Num (Java Libraries for Stochastic Simulation)

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

```
public class Num
extends java.lang.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

Fields
Modifier and Type	Field and Description
`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 10^x is representable (approximately) as a `double`.
`static int`	`DBL_MAX_EXP` Largest `int` x such that 2^x-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 2^x-1 is representable (approximately) as a normalised `double`.
`static double`	`EBASE` The constant e.
`static double`	`EULER` The Euler-Mascheroni constant.
`static double`	`ILN2` The values of 1/ln 2.
`static double`	`IRAC2` The value of 1/(2)^1/2.
`static double`	`LN_DBL_MIN` Natural logarithm of `DBL_MIN`.
`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[]`	`TEN_NEG_POW` Contains the precomputed negative powers of 10.
`static double[]`	`TWOEXP` Contains the precomputed positive powers of 2.

Method Summary

Methods
Modifier and Type	Method and Description
`static double`	`bernoulliPoly(int n, double x)` Evaluates the Bernoulli polynomial B_n(x) of degree n at x.
`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.
`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`	`digamma(double x)` Returns the value of the logarithmic derivative of the Gamma function ψ(x) = Γ'(x)/Γ(x).
`static double`	`erf(double x)` Returns the value of erf(x), the error function.
`static double`	`erfc(double x)` Returns the value of erfc(x), the complementary error function.
`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].
`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].
`static double`	`factorial(int n)` Returns the value of factorial n.
`static double`	`gammaRatioHalf(double x)` Returns the value of the ratio Γ(x + 1/2)/Γ(x) of two gamma functions, evaluated in a numerically stable way.
`static int`	`gcd(int x, int y)` Returns the greatest common divisor (gcd) of x and y.
`static long`	`gcd(long x, long y)` Returns the greatest common divisor (gcd) of x and y.
`static double`	`harmonic(long n)` Computes the n-th harmonic number H_n = ∑_j=1ⁿ1/j.
`static double`	`harmonic2(long n)` .
`static double`	`lnFactorial(int n)` Returns the value of the natural logarithm of factorial n.
`static double`	`lnGamma(double x)` Returns the natural logarithm of the gamma function Γ(x) evaluated at `x`.
`static double`	`log1p(double x)` Deprecated.
`static double`	`log2(double x)` Returns log₂(`x`).
`static double`	`multMod(double a, double s, double c, double m)` Deprecated.
`static int`	`multMod(int a, int s, int c, int m)` Deprecated.
`static long`	`multMod(long a, long s, long c, long m)` Deprecated.
`static double`	`tetragamma(double x)` Returns the value of the tetragamma function d²ψ(x)/d²x, the second derivative of the digamma function, evaluated at x.
`static double`	`trigamma(double x)` Returns the value of the trigamma function dψ(x)/dx, the derivative of the digamma function, evaluated at x.
`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.

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.
    
    See Also:
    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.
    
    See Also:
    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.
    
    See Also:
    Constant Field Values
  - LN_DBL_MIN
```
public static final double LN_DBL_MIN
```
    Natural logarithm of DBL_MIN.
    
    See Also:
    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
  - EULER
```
public static final double EULER
```
    The Euler-Mascheroni constant.
    
    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.
    
    See Also:
    Constant Field Values
  - LN2
```
public static final double LN2
```
    The values of ln 2.
    
    See Also:
    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
```
    Contains the precomputed positive powers of 2. One has TWOEXP[j]= 2^j, for j = 0,..., 64.
  - TEN_NEG_POW
```
public static final double[] TEN_NEG_POW
```
    Contains the precomputed negative powers of 10. One has TEN_NEG_POW[j]= 10^-j, for j = 0,…, 16.
- Method Detail
  - gcd
```
public static int gcd(int x,
      int y)
```
    Returns the greatest common divisor (gcd) of x and y.
    
    Parameters:
    x - integer
    y - integer
    
    Returns:
    the GCD of x and y
  - gcd
```
public static long gcd(long x,
       long y)
```
    Returns the greatest common divisor (gcd) of x and y.
    
    Parameters:
    x - integer
    y - integer
    
    Returns:
    the GCD of x and y
  - 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
  - factorial
```
public static double factorial(int n)
```
    Returns the value of factorial n.
    
    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 factorial n. Gives 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
  - 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
  - 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
```
@Deprecated
public static double log1p(double x)
```
    Deprecated.
    
