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Java example source code file (Complex.java)

This example Java source code file (Complex.java) is included in the alvinalexander.com "Java Source Code Warehouse" project. The intent of this project is to help you "Learn Java by Example" TM.

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Java - Java tags/keywords

arraylist, complex, fieldelement, inf, list, nan, notpositiveexception, nullargumentexception, object, one, override, serializable, string, util, zero

The Complex.java Java example source code

/*
 * 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.complex;

import java.io.Serializable;
import java.util.ArrayList;
import java.util.List;

import org.apache.commons.math3.FieldElement;
import org.apache.commons.math3.exception.NotPositiveException;
import org.apache.commons.math3.exception.NullArgumentException;
import org.apache.commons.math3.exception.util.LocalizedFormats;
import org.apache.commons.math3.util.FastMath;
import org.apache.commons.math3.util.MathUtils;
import org.apache.commons.math3.util.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:
 * <ul>
 *  <li>{@code 1 + NaNi}
 *  <li>{@code NaN + i}
 *  <li>{@code NaN + NaNi}
 * </ul>

* 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.math3.util.Precision#equals(double,double,int) * equals for primitive double} in {@link org.apache.commons.math3.util.Precision} * conforms with IEEE-754 while this class conforms with the standard behavior * for Java object types.</p> * */ public class Complex implements FieldElement<Complex>, Serializable { /** The square root of -1. A number representing "0.0 + 1.0i" */ public static final Complex I = new Complex(0.0, 1.0); // CHECKSTYLE: stop ConstantName /** A complex number representing "NaN + NaNi" */ public static final Complex NaN = new Complex(Double.NaN, Double.NaN); // CHECKSTYLE: resume ConstantName /** A complex number representing "+INF + INFi" */ public static final Complex INF = new Complex(Double.POSITIVE_INFINITY, Double.POSITIVE_INFINITY); /** A complex number representing "1.0 + 0.0i" */ public static final Complex ONE = new Complex(1.0, 0.0); /** A complex number representing "0.0 + 0.0i" */ public static final Complex ZERO = new Complex(0.0, 0.0); /** Serializable version identifier */ private static final long serialVersionUID = -6195664516687396620L; /** The imaginary part. */ private final double imaginary; /** The real part. */ private final double real; /** Record whether this complex number is equal to NaN. */ private final transient boolean isNaN; /** Record whether this complex number is infinite. */ private final transient boolean isInfinite; /** * Create a complex number given only the real part. * * @param real Real part. */ public Complex(double real) { this(real, 0.0); } /** * Create a complex number given the real and imaginary parts. * * @param real Real part. * @param imaginary Imaginary part. */ public Complex(double real, double imaginary) { this.real = real; this.imaginary = imaginary; isNaN = Double.isNaN(real) || Double.isNaN(imaginary); isInfinite = !isNaN && (Double.isInfinite(real) || Double.isInfinite(imaginary)); } /** * Return the absolute value of this complex number. * Returns {@code NaN} if either real or imaginary part is {@code NaN} * and {@code Double.POSITIVE_INFINITY} if neither part is {@code NaN}, * but at least one part is infinite. * * @return the absolute value. */ public double abs() { if (isNaN) { return Double.NaN; } if (isInfinite()) { return Double.POSITIVE_INFINITY; } if (FastMath.abs(real) < FastMath.abs(imaginary)) { if (imaginary == 0.0) { return FastMath.abs(real); } double q = real / imaginary; return FastMath.abs(imaginary) * FastMath.sqrt(1 + q * q); } else { if (real == 0.0) { return FastMath.abs(imaginary); } double q = imaginary / real; return FastMath.abs(real) * FastMath.sqrt(1 + q * q); } } /** * Returns a {@code Complex} whose value is * {@code (this + addend)}. * Uses the definitional formula * <p> * {@code (a + bi) + (c + di) = (a+c) + (b+d)i} * </p> * If either {@code this} or {@code addend} has a {@code NaN} value in * either part, {@link #NaN} is returned; otherwise {@code Infinite} * and {@code NaN} values are returned in the parts of the result * according to the rules for {@link java.lang.Double} arithmetic. * * @param addend Value to be added to this {@code Complex}. * @return {@code this + addend}. * @throws NullArgumentException if {@code addend} is {@code null}. */ public Complex add(Complex addend) throws NullArgumentException { MathUtils.checkNotNull(addend); if (isNaN || addend.isNaN) { return NaN; } return createComplex(real + addend.getReal(), imaginary + addend.getImaginary()); } /** * 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) { if (isNaN || Double.isNaN(addend)) { return NaN; } return createComplex(real + addend, imaginary); } /** * Returns the conjugate of this complex number. * The conjugate of {@code a + bi} is {@code a - bi}. * <p> * {@link #NaN} is returned if either the real or imaginary * part of this Complex number equals {@code Double.NaN}. * </p>

