Package de.tilman_neumann.jml

Class HarmonicNumbers

java.lang.Object
de.tilman_neumann.jml.HarmonicNumbers
public class HarmonicNumbers
extends java.lang.Object
Computation of harmonic and "hyper-harmonic" numbers.

  • Method Summary

    Modifier and Type Method Description
    static BigRational harmonic​(int n)
    Simple series computation of harmonic numbers H_{n} = 1/1 + 1/2 + 1/3 + ...
    static double harmonic_dbl​(int n)
    Simple series computation of harmonic numbers H_{n} = 1/1 + 1/2 + 1/3 + ...
    static java.math.BigDecimal harmonic_lowerBound​(java.math.BigInteger n, Scale scale)
    Lower bound for the harmonic number H_n.
    static java.math.BigDecimal harmonic_upperBound​(java.math.BigInteger n, Scale scale)
    Upper bound for the harmonic number H_n.
    static BigRational harmonicPower​(int n, int r)
    Harmonic power series H_{n,r} = sum_{i=1..n} 1/i^r.
    static BigRational hyperharmonic_closedForm​(int n, int r)
    Closed-form evaluation of "hyper-harmonic numbers" defined by
    H_{n,1} = sum_{i=1..n} 1/i
    H_{n,r} = sum_{i=1..n} H_{i,r-1}; r>1
    static BigRational hyperharmonic_recurrent​(int n, int r)
    Recurrent computation of "hyper-harmonic numbers" defined by
    H_{n,1} = sum_{i=1..n} 1/i
    H_{n,r} = sum_{i=1..n} H_{i,r-1}; r>1
    static void main​(java.lang.String[] args)  

    Methods inherited from class java.lang.Object

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

  • Method Details

    • harmonic

      public static BigRational harmonic​(int n)
      Simple series computation of harmonic numbers H_{n} = 1/1 + 1/2 + 1/3 + ... + 1/n.
      Parameters:
      n -
      Returns:
      harmonic number H_{n}

    • harmonic_dbl

      public static double harmonic_dbl​(int n)
      Simple series computation of harmonic numbers H_{n} = 1/1 + 1/2 + 1/3 + ... + 1/n.
      Parameters:
      n -
      Returns:
      harmonic number H_{n}

    • harmonic_upperBound

      public static java.math.BigDecimal harmonic_upperBound​(java.math.BigInteger n, Scale scale)
      Upper bound for the harmonic number H_n. From http://fredrik-j.blogspot.com/2009/02/how-not-to-compute-harmonic-numbers.html: H_n = ln(n) + gamma + 1/2*n^-1 - 1/12*n^-2 + 1/120*n^-4 + O(n^-6)
      Parameters:
      n -
      scale - number of exact after-komma digits
      Returns:
      upper bound for the harmonic number H_n

    • harmonic_lowerBound

      public static java.math.BigDecimal harmonic_lowerBound​(java.math.BigInteger n, Scale scale)
      Lower bound for the harmonic number H_n. From https://math.stackexchange.com/questions/306371/simple-proof-of-showing-the-harmonic-number-h-n-theta-log-n: H_n = ln(n) + gamma + 1/2*n^-1 - 1/12*n^-2 + 1/120*n^-4 - 1/252*n^-6 + O(n^-8)
      Parameters:
      n -
      scale - number of exact after-komma digits
      Returns:
      lower bound for the harmonic number H_n

    • hyperharmonic_closedForm

      public static BigRational hyperharmonic_closedForm​(int n, int r)
      Closed-form evaluation of "hyper-harmonic numbers" defined by
      H_{n,1} = sum_{i=1..n} 1/i
      H_{n,r} = sum_{i=1..n} H_{i,r-1}; r>1
      Parameters:
      n - principal argument
      r - order
      Returns:
      hyper-harmonic number H_{n,r}

    • hyperharmonic_recurrent

      public static BigRational hyperharmonic_recurrent​(int n, int r)
      Recurrent computation of "hyper-harmonic numbers" defined by
      H_{n,1} = sum_{i=1..n} 1/i
      H_{n,r} = sum_{i=1..n} H_{i,r-1}; r>1
      Parameters:
      n - principal argument
      r - order
      Returns:
      hyper-harmonic number H_{n,r}

    • harmonicPower

      public static BigRational harmonicPower​(int n, int r)
      Harmonic power series H_{n,r} = sum_{i=1..n} 1/i^r. Can also be understood as a finite version of Riemanns zeta function.
      Parameters:
      n -
      r -
      Returns:
      H_{n,r}

    • main

      public static void main​(java.lang.String[] args)