| /* |
| * 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.numbers.complex; |
| |
| import java.io.Serializable; |
| import java.util.ArrayList; |
| import java.util.List; |
| import org.apache.commons.numbers.core.Precision; |
| |
| /** |
| * Representation of a Complex number, i.e. a number which has both a |
| * real and imaginary part. |
| * |
| * <p>Implementations of arithmetic operations handle {@code NaN} and |
| * infinite values according to the rules for {@link java.lang.Double}, i.e. |
| * {@link #equals} is an equivalence relation for all instances that have |
| * a {@code NaN} in either real or imaginary part, e.g. the following are |
| * considered equal:</p> |
| * <ul> |
| * <li>{@code 1 + NaNi}</li> |
| * <li>{@code NaN + i}</li> |
| * <li>{@code NaN + NaNi}</li> |
| * </ul> |
| * |
| * <p>Note that this contradicts the IEEE-754 standard for floating |
| * point numbers (according to which the test {@code x == x} must fail if |
| * {@code x} is {@code NaN}). The method |
| * {@link org.apache.commons.numbers.core.Precision#equals(double,double,int) |
| * equals for primitive double} in class {@code Precision} conforms with |
| * IEEE-754 while this class conforms with the standard behavior for Java |
| * object types.</p> |
| * |
| * <p>Arithmetic in this class conforms to the C.99 standard for complex numbers |
| * defined in ISO/IEC 9899, Annex G.<p> |
| * |
| * @see <a href="http://www.open-std.org/JTC1/SC22/WG14/www/standards"> |
| * ISO/IEC 9899 - Programming languages - C</a> |
| */ |
| public final class Complex implements Serializable { |
| /** The square root of -1, a.k.a. "i". */ |
| public static final Complex I = new Complex(0, 1); |
| /** A complex number representing one. */ |
| public static final Complex ONE = new Complex(1, 0); |
| /** A complex number representing zero. */ |
| public static final Complex ZERO = new Complex(0, 0); |
| /** A complex number representing "NaN + NaN i". */ |
| private static final Complex NAN = new Complex(Double.NaN, Double.NaN); |
| /** 3*π/4. */ |
| private static final double PI_3_OVER_4 = 0.75 * Math.PI; |
| /** π/2. */ |
| private static final double PI_OVER_2 = 0.5 * Math.PI; |
| /** π/4. */ |
| private static final double PI_OVER_4 = 0.25 * Math.PI; |
| /** Expected number of elements when parsing text: 2. */ |
| private static final int TWO_ELEMENTS = 2; |
| |
| /** Serializable version identifier. */ |
| private static final long serialVersionUID = 20180201L; |
| |
| /** {@link #toString() String representation}. */ |
| private static final String FORMAT_START = "("; |
| /** {@link #toString() String representation}. */ |
| private static final String FORMAT_END = ")"; |
| /** {@link #toString() String representation}. */ |
| private static final String FORMAT_SEP = ","; |
| |
| /** The imaginary part. */ |
| private final double imaginary; |
| /** The real part. */ |
| private final double real; |
| |
| /** |
| * Define a constructor for a Complex. |
| * This is used in functions that implement trigonomic identities. |
| */ |
| @FunctionalInterface |
| private interface ComplexConstructor { |
| /** |
| * Create a complex number given the real and imaginary parts. |
| * |
| * @param real Real part. |
| * @param imaginary Imaginary part. |
| * @return {@code Complex} object |
| */ |
| Complex create(double real, double imaginary); |
| } |
| |
| /** |
| * Private default constructor. |
| * |
| * @param real Real part. |
| * @param imaginary Imaginary part. |
| */ |
| private Complex(double real, double imaginary) { |
| this.real = real; |
| this.imaginary = imaginary; |
| } |
| |
| /** |
| * Create a complex number given the real and imaginary parts. |
| * |
| * @param real Real part. |
| * @param imaginary Imaginary part. |
| * @return {@code Complex} object |
| */ |
| public static Complex ofCartesian(double real, double imaginary) { |
| return new Complex(real, imaginary); |
| } |
| |
| /** |
| * Create a complex number given the real part. |
| * |
| * @param real Real part. |
| * @return {@code Complex} object |
| */ |
| public static Complex ofReal(double real) { |
| return new Complex(real, 0); |
| } |
| |
| /** |
| * Creates a Complex from its polar representation. |
| * |
| * <p>If {@code r} is infinite and {@code theta} is finite, infinite or NaN |
| * values may be returned in parts of the result, following the rules for |
| * double arithmetic.</p> |
| * |
| * <pre> |
| * Examples: |
| * {@code |
| * polar2Complex(INFINITY, \(\pi\)) = INFINITY + INFINITY i |
| * polar2Complex(INFINITY, 0) = INFINITY + NaN i |
| * polar2Complex(INFINITY, \(-\frac{\pi}{4}\)) = INFINITY - INFINITY i |
| * polar2Complex(INFINITY, \(5\frac{\pi}{4}\)) = -INFINITY - INFINITY i } |
| * </pre> |
| * |
| * @param r the modulus of the complex number to create |
| * @param theta the argument of the complex number to create |
| * @return {@code Complex} |
| * @throws IllegalArgumentException if {@code r} is non-positive |
| */ |
| public static Complex ofPolar(double r, double theta) { |
| if (r <= 0) { |
| throw new IllegalArgumentException("Non-positive polar modulus argument: " + r); |
| } |
| return new Complex(r * Math.cos(theta), r * Math.sin(theta)); |
| } |
| |
| /** |
| * For a real constructor argument x, returns a new Complex object c |
| * where {@code c = cos(x) + i sin (x)}. |
| * |
| * @param x {@code double} to build the cis number |
| * @return {@code Complex} |
| */ |
| public static Complex ofCis(double x) { |
| return new Complex(Math.cos(x), Math.sin(x)); |
| } |
| |
| /** |
| * Parses a string that would be produced by {@link #toString()} |
| * and instantiates the corresponding object. |
| * |
| * @param s String representation. |
| * @return an instance. |
| * @throws NumberFormatException if the string does not conform |
| * to the specification. |
| */ |
| public static Complex parse(String s) { |
| final int startParen = s.indexOf(FORMAT_START); |
| if (startParen != 0) { |
| throw new ComplexParsingException("Expected start string: " + FORMAT_START); |
| } |
| final int len = s.length(); |
| final int endParen = s.indexOf(FORMAT_END); |
| if (endParen != len - 1) { |
| throw new ComplexParsingException("Expected end string: " + FORMAT_END); |
| } |
| final String[] elements = s.substring(1, s.length() - 1).split(FORMAT_SEP); |
| if (elements.length != TWO_ELEMENTS) { |
| throw new ComplexParsingException("Incorrect number of parts: Expected 2 but was " + |
| elements.length + |
| " (separator is '" + FORMAT_SEP + "')"); |
| } |
| |
| final double re; |
| try { |
| re = Double.parseDouble(elements[0]); |
| } catch (final NumberFormatException ex) { |
| throw new ComplexParsingException("Could not parse real part" + elements[0], ex); |
| } |
| final double im; |
| try { |
| im = Double.parseDouble(elements[1]); |
| } catch (final NumberFormatException ex) { |
| throw new ComplexParsingException("Could not parse imaginary part" + elements[1], ex); |
| } |
| |
| return ofCartesian(re, im); |