* If the imaginary part is infinite, and the real part is not * {@code NaN}, the returned value has infinite imaginary part * of the opposite sign, e.g. the conjugate of * {@code 1 + POSITIVE_INFINITY i} is {@code 1 - NEGATIVE_INFINITY i}. * </p> * @return the conjugate of this Complex object. */ public Complex conjugate() { if (isNaN) { return NaN; } return createComplex(real, -imaginary); } /** * 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 + d2 * </code> * </pre> * but uses * <a href="http://doi.acm.org/10.1145/1039813.1039814"> * prescaling of operands</a> to limit the effects of overflows and * underflows in the computation. * <p> * {@code Infinite} and {@code NaN} values are handled according to the * following rules, applied in the order presented: * <ul> * <li>If either {@code this} or {@code divisor} has a {@code NaN} value * in either part, {@link #NaN} is returned. * </li> * <li>If {@code divisor} equals {@link #ZERO}, {@link #NaN} is returned. * </li> * <li>If {@code this} and {@code divisor} are both infinite, * {@link #NaN} is returned. * </li> * <li>If {@code this} is finite (i.e., has no {@code Infinite} or * {@code NaN} parts) and {@code divisor} is infinite (one or both parts * infinite), {@link #ZERO} is returned. * </li> * <li>If {@code this} is infinite and {@code divisor} is finite, * {@code NaN} values are returned in the parts of the result if the * {@link java.lang.Double} rules applied to the definitional formula * force {@code NaN} results. * </li> * </ul> * * @param divisor Value by which this {@code Complex} is to be divided. * @return {@code this / divisor}. * @throws NullArgumentException if {@code divisor} is {@code null}. */ public Complex divide(Complex divisor) throws NullArgumentException { MathUtils.checkNotNull(divisor); if (isNaN || divisor.isNaN) { return NaN; } final double c = divisor.getReal(); final double d = divisor.getImaginary(); if (c == 0.0 && d == 0.0) { return NaN; } if (divisor.isInfinite() && !isInfinite()) { return ZERO; } if (FastMath.abs(c) < FastMath.abs(d)) { double q = c / d; double denominator = c * q + d; return createComplex((real * q + imaginary) / denominator, (imaginary * q - real) / denominator); } else { double q = d / c; double denominator = d * q + c; return createComplex((imaginary * q + real) / denominator, (imaginary - real * q) / denominator); } } /** * 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) { if (isNaN || Double.isNaN(divisor)) { return NaN; } if (divisor == 0d) { return NaN; } if (Double.isInfinite(divisor)) { return !isInfinite() ? ZERO : NaN; } return createComplex(real / divisor, imaginary / divisor); } /** {@inheritDoc} */ public Complex reciprocal() { if (isNaN) { return NaN; } if (real == 0.0 && imaginary == 0.0) { return INF; } if (isInfinite) { return ZERO; } if (FastMath.abs(real) < FastMath.abs(imaginary)) { double q = real / imaginary; double scale = 1. / (real * q + imaginary); return createComplex(scale * q, -scale); } else { double q = imaginary / real; double scale = 1. / (imaginary * q + real); return createComplex(scale, -scale * q); } } /** * Test for equality with another object. * If both the real and imaginary parts of two complex numbers * are exactly the same, and neither is {@code Double.NaN}, the two * Complex objects are considered to be equal. * The behavior is the same as for JDK's {@link Double#equals(Object) * Double}: * <ul> * <li>All {@code NaN} values are considered to be equal, * i.e, if either (or both) real and imaginary parts of the complex * number are equal to {@code Double.NaN}, the complex number is equal * to {@code NaN}. * </li> * <li> * Instances constructed with different representations of zero (i.e. * either "0" or "-0") are <em>not considered to be equal. * </li> * </ul> * * @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. */ @Override public boolean equals(Object other) { if (this == other) { return true; } if (other instanceof Complex){ Complex c = (Complex) other; if (c.isNaN) { return isNaN; } else { return MathUtils.equals(real, c.real) && MathUtils.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) * @since 3.3 */ 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. * * @since 3.3 */ 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) * @since 3.3 */ 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) * @since 3.3 */ 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 hashCode for the complex number. * Any {@code Double.NaN} value in real or imaginary part produces * the same hash code {@code 7}. * * @return a hash code value for this object. */ @Override public int hashCode() { if (isNaN) { return 7; } return 37 * (17 * MathUtils.hash(imaginary) + MathUtils.hash(real)); } /** * Access the imaginary part. * * @return the imaginary part. */ public double getImaginary() { return imaginary; } /** * Access the real part. * * @return the real part. */ public double getReal() { return real; } /** * Checks whether either or both parts of this complex number is * {@code NaN}. * * @return true if either or both parts of this complex number is * {@code NaN}; false otherwise. */ public boolean isNaN() { return isNaN; } /** * Checks whether either the real or imaginary part of this complex number * takes an infinite value (either {@code Double.POSITIVE_INFINITY} or * {@code Double.NEGATIVE_INFINITY}) and neither part * is {@code NaN}. * * @return true if one or both parts of this complex number are infinite * and neither part is {@code NaN}. */ public boolean isInfinite() { return isInfinite; } /** * Returns a {@code Complex} whose value is {@code this * factor}. * Implements preliminary checks for {@code NaN} and infinity followed by * the definitional formula: * <p> * {@code (a + bi)(c + di) = (ac - bd) + (ad + bc)i} * </p> * Returns {@link #NaN} if either {@code this} or {@code factor} has one or * more {@code NaN} parts. * <p> * Returns {@link #INF} if neither {@code this} nor {@code factor} has one * or more {@code NaN} parts and if either {@code this} or {@code factor} * has one or more infinite parts (same result is returned regardless of * the sign of the components). * </p>

