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From dblev...@apache.org
Subject svn commit: r1005322 [4/5] - in /openejb/branches/openejb-3.1.x/container/openejb-core: ./ src/main/java/org/apache/openejb/assembler/classic/ src/main/java/org/apache/openejb/math/ src/main/java/org/apache/openejb/math/stat/ src/main/java/org/apache/o...
Date Thu, 07 Oct 2010 03:18:27 GMT
Added: openejb/branches/openejb-3.1.x/container/openejb-core/src/main/java/org/apache/openejb/math/util/MathUtils.java
URL: http://svn.apache.org/viewvc/openejb/branches/openejb-3.1.x/container/openejb-core/src/main/java/org/apache/openejb/math/util/MathUtils.java?rev=1005322&view=auto
==============================================================================
--- openejb/branches/openejb-3.1.x/container/openejb-core/src/main/java/org/apache/openejb/math/util/MathUtils.java (added)
+++ openejb/branches/openejb-3.1.x/container/openejb-core/src/main/java/org/apache/openejb/math/util/MathUtils.java Thu Oct  7 03:18:24 2010
@@ -0,0 +1,1833 @@
+/*
+ * Licensed to the Apache Software Foundation (ASF) under one or more
+ * contributor license agreements.  See the NOTICE file distributed with
+ * this work for additional information regarding copyright ownership.
+ * The ASF licenses this file to You under the Apache License, Version 2.0
+ * (the "License"); you may not use this file except in compliance with
+ * the License.  You may obtain a copy of the License at
+ *
+ *      http://www.apache.org/licenses/LICENSE-2.0
+ *
+ * Unless required by applicable law or agreed to in writing, software
+ * distributed under the License is distributed on an "AS IS" BASIS,
+ * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
+ * See the License for the specific language governing permissions and
+ * limitations under the License.
+ */
+
+package org.apache.openejb.math.util;
+
+import java.math.BigDecimal;
+import java.math.BigInteger;
+import java.util.Arrays;
+
+import org.apache.openejb.math.MathRuntimeException;
+
+/**
+ * Some useful additions to the built-in functions in {@link Math}.
+ * @version $Revision: 927249 $ $Date: 2010-03-24 18:06:51 -0700 (Wed, 24 Mar 2010) $
+ */
+public final class MathUtils {
+
+    /** Smallest positive number such that 1 - EPSILON is not numerically equal to 1. */
+    public static final double EPSILON = 0x1.0p-53;
+
+    /** Safe minimum, such that 1 / SAFE_MIN does not overflow.
+     * <p>In IEEE 754 arithmetic, this is also the smallest normalized
+     * number 2<sup>-1022</sup>.</p>
+     */
+    public static final double SAFE_MIN = 0x1.0p-1022;
+
+    /**
+     * 2 &pi;.
+     * @since 2.1
+     */
+    public static final double TWO_PI = 2 * Math.PI;
+
+    /** -1.0 cast as a byte. */
+    private static final byte  NB = (byte)-1;
+
+    /** -1.0 cast as a short. */
+    private static final short NS = (short)-1;
+
+    /** 1.0 cast as a byte. */
+    private static final byte  PB = (byte)1;
+
+    /** 1.0 cast as a short. */
+    private static final short PS = (short)1;
+
+    /** 0.0 cast as a byte. */
+    private static final byte  ZB = (byte)0;
+
+    /** 0.0 cast as a short. */
+    private static final short ZS = (short)0;
+
+    /** Gap between NaN and regular numbers. */
+    private static final int NAN_GAP = 4 * 1024 * 1024;
+
+    /** Offset to order signed double numbers lexicographically. */
+    private static final long SGN_MASK = 0x8000000000000000L;
+
+    /** All long-representable factorials */
+    private static final long[] FACTORIALS = new long[] {
+                       1l,                  1l,                   2l,
+                       6l,                 24l,                 120l,
+                     720l,               5040l,               40320l,
+                  362880l,            3628800l,            39916800l,
+               479001600l,         6227020800l,         87178291200l,
+           1307674368000l,     20922789888000l,     355687428096000l,
+        6402373705728000l, 121645100408832000l, 2432902008176640000l };
+
+    /**
+     * Private Constructor
+     */
+    private MathUtils() {
+        super();
+    }
+
+    /**
+     * Add two integers, checking for overflow.
+     *
+     * @param x an addend
+     * @param y an addend
+     * @return the sum <code>x+y</code>
+     * @throws ArithmeticException if the result can not be represented as an
+     *         int
+     * @since 1.1
+     */
+    public static int addAndCheck(int x, int y) {
+        long s = (long)x + (long)y;
+        if (s < Integer.MIN_VALUE || s > Integer.MAX_VALUE) {
+            throw new ArithmeticException("overflow: add");
+        }
+        return (int)s;
+    }
+
+    /**
+     * Add two long integers, checking for overflow.
+     *
+     * @param a an addend
+     * @param b an addend
+     * @return the sum <code>a+b</code>
+     * @throws ArithmeticException if the result can not be represented as an
+     *         long
+     * @since 1.2
+     */
+    public static long addAndCheck(long a, long b) {
+        return addAndCheck(a, b, "overflow: add");
+    }
+
+    /**
+     * Add two long integers, checking for overflow.
+     *
+     * @param a an addend
+     * @param b an addend
+     * @param msg the message to use for any thrown exception.
+     * @return the sum <code>a+b</code>
+     * @throws ArithmeticException if the result can not be represented as an
+     *         long
+     * @since 1.2
+     */
+    private static long addAndCheck(long a, long b, String msg) {
+        long ret;
+        if (a > b) {
+            // use symmetry to reduce boundary cases
+            ret = addAndCheck(b, a, msg);
+        } else {
+            // assert a <= b
+
+            if (a < 0) {
+                if (b < 0) {
+                    // check for negative overflow
+                    if (Long.MIN_VALUE - b <= a) {
+                        ret = a + b;
+                    } else {
+                        throw new ArithmeticException(msg);
+                    }
+                } else {
+                    // opposite sign addition is always safe
+                    ret = a + b;
+                }
+            } else {
+                // assert a >= 0
+                // assert b >= 0
+
+                // check for positive overflow
+                if (a <= Long.MAX_VALUE - b) {
+                    ret = a + b;
+                } else {
+                    throw new ArithmeticException(msg);
+                }
+            }
+        }
+        return ret;
+    }
+
+    /**
+     * Returns an exact representation of the <a
+     * href="http://mathworld.wolfram.com/BinomialCoefficient.html"> Binomial
+     * Coefficient</a>, "<code>n choose k</code>", the number of
+     * <code>k</code>-element subsets that can be selected from an
+     * <code>n</code>-element set.
+     * <p>
+     * <Strong>Preconditions</strong>:
+     * <ul>
+     * <li> <code>0 <= k <= n </code> (otherwise
+     * <code>IllegalArgumentException</code> is thrown)</li>
+     * <li> The result is small enough to fit into a <code>long</code>. The
+     * largest value of <code>n</code> for which all coefficients are
+     * <code> < Long.MAX_VALUE</code> is 66. If the computed value exceeds
+     * <code>Long.MAX_VALUE</code> an <code>ArithMeticException</code> is
+     * thrown.</li>
+     * </ul></p>
+     *
+     * @param n the size of the set
+     * @param k the size of the subsets to be counted
+     * @return <code>n choose k</code>
+     * @throws IllegalArgumentException if preconditions are not met.
+     * @throws ArithmeticException if the result is too large to be represented
+     *         by a long integer.
+     */
+    public static long binomialCoefficient(final int n, final int k) {
+        checkBinomial(n, k);
+        if ((n == k) || (k == 0)) {
+            return 1;
+        }
+        if ((k == 1) || (k == n - 1)) {
+            return n;
+        }
+        // Use symmetry for large k
+        if (k > n / 2)
+            return binomialCoefficient(n, n - k);
+
+        // We use the formula
+        // (n choose k) = n! / (n-k)! / k!
+        // (n choose k) == ((n-k+1)*...*n) / (1*...*k)
+        // which could be written
+        // (n choose k) == (n-1 choose k-1) * n / k
+        long result = 1;
+        if (n <= 61) {
+            // For n <= 61, the naive implementation cannot overflow.
