This class provides a few constants and some methods to compute numerical quantities such as factorials, combinations, gamma functions, and so on. More...

Static Public Member Functions
static int	gcd (int x, int y)
	Returns the greatest common divisor (gcd) of $x$ and $y$. More...

static long	gcd (long x, long y)
	Returns the greatest common divisor (gcd) of $x$ and $y$. More...

static double	combination (int n, int s)
	Returns the value of $\binom{n}{s}$, the number of different combinations of $s$ objects amongst $n$. More...

static double	lnCombination (int n, int s)
	Returns the natural logarithm of $\binom{n}{s}$, the number of different combinations of $s$ objects amongst $n$. More...

static double	factorial (int n)
	Returns the value of $n!$. More...

static double	lnFactorial (int n)
	Returns the value of $\ln(n!)$, the natural logarithm of factorial $n$. More...

static double	lnFactorial (long n)
	Returns the value of $\ln(n!)$, the natural logarithm of factorial $n$. More...

static double	factoPow (int n)
	Returns the value of $n!/n^n$. More...

static double [][]	calcMatStirling (int m, int n)
	Computes and returns the Stirling numbers of the second kind. More...

static double	log2 (double x)
	Returns $\log_2 ($`x` $)$. More...

static double	lnGamma (double x)
	Returns the natural logarithm of the gamma function $\Gamma(x)$ evaluated at `x`. More...

static double	lnBeta (double lam, double nu)
	Computes the natural logarithm of the Beta function $B(\lambda, \nu)$. More...

static double	digamma (double x)
	Returns the value of the logarithmic derivative of the Gamma function $\psi(x) = \Gamma’(x) / \Gamma(x)$.

static double	trigamma (double x)
	Returns the value of the trigamma function $d\psi(x)/dx$, the derivative of the digamma function, evaluated at $x$.

static double	tetragamma (double x)
	Returns the value of the tetragamma function $d^2\psi(x)/d^2x$, the second derivative of the digamma function, evaluated at $x$.

static double	gammaRatioHalf (double x)
	Returns the value of the ratio $\Gamma(x+1/2)/\Gamma(x)$ of two gamma functions. More...

static double	sumKahan (double[] A, int n)
	Implementation of the Kahan summation algorithm. More...

static double	harmonic (long n)
	Computes the $n$-th harmonic number $H_n = \sum_{j=1}^n 1/j$.

static double	harmonic2 (long n)
	Computes the sum \[ \sideset{}’\sum_{-n/2<j\le n/2}\; \frac{1}{\|j\|}, \] where the symbol $\sum^{\prime}$ means that the term with $j=0$ is excluded from the sum.

static double	volumeSphere (double p, int t)
	Returns the volume $V$ of a sphere of radius 1 in $t$ dimensions using the norm $L_p$. More...

static double	bernoulliPoly (int n, double x)
	Evaluates the Bernoulli polynomial $B_n(x)$ of degree $n$ at $x$. More...

static double	evalCheby (double a[], int n, double x)
	Evaluates a series of Chebyshev polynomials $T_j$ at $x$ over the basic interval $[-1, \;1]$, using the method of Clenshaw [35] , i.e., computes and returns \[ y = \frac{a_0}{2} + \sum_{j=1}^n a_j T_j (x). \] . More...

static double	evalChebyStar (double a[], int n, double x)
	Evaluates a series of shifted Chebyshev polynomials $T_j^$ at $x$ over the basic interval $ [0, \;1]$, using the method of Clenshaw [35] , i.e., computes and returns \[ y = \frac{a_0}{2} + \sum_{j=1}^n a_j T_j^ (x). \] . More...

static double	besselK025 (double x)
	Returns the value of $K_{1/4}(x)$, where $K_{\nu}$ is the modified Bessel’s function of the second kind. More...

static double	expBesselK1 (double x, double y)
	Returns the value of $e^x K_1(y)$, where $ K_1$ is the modified Bessel function of the second kind of order 1. More...

static double	erf (double x)
	Returns the value of `erf`( $x$), the error function. More...

static double	erfc (double x)
	Returns the value of `erfc`( $x$), the complementary error function. More...

static double	erfInv (double u)
	Returns the value of `erf` ${}^{-1}(u)$, the inverse of the error function. More...

static double	erfcInv (double u)
	Returns the value of `erfc` ${}^{-1}(u)$, the inverse of the complementary error function. More...