* Returns finite values in components of the result per the definitional * formula in all remaining cases.</p> * * @param factor value to be multiplied by this {@code Complex}. * @return {@code this * factor}. * @throws NullArgumentException if {@code factor} is {@code null}. */ public Complex multiply(Complex factor) throws NullArgumentException { MathUtils.checkNotNull(factor); if (isNaN || factor.isNaN) { return NaN; } if (Double.isInfinite(real) || Double.isInfinite(imaginary) || Double.isInfinite(factor.real) || Double.isInfinite(factor.imaginary)) { // we don't use isInfinite() to avoid testing for NaN again return INF; } return createComplex(real * factor.real - imaginary * factor.imaginary, real * factor.imaginary + imaginary * factor.real); } /** * 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) { if (isNaN) { return NaN; } if (Double.isInfinite(real) || Double.isInfinite(imaginary)) { return INF; } return createComplex(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) { if (isNaN || Double.isNaN(factor)) { return NaN; } if (Double.isInfinite(real) || Double.isInfinite(imaginary) || Double.isInfinite(factor)) { // we don't use isInfinite() to avoid testing for NaN again return INF; } return createComplex(real * factor, imaginary * factor); } /** * Returns a {@code Complex} whose value is {@code (-this)}. * Returns {@code NaN} if either real or imaginary * part of this Complex number is {@code Double.NaN}. * * @return {@code -this}. */ public Complex negate() { if (isNaN) { return NaN; } return createComplex(-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> * If either {@code this} or {@code subtrahend} has a {@code NaN]} value in either part, * {@link #NaN} is returned; otherwise infinite and {@code NaN} values are * returned in the parts of the result according to the rules for * {@link java.lang.Double} arithmetic. * * @param subtrahend value to be subtracted from this {@code Complex}. * @return {@code this - subtrahend}. * @throws NullArgumentException if {@code subtrahend} is {@code null}. */ public Complex subtract(Complex subtrahend) throws NullArgumentException { MathUtils.checkNotNull(subtrahend); if (isNaN || subtrahend.isNaN) { return NaN; } return createComplex(real - subtrahend.getReal(), imaginary - subtrahend.getImaginary()); } /** * 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) { if (isNaN || Double.isNaN(subtrahend)) { return NaN; } return createComplex(real - subtrahend, imaginary); } /** * Compute the * <a href="http://mathworld.wolfram.com/InverseCosine.html" TARGET="_top"> * inverse cosine</a> of this complex number. * Implements the formula: * <p> * {@code acos(z) = -i (log(z + i (sqrt(1 - z<sup>2))))} * </p> * Returns {@link Complex#NaN} if either real or imaginary part of the * input argument is {@code NaN} or infinite. * * @return the inverse cosine of this complex number. * @since 1.2 */ public Complex acos() { if (isNaN) { return NaN; } return this.add(this.sqrt1z().multiply(I)).log().multiply(I.negate()); } /** * Compute the * <a href="http://mathworld.wolfram.com/InverseSine.html" TARGET="_top"> * inverse sine</a> of this complex number. * Implements the formula: * <p> * {@code asin(z) = -i (log(sqrt(1 - z<sup>2) + iz))} * </p>

