001 /*
002 * Licensed to the Apache Software Foundation (ASF) under one or more
003 * contributor license agreements. See the NOTICE file distributed with
004 * this work for additional information regarding copyright ownership.
005 * The ASF licenses this file to You under the Apache License, Version 2.0
006 * (the "License"); you may not use this file except in compliance with
007 * the License. You may obtain a copy of the License at
008 *
009 * http://www.apache.org/licenses/LICENSE-2.0
010 *
011 * Unless required by applicable law or agreed to in writing, software
012 * distributed under the License is distributed on an "AS IS" BASIS,
013 * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
014 * See the License for the specific language governing permissions and
015 * limitations under the License.
016 */
017 package org.apache.commons.math3.fraction;
018
019 import java.io.Serializable;
020 import java.math.BigInteger;
021
022 import org.apache.commons.math3.FieldElement;
023 import org.apache.commons.math3.exception.util.LocalizedFormats;
024 import org.apache.commons.math3.exception.MathArithmeticException;
025 import org.apache.commons.math3.exception.NullArgumentException;
026 import org.apache.commons.math3.util.ArithmeticUtils;
027 import org.apache.commons.math3.util.FastMath;
028
029 /**
030 * Representation of a rational number.
031 *
032 * implements Serializable since 2.0
033 *
034 * @since 1.1
035 * @version $Id: Fraction.java 1416643 2012-12-03 19:37:14Z tn $
036 */
037 public class Fraction
038 extends Number
039 implements FieldElement<Fraction>, Comparable<Fraction>, Serializable {
040
041 /** A fraction representing "2 / 1". */
042 public static final Fraction TWO = new Fraction(2, 1);
043
044 /** A fraction representing "1". */
045 public static final Fraction ONE = new Fraction(1, 1);
046
047 /** A fraction representing "0". */
048 public static final Fraction ZERO = new Fraction(0, 1);
049
050 /** A fraction representing "4/5". */
051 public static final Fraction FOUR_FIFTHS = new Fraction(4, 5);
052
053 /** A fraction representing "1/5". */
054 public static final Fraction ONE_FIFTH = new Fraction(1, 5);
055
056 /** A fraction representing "1/2". */
057 public static final Fraction ONE_HALF = new Fraction(1, 2);
058
059 /** A fraction representing "1/4". */
060 public static final Fraction ONE_QUARTER = new Fraction(1, 4);
061
062 /** A fraction representing "1/3". */
063 public static final Fraction ONE_THIRD = new Fraction(1, 3);
064
065 /** A fraction representing "3/5". */
066 public static final Fraction THREE_FIFTHS = new Fraction(3, 5);
067
068 /** A fraction representing "3/4". */
069 public static final Fraction THREE_QUARTERS = new Fraction(3, 4);
070
071 /** A fraction representing "2/5". */
072 public static final Fraction TWO_FIFTHS = new Fraction(2, 5);
073
074 /** A fraction representing "2/4". */
075 public static final Fraction TWO_QUARTERS = new Fraction(2, 4);
076
077 /** A fraction representing "2/3". */
078 public static final Fraction TWO_THIRDS = new Fraction(2, 3);
079
080 /** A fraction representing "-1 / 1". */
081 public static final Fraction MINUS_ONE = new Fraction(-1, 1);
082
083 /** Serializable version identifier */
084 private static final long serialVersionUID = 3698073679419233275L;
085
086 /** The denominator. */
087 private final int denominator;
088
089 /** The numerator. */
090 private final int numerator;
091
092 /**
093 * Create a fraction given the double value.
094 * @param value the double value to convert to a fraction.
095 * @throws FractionConversionException if the continued fraction failed to
096 * converge.
097 */
098 public Fraction(double value) throws FractionConversionException {
099 this(value, 1.0e-5, 100);
100 }
101
102 /**
103 * Create a fraction given the double value and maximum error allowed.
