A math utility class with static methods. : Math « Development Class « Java






A math utility class with static methods.

      
/*
 * LingPipe v. 3.9
 * Copyright (C) 2003-2010 Alias-i
 *
 * This program is licensed under the Alias-i Royalty Free License
 * Version 1 WITHOUT ANY WARRANTY, without even the implied warranty of
 * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the Alias-i
 * Royalty Free License Version 1 for more details.
 *
 * You should have received a copy of the Alias-i Royalty Free License
 * Version 1 along with this program; if not, visit
 * http://alias-i.com/lingpipe/licenses/lingpipe-license-1.txt or contact
 * Alias-i, Inc. at 181 North 11th Street, Suite 401, Brooklyn, NY 11211,
 * +1 (718) 290-9170.
 */

//package com.aliasi.util;

/**
 * A math utility class with static methods.
 *
 * @author  Bob Carpenter
 * @version 4.0.0
 * @since   LingPipe1.0
 */
public class Math {

    // forbid instances
    private Math() {
        /* no instances */
    }

    /**
     * The value of the golden ratio.  The golden ratio is defined to
     * be the value φ such that:
     *
     * <blockquote>
     * &phi; = (&phi; + 1) / &phi;
     * </blockquote>
     *
     * Note that this is a quadratic equation (multiply both sides by
     * &phi;) with the solution roughly <code>1.61803399</code>.
     *
     * <p>See the following for a fascinating tour of the properties
     * of the golden ratio:
     *
     * <ul>
     * <li><a href="http://mathworld.wolfram.com/GoldenRatio.html"
     *>Mathworld: Golden Ratio</a></li>
     * </ul>
     */
    public static final double GOLDEN_RATIO =  (1.0 + java.lang.Math.sqrt(5))/2.0;

    /**
     * The natural logarithm of 2.
     */
    public static final double LN_2 = java.lang.Math.log(2.0);

    static final double INV_LN_2 = 1.0/LN_2;


    /**
     * An array of the Fibonacci sequence up the maximum value
     * representable as a long integer.  The array is defined as
     * follows:
     *
     * <blockquote><pre>
     * FIBONACCI_SEQUENCE[0] = 1
     * FIBONACCI_SEQUENCE[1] = 2
     * FIBONACCI_SEQUENCE[n+2] = FIBONACCI_SEQUENCE[n+1] + FIBONACCI_SEQUENCE[n]
     * </pre></blockquote>
     *
     * So <code>FIBONACCI_SEQUENCE[0]</code> represents the second
     * Fibonacci number in the traditional numbering.  The inital entries
     * are:
     *
     * <blockquote><code>
     * 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597,
     * 2584, ...
     * </code></blockquote>
     *
     * The length of the array is 91, and the largest value is:
     *
     * <blockquote><code>
     * FIBONACCI_SEQUENCE[90] = 7540113804746346429
     *
     * </code></blockquote>
     *
     * <P>See the following references for more information on
     * the fascinating properties of Fibonacci numbers:
     *
     * <UL>
     * <LI> <a href="http://en.wikipedia.org/wiki/Fibonacci_number">Wikipedia: Fibonacci Number</a>
     * <LI> <a href="http://mathworld.wolfram.com/FibonacciNumber.html">Mathworld: Fibonacci Number</a>
     */
    public static final long[] FIBONACCI_SEQUENCE = new long[] {
    1l,
    2l,
    3l,
    5l,
    8l,
    13l,
    21l,
    34l,
    55l,
    89l,
    144l,
    233l,
    377l,
    610l,
    987l,
    1597l,
    2584l,
    4181l,
    6765l,
    10946l,
    17711l,
    28657l,
    46368l,
    75025l,
    121393l,
    196418l,
    317811l,
    514229l,
    832040l,
    1346269l,
    2178309l,
    3524578l,
    5702887l,
    9227465l,
    14930352l,
    24157817l,
    39088169l,
    63245986l,
    102334155l,
    165580141l,
    267914296l,
    433494437l,
    701408733l,
    1134903170l,
    1836311903l,
    2971215073l,
    4807526976l,
    7778742049l,
    12586269025l,
    20365011074l,
    32951280099l,
    53316291173l,
    86267571272l,
    139583862445l,
    225851433717l,
    365435296162l,
    591286729879l,
    956722026041l,
    1548008755920l,
    2504730781961l,
    4052739537881l,
    6557470319842l,
    10610209857723l,
    17167680177565l,
    27777890035288l,
    44945570212853l,
    72723460248141l,
    117669030460994l,
    190392490709135l,
    308061521170129l,
    498454011879264l,
    806515533049393l,
    1304969544928657l,
    2111485077978050l,
    3416454622906707l,
    5527939700884757l,
    8944394323791464l,
    14472334024676221l,
    23416728348467685l,
    37889062373143906l,
    61305790721611591l,
    99194853094755497l,
    160500643816367088l,
    259695496911122585l,
    420196140727489673l,
    679891637638612258l,
    1100087778366101931l,
    1779979416004714189l,
    2880067194370816120l,
    4660046610375530309l,
    7540113804746346429l
    };

