Size n, const Element *a, Size k, Size *work)
``` Returns k_th order statistic of the array a of size n
(k_th smallest element out of n elements).

C-convention is used for array indexing, so if you want
the second smallest element, call KOrdStat(n, a, 1).

If work is supplied, it is used to store the sorting index and
assumed to be >= n. If work=0, local storage is used, either on
the stack if n < kWorkMax or on the heap for n >= kWorkMax.
Note that the work index array will not contain the sorted indices but
all indeces of the smaller element in arbitrary order in work[0,...,k-1] and
all indeces of the larger element in arbitrary order in work[k+1,..,n-1]
work[k] will contain instead the index of the returned element.

Taken from "Numerical Recipes in C++" without the index array
implemented by Anna Khreshuk.

See also the declarations at the top of this file
```