Java tutorial
/* * 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.commons.math3.fraction; import java.io.Serializable; import java.math.BigInteger; import org.apache.commons.math3.FieldElement; import org.apache.commons.math3.exception.util.LocalizedFormats; import org.apache.commons.math3.exception.MathArithmeticException; import org.apache.commons.math3.exception.NullArgumentException; import org.apache.commons.math3.util.ArithmeticUtils; import org.apache.commons.math3.util.FastMath; /** * Representation of a rational number. * * implements Serializable since 2.0 * * @since 1.1 * @version $Id: Fraction.java 1416643 2012-12-03 19:37:14Z tn $ */ public class Fraction extends Number implements FieldElement<Fraction>, Comparable<Fraction>, Serializable { /** A fraction representing "2 / 1". */ public static final Fraction TWO = new Fraction(2, 1); /** A fraction representing "1". */ public static final Fraction ONE = new Fraction(1, 1); /** A fraction representing "0". */ public static final Fraction ZERO = new Fraction(0, 1); /** A fraction representing "4/5". */ public static final Fraction FOUR_FIFTHS = new Fraction(4, 5); /** A fraction representing "1/5". */ public static final Fraction ONE_FIFTH = new Fraction(1, 5); /** A fraction representing "1/2". */ public static final Fraction ONE_HALF = new Fraction(1, 2); /** A fraction representing "1/4". */ public static final Fraction ONE_QUARTER = new Fraction(1, 4); /** A fraction representing "1/3". */ public static final Fraction ONE_THIRD = new Fraction(1, 3); /** A fraction representing "3/5". */ public static final Fraction THREE_FIFTHS = new Fraction(3, 5); /** A fraction representing "3/4". */ public static final Fraction THREE_QUARTERS = new Fraction(3, 4); /** A fraction representing "2/5". */ public static final Fraction TWO_FIFTHS = new Fraction(2, 5); /** A fraction representing "2/4". */ public static final Fraction TWO_QUARTERS = new Fraction(2, 4); /** A fraction representing "2/3". */ public static final Fraction TWO_THIRDS = new Fraction(2, 3); /** A fraction representing "-1 / 1". */ public static final Fraction MINUS_ONE = new Fraction(-1, 1); /** Serializable version identifier */ private static final long serialVersionUID = 3698073679419233275L; /** The denominator. */ private final int denominator; /** The numerator. */ private final int numerator; /** * Create a fraction given the double value. * @param value the double value to convert to a fraction. * @throws FractionConversionException if the continued fraction failed to * converge. */ public Fraction(double value) throws FractionConversionException { this(value, 1.0e-5, 100); } /** * Create a fraction given the double value and maximum error allowed. * <p> * References: * <ul> * <li><a href="http://mathworld.wolfram.com/ContinuedFraction.html"> * Continued Fraction</a> equations (11) and (22)-(26)</li> * </ul> * </p> * @param value the double value to convert to a fraction. * @param epsilon maximum error allowed. The resulting fraction is within * {@code epsilon} of {@code value}, in absolute terms. * @param maxIterations maximum number of convergents * @throws FractionConversionException if the continued fraction failed to * converge. */ public Fraction(double value, double epsilon, int maxIterations) throws FractionConversionException { this(value, epsilon, Integer.MAX_VALUE, maxIterations); } /** * Create a fraction given the double value and maximum denominator. * <p> * References: * <ul> * <li><a href="http://mathworld.wolfram.com/ContinuedFraction.html"> * Continued Fraction</a> equations (11) and (22)-(26)</li> * </ul> * </p> * @param value the double value to convert to a fraction. * @param maxDenominator The maximum allowed value for denominator * @throws FractionConversionException if the continued fraction failed to * converge */ public Fraction(double value, int maxDenominator) throws FractionConversionException { this(value, 0, maxDenominator, 100); } /** * Create a fraction given the double value and either the maximum error * allowed or the maximum number of denominator digits. * <p> * * NOTE: This constructor is called with EITHER * - a valid epsilon value and the maxDenominator set to Integer.MAX_VALUE * (that way the maxDenominator has no effect). * OR * - a valid maxDenominator value and the epsilon value set to zero * (that way epsilon only has effect if there is an exact match before * the maxDenominator value is reached). * </p><p> * * It has been done this way so that the same code can be (re)used for both * scenarios. However this could be confusing to users if it were part of * the public API and