* Returns {@link Complex#NaN} if either real or imaginary part of the * input argument is {@code NaN} or infinite.</p> * * @return the inverse sine of this complex number. * @since 1.2 */ public Complex asin() { if (isNaN) { return NaN; } return sqrt1z().add(this.multiply(I)).log().multiply(I.negate()); } /** * Compute the * <a href="http://mathworld.wolfram.com/InverseTangent.html" TARGET="_top"> * inverse tangent</a> of this complex number. * Implements the formula: * <p> * {@code atan(z) = (i/2) log((i + z)/(i - z))} * </p>

* Returns {@link Complex#NaN} if either real or imaginary part of the * input argument is {@code NaN} or infinite.</p> * * @return the inverse tangent of this complex number * @since 1.2 */ public Complex atan() { if (isNaN) { return NaN; } return this.add(I).divide(I.subtract(this)).log() .multiply(I.divide(createComplex(2.0, 0.0))); } /** * Compute the * <a href="http://mathworld.wolfram.com/Cosine.html" TARGET="_top"> * cosine</a> of this complex number. * Implements the formula: * <p> * {@code cos(a + bi) = cos(a)cosh(b) - sin(a)sinh(b)i} * </p>

* where the (real) functions on the right-hand side are * {@link FastMath#sin}, {@link FastMath#cos}, * {@link FastMath#cosh} and {@link FastMath#sinh}. * </p>

* Returns {@link Complex#NaN} if either real or imaginary part of the * input argument is {@code NaN}. * </p>