+            int i = n - k + 1;
+            for (int j = 1; j <= k; j++) {
+                result = result * i / j;
+                i++;
+            }
+        } else if (n <= 66) {
+            // For n > 61 but n <= 66, the result cannot overflow,
+            // but we must take care not to overflow intermediate values.
+            int i = n - k + 1;
+            for (int j = 1; j <= k; j++) {
+                // We know that (result * i) is divisible by j,
+                // but (result * i) may overflow, so we split j:
+                // Filter out the gcd, d, so j/d and i/d are integer.
+                // result is divisible by (j/d) because (j/d)
+                // is relative prime to (i/d) and is a divisor of
+                // result * (i/d).
+                final long d = gcd(i, j);
+                result = (result / (j / d)) * (i / d);
+                i++;
+            }
+        } else {
+            // For n > 66, a result overflow might occur, so we check
+            // the multiplication, taking care to not overflow
+            // unnecessary.
+            int i = n - k + 1;
+            for (int j = 1; j <= k; j++) {
+                final long d = gcd(i, j);
+                result = mulAndCheck(result / (j / d), i / d);
+                i++;
+            }
+        }
+        return result;
+    }
+
+    /**
+     * Returns a <code>double</code> representation of the <a
+     * href="http://mathworld.wolfram.com/BinomialCoefficient.html"> Binomial
+     * Coefficient</a>, "<code>n choose k</code>", the number of
+     * <code>k</code>-element subsets that can be selected from an
+     * <code>n</code>-element set.
+     * <p>
+     * <Strong>Preconditions</strong>:
+     * <ul>
+     * <li> <code>0 <= k <= n </code> (otherwise
+     * <code>IllegalArgumentException</code> is thrown)</li>
+     * <li> The result is small enough to fit into a <code>double</code>. The
+     * largest value of <code>n</code> for which all coefficients are <
+     * Double.MAX_VALUE is 1029. If the computed value exceeds Double.MAX_VALUE,
+     * Double.POSITIVE_INFINITY is returned</li>
+     * </ul></p>
+     *
+     * @param n the size of the set
+     * @param k the size of the subsets to be counted
+     * @return <code>n choose k</code>
+     * @throws IllegalArgumentException if preconditions are not met.
+     */
+    public static double binomialCoefficientDouble(final int n, final int k) {
+        checkBinomial(n, k);
+        if ((n == k) || (k == 0)) {
+            return 1d;
+        }
+        if ((k == 1) || (k == n - 1)) {
+            return n;
+        }
+        if (k > n/2) {
+            return binomialCoefficientDouble(n, n - k);
+        }
+        if (n < 67) {
+            return binomialCoefficient(n,k);
+        }
+
+        double result = 1d;
+        for (int i = 1; i <= k; i++) {
+             result *= (double)(n - k + i) / (double)i;
+        }
+
+        return Math.floor(result + 0.5);
+    }
+
+    /**
+     * Returns the natural <code>log</code> of the <a
+     * href="http://mathworld.wolfram.com/BinomialCoefficient.html"> Binomial
+     * Coefficient</a>, "<code>n choose k</code>", the number of
+     * <code>k</code>-element subsets that can be selected from an
+     * <code>n</code>-element set.
+     * <p>
+     * <Strong>Preconditions</strong>:
+     * <ul>
+     * <li> <code>0 <= k <= n </code> (otherwise
+     * <code>IllegalArgumentException</code> is thrown)</li>
+     * </ul></p>
+     *
+     * @param n the size of the set
+     * @param k the size of the subsets to be counted
+     * @return <code>n choose k</code>
+     * @throws IllegalArgumentException if preconditions are not met.
+     */
+    public static double binomialCoefficientLog(final int n, final int k) {
+        checkBinomial(n, k);
+        if ((n == k) || (k == 0)) {
+            return 0;
+        }
+        if ((k == 1) || (k == n - 1)) {
+            return Math.log(n);
+        }
+
+        /*
+         * For values small enough to do exact integer computation,
+         * return the log of the exact value
+         */
+        if (n < 67) {
+            return Math.log(binomialCoefficient(n,k));
+        }
+
+        /*
+         * Return the log of binomialCoefficientDouble for values that will not
+         * overflow binomialCoefficientDouble
+         */
+        if (n < 1030) {
+            return Math.log(binomialCoefficientDouble(n, k));
+        }
+
+        if (k > n / 2) {
+            return binomialCoefficientLog(n, n - k);
+        }
+
+        /*
+         * Sum logs for values that could overflow
+         */
+        double logSum = 0;
+
+        // n!/(n-k)!
+        for (int i = n - k + 1; i <= n; i++) {
+            logSum += Math.log(i);
+        }
+
+        // divide by k!
+        for (int i = 2; i <= k; i++) {
+            logSum -= Math.log(i);
+        }
+
+        return logSum;
+    }
+
+    /**
+     * Check binomial preconditions.
+     * @param n the size of the set
+     * @param k the size of the subsets to be counted
+     * @exception IllegalArgumentException if preconditions are not met.
+     */
+    private static void checkBinomial(final int n, final int k)
+        throws IllegalArgumentException {
+        if (n < k) {
+            throw MathRuntimeException.createIllegalArgumentException(
+                "must have n >= k for binomial coefficient (n,k), got n = {0}, k = {1}",
+                n, k);
+        }
+        if (n < 0) {
+            throw MathRuntimeException.createIllegalArgumentException(
+                  "must have n >= 0 for binomial coefficient (n,k), got n = {0}",
+                  n);
+        }
+    }
+
+    /**
+     * Compares two numbers given some amount of allowed error.
+     *
+     * @param x the first number
+     * @param y the second number
+     * @param eps the amount of error to allow when checking for equality
+     * @return <ul><li>0 if  {@link #equals(double, double, double) equals(x, y, eps)}</li>
+     *       <li>&lt; 0 if !{@link #equals(double, double, double) equals(x, y, eps)} &amp;&amp; x &lt; y</li>
+     *       <li>> 0 if !{@link #equals(double, double, double) equals(x, y, eps)} &amp;&amp; x > y</li></ul>
+     */
+    public static int compareTo(double x, double y, double eps) {
+        if (equals(x, y, eps)) {
+            return 0;
+        } else if (x < y) {
+          return -1;
+        }
+        return 1;
+    }
+
+    /**
+     * Returns the <a href="http://mathworld.wolfram.com/HyperbolicCosine.html">
+     * hyperbolic cosine</a> of x.
+     *
+     * @param x double value for which to find the hyperbolic cosine
+     * @return hyperbolic cosine of x
+     */
+    public static double cosh(double x) {
+        return (Math.exp(x) + Math.exp(-x)) / 2.0;
+    }
+
+    /**
+     * Returns true iff both arguments are NaN or neither is NaN and they are
+     * equal
+     *
+     * @param x first value
+     * @param y second value
+     * @return true if the values are equal or both are NaN
+     */
+    public static boolean equals(double x, double y) {
+        return (Double.isNaN(x) && Double.isNaN(y)) || x == y;
+    }
+
+    /**
+     * Returns true iff both arguments are equal or within the range of allowed
+     * error (inclusive).
+     * <p>
+     * Two NaNs are considered equals, as are two infinities with same sign.
+     * </p>
+     *
+     * @param x first value
+     * @param y second value
+     * @param eps the amount of absolute error to allow
+     * @return true if the values are equal or within range of each other
+     */
+    public static boolean equals(double x, double y, double eps) {
+      return equals(x, y) || (Math.abs(y - x) <= eps);
+    }
+
+    /**
+     * Returns true iff both arguments are equal or within the range of allowed
+     * error (inclusive).
+     * Adapted from <a
+     * href="http://www.cygnus-software.com/papers/comparingfloats/comparingfloats.htm">
+     * Bruce Dawson</a>
+     *
+     * @param x first value
+     * @param y second value
+     * @param maxUlps {@code (maxUlps - 1)} is the number of floating point
+     * values between {@code x} and {@code y}.
+     * @return {@code true} if there are less than {@code maxUlps} floating
+     * point values between {@code x} and {@code y}
+     */
+    public static boolean equals(double x, double y, int maxUlps) {
+        // Check that "maxUlps" is non-negative and small enough so that the
+        // default NAN won't compare as equal to anything.
