Maths

java.lang.Object
- la.maths.Maths

```
public class Maths
extends java.lang.Object
```
Useful mathematical constants and functions. Also see README.

Field Summary

Fields
Modifier and Type Field and Description

static double epsilon
A tiny amount, epsilon, to allow for rounding error where necessary.

static double twoPI
twoPI = 2π.

Fields
Modifier and Type	Field and Description
`static double`	`epsilon` A tiny amount, epsilon, to allow for rounding error where necessary.
`static double`	`twoPI` twoPI = 2π.

Constructor Summary

Constructors
Constructor and Description

Maths()

Constructors
Constructor and Description
`Maths()`

Method Summary

All Methods Static Methods Concrete Methods
Modifier and Type	Method and Description
`static double`	`Bessel_I_asymptotic(double alpha, double x)` For "large" x, uses Hankel's asymptotic expansion.
`static double`	`Bessel_I_series(double alpha, double x)` For "small" x, uses the series expansion.
`static double`	`Bessel_I(double alpha, double x)` I_α(x), the modified Bessel function of the 1st kind; x ≥ 0, and the order α ≥ 0, are reals here.
`static double`	`Bessel_I2(double alpha, double x)` Uses `Bessel_I_series(,)` for "small" x, `Bessel_I_asymptotic(,)` for "large" x, and both at the cross-over (continuity).
`static long`	`catalan(int N)` Return the N^th Catalan number, N ≥ 0.
`static long`	`cummulativeCatalans(int N)` Return ∑_i=0..N `catalan(i)`, N≥0.
`static double`	`Gamma(double x)` Math.exp(`logGamma(x)`), for real x.
`static double`	`log_nCk(int n, int k)` log_e(ⁿC_k).
`static double`	`logBeta(double x, double y)` The log Beta (β) function, log(Β(x,y)), returns `logΓ`(x)+logΓ(y)-logΓ(x+y).
`static double`	`logFactorial(double x)` Implemented as `logGamma(x+1)`.
`static double`	`logFactorial(int n)` Implemented as `logGamma(n+1)`.
`static double`	`logGamma(double x)` log(Γ(x)), real x>0; note, Γ n = (n-1)!, for int n≥1.
`static double`	`logSum(double[] nlXs)` 'logSum', aka 'logPlus', logSum_i( - log(p[i])) = - log( ∑_i exp( log(p[i]) ) = - log( ∑_i p[i] ).
`static double`	`logSum(double nlX, double nlY)` Equivalent to `logSum({nlX, nlY})`, but a little quicker.
`static void`	`main(java.lang.String[] args)`
`static boolean`	`nearlyEqual(double x, double y)` Is \|x-y\| ≤ a standard '`epsilon`'-fraction of max(\|x\|, \|y\|) ?
`static boolean`	`nearlyEqual(double x, double y, double epsilon)` Is \|x-y\| ≤ an epsilon-fraction of max(\|x\|, \|y\|) ?
`static double[]`	`nlPrs2odds(double[] nlPr)` Convert an array, nlPr, of message lengths (negative log probabilities) into a normalised array of probabilities.
`static void`	`nlPrs2odds(double[] nlPr, double[] pr)` Convert an array, nlPr, of negative log probabilities into a normalised array of probabilities, pr.

Methods inherited from class java.lang.Object
clone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait

- Field Detail
  - epsilon
```
public static final double epsilon
```
    A tiny amount, epsilon, to allow for rounding error where necessary.
    
    See Also:
    
    Constant Field Values
  - twoPI
```
public static final double twoPI
```
    twoPI = 2π.
    
    See Also:
    