* Infinite values in real or imaginary parts of the input may result in * infinite or NaN values returned in parts of the result.</p> * <pre> * Examples: * <code> * cos(1 ± INFINITY i) = 1 \u2213 INFINITY i * cos(±INFINITY + i) = NaN + NaN i * cos(±INFINITY ± INFINITY i) = NaN + NaN i * </code> * </pre> * * @return the cosine of this complex number. * @since 1.2 */ public Complex cos() { if (isNaN) { return NaN; } return createComplex(FastMath.cos(real) * FastMath.cosh(imaginary), -FastMath.sin(real) * FastMath.sinh(imaginary)); } /** * Compute the * <a href="http://mathworld.wolfram.com/HyperbolicCosine.html" TARGET="_top"> * hyperbolic cosine</a> of this complex number. * Implements the formula: * <pre> * <code> * cosh(a + bi) = cosh(a)cos(b) + sinh(a)sin(b)i * </code> * </pre> * where the (real) functions on the right-hand side are * {@link FastMath#sin}, {@link FastMath#cos}, * {@link FastMath#cosh} and {@link FastMath#sinh}. * <p> * Returns {@link Complex#NaN} if either real or imaginary part of the * input argument is {@code NaN}. * </p> * Infinite values in real or imaginary parts of the input may result in * infinite or NaN values returned in parts of the result. * <pre> * Examples: * <code> * cosh(1 ± INFINITY i) = NaN + NaN i * cosh(±INFINITY + i) = INFINITY ± INFINITY i * cosh(±INFINITY ± INFINITY i) = NaN + NaN i * </code> * </pre> * * @return the hyperbolic cosine of this complex number. * @since 1.2 */ public Complex cosh() { if (isNaN) { return NaN; } return createComplex(FastMath.cosh(real) * FastMath.cos(imaginary), FastMath.sinh(real) * FastMath.sin(imaginary)); } /** * Compute the * <a href="http://mathworld.wolfram.com/ExponentialFunction.html" TARGET="_top"> * exponential function</a> of this complex number. * Implements the formula: * <pre> * <code> * exp(a + bi) = exp(a)cos(b) + exp(a)sin(b)i * </code> * </pre> * where the (real) functions on the right-hand side are * {@link FastMath#exp}, {@link FastMath#cos}, and * {@link FastMath#sin}. * <p> * Returns {@link Complex#NaN} if either real or imaginary part of the * input argument is {@code NaN}. * </p> * Infinite values in real or imaginary parts of the input may result in * infinite or NaN values returned in parts of the result. * <pre> * Examples: * <code> * exp(1 ± INFINITY i) = NaN + NaN i * exp(INFINITY + i) = INFINITY + INFINITY i * exp(-INFINITY + i) = 0 + 0i * exp(±INFINITY ± INFINITY i) = NaN + NaN i * </code> * </pre> * * @return <code>ethis. * @since 1.2 */ public Complex exp() { if (isNaN) { return NaN; } double expReal = FastMath.exp(real); return createComplex(expReal * FastMath.cos(imaginary), expReal * FastMath.sin(imaginary)); } /** * Compute the * <a href="http://mathworld.wolfram.com/NaturalLogarithm.html" TARGET="_top"> * natural logarithm</a> of this complex number. * Implements the formula: * <pre> * <code> * log(a + bi) = ln(|a + bi|) + arg(a + bi)i * </code> * </pre> * where ln on the right hand side is {@link FastMath#log}, * {@code |a + bi|} is the modulus, {@link Complex#abs}, and * {@code arg(a + bi) = }{@link FastMath#atan2}(b, a). * <p> * Returns {@link Complex#NaN} if either real or imaginary part of the * input argument is {@code NaN}. * </p> * Infinite (or critical) values in real or imaginary parts of the input may * result in infinite or NaN values returned in parts of the result. * <pre> * Examples: * <code> * log(1 ± INFINITY i) = INFINITY ± (π/2)i * log(INFINITY + i) = INFINITY + 0i * log(-INFINITY + i) = INFINITY + πi * log(INFINITY ± INFINITY i) = INFINITY ± (π/4)i * log(-INFINITY ± INFINITY i) = INFINITY ± (3π/4)i * log(0 + 0i) = -INFINITY + 0i * </code> * </pre> * * @return the value <code>ln   this, the natural logarithm * of {@code this}. * @since 1.2 */ public Complex log() { if (isNaN) { return NaN; } return createComplex(FastMath.log(abs()), FastMath.atan2(imaginary, real)); } /** * Returns of value of this complex number raised to the power of {@code x}. * Implements the formula: * <pre> * <code> * y<sup>x = exp(x·log(y)) * </code> * </pre> * where {@code exp} and {@code log} are {@link #exp} and * {@link #log}, respectively. * <p> * Returns {@link Complex#NaN} if either real or imaginary part of the * input argument is {@code NaN} or infinite, or if {@code y} * equals {@link Complex#ZERO}.</p> * * @param x exponent to which this {@code Complex} is to be raised. * @return <code> thisx. * @throws NullArgumentException if x is {@code null}. * @since 1.2 */ public Complex pow(Complex x) throws NullArgumentException { MathUtils.checkNotNull(x); return this.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>thisx. * @see #pow(Complex) */ public Complex pow(double x) { return this.log().multiply(x).exp(); } /** * Compute the * <a href="http://mathworld.wolfram.com/Sine.html" TARGET="_top"> * sine</a> * of this complex number. * Implements the formula: * <pre> * <code> * sin(a + bi) = sin(a)cosh(b) - cos(a)sinh(b)i * </code> * </pre> * where the (real) functions on the right-hand side are * {@link FastMath#sin}, {@link FastMath#cos}, * {@link FastMath#cosh} and {@link FastMath#sinh}. * <p> * Returns {@link Complex#NaN} if either real or imaginary part of the * input argument is {@code NaN}. * </p>