+        assert maxUlps > 0 && maxUlps < NAN_GAP;
+
+        long xInt = Double.doubleToLongBits(x);
+        long yInt = Double.doubleToLongBits(y);
+
+        // Make lexicographically ordered as a two's-complement integer.
+        if (xInt < 0) {
+            xInt = SGN_MASK - xInt;
+        }
+        if (yInt < 0) {
+            yInt = SGN_MASK - yInt;
+        }
+
+        return Math.abs(xInt - yInt) <= maxUlps;
+    }
+
+    /**
+     * Returns true iff both arguments are null or have same dimensions
+     * and all their elements are {@link #equals(double,double) equals}
+     *
+     * @param x first array
+     * @param y second array
+     * @return true if the values are both null or have same dimension
+     * and equal elements
+     * @since 1.2
+     */
+    public static boolean equals(double[] x, double[] y) {
+        if ((x == null) || (y == null)) {
+            return !((x == null) ^ (y == null));
+        }
+        if (x.length != y.length) {
+            return false;
+        }
+        for (int i = 0; i < x.length; ++i) {
+            if (!equals(x[i], y[i])) {
+                return false;
+            }
+        }
+        return true;
+    }
+
+    /**
+     * Returns n!. Shorthand for <code>n</code> <a
+     * href="http://mathworld.wolfram.com/Factorial.html"> Factorial</a>, the
+     * product of the numbers <code>1,...,n</code>.
+     * <p>
+     * <Strong>Preconditions</strong>:
+     * <ul>
+     * <li> <code>n >= 0</code> (otherwise
+     * <code>IllegalArgumentException</code> is thrown)</li>
+     * <li> The result is small enough to fit into a <code>long</code>. The
+     * largest value of <code>n</code> for which <code>n!</code> <
+     * Long.MAX_VALUE</code> is 20. If the computed value exceeds <code>Long.MAX_VALUE</code>
+     * an <code>ArithMeticException </code> is thrown.</li>
+     * </ul>
+     * </p>
+     *
+     * @param n argument
+     * @return <code>n!</code>
+     * @throws ArithmeticException if the result is too large to be represented
+     *         by a long integer.
+     * @throws IllegalArgumentException if n < 0
+     */
+    public static long factorial(final int n) {
+        if (n < 0) {
+            throw MathRuntimeException.createIllegalArgumentException(
+                  "must have n >= 0 for n!, got n = {0}",
+                  n);
+        }
+        if (n > 20) {
+            throw new ArithmeticException(
+                    "factorial value is too large to fit in a long");
+        }
+        return FACTORIALS[n];
+    }
+
+    /**
+     * Returns n!. Shorthand for <code>n</code> <a
+     * href="http://mathworld.wolfram.com/Factorial.html"> Factorial</a>, the
+     * product of the numbers <code>1,...,n</code> as a <code>double</code>.
+     * <p>
+     * <Strong>Preconditions</strong>:
+     * <ul>
+     * <li> <code>n >= 0</code> (otherwise
+     * <code>IllegalArgumentException</code> is thrown)</li>
+     * <li> The result is small enough to fit into a <code>double</code>. The
+     * largest value of <code>n</code> for which <code>n!</code> <
+     * Double.MAX_VALUE</code> is 170. If the computed value exceeds
+     * Double.MAX_VALUE, Double.POSITIVE_INFINITY is returned</li>
+     * </ul>
+     * </p>
+     *
+     * @param n argument
+     * @return <code>n!</code>
+     * @throws IllegalArgumentException if n < 0
+     */
+    public static double factorialDouble(final int n) {
+        if (n < 0) {
+            throw MathRuntimeException.createIllegalArgumentException(
+                  "must have n >= 0 for n!, got n = {0}",
+                  n);
+        }
+        if (n < 21) {
+            return factorial(n);
+        }
+        return Math.floor(Math.exp(factorialLog(n)) + 0.5);
+    }
+
+    /**
+     * Returns the natural logarithm of n!.
+     * <p>
+     * <Strong>Preconditions</strong>:
+     * <ul>
+     * <li> <code>n >= 0</code> (otherwise
+     * <code>IllegalArgumentException</code> is thrown)</li>
+     * </ul></p>
+     *
+     * @param n argument
+     * @return <code>n!</code>
+     * @throws IllegalArgumentException if preconditions are not met.
+     */
+    public static double factorialLog(final int n) {
+        if (n < 0) {
+            throw MathRuntimeException.createIllegalArgumentException(
+                  "must have n >= 0 for n!, got n = {0}",
+                  n);
+        }
+        if (n < 21) {
+            return Math.log(factorial(n));
+        }
+        double logSum = 0;
+        for (int i = 2; i <= n; i++) {
+            logSum += Math.log(i);
+        }
+        return logSum;
+    }
+
+    /**
+     * <p>
+     * Gets the greatest common divisor of the absolute value of two numbers,
+     * using the "binary gcd" method which avoids division and modulo
+     * operations. See Knuth 4.5.2 algorithm B. This algorithm is due to Josef
+     * Stein (1961).
+     * </p>
+     * Special cases:
+     * <ul>
+     * <li>The invocations
+     * <code>gcd(Integer.MIN_VALUE, Integer.MIN_VALUE)</code>,
+     * <code>gcd(Integer.MIN_VALUE, 0)</code> and
+     * <code>gcd(0, Integer.MIN_VALUE)</code> throw an
+     * <code>ArithmeticException</code>, because the result would be 2^31, which
+     * is too large for an int value.</li>
+     * <li>The result of <code>gcd(x, x)</code>, <code>gcd(0, x)</code> and
+     * <code>gcd(x, 0)</code> is the absolute value of <code>x</code>, except
+     * for the special cases above.
+     * <li>The invocation <code>gcd(0, 0)</code> is the only one which returns
+     * <code>0</code>.</li>
+     * </ul>
+     *
+     * @param p any number
+     * @param q any number
+     * @return the greatest common divisor, never negative
+     * @throws ArithmeticException if the result cannot be represented as a
+     * nonnegative int value
+     * @since 1.1
+     */
+    public static int gcd(final int p, final int q) {
+        int u = p;
+        int v = q;
+        if ((u == 0) || (v == 0)) {
+            if ((u == Integer.MIN_VALUE) || (v == Integer.MIN_VALUE)) {
+                throw MathRuntimeException.createArithmeticException(
+                        "overflow: gcd({0}, {1}) is 2^31",
+                        p, q);
+            }
+            return Math.abs(u) + Math.abs(v);
+        }
+        // keep u and v negative, as negative integers range down to
+        // -2^31, while positive numbers can only be as large as 2^31-1
+        // (i.e. we can't necessarily negate a negative number without
+        // overflow)
+        /* assert u!=0 && v!=0; */
+        if (u > 0) {
+            u = -u;
+        } // make u negative
+        if (v > 0) {
+            v = -v;
+        } // make v negative
+        // B1. [Find power of 2]
+        int k = 0;
+        while ((u & 1) == 0 && (v & 1) == 0 && k < 31) { // while u and v are
+                                                            // both even...
+            u /= 2;
+            v /= 2;
+            k++; // cast out twos.
+        }
+        if (k == 31) {
+            throw MathRuntimeException.createArithmeticException(
+                    "overflow: gcd({0}, {1}) is 2^31",
+                    p, q);
+        }
+        // B2. Initialize: u and v have been divided by 2^k and at least
+        // one is odd.
+        int t = ((u & 1) == 1) ? v : -(u / 2)/* B3 */;
+        // t negative: u was odd, v may be even (t replaces v)
+        // t positive: u was even, v is odd (t replaces u)
+        do {
+            /* assert u<0 && v<0; */
+            // B4/B3: cast out twos from t.
+            while ((t & 1) == 0) { // while t is even..
+                t /= 2; // cast out twos
+            }
+            // B5 [reset max(u,v)]
+            if (t > 0) {
+                u = -t;
+            } else {
+                v = t;
+            }
+            // B6/B3. at this point both u and v should be odd.