    Use Math.log1p instead. 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)
  - lnGamma
```
public static double lnGamma(double x)
```
    Returns the natural logarithm of the gamma function Γ(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
  - digamma
```
public static double digamma(double x)
```
    Returns the value of the logarithmic derivative of the Gamma function ψ(x) = Γ'(x)/Γ(x).
  - trigamma
```
public static double trigamma(double x)
```
    Returns the value of the trigamma function dψ(x)/dx, the derivative of the digamma function, evaluated at x.
  - tetragamma
```
public static double tetragamma(double x)
```
    Returns the value of the tetragamma function d²ψ(x)/d²x, the second derivative of the digamma function, evaluated at x.
  - gammaRatioHalf
```
public static double gammaRatioHalf(double x)
```
    Returns the value of the ratio Γ(x + 1/2)/Γ(x) of two gamma functions, evaluated in a numerically stable way. Restriction: x > 0.
  - harmonic
```
public static double harmonic(long n)
```
    Computes the n-th harmonic number H_n = ∑_j=1ⁿ1/j.
  - harmonic2
```
public static double harmonic2(long n)
```
    . Computes the sum
    
    $\displaystyle \sideset{}{'}\htsum_{{-n/2<j\le n/2}}^{}$ $\displaystyle {\frac{1}{{\vert j\vert}}}$ ,
    where the symbol ∑^′ means that the term with j = 0 is excluded from the sum.
  - 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
  - bernoulliPoly
```
public static double bernoulliPoly(int n,
                   double x)
```
    Evaluates the Bernoulli polynomial B_n(x) of degree n at x. Only degrees n <= 8 are programmed for now. The first Bernoulli polynomials of even degree are:
    
    B₀(x) = 1
    
    B₂(x) = x² - x + 1/6
    
    B₄(x) = x⁴ -2x³ + x² - 1/30
    
    B₆(x) = x⁶ -3x⁵ +5x⁴/2 - x²/2 + 1/42
    
    B₈(x) = x⁸ -4x⁷ +14x⁶/3 - 7x⁴/3 + 2x²/3 - 1/30.
  - evalCheby
```
public 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]. It uses the method of Clenshaw, i.e., computes and returns
    
    y = $\displaystyle {\frac{{a_0}}{2}}$ + ∑_j=1ⁿa_jT_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
```
public 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]. It uses the method of Clenshaw, i.e., computes and returns
    
    y = [tex2html_wrap_indisplay721] + ∑_j=1ⁿa_jT_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^*.
  - 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)
  - erf
```
public static double erf(double x)
```
    Returns the value of erf(x), the error function. It is defined as
    
    erf(x) = 2/[(π)^1/2]∫₀^xdt e^-t².
    
    Parameters:
    x - value at which the function is calculated
    
    Returns:
    the value of erf(x)
  - erfc
```
public static double erfc(double x)
```
    Returns the value of erfc(x), the complementary error function. It is defined as
    
    erfc(x) = 2/[(π)^1/2]∫_x^∞dt e^-t².
    
    Parameters:
    x - value at which the function is calculated
    
    Returns:
    the value of erfc(x)
  - multMod
```
@Deprecated
public static int multMod(int a,
                     int s,
                     int c,
                     int m)
```
    Deprecated.
    
    Use ArithmeticMod.multModM(a, s, c, 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
```
@Deprecated
public static long multMod(long a,
                      long s,
                      long c,
                      long m)
```
    Deprecated.
    
    Use ArithmeticMod.multModM(a, s, c, 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
```
@Deprecated
public static double multMod(double a,
                        double s,
                        double c,
                        double m)
```
    Deprecated.
    
    Use ArithmeticMod.multModM(a, s, c, 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.

B₀(x)	=	1
B₂(x)	=	x² - x + 1/6
B₄(x)	=	x⁴ -2x³ + x² - 1/30
B₆(x)	=	x⁶ -3x⁵ +5x⁴/2 - x²/2 + 1/42
B₈(x)	=	x⁸ -4x⁷ +14x⁶/3 - 7x⁴/3 + 2x²/3 - 1/30.

Class Num

Field Summary

Method Summary

Methods inherited from class java.lang.Object

Field Detail

DBL_EPSILON

DBL_MAX_EXP

DBL_MIN_EXP

DBL_MAX_10_EXP

DBL_MIN

LN_DBL_MIN

DBL_DIG

EBASE

EULER

RAC2

IRAC2

LN2

ILN2

MAXINTDOUBLE

MAXTWOEXP

TWOEXP

TEN_NEG_POW

Method Detail

gcd

gcd

combination

factorial

lnFactorial

calcMatStirling

log2

log1p

lnGamma

digamma

trigamma

tetragamma

gammaRatioHalf

harmonic

harmonic2

volumeSphere

bernoulliPoly

evalCheby

evalChebyStar

besselK025

erf

erfc

multMod

multMod

multMod