* Infinite values in real or imaginary parts of the input may result in * infinite or {@code NaN} values returned in parts of the result. * <pre> * Examples: * <code> * sin(1 ± INFINITY i) = 1 ± INFINITY i * sin(±INFINITY + i) = NaN + NaN i * sin(±INFINITY ± INFINITY i) = NaN + NaN i * </code> * </pre> * * @return the sine of this complex number. * @since 1.2 */ public Complex sin() { if (isNaN) { return NaN; } return createComplex(FastMath.sin(real) * FastMath.cosh(imaginary), FastMath.cos(real) * FastMath.sinh(imaginary)); } /** * Compute the * <a href="http://mathworld.wolfram.com/HyperbolicSine.html" TARGET="_top"> * hyperbolic sine</a> of this complex number. * Implements the formula: * <pre> * <code> * sinh(a + bi) = sinh(a)cos(b)) + cosh(a)sin(b)i * </code> * </pre> * where the (real) functions on the right-hand side are * {@link FastMath#sin}, {@link FastMath#cos}, * {@link FastMath#cosh} and {@link FastMath#sinh}. * <p> * Returns {@link Complex#NaN} if either real or imaginary part of the * input argument is {@code NaN}. * </p>

* Infinite values in real or imaginary parts of the input may result in * infinite or NaN values returned in parts of the result. * <pre> * Examples: * <code> * sinh(1 ± INFINITY i) = NaN + NaN i * sinh(±INFINITY + i) = ± INFINITY + INFINITY i * sinh(±INFINITY ± INFINITY i) = NaN + NaN i * </code> * </pre> * * @return the hyperbolic sine of {@code this}. * @since 1.2 */ public Complex sinh() { if (isNaN) { return NaN; } return createComplex(FastMath.sinh(real) * FastMath.cos(imaginary), FastMath.cosh(real) * FastMath.sin(imaginary)); } /** * Compute the * <a href="http://mathworld.wolfram.com/SquareRoot.html" TARGET="_top"> * square root</a> of this complex number. * Implements the following algorithm to compute {@code sqrt(a + bi)}: * <ol>

  • Let {@code t = sqrt((|a| + |a + bi|) / 2)}
  • * <li>
    if {@code  a ≥ 0} return {@code t + (b/2t)i}
         *  else return {@code |b|/2t + sign(b)t i }</pre>
         * </ol>
         * where <ul>
         * <li>{@code |a| = }{@link FastMath#abs}(a)
         * <li>{@code |a + bi| = }{@link Complex#abs}(a + bi)
         * <li>{@code sign(b) =  }{@link FastMath#copySign(double,double) copySign(1d, b)}
         * </ul>
         * <p>
         * Returns {@link Complex#NaN} if either real or imaginary part of the
         * input argument is {@code NaN}.
         * </p>
         * Infinite values in real or imaginary parts of the input may result in
         * infinite or NaN values returned in parts of the result.
         * <pre>
         *  Examples:
         *  <code>
         *   sqrt(1 ± INFINITY i) = INFINITY + NaN i
         *   sqrt(INFINITY + i) = INFINITY + 0i
         *   sqrt(-INFINITY + i) = 0 + INFINITY i
         *   sqrt(INFINITY ± INFINITY i) = INFINITY + NaN i
         *   sqrt(-INFINITY ± INFINITY i) = NaN ± INFINITY i
         *  </code>
         * </pre>
         *
         * @return the square root of {@code this}.
         * @since 1.2
         */
        public Complex sqrt() {
            if (isNaN) {
                return NaN;
            }
    
            if (real == 0.0 && imaginary == 0.0) {
                return createComplex(0.0, 0.0);
            }
    
            double t = FastMath.sqrt((FastMath.abs(real) + abs()) / 2.0);
            if (real >= 0.0) {
                return createComplex(t, imaginary / (2.0 * t));
            } else {
                return createComplex(FastMath.abs(imaginary) / (2.0 * t),
                                     FastMath.copySign(1d, imaginary) * t);
            }
        }
    