+            t = (v - u) / 2;
+            // |u| larger: t positive (replace u)
+            // |v| larger: t negative (replace v)
+        } while (t != 0);
+        return -u * (1 << k); // gcd is u*2^k
+    }
+
+    /**
+     * <p>
+     * Gets the greatest common divisor of the absolute value of two numbers,
+     * using the "binary gcd" method which avoids division and modulo
+     * operations. See Knuth 4.5.2 algorithm B. This algorithm is due to Josef
+     * Stein (1961).
+     * </p>
+     * Special cases:
+     * <ul>
+     * <li>The invocations
+     * <code>gcd(Long.MIN_VALUE, Long.MIN_VALUE)</code>,
+     * <code>gcd(Long.MIN_VALUE, 0L)</code> and
+     * <code>gcd(0L, Long.MIN_VALUE)</code> throw an
+     * <code>ArithmeticException</code>, because the result would be 2^63, which
+     * is too large for a long value.</li>
+     * <li>The result of <code>gcd(x, x)</code>, <code>gcd(0L, x)</code> and
+     * <code>gcd(x, 0L)</code> is the absolute value of <code>x</code>, except
+     * for the special cases above.
+     * <li>The invocation <code>gcd(0L, 0L)</code> is the only one which returns
+     * <code>0L</code>.</li>
+     * </ul>
+     *
+     * @param p any number
+     * @param q any number
+     * @return the greatest common divisor, never negative
+     * @throws ArithmeticException if the result cannot be represented as a nonnegative long
+     * value
+     * @since 2.1
+     */
+    public static long gcd(final long p, final long q) {
+        long u = p;
+        long v = q;
+        if ((u == 0) || (v == 0)) {
+            if ((u == Long.MIN_VALUE) || (v == Long.MIN_VALUE)){
+                throw MathRuntimeException.createArithmeticException(
+                        "overflow: gcd({0}, {1}) is 2^63",
+                        p, q);
+            }
+            return Math.abs(u) + Math.abs(v);
+        }
+        // keep u and v negative, as negative integers range down to
+        // -2^63, while positive numbers can only be as large as 2^63-1
+        // (i.e. we can't necessarily negate a negative number without
+        // overflow)
+        /* assert u!=0 && v!=0; */
+        if (u > 0) {
+            u = -u;
+        } // make u negative
+        if (v > 0) {
+            v = -v;
+        } // make v negative
+        // B1. [Find power of 2]
+        int k = 0;
+        while ((u & 1) == 0 && (v & 1) == 0 && k < 63) { // while u and v are
+                                                            // both even...
+            u /= 2;
+            v /= 2;
+            k++; // cast out twos.
+        }
+        if (k == 63) {
+            throw MathRuntimeException.createArithmeticException(
+                    "overflow: gcd({0}, {1}) is 2^63",
+                    p, q);
+        }
+        // B2. Initialize: u and v have been divided by 2^k and at least
+        // one is odd.
+        long t = ((u & 1) == 1) ? v : -(u / 2)/* B3 */;
+        // t negative: u was odd, v may be even (t replaces v)
+        // t positive: u was even, v is odd (t replaces u)
+        do {
+            /* assert u<0 && v<0; */
+            // B4/B3: cast out twos from t.
+            while ((t & 1) == 0) { // while t is even..
+                t /= 2; // cast out twos
+            }
+            // B5 [reset max(u,v)]
+            if (t > 0) {
+                u = -t;
+            } else {
+                v = t;
+            }
+            // B6/B3. at this point both u and v should be odd.
+            t = (v - u) / 2;
+            // |u| larger: t positive (replace u)
+            // |v| larger: t negative (replace v)
+        } while (t != 0);
+        return -u * (1L << k); // gcd is u*2^k
+    }
+
+    /**
+     * Returns an integer hash code representing the given double value.
+     *
+     * @param value the value to be hashed
+     * @return the hash code
+     */
+    public static int hash(double value) {
+        return new Double(value).hashCode();
+    }
+
+    /**
+     * Returns an integer hash code representing the given double array.
+     *
+     * @param value the value to be hashed (may be null)
+     * @return the hash code
+     * @since 1.2
+     */
+    public static int hash(double[] value) {
+        return Arrays.hashCode(value);
+    }
+
+    /**
+     * For a byte value x, this method returns (byte)(+1) if x >= 0 and
+     * (byte)(-1) if x < 0.
+     *
+     * @param x the value, a byte
+     * @return (byte)(+1) or (byte)(-1), depending on the sign of x
+     */
+    public static byte indicator(final byte x) {
+        return (x >= ZB) ? PB : NB;
+    }
+
+    /**
+     * For a double precision value x, this method returns +1.0 if x >= 0 and
+     * -1.0 if x < 0. Returns <code>NaN</code> if <code>x</code> is
+     * <code>NaN</code>.
+     *
+     * @param x the value, a double
+     * @return +1.0 or -1.0, depending on the sign of x
+     */
+    public static double indicator(final double x) {
+        if (Double.isNaN(x)) {
+            return Double.NaN;
+        }
+        return (x >= 0.0) ? 1.0 : -1.0;
+    }
+
+    /**
+     * For a float value x, this method returns +1.0F if x >= 0 and -1.0F if x <
+     * 0. Returns <code>NaN</code> if <code>x</code> is <code>NaN</code>.
+     *
+     * @param x the value, a float
+     * @return +1.0F or -1.0F, depending on the sign of x
+     */
+    public static float indicator(final float x) {
+        if (Float.isNaN(x)) {
+            return Float.NaN;
+        }
+        return (x >= 0.0F) ? 1.0F : -1.0F;
+    }
+
+    /**
+     * For an int value x, this method returns +1 if x >= 0 and -1 if x < 0.
+     *
+     * @param x the value, an int
+     * @return +1 or -1, depending on the sign of x
+     */
+    public static int indicator(final int x) {
+        return (x >= 0) ? 1 : -1;
+    }
+
+    /**
+     * For a long value x, this method returns +1L if x >= 0 and -1L if x < 0.
+     *
+     * @param x the value, a long
+     * @return +1L or -1L, depending on the sign of x
+     */
+    public static long indicator(final long x) {
+        return (x >= 0L) ? 1L : -1L;
+    }
+
+    /**
+     * For a short value x, this method returns (short)(+1) if x >= 0 and
+     * (short)(-1) if x < 0.
+     *
+     * @param x the value, a short
+     * @return (short)(+1) or (short)(-1), depending on the sign of x
+     */
+    public static short indicator(final short x) {
+        return (x >= ZS) ? PS : NS;
+    }
+
+    /**
+     * <p>
+     * Returns the least common multiple of the absolute value of two numbers,
+     * using the formula <code>lcm(a,b) = (a / gcd(a,b)) * b</code>.
+     * </p>
+     * Special cases:
+     * <ul>
+     * <li>The invocations <code>lcm(Integer.MIN_VALUE, n)</code> and
+     * <code>lcm(n, Integer.MIN_VALUE)</code>, where <code>abs(n)</code> is a
+     * power of 2, throw an <code>ArithmeticException</code>, because the result
+     * would be 2^31, which is too large for an int value.</li>
+     * <li>The result of <code>lcm(0, x)</code> and <code>lcm(x, 0)</code> is
+     * <code>0</code> for any <code>x</code>.
+     * </ul>
+     *
+     * @param a any number
+     * @param b any number
+     * @return the least common multiple, never negative
+     * @throws ArithmeticException
+     *             if the result cannot be represented as a nonnegative int
+     *             value
+     * @since 1.1
+     */
+    public static int lcm(int a, int b) {
+        if (a==0 || b==0){
+            return 0;
+        }
+        int lcm = Math.abs(mulAndCheck(a / gcd(a, b), b));
+        if (lcm == Integer.MIN_VALUE) {
+            throw MathRuntimeException.createArithmeticException(
+                "overflow: lcm({0}, {1}) is 2^31",
+                a, b);
+        }
+        return lcm;
+    }
+
+    /**
+     * <p>
+     * Returns the least common multiple of the absolute value of two numbers,
+     * using the formula <code>lcm(a,b) = (a / gcd(a,b)) * b</code>.
