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.dfp; import org.apache.commons.math3.Field; import org.apache.commons.math3.FieldElement; /** Field for Decimal floating point instances. * @version $Id: DfpField.java 1416643 2012-12-03 19:37:14Z tn $ * @since 2.2 */ public class DfpField implements Field<Dfp> { /** Enumerate for rounding modes. */ public enum RoundingMode { /** Rounds toward zero (truncation). */ ROUND_DOWN, /** Rounds away from zero if discarded digit is non-zero. */ ROUND_UP, /** Rounds towards nearest unless both are equidistant in which case it rounds away from zero. */ ROUND_HALF_UP, /** Rounds towards nearest unless both are equidistant in which case it rounds toward zero. */ ROUND_HALF_DOWN, /** Rounds towards nearest unless both are equidistant in which case it rounds toward the even neighbor. * This is the default as specified by IEEE 854-1987 */ ROUND_HALF_EVEN, /** Rounds towards nearest unless both are equidistant in which case it rounds toward the odd neighbor. */ ROUND_HALF_ODD, /** Rounds towards positive infinity. */ ROUND_CEIL, /** Rounds towards negative infinity. */ ROUND_FLOOR; } /** IEEE 854-1987 flag for invalid operation. */ public static final int FLAG_INVALID = 1; /** IEEE 854-1987 flag for division by zero. */ public static final int FLAG_DIV_ZERO = 2; /** IEEE 854-1987 flag for overflow. */ public static final int FLAG_OVERFLOW = 4; /** IEEE 854-1987 flag for underflow. */ public static final int FLAG_UNDERFLOW = 8; /** IEEE 854-1987 flag for inexact result. */ public static final int FLAG_INEXACT = 16; /** High precision string representation of √2. */ private static String sqr2String; // Note: the static strings are set up (once) by the ctor and @GuardedBy("DfpField.class") /** High precision string representation of √2 / 2. */ private static String sqr2ReciprocalString; /** High precision string representation of √3. */ private static String sqr3String; /** High precision string representation of √3 / 3. */ private static String sqr3ReciprocalString; /** High precision string representation of π. */ private static String piString; /** High precision string representation of e. */ private static String eString; /** High precision string representation of ln(2). */ private static String ln2String; /** High precision string representation of ln(5). */ private static String ln5String; /** High precision string representation of ln(10). */ private static String ln10String; /** The number of radix digits. * Note these depend on the radix which is 10000 digits, * so each one is equivalent to 4 decimal digits. */ private final int radixDigits; /** A {@link Dfp} with value 0. */ private final Dfp zero; /** A {@link Dfp} with value 1. */ private final Dfp one; /** A {@link Dfp} with value 2. */ private final Dfp two; /** A {@link Dfp} with value √2. */ private final Dfp sqr2; /** A two elements {@link Dfp} array with value √2 split in two pieces. */ private final Dfp[] sqr2Split; /** A {@link Dfp} with value √2 / 2. */ private final Dfp sqr2Reciprocal; /** A {@link Dfp} with value √3. */ private final Dfp sqr3; /** A {@link Dfp} with value √3 / 3. */ private final Dfp sqr3Reciprocal; /** A {@link Dfp} with value π. */ private final Dfp pi; /** A two elements {@link Dfp} array with value π split in two pieces. */ private final Dfp[] piSplit; /** A {@link Dfp} with value e. */ private final Dfp e; /** A two elements {@link Dfp} array with value e split in two pieces. */ private final Dfp[] eSplit; /** A {@link Dfp} with value ln(2). */ private final Dfp ln2; /** A two elements {@link Dfp} array with value ln(2) split in two pieces. */ private final Dfp[] ln2Split; /** A {@link Dfp} with value ln(5). */ private final Dfp ln5; /** A two elements {@link Dfp} array with value ln(5) split in two pieces. */ private final Dfp[] ln5Split; /** A {@link Dfp} with value ln(10). */ private final Dfp ln10; /** Current rounding mode. */ private RoundingMode rMode; /** IEEE 854-1987 signals. */ private int ieeeFlags; /** Create a factory for the specified number of radix digits. * <p> * Note that since the {@link Dfp} class uses 10000 as its radix, each radix * digit is equivalent to 4 decimal digits. This implies that asking for * 13, 14, 15 or 16 decimal digits will really lead to a 4 radix 10000 digits in * all cases. * </p> * @param decimalDigits minimal number of decimal digits. */ public DfpField(final int decimalDigits) { this(decimalDigits, true); } /** Create a factory for the specified number of radix digits. * <p> * Note that since the {@link Dfp} class uses 10000 as its radix, each radix * digit is equivalent to 4 decimal digits. This implies that asking for * 13, 14, 15 or 16 decimal digits will really lead to a 4 radix 10000 digits in * all cases. * </p> * @param decimalDigits minimal number of decimal digits * @param computeConstants if true, the transcendental constants for the given precision * must be computed (setting this flag to false is RESERVED for the internal recursive call) */ private DfpField(final int decimalDigits, final boolean computeConstants) { this.radixDigits = (decimalDigits < 13) ? 4 : (decimalDigits + 3) / 4; this.rMode = RoundingMode.ROUND_HALF_EVEN; this.ieeeFlags = 0; this.zero = new Dfp(this, 0); this.one = new Dfp(this, 1); this.two = new Dfp(this, 2); if (computeConstants) { // set up transcendental constants synchronized (DfpField.class) { // as a heuristic to circumvent Table-Maker's Dilemma, we set the string // representation of the constants to be at least 3 times larger than the // number of decimal digits, also as an attempt to really compute these // constants only once, we set a minimum number of digits computeStringConstants((decimalDigits < 67) ? 200 : (3 * decimalDigits)); // set up the constants at current field accuracy sqr2 = new Dfp(this, sqr2String); sqr2Split = split(sqr2String); sqr2Reciprocal = new Dfp(this, sqr2ReciprocalString); sqr3 = new Dfp(this, sqr3String); sqr3Reciprocal = new Dfp(this, sqr3ReciprocalString); pi = new Dfp(this, piString); piSplit = split(piString); e = new Dfp(this, eString); eSplit = split(eString); ln2 = new Dfp(this, ln2String); ln2Split = split(ln2String); ln5 = new Dfp(this, ln5String); ln5Split = split(ln5String); ln10 = new Dfp(this, ln10String); } } else { // dummy settings for unused constants sqr2 = null; sqr2Split = null; sqr2Reciprocal = null; sqr3 = null; sqr3Reciprocal = null; pi = null; piSplit = null; e = null; eSplit = null; ln2 = null; ln2Split = null; ln5 = null; ln5Split = null; ln10 = null; } } /** Get the number of radix digits of the {@link Dfp} instances built by this factory. * @return number of radix digits */ public int getRadixDigits() { return radixDigits; } /** Set the rounding mode. * If not set, the default value is {@link RoundingMode#ROUND_HALF_EVEN}. * @param mode desired rounding mode * Note that the rounding mode is common to all {@link Dfp} instances * belonging to the current {@link DfpField} in the system and will * affect all future calculations. */ public void setRoundingMode(final RoundingMode mode) { rMode = mode; } /** Get the current rounding mode. * @return current rounding mode */ public RoundingMode getRoundingMode() { return rMode; } /** Get the IEEE 854 status flags. * @return IEEE 854 status flags * @see #clearIEEEFlags() * @see #setIEEEFlags(int) * @see #setIEEEFlagsBits(int) * @see #FLAG_INVALID * @see #FLAG_DIV_ZERO * @see #FLAG_OVERFLOW * @see #FLAG_UNDERFLOW * @see #FLAG_INEXACT */ public int