        /**
         * Compute the
         * <a href="http://mathworld.wolfram.com/SquareRoot.html" TARGET="_top">
         * square root</a> of 1 - this2 for this complex
         * number.
         * Computes the result directly as
         * {@code sqrt(ONE.subtract(z.multiply(z)))}.
         * <p>
         * Returns {@link Complex#NaN} if either real or imaginary part of the
         * input argument is {@code NaN}.
         * </p>
         * Infinite values in real or imaginary parts of the input may result in
         * infinite or NaN values returned in parts of the result.
         *
         * @return the square root of <code>1 - this2.
         * @since 1.2
         */
        public Complex sqrt1z() {
            return createComplex(1.0, 0.0).subtract(this.multiply(this)).sqrt();
        }
    
        /**
         * Compute the
         * <a href="http://mathworld.wolfram.com/Tangent.html" TARGET="_top">
         * tangent</a> of this complex number.
         * Implements the formula:
         * <pre>
         *  <code>
         *   tan(a + bi) = sin(2a)/(cos(2a)+cosh(2b)) + [sinh(2b)/(cos(2a)+cosh(2b))]i
         *  </code>
         * </pre>
         * where the (real) functions on the right-hand side are
         * {@link FastMath#sin}, {@link FastMath#cos}, {@link FastMath#cosh} and
         * {@link FastMath#sinh}.
         * <p>
         * Returns {@link Complex#NaN} if either real or imaginary part of the
         * input argument is {@code NaN}.
         * </p>
         * Infinite (or critical) values in real or imaginary parts of the input may
         * result in infinite or NaN values returned in parts of the result.
         * <pre>
         *  Examples:
         *  <code>
         *   tan(a ± INFINITY i) = 0 ± i
         *   tan(±INFINITY + bi) = NaN + NaN i
         *   tan(±INFINITY ± INFINITY i) = NaN + NaN i
         *   tan(±π/2 + 0 i) = ±INFINITY + NaN i
         *  </code>
         * </pre>
         *
         * @return the tangent of {@code this}.
         * @since 1.2
         */
        public Complex tan() {
            if (isNaN || Double.isInfinite(real)) {
                return NaN;
            }
            if (imaginary > 20.0) {
                return createComplex(0.0, 1.0);
            }
            if (imaginary < -20.0) {
                return createComplex(0.0, -1.0);
            }
    
            double real2 = 2.0 * real;
            double imaginary2 = 2.0 * imaginary;
            double d = FastMath.cos(real2) + FastMath.cosh(imaginary2);
    
            return createComplex(FastMath.sin(real2) / d,
                                 FastMath.sinh(imaginary2) / d);
        }
    
        /**
         * Compute the
         * <a href="http://mathworld.wolfram.com/HyperbolicTangent.html" TARGET="_top">
         * hyperbolic tangent</a> of this complex number.
         * Implements the formula:
         * <pre>
         *  <code>
         *   tan(a + bi) = sinh(2a)/(cosh(2a)+cos(2b)) + [sin(2b)/(cosh(2a)+cos(2b))]i
         *  </code>
         * </pre>
         * where the (real) functions on the right-hand side are
         * {@link FastMath#sin}, {@link FastMath#cos}, {@link FastMath#cosh} and
         * {@link FastMath#sinh}.
         * <p>
         * Returns {@link Complex#NaN} if either real or imaginary part of the
         * input argument is {@code NaN}.
         * </p>
         * Infinite values in real or imaginary parts of the input may result in
         * infinite or NaN values returned in parts of the result.
         * <pre>
         *  Examples:
         *  <code>
         *   tanh(a ± INFINITY i) = NaN + NaN i
         *   tanh(±INFINITY + bi) = ±1 + 0 i
         *   tanh(±INFINITY ± INFINITY i) = NaN + NaN i
         *   tanh(0 + (π/2)i) = NaN + INFINITY i
         *  </code>
         * </pre>
         *
         * @return the hyperbolic tangent of {@code this}.
         * @since 1.2
         */
        public Complex tanh() {
            if (isNaN || Double.isInfinite(imaginary)) {
                return NaN;
            }
            if (real > 20.0) {
                return createComplex(1.0, 0.0);
            }
            if (real < -20.0) {
                return createComplex(-1.0, 0.0);
            }
            double real2 = 2.0 * real;
            double imaginary2 = 2.0 * imaginary;
            double d = FastMath.cosh(real2) + FastMath.cos(imaginary2);
    
            return createComplex(FastMath.sinh(real2) / d,
                                 FastMath.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.
         *
         * @return the argument of {@code this}.
         */
        public double getArgument() {
            return FastMath.atan2(getImaginary(), getReal());
        }
    