    /**
     * Returns <code>true</code> if the specified number is prime.  A
     * prime is a positive number greater than <code>1</code> with no
     * divisors other than <code>1</code> and itself, thus
     * <code>{2,3,5,7,11,13,...}</code>.
     *
     * @param num Number to test for primality.
     * @return <code>true</code> if the specified number is prime.
     */
    public static boolean isPrime(int num) {
        if (num < 2) return false;
        for (int i = 2; i <= num/2; ++i)
            if (num % i == 0) return false;
        return true;
    }

    /**
     * Returns the smallest prime number that is strictly larger than
     * the specified integer.  See {@link #isPrime(int)} for the
     * definition of primality.
     *
     * @param num Base from which to look for the next prime.
     * @return Smallest prime number strictly larget than specified
     * number.
     */
    public static int nextPrime(int num) {
        if (num < 2) return 2;
        for (int i = num + 1; ; ++i)
            if (isPrime(i)) return i;
    }

    /**
     * Converts a natural logarithm to a base 2 logarithm.
     * This inverts the operation of {@link #logBase2ToNaturalLog(double)}.

     * <p>If the input is <code><i>x</i> = ln <i>z</i></code>, then
     * the return value is <code>log<sub>2</sub> <i>z</i></code>.
     * Recall that <code>log<sub>2</sub> <i>z</i> = ln <i>z</i> / ln 2.
     *
     * @param x Natural log of value.
     * @return Log base 2 of value.
     */
    public static double naturalLogToBase2Log(double x) {
        return x * INV_LN_2;
    }


    /**
     * Returns the log base 2 of the specivied value.
     *
     * @param x Value whose log is taken.
     * @return Log of specified value.
     */
    public static double log2(double x) {
        return naturalLogToBase2Log(java.lang.Math.log(x));
    }


    /**
     * Returns the integer value of reading the specified byte as an
     * unsigned value.  The computation is carried out by subtracting
     * the minimum value, as defined by the constant {@link
     * Byte#MIN_VALUE}.
     *
     * @param b Byte to convert.
     * @return Unsigned value of specified byte.
     */
    public static int byteAsUnsigned(byte b) {
        return (b >= 0) ? (int)b : (256+(int)b);
    }

    /**
     * Returns the log (base 2) of the factorial of the specified long
     * integer.  The factorial of <code>n</code> is defined for
     * <code>n > 0</code> by:
     *
     * <blockquote><code>
     *  n!
     *  = <big><big>&Pi;</big></big><sub><sub>i < 0 <= n</sub></sub> i
     * </code></blockquote>
     *
     * Taking logs of both sides gives:
     *
     * <blockquote><code>
     *  log<sub><sub>2</sub></sub> n!
     *  = <big><big>&Sigma;</big></big><sub><sub>i < 0 <= n</sub></sub>
     *    log<sub><sub>2</sub></sub> i
     * </code></blockquote>
     *
     * By convention, 0! is taken to be 1, and hence <code>ln 0! = 0</code>.
     *
     * @param n Specified long integer.
     * @return Log of factorial of specified integer.
     * @throws IllegalArgumentException If the argument is negative.
     */
    public static double log2Factorial(long n) {
        if (n < 0) {
            String msg = "Factorials only defined for non-negative arguments."
                + " Found argument=" + n;
            throw new IllegalArgumentException(msg);
        }
        double sum = 0.0;
        for (long i = 1; i <= n; ++i)
            sum += log2(i);
        return sum;
    }