+     * </p>
+     * Special cases:
+     * <ul>
+     * <li>The invocations <code>lcm(Long.MIN_VALUE, n)</code> and
+     * <code>lcm(n, Long.MIN_VALUE)</code>, where <code>abs(n)</code> is a
+     * power of 2, throw an <code>ArithmeticException</code>, because the result
+     * would be 2^63, which is too large for an int value.</li>
+     * <li>The result of <code>lcm(0L, x)</code> and <code>lcm(x, 0L)</code> is
+     * <code>0L</code> for any <code>x</code>.
+     * </ul>
+     *
+     * @param a any number
+     * @param b any number
+     * @return the least common multiple, never negative
+     * @throws ArithmeticException if the result cannot be represented as a nonnegative long
+     * value
+     * @since 2.1
+     */
+    public static long lcm(long a, long b) {
+        if (a==0 || b==0){
+            return 0;
+        }
+        long lcm = Math.abs(mulAndCheck(a / gcd(a, b), b));
+        if (lcm == Long.MIN_VALUE){
+            throw MathRuntimeException.createArithmeticException(
+                "overflow: lcm({0}, {1}) is 2^63",
+                a, b);
+        }
+        return lcm;
+    }
+
+    /**
+     * <p>Returns the
+     * <a href="http://mathworld.wolfram.com/Logarithm.html">logarithm</a>
+     * for base <code>b</code> of <code>x</code>.
+     * </p>
+     * <p>Returns <code>NaN<code> if either argument is negative.  If
+     * <code>base</code> is 0 and <code>x</code> is positive, 0 is returned.
+     * If <code>base</code> is positive and <code>x</code> is 0,
+     * <code>Double.NEGATIVE_INFINITY</code> is returned.  If both arguments
+     * are 0, the result is <code>NaN</code>.</p>
+     *
+     * @param base the base of the logarithm, must be greater than 0
+     * @param x argument, must be greater than 0
+     * @return the value of the logarithm - the number y such that base^y = x.
+     * @since 1.2
+     */
+    public static double log(double base, double x) {
+        return Math.log(x)/Math.log(base);
+    }
+
+    /**
+     * Multiply two integers, checking for overflow.
+     *
+     * @param x a factor
+     * @param y a factor
+     * @return the product <code>x*y</code>
+     * @throws ArithmeticException if the result can not be represented as an
+     *         int
+     * @since 1.1
+     */
+    public static int mulAndCheck(int x, int y) {
+        long m = ((long)x) * ((long)y);
+        if (m < Integer.MIN_VALUE || m > Integer.MAX_VALUE) {
+            throw new ArithmeticException("overflow: mul");
+        }
+        return (int)m;
+    }
+
+    /**
+     * Multiply two long integers, checking for overflow.
+     *
+     * @param a first value
+     * @param b second value
+     * @return the product <code>a * b</code>
+     * @throws ArithmeticException if the result can not be represented as an
+     *         long
+     * @since 1.2
+     */
+    public static long mulAndCheck(long a, long b) {
+        long ret;
+        String msg = "overflow: multiply";
+        if (a > b) {
+            // use symmetry to reduce boundary cases
+            ret = mulAndCheck(b, a);
+        } else {
+            if (a < 0) {
+                if (b < 0) {
+                    // check for positive overflow with negative a, negative b
+                    if (a >= Long.MAX_VALUE / b) {
+                        ret = a * b;
+                    } else {
+                        throw new ArithmeticException(msg);
+                    }
+                } else if (b > 0) {
+                    // check for negative overflow with negative a, positive b
+                    if (Long.MIN_VALUE / b <= a) {
+                        ret = a * b;
+                    } else {
+                        throw new ArithmeticException(msg);
+
+                    }
+                } else {
+                    // assert b == 0
+                    ret = 0;
+                }
+            } else if (a > 0) {
+                // assert a > 0
+                // assert b > 0
+
+                // check for positive overflow with positive a, positive b
+                if (a <= Long.MAX_VALUE / b) {
+                    ret = a * b;
+                } else {
+                    throw new ArithmeticException(msg);
+                }
+            } else {
+                // assert a == 0
+                ret = 0;
+            }
+        }
+        return ret;
+    }
+
+    /**
+     * Get the next machine representable number after a number, moving
+     * in the direction of another number.
+     * <p>
+     * If <code>direction</code> is greater than or equal to<code>d</code>,
+     * the smallest machine representable number strictly greater than
+     * <code>d</code> is returned; otherwise the largest representable number
+     * strictly less than <code>d</code> is returned.</p>
+     * <p>
+     * If <code>d</code> is NaN or Infinite, it is returned unchanged.</p>
+     *
+     * @param d base number
+     * @param direction (the only important thing is whether
+     * direction is greater or smaller than d)
+     * @return the next machine representable number in the specified direction
+     * @since 1.2
+     */
+    public static double nextAfter(double d, double direction) {
+
+        // handling of some important special cases
+        if (Double.isNaN(d) || Double.isInfinite(d)) {
+                return d;
+        } else if (d == 0) {
+                return (direction < 0) ? -Double.MIN_VALUE : Double.MIN_VALUE;
+        }
+        // special cases MAX_VALUE to infinity and  MIN_VALUE to 0
+        // are handled just as normal numbers
+
+        // split the double in raw components
+        long bits     = Double.doubleToLongBits(d);
+        long sign     = bits & 0x8000000000000000L;
+        long exponent = bits & 0x7ff0000000000000L;
+        long mantissa = bits & 0x000fffffffffffffL;
+
+        if (d * (direction - d) >= 0) {
+                // we should increase the mantissa
+                if (mantissa == 0x000fffffffffffffL) {
+                        return Double.longBitsToDouble(sign |
+                                        (exponent + 0x0010000000000000L));
+                } else {
+                        return Double.longBitsToDouble(sign |
+                                        exponent | (mantissa + 1));
+                }
+        } else {
+                // we should decrease the mantissa
+                if (mantissa == 0L) {
+                        return Double.longBitsToDouble(sign |
+                                        (exponent - 0x0010000000000000L) |
+                                        0x000fffffffffffffL);
+                } else {
+                        return Double.longBitsToDouble(sign |
+                                        exponent | (mantissa - 1));
+                }
+        }
+
+    }
+
+    /**
+     * Scale a number by 2<sup>scaleFactor</sup>.
+     * <p>If <code>d</code> is 0 or NaN or Infinite, it is returned unchanged.</p>
+     *
+     * @param d base number
+     * @param scaleFactor power of two by which d sould be multiplied
+     * @return d &times; 2<sup>scaleFactor</sup>
+     * @since 2.0
+     */
+    public static double scalb(final double d, final int scaleFactor) {
+
+        // handling of some important special cases
+        if ((d == 0) || Double.isNaN(d) || Double.isInfinite(d)) {
+            return d;
+        }
+
+        // split the double in raw components
+        final long bits     = Double.doubleToLongBits(d);
+        final long exponent = bits & 0x7ff0000000000000L;
+        final long rest     = bits & 0x800fffffffffffffL;
+
+        // shift the exponent
+        final long newBits = rest | (exponent + (((long) scaleFactor) << 52));
+        return Double.longBitsToDouble(newBits);
+
+    }
+
+    /**
+     * Normalize an angle in a 2&pi wide interval around a center value.
+     * <p>This method has three main uses:</p>
+     * <ul>
+     *   <li>normalize an angle between 0 and 2&pi;:<br/>
+     *       <code>a = MathUtils.normalizeAngle(a, Math.PI);</code></li>
+     *   <li>normalize an angle between -&pi; and +&pi;<br/>
+     *       <code>a = MathUtils.normalizeAngle(a, 0.0);</code></li>
+     *   <li>compute the angle between two defining angular positions:<br>
+     *       <code>angle = MathUtils.normalizeAngle(end, start) - start;</code></li>
+     * </ul>
+     * <p>Note that due to numerical accuracy and since &pi; cannot be represented
+     * exactly, the result interval is <em>closed</em>, it cannot be half-closed
+     * as would be more satisfactory in a purely mathematical view.</p>
+     * @param a angle to normalize
+     * @param center center of the desired 2&pi; interval for the result
+     * @return a-2k&pi; with integer k and center-&pi; &lt;= a-2k&pi; &lt;= center+&pi;
+     * @since 1.2
+     */
+     public static double normalizeAngle(double a, double center) {
+         return a - TWO_PI * Math.floor((a + Math.PI - center) / TWO_PI);
+     }
+
+     /**
+      * <p>Normalizes an array to make it sum to a specified value.