this constructor should therefore remain PRIVATE. * </p> * * See JIRA issue ticket MATH-181 for more details: * * https://issues.apache.org/jira/browse/MATH-181 * * @param value the double value to convert to a fraction. * @param epsilon maximum error allowed. The resulting fraction is within * {@code epsilon} of {@code value}, in absolute terms. * @param maxDenominator maximum denominator value allowed. * @param maxIterations maximum number of convergents * @throws FractionConversionException if the continued fraction failed to * converge. */ private Fraction(double value, double epsilon, int maxDenominator, int maxIterations) throws FractionConversionException { long overflow = Integer.MAX_VALUE; double r0 = value; long a0 = (long) FastMath.floor(r0); if (FastMath.abs(a0) > overflow) { throw new FractionConversionException(value, a0, 1l); } // check for (almost) integer arguments, which should not go // to iterations. if (FastMath.abs(a0 - value) < epsilon) { this.numerator = (int) a0; this.denominator = 1; return; } long p0 = 1; long q0 = 0; long p1 = a0; long q1 = 1; long p2 = 0; long q2 = 1; int n = 0; boolean stop = false; do { ++n; double r1 = 1.0 / (r0 - a0); long a1 = (long) FastMath.floor(r1); p2 = (a1 * p1) + p0; q2 = (a1 * q1) + q0; if ((FastMath.abs(p2) > overflow) || (FastMath.abs(q2) > overflow)) { throw new FractionConversionException(value, p2, q2); } double convergent = (double) p2 / (double) q2; if (n < maxIterations && FastMath.abs(convergent - value) > epsilon && q2 < maxDenominator) { p0 = p1; p1 = p2; q0 = q1; q1 = q2; a0 = a1; r0 = r1; } else { stop = true; } } while (!stop); if (n >= maxIterations) { throw new FractionConversionException(value, maxIterations); } if (q2 < maxDenominator) { this.numerator = (int) p2; this.denominator = (int) q2; } else { this.numerator = (int) p1; this.denominator = (int) q1; } } /** * Create a fraction from an int. * The fraction is num / 1. * @param num the numerator. */ public Fraction(int num) { this(num, 1); } /** * Create a fraction given the numerator and denominator. The fraction is * reduced to lowest terms. * @param num the numerator. * @param den the denominator. * @throws MathArithmeticException if the denominator is {@code zero} */ public Fraction(int num, int den) { if (den == 0) { throw new MathArithmeticException(LocalizedFormats.ZERO_DENOMINATOR_IN_FRACTION, num, den); } if (den < 0) { if (num == Integer.MIN_VALUE || den == Integer.MIN_VALUE) { throw new MathArithmeticException(LocalizedFormats.OVERFLOW_IN_FRACTION, num, den); } num = -num; den = -den; } // reduce numerator and denominator by greatest common denominator. final int d = ArithmeticUtils.gcd(num, den); if (d > 1) { num /= d; den /= d; } // move sign to numerator. if (den < 0) { num = -num; den = -den; } this.numerator = num; this.denominator = den; } /** * Returns the absolute value of this fraction. * @return the absolute value. */ public Fraction abs() { Fraction ret; if (numerator >= 0) { ret = this; } else { ret = negate(); } return ret; } /** * Compares this object to another based on size. * @param object the object to compare to * @return -1 if this is less than <tt>object</tt>, +1 if this is greater * than <tt>object</tt>, 0 if they are equal. */ public int compareTo(Fraction object) { long nOd = ((long) numerator) * object.denominator; long dOn = ((long) denominator) * object.numerator; return (nOd < dOn) ? -1 : ((nOd > dOn) ? +1 : 0); } /** * Gets the fraction as a <tt>double</tt>. This calculates the fraction as * the numerator divided by denominator. * @return the fraction as a <tt>double</tt> */ @Override public double doubleValue() { return (double) numerator / (double) denominator; } /** * Test for the equality of two fractions. If the lowest term * numerator and denominators are the same for both fractions, the two * fractions are considered to be equal. * @param other fraction to test for equality to this fraction * @return true if two fractions are equal, false if object is * <tt>null</tt>, not an instance of {@link Fraction}, or not equal * to this fraction instance. */ @Override public boolean equals(Object other) { if (this == other) { return true; } if (other instanceof Fraction) { // since fractions are always in lowest terms, numerators and // denominators can be compared directly for equality. Fraction rhs = (Fraction) other; return (numerator == rhs.numerator) && (denominator == rhs.denominator); } return false; } /** * Gets the fraction as a <tt>float</tt>. This calculates the fraction as * the numerator divided by denominator. * @return the fraction as a <tt>float</tt> */ @Override public float