    Constant Field Values
- Constructor Detail
  - Maths
```
public Maths()
```
- Method Detail
  - nearlyEqual
```
public static boolean nearlyEqual(double x,
                                  double y)
```
    Is |x-y| ≤ a standard 'epsilon'-fraction of max(|x|, |y|) ?
  - nearlyEqual
```
public static boolean nearlyEqual(double x,
                                  double y,
                                  double epsilon)
```
    Is |x-y| ≤ an epsilon-fraction of max(|x|, |y|) ?
  - log_nCk
```
public static double log_nCk(int n,
                             int k)
```
    log_e(ⁿC_k). n choose k, ⁿC_k = n! / (k! (n-k)!). Calls logFactorial.
  - logSum
```
public static double logSum(double[] nlXs)
```
    'logSum', aka 'logPlus',
    logSum_i( - log(p[i])) = - log( ∑_i exp( log(p[i]) ) = - log( ∑_i p[i] ). (But it is not implemented that way.) For example, logSum({10, 10}) = 10 - log(2) nits.
  - logSum
```
public static double logSum(double nlX,
                            double nlY)
```
    Equivalent to logSum({nlX, nlY}), but a little quicker.
  - nlPrs2odds
```
public static double[] nlPrs2odds(double[] nlPr)
```
    Convert an array, nlPr, of message lengths (negative log probabilities) into a normalised array of probabilities. Also see nlPrs2odds(double[],double[]).
  - nlPrs2odds
```
public static void nlPrs2odds(double[] nlPr,
                              double[] pr)
```
    Convert an array, nlPr, of negative log probabilities into a normalised array of probabilities, pr. Afterwards, ∑_i pr[i] = 1.
  - catalan
```
public static long catalan(int N)
```
    Return the N^th Catalan number, N ≥ 0. This is ^2NC_N / (N+1). The series starts 1, 1, 2, 5, 14, 42, 132, ... Takes O(N)-time. The N-th Catalan number is the number of full binary trees with N internal nodes, equivalently with N+1 leaves. Also see cummulativeCatalans(N).
  - cummulativeCatalans
```
public static long cummulativeCatalans(int N)
```
    Return ∑_i=0..N catalan(i), N≥0. Takes O(N)-time.
  - Gamma
```
public static double Gamma(double x)
```
    Math.exp(logGamma(x)), for real x. Note, Γ n = (n-1)!, for int n≥1, and Γ (x+1) = x Γ x. Also defined for non-integer x<0 but there will be numerical problems for large negative values. (Recall that for int n<0, n! is infinite.)
  - logFactorial
```
public static double logFactorial(int n)
```
    Implemented as logGamma(n+1).
  - logFactorial
```
public static double logFactorial(double x)
```
    Implemented as logGamma(x+1).
  - logGamma
```
public static double logGamma(double x)
```
    log(Γ(x)), real x>0; note, Γ n = (n-1)!, for int n≥1. Also, n! = Γ(n+1), and Γ (x+1) = x Γ x. See [www.allisons.org/ll/AlgDS/Numerical/Stirling/] and logFactorial.
  - logBeta
```
public static double logBeta(double x,
                             double y)
```
    The log Beta (β) function, log(Β(x,y)), returns logΓ(x)+logΓ(y)-logΓ(x+y). For x, y > 0, B(x,y) = ₀∫¹ t^x-1 (1-t)^y-1 dt.
  - Bessel_I
```
public static double Bessel_I(double alpha,
                              double x)
```
    I_α(x), the modified Bessel function of the 1st kind; x ≥ 0, and the order α ≥ 0, are reals here. Also see [wikip] ... Note, the derivative I'_ν(z) = (I_ν-1(z) + I_ν+1(z)) / 2; I'₀ = I₁(z). Also note, I(a,x) / I(a-1,x) = 1 / (a/x + 1/((a+2)/x + ...)), a continued fraction (Watson1995).
  - Bessel_I2
```
public static double Bessel_I2(double alpha,
                               double x)
```
    Uses Bessel_I_series(,) for "small" x, Bessel_I_asymptotic(,) for "large" x, and both at the cross-over (continuity).
  - Bessel_I_series
```
public static double Bessel_I_series(double alpha,
                                     double x)
```
    For "small" x, uses the series expansion. Is accurate, but slow for large x.
  - Bessel_I_asymptotic
```
public static double Bessel_I_asymptotic(double alpha,
                                         double x)
```
    For "large" x, uses Hankel's asymptotic expansion. See [www] and [www].
  - main
```
public static void main(java.lang.String[] args)
```

Class Maths

Field Summary

Constructor Summary

Method Summary

Methods inherited from class java.lang.Object

Field Detail

epsilon

twoPI

Constructor Detail

Maths

Method Detail

nearlyEqual

nearlyEqual

log_nCk

logSum

logSum

nlPrs2odds

nlPrs2odds

catalan

cummulativeCatalans

Gamma

logFactorial

logFactorial

logGamma

logBeta

Bessel_I

Bessel_I2

Bessel_I_series

Bessel_I_asymptotic

main