        /**
         * Computes the n-th roots of this complex number.
         * The nth roots are defined by the formula:
         * <pre>
         *  <code>
         *   z<sub>k = abs1/n (cos(phi + 2πk/n) + i (sin(phi + 2πk/n))
         *  </code>
         * </pre>
         * for <i>{@code k=0, 1, ..., n-1}, 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 just
         * one element, {@link #NaN} is returned.
         * if neither part is NaN, but at least one part is infinite, the result
         * is a one-element list containing {@link #INF}.
         *
         * @param n Degree of root.
         * @return a List of all {@code n}-th roots of {@code this}.
         * @throws NotPositiveException if {@code n <= 0}.
         * @since 2.0
         */
        public List<Complex> nthRoot(int n) throws NotPositiveException {
    
            if (n <= 0) {
                throw new NotPositiveException(LocalizedFormats.CANNOT_COMPUTE_NTH_ROOT_FOR_NEGATIVE_N,
                                               n);
            }
    
            final List<Complex> result = new ArrayList();
    
            if (isNaN) {
                result.add(NaN);
                return result;
            }
            if (isInfinite()) {
                result.add(INF);
                return result;
            }
    
            // nth root of abs -- faster / more accurate to use a solver here?
            final double nthRootOfAbs = FastMath.pow(abs(), 1.0 / n);
    
            // Compute nth roots of complex number with k = 0, 1, ... n-1
            final double nthPhi = getArgument() / n;
            final double slice = 2 * FastMath.PI / n;
            double innerPart = nthPhi;
            for (int k = 0; k < n ; k++) {
                // inner part
                final double realPart = nthRootOfAbs *  FastMath.cos(innerPart);
                final double imaginaryPart = nthRootOfAbs *  FastMath.sin(innerPart);
                result.add(createComplex(realPart, imaginaryPart));
                innerPart += slice;
            }
    
            return result;
        }
    
        /**
         * Create a complex number given the real and imaginary parts.
         *
         * @param realPart Real part.
         * @param imaginaryPart Imaginary part.
         * @return a new complex number instance.
         * @since 1.2
         * @see #valueOf(double, double)
         */
        protected Complex createComplex(double realPart,
                                        double imaginaryPart) {
            return new Complex(realPart, imaginaryPart);
        }
    
        /**
         * Create a complex number given the real and imaginary parts.
         *
         * @param realPart Real part.
         * @param imaginaryPart Imaginary part.
         * @return a Complex instance.
         */
        public static Complex valueOf(double realPart,
                                      double imaginaryPart) {
            if (Double.isNaN(realPart) ||
                Double.isNaN(imaginaryPart)) {
                return NaN;
            }
            return new Complex(realPart, imaginaryPart);
        }
    
        /**
         * Create a complex number given only the real part.
         *
         * @param realPart Real part.
         * @return a Complex instance.
         */
        public static Complex valueOf(double realPart) {
            if (Double.isNaN(realPart)) {
                return NaN;
            }
            return new Complex(realPart);
        }
    
        /**
         * Resolve the transient fields in a deserialized Complex Object.
         * Subclasses will need to override {@link #createComplex} to
         * deserialize properly.
         *
         * @return A Complex instance with all fields resolved.
         * @since 2.0
         */
        protected final Object readResolve() {
            return createComplex(real, imaginary);
        }
    
        /** {@inheritDoc} */
        public ComplexField getField() {
            return ComplexField.getInstance();
        }
    
        /** {@inheritDoc} */
        @Override
        public String toString() {
            return "(" + real + ", " + imaginary + ")";
        }
    
    }
    

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