104 * <p>
105 * References:
106 * <ul>
107 * <li><a href="http://mathworld.wolfram.com/ContinuedFraction.html">
108 * Continued Fraction</a> equations (11) and (22)-(26)</li>
109 * </ul>
110 * </p>
111 * @param value the double value to convert to a fraction.
112 * @param epsilon maximum error allowed. The resulting fraction is within
113 * {@code epsilon} of {@code value}, in absolute terms.
114 * @param maxIterations maximum number of convergents
115 * @throws FractionConversionException if the continued fraction failed to
116 * converge.
117 */
118 public Fraction(double value, double epsilon, int maxIterations)
119 throws FractionConversionException
120 {
121 this(value, epsilon, Integer.MAX_VALUE, maxIterations);
122 }
123
124 /**
125 * Create a fraction given the double value and maximum denominator.
126 * <p>
127 * References:
128 * <ul>
129 * <li><a href="http://mathworld.wolfram.com/ContinuedFraction.html">
130 * Continued Fraction</a> equations (11) and (22)-(26)</li>
131 * </ul>
132 * </p>
133 * @param value the double value to convert to a fraction.
134 * @param maxDenominator The maximum allowed value for denominator
135 * @throws FractionConversionException if the continued fraction failed to
136 * converge
137 */
138 public Fraction(double value, int maxDenominator)
139 throws FractionConversionException
140 {
141 this(value, 0, maxDenominator, 100);
142 }
143
144 /**
145 * Create a fraction given the double value and either the maximum error
146 * allowed or the maximum number of denominator digits.
147 * <p>
148 *
149 * NOTE: This constructor is called with EITHER
150 * - a valid epsilon value and the maxDenominator set to Integer.MAX_VALUE
151 * (that way the maxDenominator has no effect).
152 * OR
153 * - a valid maxDenominator value and the epsilon value set to zero
154 * (that way epsilon only has effect if there is an exact match before
155 * the maxDenominator value is reached).
156 * </p><p>
157 *
158 * It has been done this way so that the same code can be (re)used for both
159 * scenarios. However this could be confusing to users if it were part of
160 * the public API and this constructor should therefore remain PRIVATE.
161 * </p>
162 *
163 * See JIRA issue ticket MATH-181 for more details:
164 *
165 * https://issues.apache.org/jira/browse/MATH-181
166 *
167 * @param value the double value to convert to a fraction.
168 * @param epsilon maximum error allowed. The resulting fraction is within
169 * {@code epsilon} of {@code value}, in absolute terms.
170 * @param maxDenominator maximum denominator value allowed.
171 * @param maxIterations maximum number of convergents
172 * @throws FractionConversionException if the continued fraction failed to
173 * converge.
174 */
175 private Fraction(double value, double epsilon, int maxDenominator, int maxIterations)
176 throws FractionConversionException
177 {
178 long overflow = Integer.MAX_VALUE;
179 double r0 = value;
180 long a0 = (long)FastMath.floor(r0);
181 if (FastMath.abs(a0) > overflow) {
182 throw new FractionConversionException(value, a0, 1l);
183 }
184
185 // check for (almost) integer arguments, which should not go
186 // to iterations.
187 if (FastMath.abs(a0 - value) < epsilon) {
188 this.numerator = (int) a0;
189 this.denominator = 1;
190 return;
191 }
192
193 long p0 = 1;
194 long q0 = 0;
195 long p1 = a0;
196 long q1 = 1;
197
198 long p2 = 0;
199 long q2 = 1;
200
201 int n = 0;
202 boolean stop = false;
203 do {
204 ++n;
205 double r1 = 1.0 / (r0 - a0);
206 long a1 = (long)FastMath.floor(r1);
207 p2 = (a1 * p1) + p0;
208 q2 = (a1 * q1) + q0;
209 if ((FastMath.abs(p2) > overflow) || (FastMath.abs(q2) > overflow)) {
210 throw new FractionConversionException(value, p2, q2);
211 }
212
213 double convergent = (double)p2 / (double)q2;
214 if (n < maxIterations && FastMath.abs(convergent - value) > epsilon && q2 < maxDenominator) {
215 p0 = p1;
216 p1 = p2;
217 q0 = q1;
218 q1 = q2;
219 a0 = a1;
220 r0 = r1;
221 } else {
222 stop = true;
223 }
224 } while (!stop);
225
226 if (n >= maxIterations) {
227 throw new FractionConversionException(value, maxIterations);
228 }
229
230 if (q2 < maxDenominator) {
231 this.numerator = (int) p2;
232 this.denominator = (int) q2;
233 } else {
234 this.numerator = (int) p1;
235 this.denominator = (int) q1;
236 }
237
238 }
239
240 /**
241 * Create a fraction from an int.