| } |
| |
| /** |
| * Returns true if either the real <em>or</em> imaginary component of the Complex is NaN |
| * <em>and</em> the Complex is not infinite. |
| * |
| * <p>Note that in contrast to {@link Double#isNaN()}: |
| * <ul> |
| * <li>There is more than one complex number that can return {@code true}. |
| * <li>Different representations of NaN can be distinguished by the |
| * {@link #equals(Object) Complex.equals(Object)} method. |
| * </ul> |
| * |
| * @return {@code true} if this instance contains NaN and no infinite parts. |
| * @see Double#isNaN(double) |
| * @see #isInfinite() |
| * @see #equals(Object) Complex.equals(Object) |
| */ |
| public boolean isNaN() { |
| if (Double.isNaN(real) || Double.isNaN(imaginary)) { |
| return !isInfinite(); |
| } |
| return false; |
| } |
| |
| /** |
| * Returns true if either real or imaginary component of the Complex is infinite. |
| * |
| * <p>Note: A complex or imaginary value with at least one infinite part is regarded |
| * as an infinity (even if its other part is a NaN).</p> |
| * |
| * @return {@code true} if this instance contains an infinite value. |
| * @see Double#isInfinite(double) |
| */ |
| public boolean isInfinite() { |
| return Double.isInfinite(real) || Double.isInfinite(imaginary); |
| } |
| |
| /** |
| * Returns true if both real and imaginary component of the Complex are finite. |
| * |
| * @return {@code true} if this instance contains finite values. |
| * @see Double#isFinite(double) |
| */ |
| public boolean isFinite() { |
| return Double.isFinite(real) && Double.isFinite(imaginary); |
| } |
| |
| /** |
| * Returns projection of this complex number onto the Riemann sphere, |
| * i.e. all infinities (including those with an NaN component) |
| * project onto real infinity, as described in the |
| * <a href="http://pubs.opengroup.org/onlinepubs/9699919799/functions/cproj.html"> |
| * IEEE and ISO C standards</a>. |
| * |
| * @return {@code Complex} projected onto the Riemann sphere. |
| */ |
| public Complex proj() { |
| if (Double.isInfinite(real) || |
| Double.isInfinite(imaginary)) { |
| return new Complex(Double.POSITIVE_INFINITY, 0); |
| } |
| return this; |
| } |
| |
| /** |
| * Return the absolute value of this complex number. |
| * This code follows the <a href="http://www.iso-9899.info/wiki/The_Standard">ISO C Standard</a>, Annex G, |
| * in calculating the returned value (i.e. the hypot(x,y) method) |
| * and in handling of NaNs. |
| * |
| * @return the absolute value. |
| */ |
| public double abs() { |
| // Delegate |
| return Math.hypot(real, imaginary); |
| } |
| |
| /** |
| * Returns a {@code Complex} whose value is |
| * {@code (this + addend)}. |
| * Uses the definitional formula |
| * <pre> |
| * (a + bi) + (c + di) = (a+c) + (b+d)i |
| * </pre> |
| * |
| * @param addend Value to be added to this {@code Complex}. |
| * @return {@code this + addend}. |
| */ |
| public Complex add(Complex addend) { |
| return new Complex(real + addend.real, |
| imaginary + addend.imaginary); |
| } |
| |
| /** |
| * Returns a {@code Complex} whose value is {@code (this + addend)}, |
| * with {@code addend} interpreted as a real number. |
| * |
| * @param addend Value to be added to this {@code Complex}. |
| * @return {@code this + addend}. |
| * @see #add(Complex) |
| */ |
| public Complex add(double addend) { |
| return new Complex(real + addend, imaginary); |
| } |
| |
| /** |
| * Returns the conjugate of this complex number. |
| * The conjugate of {@code a + bi} is {@code a - bi}. |
| * |
| * @return the conjugate of this complex object. |
| */ |
| public Complex conjugate() { |
| return new Complex(real, -imaginary); |
| } |
| |
| /** |
| * Returns the conjugate of this complex number |
| * (C++11 grammar). |
| * |
| * @return the conjugate of this complex object. |
| * @see #conjugate() |
| */ |
| public Complex conj() { |
| return conjugate(); |
| } |
| |
| /** |
| * Returns a {@code Complex} whose value is |
| * {@code (this / divisor)}. |
| * Implements the definitional formula |
| * <pre> |
| * <code> |
| * a + bi ac + bd + (bc - ad)i |
| * ------ = -------------------- |
| * c + di c<sup>2</sup> + d<sup>2</sup> |
| * </code> |
| * </pre> |
| * |
| * <p>Recalculates to recover infinities as specified in C.99 |
| * standard G.5.1. Method is fully in accordance with |
| * C++11 standards for complex numbers.</p> |
| * |
| * @param divisor Value by which this {@code Complex} is to be divided. |
| * @return {@code this / divisor}. |
| */ |
| public Complex divide(Complex divisor) { |
| double a = real; |
| double b = imaginary; |
| double c = divisor.getReal(); |
| double d = divisor.getImaginary(); |
| int ilogbw = 0; |
| final double logbw = Math.log(Math.max(Math.abs(c), Math.abs(d))) / Math.log(2); |
| if (Double.isFinite(logbw)) { |
| ilogbw = (int)logbw; |
| c = Math.scalb(c, -ilogbw); |
| d = Math.scalb(d, -ilogbw); |
| } |
| final double denom = c * c + d * d; |
| double x = Math.scalb((a * c + b * d) / denom, -ilogbw); |
| double y = Math.scalb((b * c - a * d) / denom, -ilogbw); |
| // Recover infinities and zeros that computed as NaN+iNaN |
| // the only cases are nonzero/zero, infinite/finite, and finite/infinite, ... |
| // -------------- |
| // Modification from the listing in ISO C.99 G.5.1 (8): |
| // Prevent overflow in (a * c + b * d) and (b * c - a * d). |
| // It is only the sign that is important. not the magnitude. |
| // -------------- |
| if (Double.isNaN(x) && Double.isNaN(y)) { |
| if ((denom == 0.0) && |
| (!Double.isNaN(a) || !Double.isNaN(b))) { |
| // nonzero/zero |
| x = Math.copySign(Double.POSITIVE_INFINITY, c) * a; |
| y = Math.copySign(Double.POSITIVE_INFINITY, c) * b; |
| } else if ((Double.isInfinite(a) || Double.isInfinite(b)) && |
| Double.isFinite(c) && Double.isFinite(d)) { |
| // infinite/finite |
| a = boxInfinity(a); |
| b = boxInfinity(b); |
| x = Double.POSITIVE_INFINITY * computeACplusBD(a, b, c, d); |
| y = Double.POSITIVE_INFINITY * computeBCminusAD(a, b, c, d); |
| } else if (divisor.isInfinite() && |
| Double.isFinite(a) && Double.isFinite(b)) { |
| // finite/infinite |
| c = boxInfinity(c); |
| d = boxInfinity(d); |
| x = 0.0 * computeACplusBD(a, b, c, d); |
| y = 0.0 * computeBCminusAD(a, b, c, d); |
| } |
| } |
| return new Complex(x, y); |
| } |
| |
| /** |
| * Compute {@code a*c + b*d} without overflow. |
| * It is assumed: either {@code a} and {@code b} or {@code c} and {@code d} are |
| * either zero or one (i.e. a boxed infinity); and the sign of the result is important, |
| * not the value. |
| * |
| * @param a the a |
| * @param b the b |
| * @param c the c |
| * @param d the d |
| * @return the result |
| */ |
| private static double computeACplusBD(double a, double b, double c, double d) { |
| final double ac = a * c; |
| final double bd = b * d; |
| final double result = ac + bd; |
| return Double.isFinite(result) ? |
| result : |
| // Overflow. Just divide by 2 as it is the sign of the result that matters. |
