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diff --git a/src/main/java/org/apache/commons/math/complex/Complex.java b/src/main/java/org/apache/commons/math/complex/Complex.java new file mode 100644 index 0000000..ad2bc96 --- /dev/null +++ b/src/main/java/org/apache/commons/math/complex/Complex.java @@ -0,0 +1,1007 @@ +/* + * 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.math.complex; + +import java.io.Serializable; +import java.util.ArrayList; +import java.util.List; + +import org.apache.commons.math.FieldElement; +import org.apache.commons.math.MathRuntimeException; +import org.apache.commons.math.exception.util.LocalizedFormats; +import org.apache.commons.math.util.MathUtils; +import org.apache.commons.math.util.FastMath; + +/** + * Representation of a Complex number - a number which has both a + * real and imaginary part. + * <p> + * Implementations of arithmetic operations handle <code>NaN</code> and + * infinite values according to the rules for {@link java.lang.Double} + * arithmetic, applying definitional formulas and returning <code>NaN</code> or + * infinite values in real or imaginary parts as these arise in computation. + * See individual method javadocs for details.</p> + * <p> + * {@link #equals} identifies all values with <code>NaN</code> in either real + * or imaginary part - e.g., <pre> + * <code>1 + NaNi == NaN + i == NaN + NaNi.</code></pre></p> + * + * implements Serializable since 2.0 + * + * @version $Revision: 990655 $ $Date: 2010-08-29 23:49:40 +0200 (dim. 29 août 2010) $ + */ +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 the real and imaginary parts. + * + * @param real the real part + * @param imaginary the imaginary part + */ + public Complex(double real, double imaginary) { + super(); + 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. + * <p> + * Returns <code>NaN</code> if either real or imaginary part is + * <code>NaN</code> and <code>Double.POSITIVE_INFINITY</code> if + * neither part is <code>NaN</code>, but at least one part takes an infinite + * value.</p> + * + * @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); + } + } + + /** + * Return the sum of this complex number and the given complex number. + * <p> + * Uses the definitional formula + * <pre> + * (a + bi) + (c + di) = (a+c) + (b+d)i + * </pre></p> + * <p> + * If either this or <code>rhs</code> has a NaN value in either part, + * {@link #NaN} is returned; otherwise Inifinite and NaN values are + * returned in the parts of the result according to the rules for + * {@link java.lang.Double} arithmetic.</p> + * + * @param rhs the other complex number + * @return the complex number sum + * @throws NullPointerException if <code>rhs</code> is null + */ + public Complex add(Complex rhs) { + return createComplex(real + rhs.getReal(), + imaginary + rhs.getImaginary()); + } + + /** + * Return the conjugate of this complex number. The conjugate of + * "A + Bi" is "A - Bi". + * <p> + * {@link #NaN} is returned if either the real or imaginary + * part of this Complex number equals <code>Double.NaN</code>.</p> + * <p> + * If the imaginary part is infinite, and the real part is not NaN, + * the returned value has infinite imaginary part of the opposite + * sign - e.g. the conjugate of <code>1 + POSITIVE_INFINITY i</code> + * is <code>1 - NEGATIVE_INFINITY i</code></p> + * + * @return the conjugate of this Complex object + */ + public Complex conjugate() { + if (isNaN()) { + return NaN; + } + return createComplex(real, -imaginary); + } + + /** + * Return the quotient of this complex number and the given complex number. + * <p> + * 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> + * 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> + * <p> + * Infinite and NaN values are handled / returned according to the + * following rules, applied in the order presented: + * <ul> + * <li>If either this or <code>rhs</code> has a NaN value in either part, + * {@link #NaN} is returned.</li> + * <li>If <code>rhs</code> equals {@link #ZERO}, {@link #NaN} is returned. + * </li> + * <li>If this and <code>rhs</code> are both infinite, + * {@link #NaN} is returned.</li> + * <li>If this is finite (i.e., has no infinite or NaN parts) and + * <code>rhs</code> is infinite (one or both parts infinite), + * {@link #ZERO} is returned.</li> + * <li>If this is infinite and <code>rhs</code> is finite, NaN values are + * returned in the parts of the result if the {@link java.lang.Double} + * rules applied to the definitional formula force NaN results.