242 * The fraction is num / 1.
243 * @param num the numerator.
244 */
245 public Fraction(int num) {
246 this(num, 1);
247 }
248
249 /**
250 * Create a fraction given the numerator and denominator. The fraction is
251 * reduced to lowest terms.
252 * @param num the numerator.
253 * @param den the denominator.
254 * @throws MathArithmeticException if the denominator is {@code zero}
255 */
256 public Fraction(int num, int den) {
257 if (den == 0) {
258 throw new MathArithmeticException(LocalizedFormats.ZERO_DENOMINATOR_IN_FRACTION,
259 num, den);
260 }
261 if (den < 0) {
262 if (num == Integer.MIN_VALUE ||
263 den == Integer.MIN_VALUE) {
264 throw new MathArithmeticException(LocalizedFormats.OVERFLOW_IN_FRACTION,
265 num, den);
266 }
267 num = -num;
268 den = -den;
269 }
270 // reduce numerator and denominator by greatest common denominator.
271 final int d = ArithmeticUtils.gcd(num, den);
272 if (d > 1) {
273 num /= d;
274 den /= d;
275 }
276
277 // move sign to numerator.
278 if (den < 0) {
279 num = -num;
280 den = -den;
281 }
282 this.numerator = num;
283 this.denominator = den;
284 }
285
286 /**
287 * Returns the absolute value of this fraction.
288 * @return the absolute value.
289 */
290 public Fraction abs() {
291 Fraction ret;
292 if (numerator >= 0) {
293 ret = this;
294 } else {
295 ret = negate();
296 }
297 return ret;
298 }
299
300 /**
301 * Compares this object to another based on size.
302 * @param object the object to compare to
303 * @return -1 if this is less than <tt>object</tt>, +1 if this is greater
304 * than <tt>object</tt>, 0 if they are equal.
305 */
306 public int compareTo(Fraction object) {
307 long nOd = ((long) numerator) * object.denominator;
308 long dOn = ((long) denominator) * object.numerator;
309 return (nOd < dOn) ? -1 : ((nOd > dOn) ? +1 : 0);
310 }
311
312 /**
313 * Gets the fraction as a <tt>double</tt>. This calculates the fraction as
314 * the numerator divided by denominator.
315 * @return the fraction as a <tt>double</tt>
316 */
317 @Override
318 public double doubleValue() {
319 return (double)numerator / (double)denominator;
320 }
321
322 /**
323 * Test for the equality of two fractions. If the lowest term
324 * numerator and denominators are the same for both fractions, the two
325 * fractions are considered to be equal.
326 * @param other fraction to test for equality to this fraction
327 * @return true if two fractions are equal, false if object is
328 * <tt>null</tt>, not an instance of {@link Fraction}, or not equal
329 * to this fraction instance.
330 */
331 @Override
332 public boolean equals(Object other) {
333 if (this == other) {
334 return true;
335 }
336 if (other instanceof Fraction) {
337 // since fractions are always in lowest terms, numerators and
338 // denominators can be compared directly for equality.
339 Fraction rhs = (Fraction)other;
340 return (numerator == rhs.numerator) &&
341 (denominator == rhs.denominator);
342 }
343 return false;
344 }
345
346 /**
347 * Gets the fraction as a <tt>float</tt>. This calculates the fraction as
348 * the numerator divided by denominator.