| ac * 0.5 + bd * 0.5; |
| } |
| |
| /** |
| * Compute {@code b*c - a*d} without overflow. |
| * It is assumed: either {@code a} and {@code b} or {@code c} and {@code d} are |
| * either zero or one (i.e. a boxed infinity); and the sign of the result is important, |
| * not the value. |
| * |
| * @param a the a |
| * @param b the b |
| * @param c the c |
| * @param d the d |
| * @return the result |
| */ |
| private static double computeBCminusAD(double a, double b, double c, double d) { |
| final double bc = b * c; |
| final double ad = a * d; |
| final double result = bc - ad; |
| return Double.isFinite(result) ? |
| result : |
| // Overflow. Just divide by 2 as it is the sign of the result that matters. |
| bc * 0.5 + ad * 0.5; |
| } |
| |
| /** |
| * Returns a {@code Complex} whose value is {@code (this / divisor)}, |
| * with {@code divisor} interpreted as a real number. |
| * |
| * @param divisor Value by which this {@code Complex} is to be divided. |
| * @return {@code this / divisor}. |
| * @see #divide(Complex) |
| */ |
| public Complex divide(double divisor) { |
| return divide(new Complex(divisor, 0)); |
| } |
| |
| /** |
| * Returns the multiplicative inverse of this instance. |
| * |
| * @return {@code 1 / this}. |
| * @see #divide(Complex) |
| */ |
| public Complex reciprocal() { |
| if (Math.abs(real) < Math.abs(imaginary)) { |
| final double q = real / imaginary; |
| final double scale = 1. / (real * q + imaginary); |
| double scaleQ = 0; |
| if (q != 0 && |
| scale != 0) { |
| scaleQ = scale * q; |
| } |
| return new Complex(scaleQ, -scale); |
| } |
| final double q = imaginary / real; |
| final double scale = 1. / (imaginary * q + real); |
| double scaleQ = 0; |
| if (q != 0 && |
| scale != 0) { |
| scaleQ = scale * q; |
| } |
| return new Complex(scale, -scaleQ); |
| } |
| |
| /** |
| * Test for equality with another object. If the other object is a {@code Complex} then a |
| * comparison is made of the real and imaginary parts; otherwise {@code false} is returned. |
| * |
| * <p>If both the real and imaginary parts of two complex numbers |
| * are exactly the same the two {@code Complex} objects are considered to be equal. |
| * For this purpose, two {@code double} values are considered to be |
| * the same if and only if the method {@link Double #doubleToLongBits(double)} |
| * returns the identical {@code long} value when applied to each. |
| * |
| * <p>Note that in most cases, for two instances of class |
| * {@code Complex}, {@code c1} and {@code c2}, the |
| * value of {@code c1.equals(c2)} is {@code true} if and only if |
| * |
| * <pre> |
| * {@code c1.getReal() == c2.getReal() && c1.getImaginary() == c2.getImaginary()} |
| * </pre> |
| * |
| * <p>also has the value {@code true}. However, there are exceptions: |
| * |
| * <ul> |
| * <li> |
| * Instances that contain {@code NaN} values in the same part |
| * are considered to be equal for that part, even though {@code Double.NaN==Double.NaN} |
| * has the value {@code false}. |
| * </li> |
| * <li> |
| * Instances that share a {@code NaN} value in one part |
| * but have different values in the other part are <em>not</em> considered equal. |
| * </li> |
| * <li> |
| * Instances that contain different representations of zero in the same part |
| * are <em>not</em> considered to be equal for that part, even though {@code -0.0==0.0} |
| * has the value {@code true}. |
| * </li> |
| * </ul> |
| * |
| * <p>The behavior is the same as if the components of the two complex numbers were passed |
| * to {@link java.util.Arrays#equals(double[], double[]) Arrays.equals(double[], double[])}: |
| * |
| * <pre> |
| * <code> |
| * Arrays.equals(new double[]{c1.getReal(), c1.getImaginary()}, |
| * new double[]{c2.getReal(), c2.getImaginary()}); |
| * </code> |
| * </pre> |
| * |
| * @param other Object to test for equality with this instance. |
| * @return {@code true} if the objects are equal, {@code false} if object |
| * is {@code null}, not an instance of {@code Complex}, or not equal to |
| * this instance. |
| * @see java.lang.Double#doubleToLongBits(double) |
| * @see java.util.Arrays#equals(double[], double[]) |
| */ |
| @Override |
| public boolean equals(Object other) { |
| if (this == other) { |
| return true; |
| } |
| if (other instanceof Complex) { |
| final Complex c = (Complex) other; |
| return equals(real, c.real) && |
| equals(imaginary, c.imaginary); |
| } |
| return false; |
| } |
| |
| /** |
| * Test for the floating-point equality between Complex objects. |
| * It returns {@code true} if both arguments are equal or within the |
| * range of allowed error (inclusive). |
| * |
| * @param x First value (cannot be {@code null}). |
| * @param y Second value (cannot be {@code null}). |
| * @param maxUlps {@code (maxUlps - 1)} is the number of floating point |
| * values between the real (resp. imaginary) parts of {@code x} and |
| * {@code y}. |
| * @return {@code true} if there are fewer than {@code maxUlps} floating |
| * point values between the real (resp. imaginary) parts of {@code x} |
| * and {@code y}. |
| * |
| * @see Precision#equals(double,double,int) |
| */ |
| public static boolean equals(Complex x, |
| Complex y, |
| int maxUlps) { |
| return Precision.equals(x.real, y.real, maxUlps) && |
| Precision.equals(x.imaginary, y.imaginary, maxUlps); |
| } |
| |
| /** |
| * Returns {@code true} iff the values are equal as defined by |
| * {@link #equals(Complex,Complex,int) equals(x, y, 1)}. |
| * |
| * @param x First value (cannot be {@code null}). |
| * @param y Second value (cannot be {@code null}). |
| * @return {@code true} if the values are equal. |
| */ |
| public static boolean equals(Complex x, |
| Complex y) { |
| return equals(x, y, 1); |
| } |
| |
| /** |
| * Returns {@code true} if, both for the real part and for the imaginary |
| * part, there is no double value strictly between the arguments or the |
| * difference between them is within the range of allowed error |
| * (inclusive). Returns {@code false} if either of the arguments is NaN. |
| * |
| * @param x First value (cannot be {@code null}). |
| * @param y Second value (cannot be {@code null}). |
| * @param eps Amount of allowed absolute error. |
| * @return {@code true} if the values are two adjacent floating point |
| * numbers or they are within range of each other. |
| * |
| * @see Precision#equals(double,double,double) |
| */ |
| public static boolean equals(Complex x, |
| Complex y, |
| double eps) { |
| return Precision.equals(x.real, y.real, eps) && |
| Precision.equals(x.imaginary, y.imaginary, eps); |
| } |
| |
| /** |
| * Returns {@code true} if, both for the real part and for the imaginary |
| * part, there is no double value strictly between the arguments or the |
| * relative difference between them is smaller or equal to the given |
| * tolerance. Returns {@code false} if either of the arguments is NaN. |
| * |
| * @param x First value (cannot be {@code null}). |
| * @param y Second value (cannot be {@code null}). |