</li> + * </ul></p> + * + * @param rhs the other complex number + * @return the complex number quotient + * @throws NullPointerException if <code>rhs</code> is null + */ + public Complex divide(Complex rhs) { + if (isNaN() || rhs.isNaN()) { + return NaN; + } + + double c = rhs.getReal(); + double d = rhs.getImaginary(); + if (c == 0.0 && d == 0.0) { + return NaN; + } + + if (rhs.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); + } + } + + /** + * Test for the equality of two Complex objects. + * <p> + * If both the real and imaginary parts of two Complex numbers + * are exactly the same, and neither is <code>Double.NaN</code>, the two + * Complex objects are considered to be equal.</p> + * <p> + * All <code>NaN</code> 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</code>, the complex number is equal to + * <code>Complex.NaN</code>.</p> + * + * @param other Object to test for equality to this + * @return true if two Complex objects are equal, false if + * object is null, not an instance of Complex, or + * not equal to this Complex instance + * + */ + @Override + public boolean equals(Object other) { + if (this == other) { + return true; + } + if (other instanceof Complex){ + Complex rhs = (Complex)other; + if (rhs.isNaN()) { + return this.isNaN(); + } else { + return (real == rhs.real) && (imaginary == rhs.imaginary); + } + } + return false; + } + + /** + * Get a hashCode for the complex number. + * <p> + * All NaN values have the same hash code.</p> + * + * @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; + } + + /** + * Returns true if either or both parts of this complex number is NaN; + * false otherwise + * + * @return true if either or both parts of this complex number is NaN; + * false otherwise + */ + public boolean isNaN() { + return isNaN; + } + + /** + * Returns true if either the real or imaginary part of this complex number + * takes an infinite value (either <code>Double.POSITIVE_INFINITY</code> or + * <code>Double.NEGATIVE_INFINITY</code>) and neither part + * is <code>NaN</code>. + * + * @return true if one or both parts of this complex number are infinite + * and neither part is <code>NaN</code> + */ + public boolean isInfinite() { + return isInfinite; + } + + /** + * Return the product of this complex number and the given complex number. + * <p> + * Implements preliminary checks for NaN and infinity followed by + * the definitional formula: + * <pre><code> + * (a + bi)(c + di) = (ac - bd) + (ad + bc)i + * </code></pre> + * </p> + * <p> + * Returns {@link #NaN} if either this or <code>rhs</code> has one or more + * NaN parts. + * </p> + * Returns {@link #INF} if neither this nor <code>rhs</code> has one or more + * NaN parts and if either this or <code>rhs</code> has one or more + * infinite parts (same result is returned regardless of the sign of the + * components). + * </p> + * <p> + * Returns finite values in components of the result per the + * definitional formula in all remaining cases. + * </p> + * + * @param rhs the other complex number + * @return the complex number product + * @throws NullPointerException if <code>rhs</code> is null + */ + public Complex multiply(Complex rhs) { + if (isNaN() || rhs.isNaN()) { + return NaN; + } + if (Double.isInfinite(real) || Double.isInfinite(imaginary) || + Double.isInfinite(rhs.real)|| Double.isInfinite(rhs.imaginary)) { + // we don't use Complex.isInfinite() to avoid testing for NaN again + return INF; + } + return createComplex(real * rhs.real - imaginary * rhs.imaginary, + real * rhs.imaginary + imaginary * rhs.real); + } + + /** + * Return the product of this complex number and the given scalar number. + * <p> + * Implements preliminary checks for NaN and infinity followed by + * the definitional formula: + * <pre><code> + * c(a + bi) = (ca) + (cb)i + * </code></pre> + * </p> + * <p> + * Returns {@link #NaN} if either this or <code>rhs</code> has one or more + * NaN parts. + * </p> + * Returns {@link #INF} if neither this nor <code>rhs</code> has one or more + * NaN parts and if either this or <code>rhs</code> has one or more + * infinite parts (same result is returned regardless of the sign of the + * components). + * </p> + * <p> + * Returns finite values in components of the result per the + * definitional formula in all remaining cases. + * </p> + * + * @param rhs the scalar number + * @return the complex number product + */ + public Complex multiply(double rhs) { + if (isNaN() || Double.isNaN(rhs)) { + return NaN; + } + if (Double.isInfinite(real) || Double.isInfinite(imaginary) || + Double.isInfinite(rhs)) { + // we don't use Complex.isInfinite() to avoid testing for NaN again + return INF; + } + return createComplex(real * rhs, imaginary * rhs); + } + + /** + * Return the additive inverse of this complex number. + * <p> + * Returns <code>Complex.NaN</code> if either real or imaginary + * part of this Complex number equals <code>Double.NaN</code>.