Static Public Attributes
Constants
static final double	DBL_EPSILON = 2.2204460492503131e-16
	Difference between 1.0 and the smallest `double` greater than 1.0.

static final int	DBL_MAX_EXP = 1024
	Largest `int` $x$ such that $2^{x-1}$ is representable (approximately) as a `double`.

static final int	DBL_MIN_EXP = -1021
	Smallest `int` $x$ such that $2^{x-1}$ is representable (approximately) as a normalised `double`.

static final int	DBL_MAX_10_EXP = 308
	Largest `int` $x$ such that $10^x$ is representable (approximately) as a `double`.

static final double	DBL_MIN = 2.2250738585072014e-308
	Smallest normalized positive floating-point `double`.

static final double	LN_DBL_MIN = -708.3964185322641
	Natural logarithm of `DBL_MIN`.

static final int	DBL_DIG = 15
	Number of decimal digits of precision in a `double`.

static final double	EBASE = 2.7182818284590452354
	The constant $e$.

static final double	EULER = 0.57721566490153286
	The Euler-Mascheroni constant.

static final double	RAC2 = 1.41421356237309504880
	The value of $\sqrt{2}$.

static final double	IRAC2 = 0.70710678118654752440
	The value of $1/\sqrt{2}$.

static final double	LN2 = 0.69314718055994530941
	The values of $\ln2$.

static final double	ILN2 = 1.44269504088896340737
	The values of $1/\ln2$.

static final double	MAXINTDOUBLE = 9007199254740992.0
	Largest integer $n_0 = 2^{53}$ such that any integer $n \le n_0$ is represented exactly as a `double`.

static final double	MAXTWOEXP = 64
	Powers of 2 up to `MAXTWOEXP` are stored exactly in the array `TWOEXP`.

static final double	TWOEXP []
	Contains the precomputed positive powers of 2. More...

static final double	TEN_NEG_POW []
	Contains the precomputed negative powers of 10. More...

Detailed Description

This class provides a few constants and some methods to compute numerical quantities such as factorials, combinations, gamma functions, and so on.

Member Function Documentation

◆ bernoulliPoly()

static double bernoulliPoly	(	int	n,
		double	x
	)

static

Evaluates the Bernoulli polynomial $B_n(x)$ of degree $n$ at $x$.

Only degrees $n\le8$ are programmed for now. The first Bernoulli polynomials of even degree are:

\begin{align} B_0(x) & = 1 \nonumber \\ B_2(x) & = x^2-x+1/6 \nonumber \\ B_4(x) & = x^4-2x^3+x^2-1/30 \tag{bernoulli} \\ B_6(x) & = x^6-3x^5+5x^4/2-x^2/2+1/42 \nonumber \\ B_8(x) & = x^8-4x^7+14x^6/3 - 7x^4/3 +2x^2/3-1/30. \nonumber \end{align}

◆ besselK025()

static double besselK025 ( double x )

static

Returns the value of $K_{1/4}(x)$, where $K_{\nu}$ is the modified Bessel’s function of the second kind.

The relative error on the returned value is less than $0.5\times10^{-6}$ for $x > 10^{-300}$.

Parameters

x	value at which the function is calculated

Returns: the value of $K_{1/4}(x)$

◆ calcMatStirling()

static double [][] calcMatStirling	(	int	m,
		int	n
	)

static

Computes and returns the Stirling numbers of the second kind.

\[ M[i,j] = \left\{\begin{matrix} j \\ i \end{matrix}\right\} \quad\mbox{ for $0\le i\le m$ and $0\le i\le j\le n$}. \tag{Stirling2} \]

See [111] (Section 1.2.6). The matrix $M$ is the transpose of Knuth’s (1973).

Parameters

m	number of rows of the allocated matrix
n	number of columns of the allocated matrix

Returns: the matrix of Stirling numbers

◆ combination()

static double combination	(	int	n,
		int	s
	)

static

Returns the value of $\binom{n}{s}$, the number of different combinations of $s$ objects amongst $n$.

Parameters

n	total number of objects
s	number of chosen objects on a combination

Returns: the combination of $s$ objects amongst $n$

◆ erf()

static double erf ( double x )

static

Returns the value of erf( $x$), the error function.

It is defined as

\[ \mbox{erf}(x) = \frac{2}{\sqrt{\pi}}\int_0^x dt e^{-t^2}. \]

Parameters

x	value at which the function is calculated

Returns: the value of erf $(x)$

◆ erfc()

static double erfc ( double x )

static

Returns the value of erfc( $x$), the complementary error function.

It is defined as

\[ \mbox{erfc}(x) = \frac{2}{\sqrt{\pi}}\int_x^{\infty}dt e^{-t^2}. \]

Parameters

x	value at which the function is calculated

Returns: the value of erfc $(x)$

◆ erfcInv()

static double erfcInv ( double u )

static

Returns the value of erfc ${}^{-1}(u)$, the inverse of the complementary error function.