| * @param eps Amount of allowed relative error. |
| * @return {@code true} if the values are two adjacent floating point |
| * numbers or they are within range of each other. |
| * |
| * @see Precision#equalsWithRelativeTolerance(double,double,double) |
| */ |
| public static boolean equalsWithRelativeTolerance(Complex x, Complex y, |
| double eps) { |
| return Precision.equalsWithRelativeTolerance(x.real, y.real, eps) && |
| Precision.equalsWithRelativeTolerance(x.imaginary, y.imaginary, eps); |
| } |
| |
| /** |
| * Get a hash code for the complex number. |
| * |
| * <p>The behavior is the same as if the components of the complex number were passed |
| * to {@link java.util.Arrays#hashCode(double[]) Arrays.hashCode(double[])}: |
| * <pre> |
| * {@code Arrays.hashCode(new double[]{getReal(), getImaginary()})} |
| * </pre> |
| * |
| * @return a hash code value for this object. |
| * @see java.util.Arrays#hashCode(double[]) Arrays.hashCode(double[]) |
| */ |
| @Override |
| public int hashCode() { |
| return 31 * (31 + Double.hashCode(real)) + Double.hashCode(imaginary); |
| } |
| |
| /** |
| * Access the imaginary part. |
| * |
| * @return the imaginary part. |
| */ |
| public double getImaginary() { |
| return imaginary; |
| } |
| |
| /** |
| * Access the imaginary part (C++ grammar). |
| * |
| * @return the imaginary part. |
| */ |
| public double imag() { |
| return imaginary; |
| } |
| |
| /** |
| * Access the real part. |
| * |
| * @return the real part. |
| */ |
| public double getReal() { |
| return real; |
| } |
| |
| /** |
| * Access the real part (C++ grammar). |
| * |
| * @return the real part. |
| */ |
| public double real() { |
| return real; |
| } |
| |
| /** |
| * Returns a {@code Complex} whose value is {@code this * factor}. |
| * Implements the definitional formula: |
| * <pre> |
| * (a + bi)(c + di) = (ac - bd) + (ad + bc)i |
| * </pre> |
| * |
| * <p>Recalculates to recover infinities as specified in C.99 |
| * standard G.5.1. Method is fully in accordance with |
| * C++11 standards for complex numbers.</p> |
| * |
| * @param factor value to be multiplied by this {@code Complex}. |
| * @return {@code this * factor}. |
| */ |
| public Complex multiply(Complex factor) { |
| double a = real; |
| double b = imaginary; |
| double c = factor.getReal(); |
| double d = factor.getImaginary(); |
| final double ac = a * c; |
| final double bd = b * d; |
| final double ad = a * d; |
| final double bc = b * c; |
| double x = ac - bd; |
| double y = ad + bc; |
| |
| // -------------- |
| // NaN can occur if: |
| // - any of (a,b,c,d) are NaN (for NaN or Infinite complex numbers) |
| // - a multiplication of infinity by zero (ac,bd,ad,bc). |
| // - a subtraction of infinity from infinity (e.g. ac - bd) |
| // Note that (ac,bd,ad,bc) can be infinite due to overflow. |
| // |
| // Detect a NaN result and perform correction. |
| // |
| // Modification from the listing in ISO C.99 G.5.1 (6) |
| // Do not correct infinity multiplied by zero. This is left as NaN. |
| // -------------- |
| |
| if (Double.isNaN(x) && Double.isNaN(y)) { |
| // Recover infinities that computed as NaN+iNaN ... |
| boolean recalc = false; |
| if (isInfinite() && isNotZero(c, d)) { |
| // This complex is infinite. |
| // "Box" the infinity and change NaNs in the other factor to 0. |
| a = boxInfinity(a); |
| b = boxInfinity(b); |
| c = changeNaNtoZero(c); |
| d = changeNaNtoZero(d); |
| recalc = true; |
| } |
| // (c, d) may have been corrected so do not use factor.isInfinite(). |
| if ((Double.isInfinite(c) || Double.isInfinite(d)) && |
| isNotZero(a, b)) { |
| // This other complex is infinite. |
| // "Box" the infinity and change NaNs in the other factor to 0. |
| c = boxInfinity(c); |
| d = boxInfinity(d); |
| a = changeNaNtoZero(a); |
| b = changeNaNtoZero(b); |
| recalc = true; |
| } |
| if (!recalc && (Double.isInfinite(ac) || Double.isInfinite(bd) || |
| Double.isInfinite(ad) || Double.isInfinite(bc))) { |
| // The result overflowed to infinity. |
| // Recover infinities from overflow by changing NaNs to 0 ... |
| a = changeNaNtoZero(a); |
| b = changeNaNtoZero(b); |
| c = changeNaNtoZero(c); |
| d = changeNaNtoZero(d); |
| recalc = true; |
| } |
| if (recalc) { |
| x = Double.POSITIVE_INFINITY * (a * c - b * d); |
| y = Double.POSITIVE_INFINITY * (a * d + b * c); |
| } |
| } |
| return new Complex(x, y); |
| } |
| |
| /** |
| * Box values for the real or imaginary component of an infinite complex number. |
| * Any infinite value will be returned as one. Non-infinite values will be returned as zero. |
| * The sign is maintained. |
| * |
| * <pre> |
| * inf = 1 |
| * -inf = -1 |
| * x = 0 |
| * -x = -0 |
| * </pre> |
| * |
| * @param component the component |
| * @return the boxed value |
| */ |
| private static double boxInfinity(double component) { |
| return Math.copySign(Double.isInfinite(component) ? 1.0 : 0.0, component); |
| } |
| |
| /** |
| * Checks if the complex number is not zero. |
| * |
| * @param real the real component |
| * @param imaginary the imaginary component |
| * @return true if the complex is zero |
| */ |
| private static boolean isNotZero(double real, double imaginary) { |
| // The use of equals is deliberate. |
| // This method must distinguish NaN from zero thus ruling out: |
| // (real != 0.0 || imaginary != 0.0) |
| return !(real == 0.0 && imaginary == 0.0); |
| } |
| |
| /** |
| * Change NaN to zero preserving the sign; otherwise return the value. |
| * |
| * @param value the value |
| * @return the new value |
| */ |
| private static double changeNaNtoZero(double value) { |
| return Double.isNaN(value) ? Math.copySign(0.0, value) : value; |
| } |
| |
| /** |
| * Returns a {@code Complex} whose value is {@code this * factor}, with {@code factor} |
| * interpreted as a integer number. |
| * |
| * @param factor value to be multiplied by this {@code Complex}. |
| * @return {@code this * factor}. |
| * @see #multiply(Complex) |
| */ |
| public Complex multiply(final int factor) { |
| return new Complex(real * factor, imaginary * factor); |
| } |
| |
| /** |
| * Returns a {@code Complex} whose value is {@code this * factor}, with {@code factor} |
| * interpreted as a real number. |
| * |
| * @param factor value to be multiplied by this {@code Complex}. |
| * @return {@code this * factor}. |
| * @see #multiply(Complex) |
| */ |
| public Complex multiply(double factor) { |
| return new Complex(real * factor, imaginary * factor); |
| } |
| |
| /** |
| * Returns a {@code Complex} whose value is {@code (-this)}. |
| * |
| * @return {@code -this}. |
| */ |
| public Complex negate() { |
| return new Complex(-real, -imaginary); |
| } |
| |
| /** |
| * Returns a {@code Complex} whose value is |
| * {@code (this - subtrahend)}. |
| * Uses the definitional formula |
| * <p> |
| * {@code (a + bi) - (c + di) = (a-c) + (b-d)i} |
| * </p> |
| * |
| * @param subtrahend value to be subtracted from this {@code Complex}. |
| * @return {@code this - subtrahend}. |
| */ |