floatValue() { return (float) doubleValue(); } /** * Access the denominator. * @return the denominator. */ public int getDenominator() { return denominator; } /** * Access the numerator. * @return the numerator. */ public int getNumerator() { return numerator; } /** * Gets a hashCode for the fraction. * @return a hash code value for this object */ @Override public int hashCode() { return 37 * (37 * 17 + numerator) + denominator; } /** * Gets the fraction as an <tt>int</tt>. This returns the whole number part * of the fraction. * @return the whole number fraction part */ @Override public int intValue() { return (int) doubleValue(); } /** * Gets the fraction as a <tt>long</tt>. This returns the whole number part * of the fraction. * @return the whole number fraction part */ @Override public long longValue() { return (long) doubleValue(); } /** * Return the additive inverse of this fraction. * @return the negation of this fraction. */ public Fraction negate() { if (numerator == Integer.MIN_VALUE) { throw new MathArithmeticException(LocalizedFormats.OVERFLOW_IN_FRACTION, numerator, denominator); } return new Fraction(-numerator, denominator); } /** * Return the multiplicative inverse of this fraction. * @return the reciprocal fraction */ public Fraction reciprocal() { return new Fraction(denominator, numerator); } /** * <p>Adds the value of this fraction to another, returning the result in reduced form. * The algorithm follows Knuth, 4.5.1.</p> * * @param fraction the fraction to add, must not be {@code null} * @return a {@code Fraction} instance with the resulting values * @throws NullArgumentException if the fraction is {@code null} * @throws MathArithmeticException if the resulting numerator or denominator exceeds * {@code Integer.MAX_VALUE} */ public Fraction add(Fraction fraction) { return addSub(fraction, true /* add */); } /** * Add an integer to the fraction. * @param i the <tt>integer</tt> to add. * @return this + i */ public Fraction add(final int i) { return new Fraction(numerator + i * denominator, denominator); } /** * <p>Subtracts the value of another fraction from the value of this one, * returning the result in reduced form.</p> * * @param fraction the fraction to subtract, must not be {@code null} * @return a {@code Fraction} instance with the resulting values * @throws NullArgumentException if the fraction is {@code null} * @throws MathArithmeticException if the resulting numerator or denominator * cannot be represented in an {@code int}. */ public Fraction subtract(Fraction fraction) { return addSub(fraction, false /* subtract */); } /** * Subtract an integer from the fraction. * @param i the <tt>integer</tt> to subtract. * @return this - i */ public Fraction subtract(final int i) { return new Fraction(numerator - i * denominator, denominator); } /** * Implement add and subtract using algorithm described in Knuth 4.5.1. * * @param fraction the fraction to subtract, must not be {@code null} * @param isAdd true to add, false to subtract * @return a {@code Fraction} instance with the resulting values * @throws NullArgumentException if the fraction is {@code null} * @throws MathArithmeticException if the resulting numerator or denominator * cannot be represented in an {@code int}. */ private Fraction addSub(Fraction fraction, boolean isAdd) { if (fraction == null) { throw new NullArgumentException(LocalizedFormats.FRACTION); } // zero is identity for addition. if (numerator == 0) { return isAdd ? fraction : fraction.negate(); } if (fraction.numerator == 0) { return this; } // if denominators are randomly distributed, d1 will be 1 about 61% // of the time. int d1 = ArithmeticUtils.gcd(denominator, fraction.denominator); if (d1 == 1) { // result is ( (u*v' +/- u'v) / u'v') int uvp = ArithmeticUtils.mulAndCheck(numerator, fraction.denominator); int upv = ArithmeticUtils.mulAndCheck(fraction.numerator, denominator); return new Fraction( isAdd ? ArithmeticUtils.addAndCheck(uvp, upv) : ArithmeticUtils.subAndCheck(uvp, upv), ArithmeticUtils.mulAndCheck(denominator, fraction.denominator)); } // the quantity 't' requires 65 bits of precision; see knuth 4.5.1 // exercise 7. we're going to use a BigInteger. // t = u(v'/d1) +/- v(u'/d1) BigInteger uvp = BigInteger.valueOf(numerator).multiply(BigInteger.valueOf(fraction.denominator / d1)); BigInteger upv = BigInteger.valueOf(fraction.numerator).multiply(BigInteger.valueOf(denominator / d1)); BigInteger t = isAdd ? uvp.add(upv) : uvp.subtract(upv); // but d2 doesn't need extra precision because // d2 = gcd(t,d1) = gcd(t mod d1, d1) int tmodd1 = t.mod(BigInteger.valueOf(d1)).intValue(); int d2 = (tmodd1 == 0) ? d1 : ArithmeticUtils.gcd(tmodd1, d1); // result is (t/d2) / (u'/d1)(v'/d2) BigInteger w = t.divide(BigInteger.valueOf(d2)); if (w.bitLength() > 31) { throw new MathArithmeticException(LocalizedFormats.NUMERATOR_OVERFLOW_AFTER_MULTIPLY, w); } return new Fraction(w.intValue(), ArithmeticUtils.mulAndCheck(denominator / d1, fraction.denominator / d2)); } /** * <p>Multiplies the value of this fraction by another, returning the * result in reduced form.