349 * @return the fraction as a <tt>float</tt>
350 */
351 @Override
352 public float floatValue() {
353 return (float)doubleValue();
354 }
355
356 /**
357 * Access the denominator.
358 * @return the denominator.
359 */
360 public int getDenominator() {
361 return denominator;
362 }
363
364 /**
365 * Access the numerator.
366 * @return the numerator.
367 */
368 public int getNumerator() {
369 return numerator;
370 }
371
372 /**
373 * Gets a hashCode for the fraction.
374 * @return a hash code value for this object
375 */
376 @Override
377 public int hashCode() {
378 return 37 * (37 * 17 + numerator) + denominator;
379 }
380
381 /**
382 * Gets the fraction as an <tt>int</tt>. This returns the whole number part
383 * of the fraction.
384 * @return the whole number fraction part
385 */
386 @Override
387 public int intValue() {
388 return (int)doubleValue();
389 }
390
391 /**
392 * Gets the fraction as a <tt>long</tt>. This returns the whole number part
393 * of the fraction.
394 * @return the whole number fraction part
395 */
396 @Override
397 public long longValue() {
398 return (long)doubleValue();
399 }
400
401 /**
402 * Return the additive inverse of this fraction.
403 * @return the negation of this fraction.
404 */
405 public Fraction negate() {
406 if (numerator==Integer.MIN_VALUE) {
407 throw new MathArithmeticException(LocalizedFormats.OVERFLOW_IN_FRACTION, numerator, denominator);
408 }
409 return new Fraction(-numerator, denominator);
410 }
411
412 /**
413 * Return the multiplicative inverse of this fraction.
414 * @return the reciprocal fraction
415 */
416 public Fraction reciprocal() {
417 return new Fraction(denominator, numerator);
418 }
419
420 /**
421 * <p>Adds the value of this fraction to another, returning the result in reduced form.
422 * The algorithm follows Knuth, 4.5.1.</p>
423 *
424 * @param fraction the fraction to add, must not be {@code null}
425 * @return a {@code Fraction} instance with the resulting values
426 * @throws NullArgumentException if the fraction is {@code null}
427 * @throws MathArithmeticException if the resulting numerator or denominator exceeds
428 * {@code Integer.MAX_VALUE}
429 */
430 public Fraction add(Fraction fraction) {
431 return addSub(fraction, true /* add */);
432 }
433
434 /**
435 * Add an integer to the fraction.
436 * @param i the <tt>integer</tt> to add.
437 * @return this + i
438 */
439 public Fraction add(final int i) {
440 return new Fraction(numerator + i * denominator, denominator);
441 }
442
443 /**
444 * <p>Subtracts the value of another fraction from the value of this one,
445 * returning the result in reduced form.</p>
446 *
447 * @param fraction the fraction to subtract, must not be {@code null}
448 * @return a {@code Fraction} instance with the resulting values
449 * @throws NullArgumentException if the fraction is {@code null}
450 * @throws MathArithmeticException if the resulting numerator or denominator
451 * cannot be represented in an {@code int}.
452 */
453 public Fraction subtract(Fraction fraction) {
454 return addSub(fraction, false /* subtract */);
455 }
456
457 /**
458 * Subtract an integer from the fraction.
459 * @param i the <tt>integer</tt> to subtract.
460 * @return this - i
461 */
462 public Fraction subtract(final int i) {
463 return new Fraction(numerator - i * denominator, denominator);
464 }
465
466 /**
467 * Implement add and subtract using algorithm described in Knuth 4.5.1.
468 *
469 * @param fraction the fraction to subtract, must not be {@code null}
470 * @param isAdd true to add, false to subtract
471 * @return a {@code Fraction} instance with the resulting values
472 * @throws NullArgumentException if the fraction is {@code null}
473 * @throws MathArithmeticException if the resulting numerator or denominator
474 * cannot be represented in an {@code int}.