getIEEEFlags() { return ieeeFlags; } /** Clears the IEEE 854 status flags. * @see #getIEEEFlags() * @see #setIEEEFlags(int) * @see #setIEEEFlagsBits(int) * @see #FLAG_INVALID * @see #FLAG_DIV_ZERO * @see #FLAG_OVERFLOW * @see #FLAG_UNDERFLOW * @see #FLAG_INEXACT */ public void clearIEEEFlags() { ieeeFlags = 0; } /** Sets the IEEE 854 status flags. * @param flags desired value for the flags * @see #getIEEEFlags() * @see #clearIEEEFlags() * @see #setIEEEFlagsBits(int) * @see #FLAG_INVALID * @see #FLAG_DIV_ZERO * @see #FLAG_OVERFLOW * @see #FLAG_UNDERFLOW * @see #FLAG_INEXACT */ public void setIEEEFlags(final int flags) { ieeeFlags = flags & (FLAG_INVALID | FLAG_DIV_ZERO | FLAG_OVERFLOW | FLAG_UNDERFLOW | FLAG_INEXACT); } /** Sets some bits in the IEEE 854 status flags, without changing the already set bits. * <p> * Calling this method is equivalent to call {@code setIEEEFlags(getIEEEFlags() | bits)} * </p> * @param bits bits to set * @see #getIEEEFlags() * @see #clearIEEEFlags() * @see #setIEEEFlags(int) * @see #FLAG_INVALID * @see #FLAG_DIV_ZERO * @see #FLAG_OVERFLOW * @see #FLAG_UNDERFLOW * @see #FLAG_INEXACT */ public void setIEEEFlagsBits(final int bits) { ieeeFlags |= bits & (FLAG_INVALID | FLAG_DIV_ZERO | FLAG_OVERFLOW | FLAG_UNDERFLOW | FLAG_INEXACT); } /** Makes a {@link Dfp} with a value of 0. * @return a new {@link Dfp} with a value of 0 */ public Dfp newDfp() { return new Dfp(this); } /** Create an instance from a byte value. * @param x value to convert to an instance * @return a new {@link Dfp} with the same value as x */ public Dfp newDfp(final byte x) { return new Dfp(this, x); } /** Create an instance from an int value. * @param x value to convert to an instance * @return a new {@link Dfp} with the same value as x */ public Dfp newDfp(final int x) { return new Dfp(this, x); } /** Create an instance from a long value. * @param x value to convert to an instance * @return a new {@link Dfp} with the same value as x */ public Dfp newDfp(final long x) { return new Dfp(this, x); } /** Create an instance from a double value. * @param x value to convert to an instance * @return a new {@link Dfp} with the same value as x */ public Dfp newDfp(final double x) { return new Dfp(this, x); } /** Copy constructor. * @param d instance to copy * @return a new {@link Dfp} with the same value as d */ public Dfp newDfp(Dfp d) { return new Dfp(d); } /** Create a {@link Dfp} given a String representation. * @param s string representation of the instance * @return a new {@link Dfp} parsed from specified string */ public Dfp newDfp(final String s) { return new Dfp(this, s); } /** Creates a {@link Dfp} with a non-finite value. * @param sign sign of the Dfp to create * @param nans code of the value, must be one of {@link Dfp#INFINITE}, * {@link Dfp#SNAN}, {@link Dfp#QNAN} * @return a new {@link Dfp} with a non-finite value */ public Dfp newDfp(final byte sign, final byte nans) { return new Dfp(this, sign, nans); } /** Get the constant 0. * @return a {@link Dfp} with value 0 */ public Dfp getZero() { return zero; } /** Get the constant 1. * @return a {@link Dfp} with value 1 */ public Dfp getOne() { return one; } /** {@inheritDoc} */ public Class<? extends FieldElement<Dfp>> getRuntimeClass() { return Dfp.class; } /** Get the constant 2. * @return a {@link Dfp} with value 2 */ public Dfp getTwo() { return two; } /** Get the constant √2. * @return a {@link Dfp} with value √2 */ public Dfp getSqr2() { return sqr2; } /** Get the constant √2 split in two pieces. * @return a {@link Dfp} with value √2 split in two pieces */ public Dfp[] getSqr2Split() { return sqr2Split.clone(); } /** Get the constant √2 / 2. * @return a {@link Dfp} with value √2 / 2 */ public Dfp getSqr2Reciprocal() { return sqr2Reciprocal; } /** Get