| public Complex subtract(Complex subtrahend) { |
| return new Complex(real - subtrahend.real, |
| imaginary - subtrahend.imaginary); |
| } |
| |
| /** |
| * Returns a {@code Complex} whose value is |
| * {@code (this - subtrahend)}. |
| * |
| * @param subtrahend value to be subtracted from this {@code Complex}. |
| * @return {@code this - subtrahend}. |
| * @see #subtract(Complex) |
| */ |
| public Complex subtract(double subtrahend) { |
| return new Complex(real - subtrahend, imaginary); |
| } |
| |
| /** |
| * Compute the |
| * <a href="http://mathworld.wolfram.com/InverseCosine.html"> |
| * inverse cosine</a> of this complex number. |
| * Implements the formula: |
| * <pre> |
| * <code> |
| * acos(z) = -i (log(z + i (sqrt(1 - z<sup>2</sup>)))) |
| * </code> |
| * </pre> |
| * |
| * @return the inverse cosine of this complex number. |
| */ |
| public Complex acos() { |
| if (real == 0 && |
| Double.isNaN(imaginary)) { |
| return new Complex(PI_OVER_2, Double.NaN); |
| } else if (neitherInfiniteNorZeroNorNaN(real) && |
| imaginary == Double.POSITIVE_INFINITY) { |
| return new Complex(PI_OVER_2, Double.NEGATIVE_INFINITY); |
| } else if (real == Double.NEGATIVE_INFINITY && |
| imaginary == 1) { |
| return new Complex(Math.PI, Double.NEGATIVE_INFINITY); |
| } else if (real == Double.POSITIVE_INFINITY && |
| imaginary == 1) { |
| return new Complex(0, Double.NEGATIVE_INFINITY); |
| } else if (real == Double.NEGATIVE_INFINITY && |
| imaginary == Double.POSITIVE_INFINITY) { |
| return new Complex(PI_3_OVER_4, Double.NEGATIVE_INFINITY); |
| } else if (real == Double.POSITIVE_INFINITY && |
| imaginary == Double.POSITIVE_INFINITY) { |
| return new Complex(PI_OVER_4, Double.NEGATIVE_INFINITY); |
| } else if (Double.isInfinite(real) && |
| Double.isNaN(imaginary)) { |
| // Swap real and imaginary |
| return new Complex(Double.NaN, real); |
| } else if (Double.isNaN(real) && |
| imaginary == Double.POSITIVE_INFINITY) { |
| return new Complex(Double.NaN, Double.NEGATIVE_INFINITY); |
| } |
| return add(sqrt1z().multiplyByI()).log().multiplyByNegI(); |
| } |
| /** |
| * Compute the |
| * <a href="http://mathworld.wolfram.com/InverseSine.html"> |
| * inverse sine</a> of this complex number. |
| * <pre> |
| * <code> |
| * asin(z) = -i (log(sqrt(1 - z<sup>2</sup>) + iz)) |
| * </code> |
| * </pre> |
| * |
| * <p>As per the C.99 standard this function is computed using the trigonomic identity:</p> |
| * <pre> |
| * asin(z) = -i asinh(iz) |
| * </pre> |
| * |
| * @return the inverse sine of this complex number |
| */ |
| public Complex asin() { |
| // Define in terms of asinh |
| // asin(z) = -i asinh(iz) |
| return multiplyByI().asinh().multiplyByNegI(); |
| } |
| |
| /** |
| * Compute the |
| * <a href="http://mathworld.wolfram.com/InverseTangent.html"> |
| * inverse tangent</a> of this complex number. |
| * Implements the formula: |
| * <pre> |
| * atan(z) = (i/2) log((i + z)/(i - z)) |
| * </pre> |
| * |
| * <p>As per the C.99 standard this function is computed using the trigonomic identity:</p> |
| * <pre> |
| * atan(z) = -i atanh(iz) |
| * </pre> |
| * |
| * @return the inverse tangent of this complex number |
| */ |
| public Complex atan() { |
| // Define in terms of atanh |
| // atan(z) = -i atanh(iz) |
| return multiplyByI().atanh().multiplyByNegI(); |
| } |
| |
| /** |
| * Multiply the Complex by I. |
| * |
| * @return the result (iz) |
| */ |
| private Complex multiplyByI() { |
| return new Complex(-imaginary, real); |
| } |
| |
| /** |
| * Multiply the Complex by -I. |
| * |
| * @return the result (-iz) |
| */ |
| private Complex multiplyByNegI() { |
| return new Complex(imaginary, -real); |
| } |
| |
| /** |
| * Compute the |
| * <a href="http://mathworld.wolfram.com/InverseHyperbolicSine.html"> |
| * inverse hyperbolic sine</a> of this complex number. |
| * Implements the formula: |
| * <pre> |
| * asinh(z) = log(z+sqrt(z^2+1)) |
| * </pre> |
| * |
| * @return the inverse hyperbolic sine of this complex number |
| */ |
| public Complex asinh() { |
| if (neitherInfiniteNorZeroNorNaN(real) && |
| imaginary == Double.POSITIVE_INFINITY) { |
| return new Complex(Double.POSITIVE_INFINITY, PI_OVER_2); |
| } else if (real == Double.POSITIVE_INFINITY && |
| Double.isFinite(imaginary)) { |
| return new Complex(Double.POSITIVE_INFINITY, 0); |
| } else if (real == Double.POSITIVE_INFINITY && |
| imaginary == Double.POSITIVE_INFINITY) { |
| return new Complex(Double.POSITIVE_INFINITY, PI_OVER_4); |
| } else if (real == Double.POSITIVE_INFINITY && |
| Double.isNaN(imaginary)) { |
| return new Complex(Double.POSITIVE_INFINITY, Double.NaN); |
| } else if (Double.isNaN(real) && |
| imaginary == 0) { |
| return new Complex(Double.NaN, 0); |
| } else if (Double.isNaN(real) && |
| imaginary == Double.POSITIVE_INFINITY) { |
| return new Complex(Double.POSITIVE_INFINITY, Double.NaN); |
| } |
| return square().add(ONE).sqrt().add(this).log(); |
| } |
| |
| /** |
| * Compute the |
| * <a href="http://mathworld.wolfram.com/InverseHyperbolicTangent.html"> |
| * inverse hyperbolic tangent</a> of this complex number. |
| * Implements the formula: |
| * <pre> |
| * atanh(z) = log((1+z)/(1-z))/2 |
| * </pre> |
| * |
| * @return the inverse hyperbolic tangent of this complex number |
| */ |
| public Complex atanh() { |
| if (real == 0 && |
| Double.isNaN(imaginary)) { |
| return new Complex(0, Double.NaN); |
| } else if (neitherInfiniteNorZeroNorNaN(real) && |
| imaginary == 0) { |
| return new Complex(Double.POSITIVE_INFINITY, 0); |
| } else if (neitherInfiniteNorZeroNorNaN(real) && |
| imaginary == Double.POSITIVE_INFINITY) { |
| return new Complex(0, PI_OVER_2); |
| } else if (real == Double.POSITIVE_INFINITY && |
| neitherInfiniteNorZeroNorNaN(imaginary)) { |
| return new Complex(0, PI_OVER_2); |
| } else if (real == Double.POSITIVE_INFINITY && |
| imaginary == Double.POSITIVE_INFINITY) { |
| return new Complex(0, PI_OVER_2); |
| } else if (real == Double.POSITIVE_INFINITY && |
| Double.isNaN(imaginary)) { |
| return new Complex(0, Double.NaN); |
| } else if (Double.isNaN(real) && |
| imaginary == Double.POSITIVE_INFINITY) { |
| return new Complex(0, PI_OVER_2); |
| } |
| return add(ONE).divide(ONE.subtract(this)).log().multiply(0.5); |
| } |
| |
| /** |
| * Compute the |
| * <a href="http://mathworld.wolfram.com/InverseHyperbolicCosine.html"> |
| * inverse hyperbolic cosine</a> of this complex number. |
| * Implements the formula: |
| * <pre> |
| * acosh(z) = log(z+sqrt(z^2-1)) |
| * </pre> |
| * |
| * @return the inverse hyperbolic cosine of this complex number |
| */ |
| public Complex acosh() { |
| if (Double.isFinite(real)) { |
| if (real == 0 && |
| imaginary == 0) { |
| return new Complex(0, PI_OVER_2); |
| } |
| if (imaginary == Double.POSITIVE_INFINITY) { |
| return new Complex(Double.POSITIVE_INFINITY, PI_OVER_2); |
| } |
| if (Double.isNaN(imaginary)) { |
| return NAN; |
| } |
| } else if (Double.isInfinite(real)) { |
| if (real == Double.NEGATIVE_INFINITY && |
| positiveSignedFinite(imaginary)) { |