475 */
476 private Fraction addSub(Fraction fraction, boolean isAdd) {
477 if (fraction == null) {
478 throw new NullArgumentException(LocalizedFormats.FRACTION);
479 }
480 // zero is identity for addition.
481 if (numerator == 0) {
482 return isAdd ? fraction : fraction.negate();
483 }
484 if (fraction.numerator == 0) {
485 return this;
486 }
487 // if denominators are randomly distributed, d1 will be 1 about 61%
488 // of the time.
489 int d1 = ArithmeticUtils.gcd(denominator, fraction.denominator);
490 if (d1==1) {
491 // result is ( (u*v' +/- u'v) / u'v')
492 int uvp = ArithmeticUtils.mulAndCheck(numerator, fraction.denominator);
493 int upv = ArithmeticUtils.mulAndCheck(fraction.numerator, denominator);
494 return new Fraction
495 (isAdd ? ArithmeticUtils.addAndCheck(uvp, upv) :
496 ArithmeticUtils.subAndCheck(uvp, upv),
497 ArithmeticUtils.mulAndCheck(denominator, fraction.denominator));
498 }
499 // the quantity 't' requires 65 bits of precision; see knuth 4.5.1
500 // exercise 7. we're going to use a BigInteger.
501 // t = u(v'/d1) +/- v(u'/d1)
502 BigInteger uvp = BigInteger.valueOf(numerator)
503 .multiply(BigInteger.valueOf(fraction.denominator/d1));
504 BigInteger upv = BigInteger.valueOf(fraction.numerator)
505 .multiply(BigInteger.valueOf(denominator/d1));
506 BigInteger t = isAdd ? uvp.add(upv) : uvp.subtract(upv);
507 // but d2 doesn't need extra precision because
508 // d2 = gcd(t,d1) = gcd(t mod d1, d1)
509 int tmodd1 = t.mod(BigInteger.valueOf(d1)).intValue();
510 int d2 = (tmodd1==0)?d1:ArithmeticUtils.gcd(tmodd1, d1);
511
512 // result is (t/d2) / (u'/d1)(v'/d2)
513 BigInteger w = t.divide(BigInteger.valueOf(d2));
514 if (w.bitLength() > 31) {
515 throw new MathArithmeticException(LocalizedFormats.NUMERATOR_OVERFLOW_AFTER_MULTIPLY,
516 w);
517 }
518 return new Fraction (w.intValue(),
519 ArithmeticUtils.mulAndCheck(denominator/d1,
520 fraction.denominator/d2));
521 }
522
523 /**
524 * <p>Multiplies the value of this fraction by another, returning the
525 * result in reduced form.</p>
526 *
527 * @param fraction the fraction to multiply by, must not be {@code null}
528 * @return a {@code Fraction} instance with the resulting values
529 * @throws NullArgumentException if the fraction is {@code null}
530 * @throws MathArithmeticException if the resulting numerator or denominator exceeds
531 * {@code Integer.MAX_VALUE}
532 */
533 public Fraction multiply(Fraction fraction) {
534 if (fraction == null) {
535 throw new NullArgumentException(LocalizedFormats.FRACTION);
536 }
537 if (numerator == 0 || fraction.numerator == 0) {
538 return ZERO;
539 }
540 // knuth 4.5.1
541 // make sure we don't overflow unless the result *must* overflow.
542 int d1 = ArithmeticUtils.gcd(numerator, fraction.denominator);
543 int d2 = ArithmeticUtils.gcd(fraction.numerator, denominator);
544 return getReducedFraction
545 (ArithmeticUtils.mulAndCheck(numerator/d1, fraction.numerator/d2),
546 ArithmeticUtils.mulAndCheck(denominator/d2, fraction.denominator/d1));
547 }
548
549 /**
550 * Multiply the fraction by an integer.
551 * @param i the <tt>integer</tt> to multiply by.