+      * Returns the result of the transformation <pre>
+      *    x |-> x * normalizedSum / sum
+      * </pre>
+      * applied to each non-NaN element x of the input array, where sum is the
+      * sum of the non-NaN entries in the input array.</p>
+      *
+      * <p>Throws IllegalArgumentException if <code>normalizedSum</code> is infinite
+      * or NaN and ArithmeticException if the input array contains any infinite elements
+      * or sums to 0</p>
+      *
+      * <p>Ignores (i.e., copies unchanged to the output array) NaNs in the input array.</p>
+      *
+      * @param values input array to be normalized
+      * @param normalizedSum target sum for the normalized array
+      * @return normalized array
+      * @throws ArithmeticException if the input array contains infinite elements or sums to zero
+      * @throws IllegalArgumentException if the target sum is infinite or NaN
+      * @since 2.1
+      */
+     public static double[] normalizeArray(double[] values, double normalizedSum)
+       throws ArithmeticException, IllegalArgumentException {
+         if (Double.isInfinite(normalizedSum)) {
+             throw MathRuntimeException.createIllegalArgumentException(
+                     "Cannot normalize to an infinite value");
+         }
+         if (Double.isNaN(normalizedSum)) {
+             throw MathRuntimeException.createIllegalArgumentException(
+                     "Cannot normalize to NaN");
+         }
+         double sum = 0d;
+         final int len = values.length;
+         double[] out = new double[len];
+         for (int i = 0; i < len; i++) {
+             if (Double.isInfinite(values[i])) {
+                 throw MathRuntimeException.createArithmeticException(
+                         "Array contains an infinite element, {0} at index {1}", values[i], i);
+             }
+             if (!Double.isNaN(values[i])) {
+                 sum += values[i];
+             }
+         }
+         if (sum == 0) {
+             throw MathRuntimeException.createArithmeticException(
+                     "Array sums to zero");
+         }
+         for (int i = 0; i < len; i++) {
+             if (Double.isNaN(values[i])) {
+                 out[i] = Double.NaN;
+             } else {
+                 out[i] = values[i] * normalizedSum / sum;
+             }
+         }
+         return out;
+     }
+
+    /**
+     * Round the given value to the specified number of decimal places. The
+     * value is rounded using the {@link BigDecimal#ROUND_HALF_UP} method.
+     *
+     * @param x the value to round.
+     * @param scale the number of digits to the right of the decimal point.
+     * @return the rounded value.
+     * @since 1.1
+     */
+    public static double round(double x, int scale) {
+        return round(x, scale, BigDecimal.ROUND_HALF_UP);
+    }
+
+    /**
+     * Round the given value to the specified number of decimal places. The
+     * value is rounded using the given method which is any method defined in
+     * {@link BigDecimal}.
+     *
+     * @param x the value to round.
+     * @param scale the number of digits to the right of the decimal point.
+     * @param roundingMethod the rounding method as defined in
+     *        {@link BigDecimal}.
+     * @return the rounded value.
+     * @since 1.1
+     */
+    public static double round(double x, int scale, int roundingMethod) {
+        try {
+            return (new BigDecimal
+                   (Double.toString(x))
+                   .setScale(scale, roundingMethod))
+                   .doubleValue();
+        } catch (NumberFormatException ex) {
+            if (Double.isInfinite(x)) {
+                return x;
+            } else {
+                return Double.NaN;
+            }
+        }
+    }
+
+    /**
+     * Round the given value to the specified number of decimal places. The
+     * value is rounding using the {@link BigDecimal#ROUND_HALF_UP} method.
+     *
+     * @param x the value to round.
+     * @param scale the number of digits to the right of the decimal point.
+     * @return the rounded value.
+     * @since 1.1
+     */
+    public static float round(float x, int scale) {
+        return round(x, scale, BigDecimal.ROUND_HALF_UP);
+    }
+
+    /**
+     * Round the given value to the specified number of decimal places. The
+     * value is rounded using the given method which is any method defined in
+     * {@link BigDecimal}.
+     *
+     * @param x the value to round.
+     * @param scale the number of digits to the right of the decimal point.
+     * @param roundingMethod the rounding method as defined in
+     *        {@link BigDecimal}.
+     * @return the rounded value.
+     * @since 1.1
+     */
+    public static float round(float x, int scale, int roundingMethod) {
+        float sign = indicator(x);
+        float factor = (float)Math.pow(10.0f, scale) * sign;
+        return (float)roundUnscaled(x * factor, sign, roundingMethod) / factor;
+    }
+
+    /**
+     * Round the given non-negative, value to the "nearest" integer. Nearest is
+     * determined by the rounding method specified. Rounding methods are defined
+     * in {@link BigDecimal}.
+     *
+     * @param unscaled the value to round.
+     * @param sign the sign of the original, scaled value.
+     * @param roundingMethod the rounding method as defined in
+     *        {@link BigDecimal}.
+     * @return the rounded value.
+     * @since 1.1
+     */
+    private static double roundUnscaled(double unscaled, double sign,
+        int roundingMethod) {
+        switch (roundingMethod) {
+        case BigDecimal.ROUND_CEILING :
+            if (sign == -1) {
+                unscaled = Math.floor(nextAfter(unscaled, Double.NEGATIVE_INFINITY));
+            } else {
+                unscaled = Math.ceil(nextAfter(unscaled, Double.POSITIVE_INFINITY));
+            }
+            break;
+        case BigDecimal.ROUND_DOWN :
+            unscaled = Math.floor(nextAfter(unscaled, Double.NEGATIVE_INFINITY));
+            break;
+        case BigDecimal.ROUND_FLOOR :
+            if (sign == -1) {
+                unscaled = Math.ceil(nextAfter(unscaled, Double.POSITIVE_INFINITY));
+            } else {
+                unscaled = Math.floor(nextAfter(unscaled, Double.NEGATIVE_INFINITY));
+            }
+            break;
+        case BigDecimal.ROUND_HALF_DOWN : {
+            unscaled = nextAfter(unscaled, Double.NEGATIVE_INFINITY);
+            double fraction = unscaled - Math.floor(unscaled);
+            if (fraction > 0.5) {
+                unscaled = Math.ceil(unscaled);
+            } else {
+                unscaled = Math.floor(unscaled);
+            }
+            break;
+        }
+        case BigDecimal.ROUND_HALF_EVEN : {
+            double fraction = unscaled - Math.floor(unscaled);
+            if (fraction > 0.5) {
+                unscaled = Math.ceil(unscaled);
+            } else if (fraction < 0.5) {
+                unscaled = Math.floor(unscaled);
+            } else {
+                // The following equality test is intentional and needed for rounding purposes
+                if (Math.floor(unscaled) / 2.0 == Math.floor(Math
+                    .floor(unscaled) / 2.0)) { // even
+                    unscaled = Math.floor(unscaled);
+                } else { // odd
+                    unscaled = Math.ceil(unscaled);
+                }
+            }
+            break;
+        }
+        case BigDecimal.ROUND_HALF_UP : {
+            unscaled = nextAfter(unscaled, Double.POSITIVE_INFINITY);
+            double fraction = unscaled - Math.floor(unscaled);
+            if (fraction >= 0.5) {
+                unscaled = Math.ceil(unscaled);
+            } else {
+                unscaled = Math.floor(unscaled);
+            }
+            break;
+        }
+        case BigDecimal.ROUND_UNNECESSARY :
+            if (unscaled != Math.floor(unscaled)) {
+                throw new ArithmeticException("Inexact result from rounding");
+            }
+            break;
+        case BigDecimal.ROUND_UP :
+            unscaled = Math.ceil(nextAfter(unscaled,  Double.POSITIVE_INFINITY));
+            break;
+        default :
+            throw MathRuntimeException.createIllegalArgumentException(
+                  "invalid rounding method {0}, valid methods: {1} ({2}), {3} ({4})," +
+                  " {5} ({6}), {7} ({8}), {9} ({10}), {11} ({12}), {13} ({14}), {15} ({16})",
+                  roundingMethod,
+                  "ROUND_CEILING",     BigDecimal.ROUND_CEILING,
+                  "ROUND_DOWN",        BigDecimal.ROUND_DOWN,
+                  "ROUND_FLOOR",       BigDecimal.ROUND_FLOOR,
+                  "ROUND_HALF_DOWN",   BigDecimal.ROUND_HALF_DOWN,
+                  "ROUND_HALF_EVEN",   BigDecimal.ROUND_HALF_EVEN,
+                  "ROUND_HALF_UP",     BigDecimal.ROUND_HALF_UP,
+                  "ROUND_UNNECESSARY", BigDecimal.ROUND_UNNECESSARY,
+                  "ROUND_UP",          BigDecimal.ROUND_UP);
+        }
+        return unscaled;
+    }
+
+    /**
+     * Returns the <a href="http://mathworld.wolfram.com/Sign.html"> sign</a>
+     * for byte value <code>x</code>.