| return new Complex(Double.POSITIVE_INFINITY, Math.PI); |
| } |
| if (real == Double.POSITIVE_INFINITY && |
| positiveSignedFinite(imaginary)) { |
| return new Complex(Double.POSITIVE_INFINITY, 0); |
| } |
| if (real == Double.NEGATIVE_INFINITY && |
| imaginary == Double.POSITIVE_INFINITY) { |
| return new Complex(Double.POSITIVE_INFINITY, PI_3_OVER_4); |
| } |
| if (real == Double.POSITIVE_INFINITY && |
| imaginary == Double.POSITIVE_INFINITY) { |
| return new Complex(Double.POSITIVE_INFINITY, PI_OVER_4); |
| } |
| if (Double.isNaN(imaginary)) { |
| return new Complex(Double.POSITIVE_INFINITY, Double.NaN); |
| } |
| } else if (Double.isNaN(real)) { |
| if (imaginary == Double.POSITIVE_INFINITY) { |
| return new Complex(Double.POSITIVE_INFINITY, Double.NaN); |
| } |
| if (Double.isFinite(imaginary) || Double.isNaN(imaginary)) { |
| // optionally raises the ‘‘invalid’’ floating-point exception, for finite y. |
| // No condition for imaginary as negative infinity |
| return NAN; |
| } |
| } |
| return square().subtract(ONE).sqrt().add(this).log(); |
| } |
| |
| /** |
| * Compute the square of this complex number. |
| * |
| * @return square of this complex number |
| */ |
| public Complex square() { |
| return multiply(this); |
| } |
| |
| /** |
| * Compute the |
| * <a href="http://mathworld.wolfram.com/Cosine.html"> |
| * cosine</a> of this complex number. |
| * Implements the formula: |
| * <pre> |
| * cos(a + bi) = cos(a)cosh(b) - sin(a)sinh(b)i} |
| * </pre> |
| * where the (real) functions on the right-hand side are |
| * {@link Math#sin}, {@link Math#cos}, |
| * {@link Math#cosh} and {@link Math#sinh}. |
| * |
| * <p>As per the C.99 standard this function is computed using the trigonomic identity:</p> |
| * <pre> |
| * cos(z) = cosh(iz) |
| * </pre> |
| * |
| * @return the cosine of this complex number. |
| */ |
| public Complex cos() { |
| // Define in terms of cosh |
| // cos(z) = cosh(iz) |
| // Multiply this number by I and compute cosh. |
| return cosh(-imaginary, real, Complex::ofCartesian); |
| } |
| |
| /** |
| * Compute the |
| * <a href="http://mathworld.wolfram.com/HyperbolicCosine.html"> |
| * hyperbolic cosine</a> of this complex number. |
| * Implements the formula: |
| * <pre> |
| * cosh(a + bi) = cosh(a)cos(b) + sinh(a)sin(b)i |
| * </pre> |
| * where the (real) functions on the right-hand side are |
| * {@link Math#sin}, {@link Math#cos}, |
| * {@link Math#cosh} and {@link Math#sinh}. |
| * |
| * @return the hyperbolic cosine of this complex number. |
| */ |
| public Complex cosh() { |
| return cosh(real, imaginary, Complex::ofCartesian); |
| } |
| |
| /** |
| * Compute the hyperbolic cosine of the complex number. |
| * |
| * <p>This function exists to allow implementation of the identity |
| * {@code cos(z) = cosh(iz)}.<p> |
| * |
| * @param real Real part. |
| * @param imaginary Imaginary part. |
| * @param constructor Constructor. |
| * @return the hyperbolic cosine of this complex number |
| */ |
| private static Complex cosh(double real, double imaginary, ComplexConstructor constructor) { |
| if (real == 0 && |
| imaginary == Double.POSITIVE_INFINITY) { |
| return constructor.create(Double.NaN, 0); |
| } else if (real == 0 && |
| Double.isNaN(imaginary)) { |
| return constructor.create(Double.NaN, 0); |
| } else if (real == Double.POSITIVE_INFINITY && |
| imaginary == 0) { |
| return constructor.create(Double.POSITIVE_INFINITY, 0); |
| } else if (real == Double.POSITIVE_INFINITY && |
| (imaginary == Double.POSITIVE_INFINITY || Double.isNaN(imaginary))) { |
| return constructor.create(Double.POSITIVE_INFINITY, Double.NaN); |
| } else if (Double.isNaN(real) && |
| imaginary == 0) { |
| return constructor.create(Double.NaN, 0); |
| } |
| |
| return constructor.create(Math.cosh(real) * Math.cos(imaginary), |
| Math.sinh(real) * Math.sin(imaginary)); |
| } |
| |
| /** |
| * Compute the |
| * <a href="http://mathworld.wolfram.com/ExponentialFunction.html"> |
| * exponential function</a> of this complex number. |
| * Implements the formula: |
| * <pre> |
| * exp(a + bi) = exp(a)cos(b) + exp(a)sin(b)i |
| * </pre> |
| * where the (real) functions on the right-hand side are |
| * {@link Math#exp}, {@link Math#cos}, and |
| * {@link Math#sin}. |
| * |
| * @return <code><i>e</i><sup>this</sup></code>. |
| */ |
| public Complex exp() { |
| if (real == Double.POSITIVE_INFINITY && |
| imaginary == 0) { |
| return new Complex(Double.POSITIVE_INFINITY, 0); |
| } else if (real == Double.NEGATIVE_INFINITY && |
| imaginary == Double.POSITIVE_INFINITY) { |
| return Complex.ZERO; |
| } else if (real == Double.POSITIVE_INFINITY && |
| (imaginary == Double.POSITIVE_INFINITY || Double.isNaN(imaginary))) { |
| return new Complex(Double.POSITIVE_INFINITY, Double.NaN); |
| } else if (real == Double.NEGATIVE_INFINITY && |
| Double.isNaN(imaginary)) { |
| return Complex.ZERO; |
| } else if (Double.isNaN(real) && |
| imaginary == 0) { |
| return new Complex(Double.NaN, 0); |
| } |
| final double expReal = Math.exp(real); |
| return new Complex(expReal * Math.cos(imaginary), |
| expReal * Math.sin(imaginary)); |
| } |
| |
| /** |
| * Compute the |
| * <a href="http://mathworld.wolfram.com/NaturalLogarithm.html"> |
| * natural logarithm</a> of this complex number. |
| * Implements the formula: |
| * <pre> |
| * log(a + bi) = ln(|a + bi|) + arg(a + bi)i |
| * </pre> |
| * where ln on the right hand side is {@link Math#log}, |
| * {@code |a + bi|} is the modulus, {@link Complex#abs}, and |
| * {@code arg(a + bi) = }{@link Math#atan2}(b, a). |
| * |
| * @return the value <code>ln this</code>, the natural logarithm |
| * of {@code this}. |
| */ |
| public Complex log() { |
| if (real == Double.POSITIVE_INFINITY && |
| imaginary == Double.POSITIVE_INFINITY) { |
| return new Complex(Double.POSITIVE_INFINITY, PI_OVER_4); |
| } else if ((real == Double.POSITIVE_INFINITY && |
| Double.isNaN(imaginary)) || |
| (Double.isNaN(real) && |
| imaginary == Double.POSITIVE_INFINITY)) { |
| return new Complex(Double.POSITIVE_INFINITY, Double.NaN); |
| } |
| return new Complex(Math.log(abs()), |
| Math.atan2(imaginary, real)); |
| } |
| |
| /** |
| * Compute the base 10 or |
| * <a href="http://mathworld.wolfram.com/CommonLogarithm.html"> |
| * common logarithm</a> of this complex number. |
| * |
| * @return the base 10 logarithm of <code>this</code>. |
| */ |
| public Complex log10() { |
| Complex logE = log(); |
| return ofCartesian(logE.real / Math.log(10), logE.imaginary); |
| } |
| |
| /** |
| * Returns of value of this complex number raised to the power of {@code x}. |
| * Implements the formula: |
| * <pre> |
| * <code> |
| * y<sup>x</sup> = exp(x·log(y)) |
| * </code> |
| * </pre> |
| * where {@code exp} and {@code log} are {@link #exp} and |
| * {@link #log}, respectively. |
| * |
| * @param x exponent to which this {@code Complex} is to be raised. |
| * @return <code>this<sup>x</sup></code>. |
| */ |
| public Complex pow(Complex x) { |
| if (real == 0 && |