552 * @return this * i
553 */
554 public Fraction multiply(final int i) {
555 return new Fraction(numerator * i, denominator);
556 }
557
558 /**
559 * <p>Divide the value of this fraction by another.</p>
560 *
561 * @param fraction the fraction to divide by, must not be {@code null}
562 * @return a {@code Fraction} instance with the resulting values
563 * @throws IllegalArgumentException if the fraction is {@code null}
564 * @throws MathArithmeticException if the fraction to divide by is zero
565 * @throws MathArithmeticException if the resulting numerator or denominator exceeds
566 * {@code Integer.MAX_VALUE}
567 */
568 public Fraction divide(Fraction fraction) {
569 if (fraction == null) {
570 throw new NullArgumentException(LocalizedFormats.FRACTION);
571 }
572 if (fraction.numerator == 0) {
573 throw new MathArithmeticException(LocalizedFormats.ZERO_FRACTION_TO_DIVIDE_BY,
574 fraction.numerator, fraction.denominator);
575 }
576 return multiply(fraction.reciprocal());
577 }
578
579 /**
580 * Divide the fraction by an integer.
581 * @param i the <tt>integer</tt> to divide by.
582 * @return this * i
583 */
584 public Fraction divide(final int i) {
585 return new Fraction(numerator, denominator * i);
586 }
587
588 /**
589 * <p>
590 * Gets the fraction percentage as a <tt>double</tt>. This calculates the
591 * fraction as the numerator divided by denominator multiplied by 100.
592 * </p>
593 *
594 * @return the fraction percentage as a <tt>double</tt>.
595 */
596 public double percentageValue() {
597 return 100 * doubleValue();
598 }
599
600 /**
601 * <p>Creates a {@code Fraction} instance with the 2 parts
602 * of a fraction Y/Z.</p>
603 *
604 * <p>Any negative signs are resolved to be on the numerator.</p>
605 *
606 * @param numerator the numerator, for example the three in 'three sevenths'
607 * @param denominator the denominator, for example the seven in 'three sevenths'
608 * @return a new fraction instance, with the numerator and denominator reduced
609 * @throws MathArithmeticException if the denominator is {@code zero}
610 */
611 public static Fraction getReducedFraction(int numerator, int denominator) {
612 if (denominator == 0) {
613 throw new MathArithmeticException(LocalizedFormats.ZERO_DENOMINATOR_IN_FRACTION,
614 numerator, denominator);
615 }
616 if (numerator==0) {
617 return ZERO; // normalize zero.
618 }
619 // allow 2^k/-2^31 as a valid fraction (where k>0)
620 if (denominator==Integer.MIN_VALUE && (numerator&1)==0) {
621 numerator/=2; denominator/=2;
622 }
623 if (denominator < 0) {
624 if (numerator==Integer.MIN_VALUE ||
625 denominator==Integer.MIN_VALUE) {
626 throw new MathArithmeticException(LocalizedFormats.OVERFLOW_IN_FRACTION,
627 numerator, denominator);
628 }
629 numerator = -numerator;
630 denominator = -denominator;
631 }
632 // simplify fraction.
633 int gcd = ArithmeticUtils.gcd(numerator, denominator);
634 numerator /= gcd;
635 denominator /= gcd;
636 return new Fraction(numerator, denominator);
637 }
638
639 /**
640 * <p>
641 * Returns the {@code String} representing this fraction, ie
642 * "num / dem" or just "num" if the denominator is one.
643 * </p>
644 *
645 * @return a string representation of the fraction.
646 * @see java.lang.Object#toString()
647 */
648 @Override
649 public String toString() {
650 String str = null;
651 if (denominator == 1) {
652 str = Integer.toString(numerator);
653 } else if (numerator == 0) {
654 str = "0";
655 } else {
656 str = numerator + " / " + denominator;
657 }
658 return str;
659 }
660
661 /** {@inheritDoc} */
662 public FractionField getField() {
663 return FractionField.getInstance();
664 }
665
666 }