</p> + * + * @return the negation of this complex number + */ + public Complex negate() { + if (isNaN()) { + return NaN; + } + + return createComplex(-real, -imaginary); + } + + /** + * Return the difference between this complex number and the given complex + * number. + * <p> + * Uses the definitional formula + * <pre> + * (a + bi) - (c + di) = (a-c) + (b-d)i + * </pre></p> + * <p> + * If either this or <code>rhs</code> has a NaN value in either part, + * {@link #NaN} is returned; otherwise inifinite and NaN values are + * returned in the parts of the result according to the rules for + * {@link java.lang.Double} arithmetic. </p> + * + * @param rhs the other complex number + * @return the complex number difference + * @throws NullPointerException if <code>rhs</code> is null + */ + public Complex subtract(Complex rhs) { + if (isNaN() || rhs.isNaN()) { + return NaN; + } + + return createComplex(real - rhs.getReal(), + imaginary - rhs.getImaginary()); + } + + /** + * Compute the + * <a href="http://mathworld.wolfram.com/InverseCosine.html" TARGET="_top"> + * inverse cosine</a> of this complex number. + * <p> + * Implements the formula: <pre> + * <code> acos(z) = -i (log(z + i (sqrt(1 - z<sup>2</sup>))))</code></pre></p> + * <p> + * Returns {@link Complex#NaN} if either real or imaginary part of the + * input argument is <code>NaN</code> or infinite.</p> + * + * @return the inverse cosine of this complex number + * @since 1.2 + */ + public Complex acos() { + if (isNaN()) { + return Complex.NaN; + } + + return this.add(this.sqrt1z().multiply(Complex.I)).log() + .multiply(Complex.I.negate()); + } + + /** + * Compute the + * <a href="http://mathworld.wolfram.com/InverseSine.html" TARGET="_top"> + * inverse sine</a> of this complex number. + * <p> + * Implements the formula: <pre> + * <code> asin(z) = -i (log(sqrt(1 - z<sup>2</sup>) + iz)) </code></pre></p> + * <p> + * Returns {@link Complex#NaN} if either real or imaginary part of the + * input argument is <code>NaN</code> or infinite.</p> + * + * @return the inverse sine of this complex number. + * @since 1.2 + */ + public Complex asin() { + if (isNaN()) { + return Complex.NaN; + } + + return sqrt1z().add(this.multiply(Complex.I)).log() + .multiply(Complex.I.negate()); + } + + /** + * Compute the + * <a href="http://mathworld.wolfram.com/InverseTangent.html" TARGET="_top"> + * inverse tangent</a> of this complex number. + * <p> + * Implements the formula: <pre> + * <code> atan(z) = (i/2) log((i + z)/(i - z)) </code></pre></p> + * <p> + * Returns {@link Complex#NaN} if either real or imaginary part of the + * input argument is <code>NaN</code> or infinite.</p> + * + * @return the inverse tangent of this complex number + * @since 1.2 + */ + public Complex atan() { + if (isNaN()) { + return Complex.NaN; + } + + return this.add(Complex.I).divide(Complex.I.subtract(this)).log() + .multiply(Complex.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. + * <p> + * Implements the formula: <pre> + * <code> cos(a + bi) = cos(a)cosh(b) - sin(a)sinh(b)i</code></pre> + * where the (real) functions on the right-hand side are + * {@link java.lang.Math#sin}, {@link java.lang.Math#cos}, + * {@link MathUtils#cosh} and {@link MathUtils#sinh}.</p> + * <p> + * Returns {@link Complex#NaN} if either real or imaginary part of the + * input argument is <code>NaN</code>.</p> + * <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> + * cos(1 ± INFINITY i) = 1 ∓ INFINITY i + * cos(±INFINITY + i) = NaN + NaN i + * cos(±INFINITY ± INFINITY i) = NaN + NaN i</code></pre></p> + * + * @return the cosine of this complex number + * @since 1.2 + */ + public Complex cos() { + if (isNaN()) { + return Complex.NaN; + } + + return createComplex(FastMath.cos(real) * MathUtils.cosh(imaginary), + -FastMath.sin(real) * MathUtils.sinh(imaginary)); + } + + /** + * Compute the + * <a href="http://mathworld.wolfram.com/HyperbolicCosine.html" TARGET="_top"> + * hyperbolic cosine</a> of this complex number. + * <p> + * 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 java.lang.Math#sin}, {@link java.lang.Math#cos}, + * {@link MathUtils#cosh} and {@link MathUtils#sinh}.