| imaginary == 0) { |
| // This value is zero. Test the other. |
| if (x.real > 0 && |
| x.imaginary == 0) { |
| // 0 raised to positive number is 0 |
| return ZERO; |
| } |
| // 0 raised to anything else is NaN |
| return NAN; |
| } |
| return log().multiply(x).exp(); |
| } |
| |
| /** |
| * Returns of value of this complex number raised to the power of {@code x}. |
| * |
| * @param x exponent to which this {@code Complex} is to be raised. |
| * @return <code>this<sup>x</sup></code>. |
| * @see #pow(Complex) |
| */ |
| public Complex pow(double x) { |
| if (real == 0 && |
| imaginary == 0) { |
| // This value is zero. Test the other. |
| if (x > 0) { |
| // 0 raised to positive number is 0 |
| return ZERO; |
| } |
| // 0 raised to anything else is NaN |
| return NAN; |
| } |
| return log().multiply(x).exp(); |
| } |
| |
| /** |
| * Compute the |
| * <a href="http://mathworld.wolfram.com/Sine.html"> |
| * sine</a> |
| * of this complex number. |
| * Implements the formula: |
| * <pre> |
| * sin(a + bi) = sin(a)cosh(b) - cos(a)sinh(b)i |
| * </pre> |
| * where the (real) functions on the right-hand side are |
| * {@link Math#sin}, {@link Math#cos}, |
| * {@link Math#cosh} and {@link Math#sinh}. |
| * |
| * <p>As per the C.99 standard this function is computed using the trigonomic identity:</p> |
| * <pre> |
| * sin(z) = -i sinh(iz) |
| * </pre> |
| * |
| * @return the sine of this complex number. |
| */ |
| public Complex sin() { |
| // Define in terms of sinh |
| // sin(z) = -i sinh(iz) |
| // Multiply this number by I, compute cosh, then multiply by back |
| return sinh(-imaginary, real, Complex::multiplyNegativeI); |
| } |
| |
| /** |
| * Compute the |
| * <a href="http://mathworld.wolfram.com/HyperbolicSine.html"> |
| * hyperbolic sine</a> of this complex number. |
| * Implements the formula: |
| * <pre> |
| * sinh(a + bi) = sinh(a)cos(b)) + cosh(a)sin(b)i |
| * </pre> |
| * where the (real) functions on the right-hand side are |
| * {@link Math#sin}, {@link Math#cos}, |
| * {@link Math#cosh} and {@link Math#sinh}. |
| * |
| * @return the hyperbolic sine of {@code this}. |
| */ |
| public Complex sinh() { |
| return sinh(real, imaginary, Complex::ofCartesian); |
| } |
| |
| /** |
| * Compute the hyperbolic sine of the complex number. |
| * |
| * <p>This function exists to allow implementation of the identity |
| * {@code sin(z) = -i sinh(iz)}.<p> |
| * |
| * @param real Real part. |
| * @param imaginary Imaginary part. |
| * @param constructor Constructor. |
| * @return the hyperbolic sine of this complex number |
| */ |
| private static Complex sinh(double real, double imaginary, ComplexConstructor constructor) { |
| if (real == 0 && |
| imaginary == 0) { |
| // Ignore the constructor. |
| // It is used in trignomic identities to multiply the result by -i. |
| // Here the result would still be zero. |
| return Complex.ZERO; |
| } else if (real == 0 && |
| imaginary == Double.POSITIVE_INFINITY) { |
| return constructor.create(0, Double.NaN); |
| } else if (real == 0 && |
| Double.isNaN(imaginary)) { |
| return constructor.create(0, Double.NaN); |
| } else if (real == Double.POSITIVE_INFINITY && |
| imaginary == 0) { |
| return constructor.create(Double.POSITIVE_INFINITY, 0); |
| } else if (real == Double.POSITIVE_INFINITY && |
| imaginary == Double.POSITIVE_INFINITY) { |
| return constructor.create(Double.POSITIVE_INFINITY, Double.NaN); |
| } else if (real == Double.POSITIVE_INFINITY && |
| Double.isNaN(imaginary)) { |
| return constructor.create(Double.POSITIVE_INFINITY, Double.NaN); |
| } else if (Double.isNaN(real) && |
| imaginary == 0) { |
| return constructor.create(Double.NaN, 0); |
| } |
| return constructor.create(Math.sinh(real) * Math.cos(imaginary), |
| Math.cosh(real) * Math.sin(imaginary)); |
| } |
| |
| /** |
| * Compute the |
| * <a href="http://mathworld.wolfram.com/SquareRoot.html"> |
| * square root</a> of this complex number. |
| * Implements the following algorithm to compute {@code sqrt(a + bi)}: |
| * <ol><li>Let {@code t = sqrt((|a| + |a + bi|) / 2)}</li> |
| * <li><pre>if {@code a ≥ 0} return {@code t + (b/2t)i} |
| * else return {@code |b|/2t + sign(b)t i }</pre></li> |
| * </ol> |
| * where <ul> |
| * <li>{@code |a| = }{@link Math#abs}(a)</li> |
| * <li>{@code |a + bi| = }{@link Complex#abs}(a + bi)</li> |
| * <li>{@code sign(b) = }{@link Math#copySign(double,double) copySign(1d, b)} |
| * </ul> |
| * |
| * @return the square root of {@code this}. |
| */ |
| public Complex sqrt() { |
| if (real == 0 && |
| imaginary == 0) { |
| return ZERO; |
| } else if (neitherInfiniteNorZeroNorNaN(real) && |
| imaginary == Double.POSITIVE_INFINITY) { |
| return new Complex(Double.POSITIVE_INFINITY, Double.POSITIVE_INFINITY); |
| } else if (real == Double.NEGATIVE_INFINITY && |
| neitherInfiniteNorZeroNorNaN(imaginary)) { |
| return new Complex(0, Double.NaN); |
| } else if (real == Double.NEGATIVE_INFINITY && |
| Double.isNaN(imaginary)) { |
| return new Complex(Double.NaN, Double.POSITIVE_INFINITY); |
| } else if (real == Double.POSITIVE_INFINITY && |
| Double.isNaN(imaginary)) { |
| return new Complex(Double.POSITIVE_INFINITY, Double.NaN); |
| } |
| |
| final double t = Math.sqrt((Math.abs(real) + abs()) / 2); |
| if (real >= 0) { |
| return new Complex(t, imaginary / (2 * t)); |
| } |
| return new Complex(Math.abs(imaginary) / (2 * t), |
| Math.copySign(1d, imaginary) * t); |
| } |
| |
| /** |
| * Compute the |
| * <a href="http://mathworld.wolfram.com/SquareRoot.html"> |
| * square root</a> of <code>1 - this<sup>2</sup></code> for this complex |
| * number. |
| * |
| * <p>Computes the result directly as |
| * {@code sqrt(ONE.subtract(z.multiply(z)))}.</p> |
| * |
| * @return the square root of <code>1 - this<sup>2</sup></code>. |
| */ |
| private Complex sqrt1z() { |
| return ONE.subtract(square()).sqrt(); |
| } |
| |
| /** |
| * Compute the |
| * <a href="http://mathworld.wolfram.com/Tangent.html"> |
| * tangent</a> of this complex number. |
| * Implements the formula: |
| * <pre> |
| * tan(a + bi) = sin(2a)/(cos(2a)+cosh(2b)) + [sinh(2b)/(cos(2a)+cosh(2b))]i |
| * </pre> |
| * where the (real) functions on the right-hand side are |
| * {@link Math#sin}, {@link Math#cos}, {@link Math#cosh} and |
| * {@link Math#sinh}. |
| * |
| * <p>As per the C.99 standard this function is computed using the trigonomic identity:</p> |
| * <pre> |
| * tan(z) = -i tanh(iz) |
| * </pre> |
| * |
| * @return the tangent of {@code this}. |
| */ |
| public Complex tan() { |
| // Define in terms of tanh |
| // tan(z) = -i tanh(iz) |
| // Multiply this number by I, compute cosh, then multiply by back |
| return tanh(-imaginary, real, Complex::multiplyNegativeI); |
| } |
| |
| /** |
| * Compute the |
| * <a href="http://mathworld.wolfram.com/HyperbolicTangent.html"> |
| * hyperbolic tangent</a> of this complex number. |
| * Implements the formula: |
| * <pre> |
| * tan(a + bi) = sinh(2a)/(cosh(2a)+cos(2b)) + [sin(2b)/(cosh(2a)+cos(2b))]i |
| * </pre> |