    /**
     * Returns the sum of the specified array of double values.
     *
     * @param xs Variable length list of values, or an array of values.
     * @return The sum of the values.
     */
    public static double sum(double... xs) {
        double sum = 0.0;
        for (int i = 0; i < xs.length; ++i)
            sum += xs[i];
        return sum;
    }

    /**
     * Returns the minimum of the specified array of double values.
     * If the length of the array is zero, the result is {@link
     * Double#NaN}.
     *
     * @param xs Variable length list of values, or an array.
     * @return Minimum value in array.
     */
    public static double minimum(double... xs) {
        if (xs.length == 0) return Double.NaN;
        double min = xs[0];
        for (int i = 1; i < xs.length; ++i)
            if (xs[i] < min) min = xs[i];
        return min;
    }

    /**
     * Returns the maximum of the specified array of double values.
     * If the length of the array is zero, the result is {@link
     * Double#NaN}.
     *
     * @param xs Variable length list of values, or an array.
     * @return Maximum value in array.
     */
    public static double maximum(double... xs) {
        if (xs.length == 0) return Double.NaN;
        double max = xs[0];
        for (int i = 1; i < xs.length; ++i)
            if (xs[i] > max) max = xs[i];
        return max;
    }


    /**
     * Returns the log (base 2) of the binomial coefficient of the
     * specified arguments.  The binomial coefficient is equal to the
     * number of ways to choose a subset of size <code>m</code> from a
     * set of <code>n</code> objects, which is pronounced "n choose
     * m", and is given by:
     *
     * <blockquote><code>
     *   choose(n,m) = n! / ( m! * (n-m)!)
     *   <br>
     *   log<sub>2</sub> choose(n,m)
     *    = log<sub>2</sub> n - log<sub>2</sub> m
     *      - log<sub>2</sub> (n-m)
     * </code></blockquote>
     *
     * @return The log (base 2) of the binomial coefficient of the
     * specified arguments.
     */
    public static double log2BinomialCoefficient(long n, long m) {
        return log2(n) - log2(m) - log2(n-m);
    }



    static double[] LANCZOS_COEFFS = new double[] {
        0.99999999999980993,
        676.5203681218851,
        -1259.1392167224028,
        771.32342877765313,
        -176.61502916214059,
        12.507343278686905,
        -0.13857109526572012,
        9.9843695780195716e-6,
        1.5056327351493116e-7
    };

    static double SQRT_2_PI = java.lang.Math.sqrt(2.0 * java.lang.Math.PI);

    // assumes input in [0.5,1.5] inclusive
    static double lanczosGamma(double z) {
        double zMinus1 = z - 1;
        double x = LANCZOS_COEFFS[0];
        for (int i = 1; i < LANCZOS_COEFFS.length - 2; ++i)
            x += LANCZOS_COEFFS[i] / (zMinus1 + i);
        double t = zMinus1 + (LANCZOS_COEFFS.length - 2) + 0.5;
        return SQRT_2_PI
            * java.lang.Math.pow(t, zMinus1 + 0.5)
            * java.lang.Math.exp(-t) * x;
    }


    /**
     * Returns the value of the digamma function for the specified
     * value.  The returned values are accurate to at least 13
     * decimal places.
     *
     * <p>The digamma function is the derivative of the log of the
     * gamma function; see the method documentation for {@link
     * #log2Gamma(double)} for more information on the gamma function
     * itself.
     *
     * <blockquote><pre>
     * &Psi;(z)
     * = <i>d</i> log &Gamma;(z) / <i>d</i>z
     * = &Gamma;'(z) / &Gamma;(z)
     * </pre></blockquote>
     *
     * <p>The numerical approximation is derived from:
     *
     * <ul>
     * <li>Richard J. Mathar. 2005.
     * <a href="http://arxiv.org/abs/math/0403344">Chebyshev Series Expansion of Inverse Polynomials</a>.
     * <li>
     * <li>Richard J. Mathar. 2005.
     * <a href="http://www.strw.leidenuniv.nl/~mathar/progs/digamma.c">digamma.c</a>.
     * (C Program implementing algorithm.)
     * </li>
     * </ul>
     *
     * <i>Implementation Note:</i> The recursive calls in the C
     * implementation have been transformed into loops and
     * accumulators, and the recursion for values greater than three
     * replaced with a simpler reduction.  The number of loops
     * required before the fixed length expansion is approximately
     * integer value of the absolute value of the input.  Each loop
     * requires a floating point division, two additions and a local
     * variable assignment.  The fixed portion of the algorithm is
     * roughly 30 steps requiring four multiplications, three
     * additions, one static final array lookup, and four assignments per
     * loop iteration.
     *
     * @param x Value at which to evaluate the digamma function.
     * @return The value of the digamma function at the specified
     * value.
     */
    public static double digamma(double x)
    {
        if (x <= 0.0 && (x == (double)((long) x)))
            return Double.NaN;

