umontreal.iro.lecuyer.util
Class Num

java.lang.Object
  umontreal.iro.lecuyer.util.Num

public class Num
extends Object

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

Field Summary

static int	DBL_DIG Number of decimal digits of precision in a double.
static double	DBL_EPSILON Difference between 1.0 and the smallest double greater than 1.0.
static int	DBL_MAX_10_EXP Largest int x such that 10x is representable (approximately) as a double.
static int	DBL_MAX_EXP Largest int x such that 2x-1 is representable (approximately) as a double.
static double	DBL_MIN Smallest normalized positive floating-point double.
static int	DBL_MIN_EXP Smallest int x such that 2x-1 is representable (approximately) as a normalised double.
static double	EBASE The constant e.
static double	ILN2 The values of 1/ln 2.
static double	IRAC2 The value of 1/(2)1/2.
static double	LN2 The values of ln 2.
static double	MAXINTDOUBLE Largest integer n0 = 253 such that any integer n <= n0 is represented exactly as a double.
static double	MAXTWOEXP Powers of 2 up to MAXTWOEXP are stored exactly in the array TWOEXP.
static double	RAC2 The value of (2)1/2.
static double[]	TWOEXP This table contains the powers of two, precomputed.

Method Summary

static double	besselK025(double x) Returns the value of K1/4(x), where Ka is the modified Bessel's function of the second kind.
static double[][]	calcMatStirling(int m, int n) Computes and returns the Stirling numbers of the second kind
static double	combination(int n, int s) Returns the number of different combinations of s objects amongst n.
static double	evalCheby(double[] S, int N, double x) Evaluates a series of Chebyshev polynomials Tj, at point x∈[- 1, 1].
static double	factorial(int n) Returns the value of n factorial.
static double	lnFactorial(int n) Returns the value of the natural logarithm of n factorial.
static double	lnGamma(double x) Returns the natural logarithm of the gamma function evaluated at x.
static double	log1p(double x) Returns a value equivalent to log(1 + x) accurate also for small x.
static double	log2(double x) Returns log2(x).
static double	multMod(double a, double s, double c, double m) Returns (as + c)mod m.
static int	multMod(int a, int s, int c, int m) Returns (as + c)mod m.
static long	multMod(long a, long s, long c, long m) Returns (as + c)mod m.
static double	volumeSphere(double p, int t) Returns the volume V of a sphere of radius 1 in t dimensions using the norm Lp.

Methods inherited from class java.lang.Object

equals, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait

Field Detail

DBL_EPSILON

public static final double DBL_EPSILON

Difference between 1.0 and the smallest double greater than 1.0.

See Also:

Constant Field Values

DBL_MAX_EXP

public static final int DBL_MAX_EXP

Largest int x such that 2^x-1 is representable (approximately) as a double.

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Constant Field Values

DBL_MIN_EXP

public static final int DBL_MIN_EXP

Smallest int x such that 2^x-1 is representable (approximately) as a normalised double.

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Constant Field Values

DBL_MAX_10_EXP

public static final int DBL_MAX_10_EXP

Largest int x such that 10^x is representable (approximately) as a double.

See Also:

Constant Field Values

DBL_MIN

public static final double DBL_MIN

Smallest normalized positive floating-point double.

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Constant Field Values

DBL_DIG

public static final int DBL_DIG

Number of decimal digits of precision in a double.

See Also:

Constant Field Values

EBASE

public static final double EBASE

The constant e.

See Also:

Constant Field Values

RAC2

public static final double RAC2

The value of (2)^1/2.

See Also:

Constant Field Values

IRAC2

public static final double IRAC2

The value of 1/(2)^1/2.

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Constant Field Values

LN2

public static final double LN2

The values of ln 2.

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Constant Field Values

ILN2

public static final double ILN2

The values of 1/ln 2.