the constant √3. * @return a {@link Dfp} with value √3 */ public Dfp getSqr3() { return sqr3; } /** Get the constant √3 / 3. * @return a {@link Dfp} with value √3 / 3 */ public Dfp getSqr3Reciprocal() { return sqr3Reciprocal; } /** Get the constant π. * @return a {@link Dfp} with value π */ public Dfp getPi() { return pi; } /** Get the constant π split in two pieces. * @return a {@link Dfp} with value π split in two pieces */ public Dfp[] getPiSplit() { return piSplit.clone(); } /** Get the constant e. * @return a {@link Dfp} with value e */ public Dfp getE() { return e; } /** Get the constant e split in two pieces. * @return a {@link Dfp} with value e split in two pieces */ public Dfp[] getESplit() { return eSplit.clone(); } /** Get the constant ln(2). * @return a {@link Dfp} with value ln(2) */ public Dfp getLn2() { return ln2; } /** Get the constant ln(2) split in two pieces. * @return a {@link Dfp} with value ln(2) split in two pieces */ public Dfp[] getLn2Split() { return ln2Split.clone(); } /** Get the constant ln(5). * @return a {@link Dfp} with value ln(5) */ public Dfp getLn5() { return ln5; } /** Get the constant ln(5) split in two pieces. * @return a {@link Dfp} with value ln(5) split in two pieces */ public Dfp[] getLn5Split() { return ln5Split.clone(); } /** Get the constant ln(10). * @return a {@link Dfp} with value ln(10) */ public Dfp getLn10() { return ln10; } /** Breaks a string representation up into two {@link Dfp}'s. * The split is such that the sum of them is equivalent to the input string, * but has higher precision than using a single Dfp. * @param a string representation of the number to split * @return an array of two {@link Dfp Dfp} instances which sum equals a */ private Dfp[] split(final String a) { Dfp result[] = new Dfp[2]; boolean leading = true; int sp = 0; int sig = 0; char[] buf = new char[a.length()]; for (int i = 0; i < buf.length; i++) { buf[i] = a.charAt(i); if (buf[i] >= '1' && buf[i] <= '9') { leading = false; } if (buf[i] == '.') { sig += (400 - sig) % 4; leading = false; } if (sig == (radixDigits / 2) * 4) { sp = i; break; } if (buf[i] >= '0' && buf[i] <= '9' && !leading) { sig++; } } result[0] = new Dfp(this, new String(buf, 0, sp)); for (int i = 0; i < buf.length; i++) { buf[i] = a.charAt(i); if (buf[i] >= '0' && buf[i] <= '9' && i < sp) { buf[i] = '0'; } } result[1] = new Dfp(this, new String(buf)); return result; } /** Recompute the high precision string constants. * @param highPrecisionDecimalDigits precision at which the string constants mus be computed */ private static void computeStringConstants(final int highPrecisionDecimalDigits) { if (sqr2String == null || sqr2String.length() < highPrecisionDecimalDigits - 3) { // recompute the string representation of the transcendental constants final DfpField highPrecisionField = new DfpField(highPrecisionDecimalDigits, false); final Dfp highPrecisionOne = new Dfp(highPrecisionField, 1); final Dfp highPrecisionTwo = new Dfp(highPrecisionField, 2); final Dfp highPrecisionThree = new Dfp(highPrecisionField, 3); final Dfp highPrecisionSqr2 = highPrecisionTwo.sqrt(); sqr2String = highPrecisionSqr2.toString(); sqr2ReciprocalString = highPrecisionOne.divide(highPrecisionSqr2).toString(); final Dfp highPrecisionSqr3 = highPrecisionThree.sqrt(); sqr3String = highPrecisionSqr3.toString(); sqr3ReciprocalString = highPrecisionOne.divide(highPrecisionSqr3).toString(); piString = computePi(highPrecisionOne, highPrecisionTwo, highPrecisionThree).toString(); eString = computeExp(highPrecisionOne, highPrecisionOne).toString(); ln2String = computeLn(highPrecisionTwo, highPrecisionOne, highPrecisionTwo).toString(); ln5String = computeLn(new Dfp(highPrecisionField, 5), highPrecisionOne, highPrecisionTwo).toString(); ln10String = computeLn(new Dfp(highPrecisionField, 10), highPrecisionOne, highPrecisionTwo).toString(); } } /** Compute π using Jonathan and Peter Borwein quartic formula. * @param one constant with value 1 at desired precision * @param two constant with value 2 at desired precision * @param three constant with value 3 at desired precision * @return π */ private static Dfp computePi(final Dfp one, final Dfp two, final Dfp three) { Dfp sqrt2 = two.sqrt(); Dfp yk = sqrt2.subtract(one); Dfp four = two.add(two); Dfp two2kp3 = two; Dfp ak = two.multiply(three.subtract(two.multiply(sqrt2))); // The formula converges quartically. This means the number of correct // digits is multiplied by 4 at each iteration! Five iterations are // sufficient for about 160 digits, eight iterations give about // 10000 digits (this has been checked) and 20 iterations more than // 160 billions of digits (this has NOT been checked). // So the limit here is considered sufficient for most purposes ... for (int i = 1; i < 20; i++) { final Dfp ykM1 = yk; final Dfp y2 = yk.multiply(yk); final Dfp oneMinusY4 = one.subtract(y2.multiply(y2)); final Dfp s = oneMinusY4.sqrt().sqrt(); yk = one.subtract(s).divide(one.add(s)); two2kp3 = two2kp3.multiply(four); final Dfp p = one.add(yk); final Dfp p2 = p.multiply(p); ak = ak.multiply(p2.multiply(p2)) .subtract(two2kp3.multiply(yk).multiply(one.add(yk).add(yk.multiply(yk)))); if (yk.equals(ykM1)) { break; } } return one.divide(ak); } /** Compute exp(a). * @param a number for which we want the exponential * @param one constant with value 1 at desired precision * @return exp(a) */ public static Dfp computeExp(final Dfp a, final Dfp one) { Dfp y = new Dfp(one); Dfp py = new Dfp(one); Dfp f = new Dfp(one); Dfp fi = new Dfp(one); Dfp x = new Dfp(one); for (int i = 0; i < 10000; i++) { x = x.multiply(a); y = y.add(x.divide(f)); fi = fi.add(one); f = f.multiply(fi); if (y.equals(py)) { break; } py = new Dfp(y); } return y; } /** Compute ln(a). * * Let f(x) = ln(x), * * We know that f'(x) = 1/x, thus from Taylor's theorem we have: * * ----- n+1 n * f(x) = \ (-1) (x - 1) * / ---------------- for 1 <= n <= infinity * ----- n * * or * 2 3 4 * (x-1) (x-1) (x-1) * ln(x) = (x-1) - ----- + ------ - ------ + ... * 2 3 4 * * alternatively, * * 2 3 4 * x x x * ln(x+1) = x - - + - - - + ... * 2 3 4 * * This series can be used to compute ln(x), but it converges too slowly. * * If we substitute -x for x above, we get * * 2 3 4 * x x x * ln(1-x) = -x - - - - - - + ... * 2 3 4 * * Note that all terms are now negative. Because the even powered ones * absorbed the sign. Now, subtract the series above from the previous * one to get ln(x+1) - ln(1-x). Note the even terms cancel out leaving * only the odd ones * * 3 5 7 * 2x 2x 2x * ln(x+1) - ln(x-1) = 2x + --- + --- + ---- + ... * 3 5 7 * * By the property of logarithms that ln(a) - ln(b) = ln (a/b) we have: * * 3 5 7 * x+1 / x x x \ * ln ----- = 2 * | x + ---- + ---- + ---- + ... | * x-1 \ 3 5 7 / * * But now we want to find ln(a), so we need to find the value of x * such that a = (x+1)/(x-1). This is easily solved to find that * x = (a-1)/(a+1). * @param a number for which we want the exponential * @param one constant with value 1 at desired precision * @param two constant with value 2 at desired precision * @return ln(a) */ public static Dfp computeLn(final Dfp a, final Dfp one, final Dfp two) { int den = 1; Dfp x = a.add(new Dfp(a.getField(), -1)).divide(a.add(one)); Dfp y = new Dfp(x); Dfp num = new Dfp(x); Dfp py = new Dfp(y); for (int i = 0; i < 10000; i++) { num = num.multiply(x); num = num.multiply(x); den = den + 2; Dfp t = num.divide(den); y = y.add(t); if (y.equals(py)) { break; } py = new Dfp(y); } return y.multiply(two); } }