        double accum = 0.0;
        if (x < 0.0) {
            accum += java.lang.Math.PI
                / java.lang.Math.tan(java.lang.Math.PI * (1.0 - x));
            x = 1.0 - x;
        }

        if (x < 1.0 ) {
            while (x < 1.0)
                accum -= 1.0 / x++;
        }

        if (x == 1.0)
            return accum - NEGATIVE_DIGAMMA_1;

        if (x == 2.0)
            return accum + 1.0 - NEGATIVE_DIGAMMA_1;

        if (x == 3.0)
            return accum + 1.5 - NEGATIVE_DIGAMMA_1;

        // simpler recursion than Mahar to reduce recursion
        if (x > 3.0) {
            while (x > 3.0)
                accum += 1.0 / --x;
            return accum + digamma(x);
        }

        x -= 2.0;
        double tNMinus1 = 1.0;
        double tN = x;
        double digamma = DIGAMMA_COEFFS[0] + DIGAMMA_COEFFS[1] * tN;
        for (int n = 2; n < DIGAMMA_COEFFS.length; n++) {
            double tN1 = 2.0 * x * tN - tNMinus1;
            digamma += DIGAMMA_COEFFS[n] * tN1;
            tNMinus1 = tN;
            tN = tN1;
        }
        return accum + digamma;
    }


    /**
     * The &gamma; constant for computing the digamma function.
     *
     * <p>The value is defined as the negative of the digamma funtion
     * evaluated at 1:
     *
     * <blockquote><pre>
     * &gamma; = - &Psi;(1)
     *
     */
    static double NEGATIVE_DIGAMMA_1 = 0.5772156649015328606065120900824024;

    private static final double DIGAMMA_COEFFS[]
        = {
        .30459198558715155634315638246624251,
        .72037977439182833573548891941219706,
        -.12454959243861367729528855995001087,
        .27769457331927827002810119567456810e-1,
        -.67762371439822456447373550186163070e-2,
        .17238755142247705209823876688592170e-2,
        -.44817699064252933515310345718960928e-3,
        .11793660000155572716272710617753373e-3,
        -.31253894280980134452125172274246963e-4,
        .83173997012173283398932708991137488e-5,
        -.22191427643780045431149221890172210e-5,
        .59302266729329346291029599913617915e-6,
        -.15863051191470655433559920279603632e-6,
        .42459203983193603241777510648681429e-7,
        -.11369129616951114238848106591780146e-7,
        .304502217295931698401459168423403510e-8,
        -.81568455080753152802915013641723686e-9,
        .21852324749975455125936715817306383e-9,
        -.58546491441689515680751900276454407e-10,
        .15686348450871204869813586459513648e-10,
        -.42029496273143231373796179302482033e-11,
        .11261435719264907097227520956710754e-11,
        -.30174353636860279765375177200637590e-12,
        .80850955256389526647406571868193768e-13,
        -.21663779809421233144009565199997351e-13,
        .58047634271339391495076374966835526e-14,
        -.15553767189204733561108869588173845e-14,
        .41676108598040807753707828039353330e-15,
        -.11167065064221317094734023242188463e-15 };

    /**
     * Returns the relative absolute difference between the specified
     * values, defined to be:
     *
     * <blockquote><pre>
     * relAbsDiff(x,y) = abs(x-y) / (abs(x) + abs(y))</pre></blockquote>
     *
     * @param x First value.
     * @param y Second value.
     * @return The absolute relative difference between the values.
     */
    public static double relativeAbsoluteDifference(double x, double y) {
        return (Double.isInfinite(x) || Double.isInfinite(y))
            ? Double.POSITIVE_INFINITY
            : (java.lang.Math.abs(x - y)
               / (java.lang.Math.abs(x) + java.lang.Math.abs(y)));
    }