</p> * * @param fraction the fraction to multiply by, must not be {@code null} * @return a {@code Fraction} instance with the resulting values * @throws NullArgumentException if the fraction is {@code null} * @throws MathArithmeticException if the resulting numerator or denominator exceeds * {@code Integer.MAX_VALUE} */ public Fraction multiply(Fraction fraction) { if (fraction == null) { throw new NullArgumentException(LocalizedFormats.FRACTION); } if (numerator == 0 || fraction.numerator == 0) { return ZERO; } // knuth 4.5.1 // make sure we don't overflow unless the result *must* overflow. int d1 = ArithmeticUtils.gcd(numerator, fraction.denominator); int d2 = ArithmeticUtils.gcd(fraction.numerator, denominator); return getReducedFraction(ArithmeticUtils.mulAndCheck(numerator / d1, fraction.numerator / d2), ArithmeticUtils.mulAndCheck(denominator / d2, fraction.denominator / d1)); } /** * Multiply the fraction by an integer. * @param i the <tt>integer</tt> to multiply by. * @return this * i */ public Fraction multiply(final int i) { return new Fraction(numerator * i, denominator); } /** * <p>Divide the value of this fraction by another.</p> * * @param fraction the fraction to divide by, must not be {@code null} * @return a {@code Fraction} instance with the resulting values * @throws IllegalArgumentException if the fraction is {@code null} * @throws MathArithmeticException if the fraction to divide by is zero * @throws MathArithmeticException if the resulting numerator or denominator exceeds * {@code Integer.MAX_VALUE} */ public Fraction divide(Fraction fraction) { if (fraction == null) { throw new NullArgumentException(LocalizedFormats.FRACTION); } if (fraction.numerator == 0) { throw new MathArithmeticException(LocalizedFormats.ZERO_FRACTION_TO_DIVIDE_BY, fraction.numerator, fraction.denominator); } return multiply(fraction.reciprocal()); } /** * Divide the fraction by an integer. * @param i the <tt>integer</tt> to divide by. * @return this * i */ public Fraction divide(final int i) { return new Fraction(numerator, denominator * i); } /** * <p> * Gets the fraction percentage as a <tt>double</tt>. This calculates the * fraction as the numerator divided by denominator multiplied by 100. * </p> * * @return the fraction percentage as a <tt>double</tt>. */ public double percentageValue() { return 100 * doubleValue(); } /** * <p>Creates a {@code Fraction} instance with the 2 parts * of a fraction Y/Z.</p> * * <p>Any negative signs are resolved to be on the numerator.</p> * * @param numerator the numerator, for example the three in 'three sevenths' * @param denominator the denominator, for example the seven in 'three sevenths' * @return a new fraction instance, with the numerator and denominator reduced * @throws MathArithmeticException if the denominator is {@code zero} */ public static Fraction getReducedFraction(int numerator, int denominator) { if (denominator == 0) { throw new MathArithmeticException(LocalizedFormats.ZERO_DENOMINATOR_IN_FRACTION, numerator, denominator); } if (numerator == 0) { return ZERO; // normalize zero. } // allow 2^k/-2^31 as a valid fraction (where k>0) if (denominator == Integer.MIN_VALUE && (numerator & 1) == 0) { numerator /= 2; denominator /= 2; } if (denominator < 0) { if (numerator == Integer.MIN_VALUE || denominator == Integer.MIN_VALUE) { throw new MathArithmeticException(LocalizedFormats.OVERFLOW_IN_FRACTION, numerator, denominator); } numerator = -numerator; denominator = -denominator; } // simplify fraction. int gcd = ArithmeticUtils.gcd(numerator, denominator); numerator /= gcd; denominator /= gcd; return new Fraction(numerator, denominator); } /** * <p> * Returns the {@code String} representing this fraction, ie * "num / dem" or just "num" if the denominator is one. * </p> * * @return a string representation of the fraction. * @see java.lang.Object#toString() */ @Override public String toString() { String str = null; if (denominator == 1) { str = Integer.toString(numerator); } else if (numerator == 0) { str = "0"; } else { str = numerator + " / " + denominator; } return str; } /** {@inheritDoc} */ public FractionField getField() { return FractionField.getInstance(); } }