| * where the (real) functions on the right-hand side are |
| * {@link Math#sin}, {@link Math#cos}, {@link Math#cosh} and |
| * {@link Math#sinh}. |
| * |
| * @return the hyperbolic tangent of {@code this}. |
| */ |
| public Complex tanh() { |
| return tanh(real, imaginary, Complex::ofCartesian); |
| } |
| |
| /** |
| * Compute the hyperbolic tangent of this complex number. |
| * |
| * <p>This function exists to allow implementation of the identity |
| * {@code tan(z) = -i tanh(iz)}.<p> |
| * |
| * @param real Real part. |
| * @param imaginary Imaginary part. |
| * @param constructor Constructor. |
| * @return the hyperbolic tangent of this complex number |
| */ |
| private static Complex tanh(double real, double imaginary, ComplexConstructor constructor) { |
| // TODO: Should these checks be made on real2 and imaginary2? |
| // Compare to other library implementations. |
| // |
| // Math.cos and Math.sin return NaN for infinity. |
| // Math.cosh returns positive infinity for infinity. |
| // Math.sinh returns the input infinity for infinity. |
| |
| if (real == Double.POSITIVE_INFINITY && |
| (imaginary == Double.POSITIVE_INFINITY || Double.isNaN(imaginary))) { |
| return constructor.create(1, 0); |
| } else if (Double.isNaN(real) && |
| imaginary == 0) { |
| return constructor.create(Double.NaN, 0); |
| } |
| final double real2 = 2 * real; |
| final double imaginary2 = 2 * imaginary; |
| final double d = Math.cosh(real2) + Math.cos(imaginary2); |
| |
| return constructor.create(Math.sinh(real2) / d, |
| Math.sin(imaginary2) / d); |
| } |
| |
| /** |
| * Compute the argument of this complex number. |
| * The argument is the angle phi between the positive real axis and |
| * the point representing this number in the complex plane. |
| * The value returned is between -PI (not inclusive) |
| * and PI (inclusive), with negative values returned for numbers with |
| * negative imaginary parts. |
| * |
| * <p>If either real or imaginary part (or both) is NaN, NaN is returned. |
| * Infinite parts are handled as {@code Math.atan2} handles them, |
| * essentially treating finite parts as zero in the presence of an |
| * infinite coordinate and returning a multiple of pi/4 depending on |
| * the signs of the infinite parts. |
| * See the javadoc for {@code Math.atan2} for full details.</p> |
| * |
| * @return the argument of {@code this}. |
| */ |
| public double getArgument() { |
| // Delegate |
| return Math.atan2(imaginary, real); |
| } |
| |
| /** |
| * Compute the argument of this complex number |
| * (C++11 grammar). |
| * |
| * @return the argument of {@code this}. |
| * @see #getArgument() |
| */ |
| public double arg() { |
| return getArgument(); |
| } |
| |
| /** |
| * Computes the n-th roots of this complex number. |
| * The nth roots are defined by the formula: |
| * <pre> |
| * <code> |
| * z<sub>k</sub> = abs<sup>1/n</sup> (cos(phi + 2πk/n) + i (sin(phi + 2πk/n)) |
| * </code> |
| * </pre> |
| * for <i>{@code k=0, 1, ..., n-1}</i>, where {@code abs} and {@code phi} |
| * are respectively the {@link #abs() modulus} and |
| * {@link #getArgument() argument} of this complex number. |
| * |
| * <p>If one or both parts of this complex number is NaN, a list with all |
| * all elements set to {@code NaN + NaN i} is returned.</p> |
| * |
| * @param n Degree of root. |
| * @return a List of all {@code n}-th roots of {@code this}. |
| * @throws IllegalArgumentException if {@code n} is zero. |
| */ |
| public List<Complex> nthRoot(int n) { |
| if (n == 0) { |
| throw new IllegalArgumentException("cannot compute zeroth root"); |
| } |
| |
| final List<Complex> result = new ArrayList<>(); |
| |
| // nth root of abs -- faster / more accurate to use a solver here? |
| final double nthRootOfAbs = Math.pow(abs(), 1d / n); |
| |
| // Compute nth roots of complex number with k = 0, 1, ... n-1 |
| final double nthPhi = getArgument() / n; |
| final double slice = 2 * Math.PI / n; |
| double innerPart = nthPhi; |
| for (int k = 0; k < Math.abs(n); k++) { |
| // inner part |
| final double realPart = nthRootOfAbs * Math.cos(innerPart); |
| final double imaginaryPart = nthRootOfAbs * Math.sin(innerPart); |
| result.add(new Complex(realPart, imaginaryPart)); |
| innerPart += slice; |
| } |
| |
| return result; |
| } |
| |
| /** {@inheritDoc} */ |
| @Override |
| public String toString() { |
| final StringBuilder s = new StringBuilder(); |
| s.append(FORMAT_START) |
| .append(real).append(FORMAT_SEP) |
| .append(imaginary) |
| .append(FORMAT_END); |
| |
| return s.toString(); |
| } |
| |
| /** |
| * Returns {@code true} if the values are equal according to semantics of |
| * {@link Double#equals(Object)}. |
| * |
| * @param x Value |
| * @param y Value |
| * @return {@code Double.valueof(x).equals(Double.valueOf(y))} |
| */ |
| private static boolean equals(double x, double y) { |
| return Double.doubleToLongBits(x) == Double.doubleToLongBits(y); |
| } |
| |
| /** |
| * Check that a value meets all the following conditions. |
| * <ul> |
| * <li>it is not {@code NaN},</li> |
| * <li>it is not infinite,</li> |
| * <li>it is not zero,</li> |
| * </ul> |
| * |
| * @param d Value. |
| * @return {@code true} if {@code d} meets all the conditions and |
| * {@code false} otherwise. |
| */ |
| private static boolean neitherInfiniteNorZeroNorNaN(double d) { |
| return Double.isFinite(d) && d != 0; |
| } |
| |
| /** |
| * Check that a value meets all the following conditions. |
| * <ul> |
| * <li>it is not {@code NaN},</li> |
| * <li>it is not infinite,</li> |
| * <li>it is positive signed,</li> |
| * </ul> |
| * |
| * <p>Note: This is false for negative zero.</p> |
| * |
| * @param d Value. |
| * @return {@code true} if {@code d} meets all the conditions and |
| * {@code false} otherwise. |
| */ |
| private static boolean positiveSignedFinite(double d) { |
| return (Double.isFinite(d) && d > 0) || Double.doubleToLongBits(d) == 0; |
| } |
| |
| /** |
| * Create a complex number given the real and imaginary parts, then multiply by {@code -i}. |
| * This is used in functions that implement trigonomic identities. It is the functional |
| * equivalent of: |
| * |
| * <pre> |
| * z = new Complex(real, imaginary).multiply(new Complex(0, -1)); |
| * </pre> |
| * |
| * @param real Real part. |
| * @param imaginary Imaginary part. |
| * @return {@code Complex} object |
| */ |
| private static Complex multiplyNegativeI(double real, double imaginary) { |
| return new Complex(imaginary, -real); |
| } |
| |
| /** See {@link #parse(String)}. */ |
| private static class ComplexParsingException extends NumberFormatException { |
| /** Serializable version identifier. */ |
| private static final long serialVersionUID = 20180430L; |
| |
| /** |
| * @param msg Error message. |
| */ |
| ComplexParsingException(String msg) { |
| super(msg); |
| } |
| |
| /** |
| * @param msg Error message. |
| * @param cause Cause. |
| */ |
| ComplexParsingException(String msg, Throwable cause) { |
| super(msg); |
| initCause(cause); |
| } |
| } |
| } |