    /**
     * This method returns the log of the sum of the natural
     * exponentiated values in the specified array.  Mathematically,
     * the result is
     *
     * <blockquote><pre>
     * logSumOfExponentials(xs) = log <big><big>( &Sigma;</big></big><sub>i</sub> exp(xs[i]) <big><big>)</big></big></pre></blockquote>
     *
     * But the result is not calculated directly.  Instead, the
     * calculation performed is:
     *
     * <blockquote><pre>
     * logSumOfExponentials(xs) = max(xs) + log <big><big>( &Sigma;</big></big><sub>i</sub> exp(xs[i] - max(xs)) <big><big>)</big></big></pre></blockquote>
     *
     * which produces the same result, but is much more arithmetically
     * stable, because the largest value for which <code>exp()</code>
     * is calculated is 0.0.
     *
     * <p>Values of {@code Double.NEGATIVE_INFINITY} are treated as
     * having exponentials of 0 and logs of negative infinity.
     * That is, they are ignored for the purposes of this computation.
     *
     * @param xs Array of values.
     * @return The log of the sum of the exponentiated values in the
     * array.
     */
    public static double logSumOfExponentials(double[] xs) {
        if (xs.length == 1) return xs[0];
        double max = maximum(xs);
        double sum = 0.0;
        for (int i = 0; i < xs.length; ++i)
            if (xs[i] != Double.NEGATIVE_INFINITY)
                sum += java.lang.Math.exp(xs[i] - max);
        return max + java.lang.Math.log(sum);
    }

    /**
     * Returns the maximum value of an element in xs.  If any of the
     * values are {@code Double.NaN}, or if the input array is empty,
     * the result is {@code Double.NaN}.
     *
     * @param xs Array in which to find maximum.
     * @return Maximum value in array.
     */
    public static double max(double... xs) {
        if (xs.length == 0)
            return Double.NaN;
        double max = xs[0];
        for (int i = 1; i < xs.length; ++i)
            max = java.lang.Math.max(max,xs[i]);
        return max;
    }

    /**
     * Returns the maximum value of an element in the specified array.
     *
     * @param xs Array in which to find maximum.
     * @return Maximum value in the array.
     * @throws ArrayIndexOutOfBoundsException If the specified array does
     * not contai at least one element.
     */
    public static int max(int... xs) {
        int max = xs[0];
        for (int i = 1; i < xs.length; ++i)
            if (xs[i] > max)
                max = xs[i];
        return max;
    }



    /**
     * Returns the sum of the specified integer array.  Note that
     * there is no check for overflow.  If the array is of length 0,
     * the sum is defined to be 0.
     *
     * @param xs Array of integers to sum.
     * @return Sum of the array.
     */
    public static int sum(int... xs) {
        int sum = 0;
        for (int i = 0; i < xs.length; ++i)
            sum += xs[i];
        return sum;
    }

}

   
    
    
    
    
    
  








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53.Returns exact (http://mathworld.wolfram.com/BinomialCoefficient.html) Binomial Coefficient
54.Returns a double representation of the (http://mathworld.wolfram.com/BinomialCoefficient.html) Binomial Coefficient
55.Returns the natural log of the (http://mathworld.wolfram.com/BinomialCoefficient.html) Binomial Coefficient
56.Returns the hyperbolic cosine of x.
57.Math Utils
58.Implements the methods which are in the standard J2SE's Math class, but are not in in J2ME's.
59.Utility methods for mathematical problems.
60.Computes the binomial coefficient "n over k"
61.Log Gamma
62.Log Beta
63.Beta
64.Gamma
65.Factorial
66.Computes p(x;n,p) where x~B(n,p)
67.Returns the sum of two doubles expressed in log space
68.sigmod
69.sigmod rev
70.Numbers that are closer than this are considered equal
71.Returns the KL divergence, K(p1 || p2).
72.Returns the sum of two doubles expressed in log space
73.Returns the difference of two doubles expressed in log space
74.Is Prime
75.Statistical functions on arrays of numbers, namely, the mean, variance, standard deviation, covariance, min and max
76.This class calculates the Factorial of a numbers passed into the program through command line arguments.This class calculates the Factorial of a numbers passed into the program through command line arguments.
77.Calculates the Greatest Common Divisor of two numbers passed into the program through command line arguments.
78.Variance: the square of the standard deviation.
79.Population Standard Deviation
80.Returns from a static prime table the least prime that is greater than or equal to a specified value.