+     * <p>
+     * For a byte value x, this method returns (byte)(+1) if x > 0, (byte)(0) if
+     * x = 0, and (byte)(-1) if x < 0.</p>
+     *
+     * @param x the value, a byte
+     * @return (byte)(+1), (byte)(0), or (byte)(-1), depending on the sign of x
+     */
+    public static byte sign(final byte x) {
+        return (x == ZB) ? ZB : (x > ZB) ? PB : NB;
+    }
+
+    /**
+     * Returns the <a href="http://mathworld.wolfram.com/Sign.html"> sign</a>
+     * for double precision <code>x</code>.
+     * <p>
+     * For a double value <code>x</code>, this method returns
+     * <code>+1.0</code> if <code>x > 0</code>, <code>0.0</code> if
+     * <code>x = 0.0</code>, and <code>-1.0</code> if <code>x < 0</code>.
+     * Returns <code>NaN</code> if <code>x</code> is <code>NaN</code>.</p>
+     *
+     * @param x the value, a double
+     * @return +1.0, 0.0, or -1.0, depending on the sign of x
+     */
+    public static double sign(final double x) {
+        if (Double.isNaN(x)) {
+            return Double.NaN;
+        }
+        return (x == 0.0) ? 0.0 : (x > 0.0) ? 1.0 : -1.0;
+    }
+
+    /**
+     * Returns the <a href="http://mathworld.wolfram.com/Sign.html"> sign</a>
+     * for float value <code>x</code>.
+     * <p>
+     * For a float value x, this method returns +1.0F if x > 0, 0.0F if x =
+     * 0.0F, and -1.0F if x < 0. Returns <code>NaN</code> if <code>x</code>
+     * is <code>NaN</code>.</p>
+     *
+     * @param x the value, a float
+     * @return +1.0F, 0.0F, or -1.0F, depending on the sign of x
+     */
+    public static float sign(final float x) {
+        if (Float.isNaN(x)) {
+            return Float.NaN;
+        }
+        return (x == 0.0F) ? 0.0F : (x > 0.0F) ? 1.0F : -1.0F;
+    }
+
+    /**
+     * Returns the <a href="http://mathworld.wolfram.com/Sign.html"> sign</a>
+     * for int value <code>x</code>.
+     * <p>
+     * For an int value x, this method returns +1 if x > 0, 0 if x = 0, and -1
+     * if x < 0.</p>
+     *
+     * @param x the value, an int
+     * @return +1, 0, or -1, depending on the sign of x
+     */
+    public static int sign(final int x) {
+        return (x == 0) ? 0 : (x > 0) ? 1 : -1;
+    }
+
+    /**
+     * Returns the <a href="http://mathworld.wolfram.com/Sign.html"> sign</a>
+     * for long value <code>x</code>.
+     * <p>
+     * For a long value x, this method returns +1L if x > 0, 0L if x = 0, and
+     * -1L if x < 0.</p>
+     *
+     * @param x the value, a long
+     * @return +1L, 0L, or -1L, depending on the sign of x
+     */
+    public static long sign(final long x) {
+        return (x == 0L) ? 0L : (x > 0L) ? 1L : -1L;
+    }
+
+    /**
+     * Returns the <a href="http://mathworld.wolfram.com/Sign.html"> sign</a>
+     * for short value <code>x</code>.
+     * <p>
+     * For a short value x, this method returns (short)(+1) if x > 0, (short)(0)
+     * if x = 0, and (short)(-1) if x < 0.</p>
+     *
+     * @param x the value, a short
+     * @return (short)(+1), (short)(0), or (short)(-1), depending on the sign of
+     *         x
+     */
+    public static short sign(final short x) {
+        return (x == ZS) ? ZS : (x > ZS) ? PS : NS;
+    }
+
+    /**
+     * Returns the <a href="http://mathworld.wolfram.com/HyperbolicSine.html">
+     * hyperbolic sine</a> of x.
+     *
+     * @param x double value for which to find the hyperbolic sine
+     * @return hyperbolic sine of x
+     */
+    public static double sinh(double x) {
+        return (Math.exp(x) - Math.exp(-x)) / 2.0;
+    }
+
+    /**
+     * Subtract two integers, checking for overflow.
+     *
+     * @param x the minuend
+     * @param y the subtrahend
+     * @return the difference <code>x-y</code>
+     * @throws ArithmeticException if the result can not be represented as an
+     *         int
+     * @since 1.1
+     */
+    public static int subAndCheck(int x, int y) {
+        long s = (long)x - (long)y;
+        if (s < Integer.MIN_VALUE || s > Integer.MAX_VALUE) {
+            throw new ArithmeticException("overflow: subtract");
+        }
+        return (int)s;
+    }
+
+    /**
+     * Subtract two long integers, checking for overflow.
+     *
+     * @param a first value
+     * @param b second value
+     * @return the difference <code>a-b</code>
+     * @throws ArithmeticException if the result can not be represented as an
+     *         long
+     * @since 1.2
+     */
+    public static long subAndCheck(long a, long b) {
+        long ret;
+        String msg = "overflow: subtract";
+        if (b == Long.MIN_VALUE) {
+            if (a < 0) {
+                ret = a - b;
+            } else {
+                throw new ArithmeticException(msg);
+            }
+        } else {
+            // use additive inverse
+            ret = addAndCheck(a, -b, msg);
+        }
+        return ret;
+    }
+
+    /**
+     * Raise an int to an int power.
+     * @param k number to raise
+     * @param e exponent (must be positive or null)
+     * @return k<sup>e</sup>
+     * @exception IllegalArgumentException if e is negative
+     */
+    public static int pow(final int k, int e)
+        throws IllegalArgumentException {
+
+        if (e < 0) {
+            throw MathRuntimeException.createIllegalArgumentException(
+                "cannot raise an integral value to a negative power ({0}^{1})",
+                k, e);
+        }
+
+        int result = 1;
+        int k2p    = k;
+        while (e != 0) {
+            if ((e & 0x1) != 0) {
+                result *= k2p;
+            }
+            k2p *= k2p;
+            e = e >> 1;
+        }
+
+        return result;
+
+    }
+
+    /**
+     * Raise an int to a long power.
+     * @param k number to raise
+     * @param e exponent (must be positive or null)
+     * @return k<sup>e</sup>
+     * @exception IllegalArgumentException if e is negative
+     */
+    public static int pow(final int k, long e)
+        throws IllegalArgumentException {
+
+        if (e < 0) {
+            throw MathRuntimeException.createIllegalArgumentException(
+                "cannot raise an integral value to a negative power ({0}^{1})",
+                k, e);
+        }
+
+        int result = 1;
+        int k2p    = k;
+        while (e != 0) {
+            if ((e & 0x1) != 0) {
+                result *= k2p;
+            }
+            k2p *= k2p;
+            e = e >> 1;
+        }
+
+        return result;
+
+    }
+
+    /**
+     * Raise a long to an int power.