If $u = $erfc $(x)$, then $x = $erfc ${}^{-1}(u)$.

Parameters

u	value at which the function is calculated

Returns: the value of erfcInv $(u)$

◆ erfInv()

static double erfInv ( double u )

static

Returns the value of erf ${}^{-1}(u)$, the inverse of the error function.

If $u = $erf $(x)$, then $x = $erf ${}^{-1}(u)$.

Parameters

u	value at which the function is calculated

Returns: the value of erfInv $(u)$

◆ evalCheby()

static double evalCheby	(	double	a[],
		int	n,
		double	x
	)

static

Evaluates a series of Chebyshev polynomials $T_j$ at $x$ over the basic interval $[-1, \;1]$, using the method of Clenshaw [35] , i.e., computes and returns

\[ y = \frac{a_0}{2} + \sum_{j=1}^n a_j T_j (x). \]

.

Parameters

a	coefficients of the polynomials
n	largest degree of polynomials
x	the parameter of the $T_j$ functions

Returns: the value of a series of Chebyshev polynomials $T_j$.

◆ evalChebyStar()

static double evalChebyStar	(	double	a[],
		int	n,
		double	x
	)

static

Evaluates a series of shifted Chebyshev polynomials $T_j^*$ at $x$ over the basic interval $ [0, \;1]$, using the method of Clenshaw [35] , i.e., computes and returns

\[ y = \frac{a_0}{2} + \sum_{j=1}^n a_j T_j^* (x). \]

.

Parameters

a	coefficients of the polynomials
n	largest degree of polynomials
x	the parameter of the $T_j^*$ functions

Returns: the value of a series of Chebyshev polynomials $T_j^*$.

◆ expBesselK1()

static double expBesselK1	(	double	x,
		double	y
	)

static

Returns the value of $e^x K_1(y)$, where $ K_1$ is the modified Bessel function of the second kind of order 1.

Restriction: $y > 0$.

◆ factoPow()

static double factoPow ( int n )

static

Returns the value of $n!/n^n$.

Parameters

n integer

Returns: the value of $n!/n^n$

◆ factorial()

static double factorial ( int n )

static

Returns the value of $n!$.

Parameters

n	the integer for which the factorial must be computed

Returns: the value of $n!$

◆ gammaRatioHalf()

static double gammaRatioHalf ( double x )

static

Returns the value of the ratio $\Gamma(x+1/2)/\Gamma(x)$ of two gamma functions.

This ratio is evaluated in a numerically stable way. Restriction: $x>0$.

◆ gcd() [1/2]

static int gcd	(	int	x,
		int	y
	)

static

Returns the greatest common divisor (gcd) of $x$ and $y$.

Parameters

x	integer
y	integer

Returns: the GCD of $x$ and $y$

◆ gcd() [2/2]

static long gcd	(	long	x,
		long	y
	)

static

Returns the greatest common divisor (gcd) of $x$ and $y$.

Parameters

x	integer
y	integer

Returns: the GCD of $x$ and $y$

◆ lnBeta()

static double lnBeta	(	double	lam,
		double	nu
	)

static

Computes the natural logarithm of the Beta function $B(\lambda, \nu)$.

It is defined in terms of the Gamma function as

\[ B(\lambda, \nu) = \frac{\Gamma(\lambda)\Gamma(\nu)}{\Gamma(\lambda+ \nu)} \]

with lam $=\lambda$ and nu $=\nu$.

◆ lnCombination()

static double lnCombination	(	int	n,
		int	s
	)

static

Returns the natural logarithm of $\binom{n}{s}$, the number of different combinations of $s$ objects amongst $n$.

Parameters

n	total number of objects
s	number of chosen objects on a combination

Returns: the natural log of the combination

◆ lnFactorial() [1/2]

static double lnFactorial ( int n )

static

Returns the value of $\ln(n!)$, the natural logarithm of factorial $n$.

Gives 16 decimals of precision (relative error $< 0.5\times10^{-15}$).

Parameters

n	argument of the log-factorial

Returns: natural logarithm of $n$ factorial

◆ lnFactorial() [2/2]

static double lnFactorial ( long n )

static

Returns the value of $\ln(n!)$, the natural logarithm of factorial $n$.

Gives 16 decimals of precision (relative error $< 0.5\times10^{-15}$).

Parameters

n	argument of the log-factorial

Returns: natural logarithm of $n$ factorial

◆ lnGamma()

static double lnGamma ( double x )

static

Returns the natural logarithm of the gamma function $\Gamma(x)$ evaluated at x.