See Also:

Constant Field Values

MAXINTDOUBLE

public static final double MAXINTDOUBLE

Largest integer n₀ = 2⁵³ such that any integer n <= n₀ is represented exactly as a double.

See Also:

Constant Field Values

MAXTWOEXP

public static final double MAXTWOEXP

Powers of 2 up to MAXTWOEXP are stored exactly in the array TWOEXP.

See Also:

Constant Field Values

TWOEXP

public static final double[] TWOEXP

This table contains the powers of two, precomputed. One has TWOEXP[i]= 2ⁱ for i = 0,..., 64.

Method Detail

log2

public static double log2(double x)

Returns log₂(x).

Parameters:

x - the value for which the logarithm must be computed

Returns:

the value of log₂(x)

log1p

public static double log1p(double x)

Returns a value equivalent to log(1 + x) accurate also for small x.

Parameters:

x - the argument of the function

Returns:

the value of log(1 + x)

factorial

public static double factorial(int n)

Returns the value of n factorial.

Parameters:

n - the integer for which the factorial must be computed

Returns:

the value of n!

lnFactorial

public static double lnFactorial(int n)

Returns the value of the natural logarithm of n factorial. Gives at least 16 decimals of precision (relative error < 0.5×10^-15).

Parameters:

n - the integer for which the log-factorial has to be computed

Returns:

the natural logarithm of n factorial

lnGamma

public static double lnGamma(double x)

Returns the natural logarithm of the gamma function evaluated at x. Gives 16 decimals of precision. Restriction: 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

combination

public static double combination(int n,
                                 int s)

Returns the number of different combinations of s objects amongst n. Uses an algorithm that prevents overflows (when computing factorials), if possible.

Parameters:

n - total number of objects

s - number of chosen objects on a combination

Returns:

the number of different combinations of s objects amongst n

calcMatStirling

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

Computes and returns the Stirling numbers of the second kind

Parameters:

m - number of rows of the allocated matrix

n - number of columns of the allocated matrix

Returns:

the matrix of Stirling numbers

volumeSphere

public 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. It is given by the formula

V = ([2Γ(1 + 1/p)]^t)/Γ(1 + t/p), p > 0,

where Γ is the gamma function. The case of the sup norm L_∞ is obtained by choosing p = 0. Restrictions: p >= 0 and t >= 1.

Parameters:

p - index of the used norm

t - number of dimensions

Returns:

the volume of a sphere

evalCheby

public static double evalCheby(double[] S,
                               int N,
                               double x)

Evaluates a series of Chebyshev polynomials T_j, at point x∈[- 1, 1]. It uses the method of Clenshaw, i.e., computes and returns

y = S₀/2 + ∑_j=1^NS_jT_j(x).

Parameters:

S - 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.

besselK025

public static double besselK025(double x)

Returns the value of K_1/4(x), where K_a is the modified Bessel's function of the second kind. The relative error on the returned value is less than 0.5×10^-6 for x > 10^-300. .

Parameters:

x - value at which the function is calculated

Returns:

the value of K_1/4(x)

multMod

public static int multMod(int a,
                          int s,
                          int c,
                          int m)

Returns (as + c)mod m. Restriction: assumes that a, c, s < m.

Parameters:

a - first term of the product

s - second term of the product

c - shift factor

m - the used modulo

Returns:

the value of (as + c)mod m.

multMod

public static long multMod(long a,
                           long s,
                           long c,
                           long m)

Returns (as + c)mod m. Uses the class ArithmeticMod. Restriction: assumes that a, c, s < m.

Parameters:

a - first term of the product

s - second term of the product

c - shift factor

m - the used modulo

Returns:

the value of (as + c)mod m.

multMod

public static double multMod(double a,
                             double s,
                             double c,
                             double m)

Returns (as + c)mod m. Restriction: assumes that a, s, c are < m and a, s, c, m are < 2³⁵.

Parameters:

a - first term of the product

s - second term of the product

c - shift factor

m - the used modulo

Returns:

the value of (as + c)mod m.