+     * @param k number to raise
+     * @param e exponent (must be positive or null)
+     * @return k<sup>e</sup>
+     * @exception IllegalArgumentException if e is negative
+     */
+    public static long pow(final long k, int e)
+        throws IllegalArgumentException {
+
+        if (e < 0) {
+            throw MathRuntimeException.createIllegalArgumentException(
+                "cannot raise an integral value to a negative power ({0}^{1})",
+                k, e);
+        }
+
+        long result = 1l;
+        long k2p    = k;
+        while (e != 0) {
+            if ((e & 0x1) != 0) {
+                result *= k2p;
+            }
+            k2p *= k2p;
+            e = e >> 1;
+        }
+
+        return result;
+
+    }
+
+    /**
+     * Raise a long to a long power.
+     * @param k number to raise
+     * @param e exponent (must be positive or null)
+     * @return k<sup>e</sup>
+     * @exception IllegalArgumentException if e is negative
+     */
+    public static long pow(final long k, long e)
+        throws IllegalArgumentException {
+
+        if (e < 0) {
+            throw MathRuntimeException.createIllegalArgumentException(
+                "cannot raise an integral value to a negative power ({0}^{1})",
+                k, e);
+        }
+
+        long result = 1l;
+        long k2p    = k;
+        while (e != 0) {
+            if ((e & 0x1) != 0) {
+                result *= k2p;
+            }
+            k2p *= k2p;
+            e = e >> 1;
+        }
+
+        return result;
+
+    }
+
+    /**
+     * Raise a BigInteger to an int power.
+     * @param k number to raise
+     * @param e exponent (must be positive or null)
+     * @return k<sup>e</sup>
+     * @exception IllegalArgumentException if e is negative
+     */
+    public static BigInteger pow(final BigInteger k, int e)
+        throws IllegalArgumentException {
+
+        if (e < 0) {
+            throw MathRuntimeException.createIllegalArgumentException(
+                "cannot raise an integral value to a negative power ({0}^{1})",
+                k, e);
+        }
+
+        return k.pow(e);
+
+    }
+
+    /**
+     * Raise a BigInteger to a long power.
+     * @param k number to raise
+     * @param e exponent (must be positive or null)
+     * @return k<sup>e</sup>
+     * @exception IllegalArgumentException if e is negative
+     */
+    public static BigInteger pow(final BigInteger k, long e)
+        throws IllegalArgumentException {
+
+        if (e < 0) {
+            throw MathRuntimeException.createIllegalArgumentException(
+                "cannot raise an integral value to a negative power ({0}^{1})",
+                k, e);
+        }
+
+        BigInteger result = BigInteger.ONE;
+        BigInteger k2p    = k;
+        while (e != 0) {
+            if ((e & 0x1) != 0) {
+                result = result.multiply(k2p);
+            }
+            k2p = k2p.multiply(k2p);
+            e = e >> 1;
+        }
+
+        return result;
+
+    }
+
+    /**
+     * Raise a BigInteger to a BigInteger power.
+     * @param k number to raise
+     * @param e exponent (must be positive or null)
+     * @return k<sup>e</sup>
+     * @exception IllegalArgumentException if e is negative
+     */
+    public static BigInteger pow(final BigInteger k, BigInteger e)
+        throws IllegalArgumentException {
+
+        if (e.compareTo(BigInteger.ZERO) < 0) {
+            throw MathRuntimeException.createIllegalArgumentException(
+                "cannot raise an integral value to a negative power ({0}^{1})",
+                k, e);
+        }
+
+        BigInteger result = BigInteger.ONE;
+        BigInteger k2p    = k;
+        while (!BigInteger.ZERO.equals(e)) {
+            if (e.testBit(0)) {
+                result = result.multiply(k2p);
+            }
+            k2p = k2p.multiply(k2p);
+            e = e.shiftRight(1);
+        }
+
+        return result;
+
+    }
+
+    /**
+     * Calculates the L<sub>1</sub> (sum of abs) distance between two points.
+     *
+     * @param p1 the first point
+     * @param p2 the second point
+     * @return the L<sub>1</sub> distance between the two points
+     */
+    public static double distance1(double[] p1, double[] p2) {
+        double sum = 0;
+        for (int i = 0; i < p1.length; i++) {
+            sum += Math.abs(p1[i] - p2[i]);
+        }
+        return sum;
+    }
+
+    /**
+     * Calculates the L<sub>1</sub> (sum of abs) distance between two points.
+     *
+     * @param p1 the first point
+     * @param p2 the second point
+     * @return the L<sub>1</sub> distance between the two points
+     */
+    public static int distance1(int[] p1, int[] p2) {
+      int sum = 0;
+      for (int i = 0; i < p1.length; i++) {
+          sum += Math.abs(p1[i] - p2[i]);
+      }
+      return sum;
+    }
+
+    /**
+     * Calculates the L<sub>2</sub> (Euclidean) distance between two points.
+     *
+     * @param p1 the first point
+     * @param p2 the second point
+     * @return the L<sub>2</sub> distance between the two points
+     */
+    public static double distance(double[] p1, double[] p2) {
+        double sum = 0;
+        for (int i = 0; i < p1.length; i++) {
+            final double dp = p1[i] - p2[i];
+            sum += dp * dp;
+        }
+        return Math.sqrt(sum);
+    }
+
+    /**
+     * Calculates the L<sub>2</sub> (Euclidean) distance between two points.
+     *
+     * @param p1 the first point
+     * @param p2 the second point
+     * @return the L<sub>2</sub> distance between the two points
+     */
+    public static double distance(int[] p1, int[] p2) {
+      double sum = 0;
+      for (int i = 0; i < p1.length; i++) {
+          final double dp = p1[i] - p2[i];
+          sum += dp * dp;
+      }
+      return Math.sqrt(sum);
+    }
+
+    /**
+     * Calculates the L<sub>&infin;</sub> (max of abs) distance between two points.
+     *
+     * @param p1 the first point
+     * @param p2 the second point
+     * @return the L<sub>&infin;</sub> distance between the two points
+     */
+    public static double distanceInf(double[] p1, double[] p2) {
+        double max = 0;
+        for (int i = 0; i < p1.length; i++) {
+            max = Math.max(max, Math.abs(p1[i] - p2[i]));
+        }
+        return max;
+    }
+
+    /**
+     * Calculates the L<sub>&infin;</sub> (max of abs) distance between two points.
+     *
+     * @param p1 the first point
+     * @param p2 the second point
+     * @return the L<sub>&infin;</sub> distance between the two points
+     */
+    public static int distanceInf(int[] p1, int[] p2) {
+        int max = 0;
+        for (int i = 0; i < p1.length; i++) {
+            max = Math.max(max, Math.abs(p1[i] - p2[i]));
+        }
+        return max;
+    }
+
+    /**
+     * Checks that the given array is sorted.
+     *
+     * @param val Values
+     * @param dir Order direction (-1 for decreasing, 1 for increasing)
+     * @param strict Whether the order should be strict
+     * @throws IllegalArgumentException if the array is not sorted.
+     */
+    public static void checkOrder(double[] val, int dir, boolean strict) {
+        double previous = val[0];
+
+        int max = val.length;
+        for (int i = 1; i < max; i++) {
+            if (dir > 0) {
+                if (strict) {
+                    if (val[i] <= previous) {
+                        throw MathRuntimeException.createIllegalArgumentException("points {0} and {1} are not strictly increasing ({2} >= {3})",
+                                                                                  i - 1, i, previous, val[i]);
+                    }
+                } else {
+                    if (val[i] < previous) {
+                        throw MathRuntimeException.createIllegalArgumentException("points {0} and {1} are not increasing ({2} > {3})",
+                                                                                  i - 1, i, previous, val[i]);
+                    }
+                }
+            } else {
+                if (strict) {
+                    if (val[i] >= previous) {
+                        throw MathRuntimeException.createIllegalArgumentException("points {0} and {1} are not strictly decreasing ({2} <= {3})",
+                                                                                  i - 1, i, previous, val[i]);
+                    }
+                } else {
+                    if (val[i] > previous) {
+                        throw MathRuntimeException.createIllegalArgumentException("points {0} and {1} are not decreasing ({2} < {3})",
+                                                                                  i - 1, i, previous, val[i]);
+                    }
+                }
+            }
+
+            previous = val[i];
+        }
+    }
+}

Propchange: openejb/branches/openejb-3.1.x/container/openejb-core/src/main/java/org/apache/openejb/math/util/MathUtils.java
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