Gives 16 decimals of precision, but is implemented only for $x>0$.

Parameters

x	the value for which the lnGamma function must be computed

Returns: the natural logarithm of the gamma function

◆ log2()

static double log2 ( double x )

static

Returns $\log_2 ($x $)$.

Parameters

x	the value for which the logarithm must be computed

Returns: the value of $\log_2 ($x $)$

◆ sumKahan()

static double sumKahan	(	double []	A,
		int	n
	)

static

Implementation of the Kahan summation algorithm.

Sums the first $n$ elements of $A$ and returns the sum. This algorithm is more precise than the naive algorithm. See http://en.wikipedia.org/wiki/Kahan_summation_algorithm.

◆ volumeSphere()

static double volumeSphere	(	double	p,
		int	t
	)

static

Returns the volume $V$ of a sphere of radius 1 in $t$ dimensions using the norm $L_p$.

It is given by the formula

\[ V = \frac{\left[2 \Gamma(1 + 1/p)\right]^t}{\Gamma\left(1 + t/p\right)}, \qquad p > 0, \]

where $\Gamma$ is the gamma function. The case of the sup norm $L_{\infty}$ is obtained by choosing $p=0$. Restrictions: $p\ge0$ and $t\ge1$.

Parameters

p	index of the used norm
t	number of dimensions

Returns: the volume of a sphere

Member Data Documentation

◆ TEN_NEG_POW

final double TEN_NEG_POW[]

static

Initial value:

= {
      1.0, 1.0e-1, 1.0e-2, 1.0e-3, 1.0e-4, 1.0e-5, 1.0e-6, 1.0e-7, 1.0e-8,
      1.0e-9, 1.0e-10, 1.0e-11, 1.0e-12, 1.0e-13, 1.0e-14, 1.0e-15, 1.0e-16
     }

Contains the precomputed negative powers of 10.

One has TEN_NEG_POW[j] $ = 10^{-j}$, for $j=0,…,16$.

◆ TWOEXP

final double TWOEXP[]

static

Initial value:

= {
0, 2.0, 4.0, 8.0, 1.6e1, 3.2e1,
4e1, 1.28e2, 2.56e2, 5.12e2, 1.024e3,
048e3, 4.096e3, 8.192e3, 1.6384e4, 3.2768e4,
5536e4, 1.31072e5, 2.62144e5, 5.24288e5,
048576e6, 2.097152e6, 4.194304e6, 8.388608e6,
6777216e7, 3.3554432e7, 6.7108864e7,
34217728e8, 2.68435456e8, 5.36870912e8,
073741824e9, 2.147483648e9, 4.294967296e9,
589934592e9, 1.7179869184e10, 3.4359738368e10,
8719476736e10, 1.37438953472e11, 2.74877906944e11,
49755813888e11, 1.099511627776e12, 2.199023255552e12,
398046511104e12, 8.796093022208e12,
7592186044416e13, 3.5184372088832e13,
0368744177664e13, 1.40737488355328e14,
81474976710656e14, 5.62949953421312e14,
125899906842624e15, 2.251799813685248e15,
503599627370496e15, 9.007199254740992e15,
8014398509481984e16, 3.6028797018963968e16,
2057594037927936e16, 1.44115188075855872e17,
88230376151711744e17, 5.76460752303423488e17,
152921504606846976e18, 2.305843009213693952e18,
611686018427387904e18, 9.223372036854775808e18,
8446744073709551616e19
     }

Contains the precomputed positive powers of 2.

One has TWOEXP[j] $ = 2^j$, for $j=0,…,64$.

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

Num.java

a	coefficients of the polynomials
n	largest degree of polynomials
x	the parameter of the \(T_j\) functions

a	coefficients of the polynomials
n	largest degree of polynomials
x	the parameter of the \(T_j^*\) functions

Static Public Member Functions

Static Public Attributes

Detailed Description

Member Function Documentation

◆ bernoulliPoly()

◆ besselK025()

◆ calcMatStirling()

◆ combination()

◆ erf()

◆ erfc()

◆ erfcInv()

◆ erfInv()

◆ evalCheby()

◆ evalChebyStar()

◆ expBesselK1()

◆ factoPow()

◆ factorial()

◆ gammaRatioHalf()

◆ gcd() [1/2]

◆ gcd() [2/2]

◆ lnBeta()

◆ lnCombination()

◆ lnFactorial() [1/2]

◆ lnFactorial() [2/2]

◆ lnGamma()

◆ log2()

◆ sumKahan()

◆ volumeSphere()

Member Data Documentation

◆ TEN_NEG_POW

◆ TWOEXP