</p> + * <p> + * Returns {@link Complex#NaN} if either real or imaginary part of the + * input argument is <code>NaN</code>.</p> + * <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></p> + * + * @return the hyperbolic cosine of this complex number. + * @since 1.2 + */ + public Complex cosh() { + if (isNaN()) { + return Complex.NaN; + } + + return createComplex(MathUtils.cosh(real) * FastMath.cos(imaginary), + MathUtils.sinh(real) * FastMath.sin(imaginary)); + } + + /** + * Compute the + * <a href="http://mathworld.wolfram.com/ExponentialFunction.html" TARGET="_top"> + * exponential function</a> of this complex number. + * <p> + * 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 java.lang.Math#exp}, {@link java.lang.Math#cos}, and + * {@link java.lang.Math#sin}.</p> + * <p> + * Returns {@link Complex#NaN} if either real or imaginary part of the + * input argument is <code>NaN</code>.</p> + * <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></p> + * + * @return <i>e</i><sup><code>this</code></sup> + * @since 1.2 + */ + public Complex exp() { + if (isNaN()) { + return Complex.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. + * <p> + * 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 java.lang.Math#log}, + * <code>|a + bi|</code> is the modulus, {@link Complex#abs}, and + * <code>arg(a + bi) = {@link java.lang.Math#atan2}(b, a)</code></p> + * <p> + * Returns {@link Complex#NaN} if either real or imaginary part of the + * input argument is <code>NaN</code>.</p> + * <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></p> + * + * @return ln of this complex number. + * @since 1.2 + */ + public Complex log() { + if (isNaN()) { + return Complex.NaN; + } + + return createComplex(FastMath.log(abs()), + FastMath.atan2(imaginary, real)); + } + + /** + * Returns of value of this complex number raised to the power of <code>x</code>. + * <p> + * Implements the formula: <pre> + * <code> y<sup>x</sup> = exp(x·log(y))</code></pre> + * where <code>exp</code> and <code>log</code> are {@link #exp} and + * {@link #log}, respectively.</p> + * <p> + * Returns {@link Complex#NaN} if either real or imaginary part of the + * input argument is <code>NaN</code> or infinite, or if <code>y</code> + * equals {@link Complex#ZERO}.</p> + * + * @param x the exponent. + * @return <code>this</code><sup><code>x</code></sup> + * @throws NullPointerException if x is null + * @since 1.2 + */ + public Complex pow(Complex x) { + if (x == null) { + throw new NullPointerException(); + } + return this.log().multiply(x).exp(); + } + + /** + * Compute the + * <a href="http://mathworld.wolfram.com/Sine.html" TARGET="_top"> + * sine</a> + * of this complex number. + * <p> + * 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 java.lang.Math#sin}, {@link java.lang.Math#cos}, + * {@link MathUtils#cosh} and {@link MathUtils#sinh}.</p> + * <p> + * Returns {@link Complex#NaN} if either real or imaginary part of the + * input argument is <code>NaN</code>.</p> + * <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> + * sin(1 ± INFINITY i) = 1 ± INFINITY i + * sin(±INFINITY + i) = NaN + NaN i + * sin(±INFINITY ± INFINITY i) = NaN + NaN i</code></pre></p> + * + * @return the sine of this complex number. + * @since 1.2 + */ + public Complex sin() { + if (isNaN()) { + return Complex.NaN; + } + + return createComplex(FastMath.sin(real) * MathUtils.cosh(imaginary), + FastMath.cos(real) * MathUtils.sinh(imaginary)); + } + + /** + * Compute the + * <a href="http://mathworld.wolfram.com/HyperbolicSine.html" TARGET="_top"> + * hyperbolic sine</a> of this complex number. + * <p> + * 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 java.lang.Math#sin}, {@link java.lang.Math#cos}, + * {@link MathUtils#cosh} and {@link MathUtils#sinh}.</p> + * <p> + * Returns {@link Complex#NaN} if either real or imaginary part of the + * input argument is <code>NaN</code>.</p> + * <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></p> + * + * @return the hyperbolic sine of this complex number + * @since 1.2 + */ + public Complex sinh() { + if (isNaN()) { + return Complex.NaN; + } + + return createComplex(MathUtils.sinh(real) * FastMath.cos(imaginary), + MathUtils.cosh(real) * FastMath.sin(imaginary)); + } + + /** + * Compute the + * <a href="http://mathworld.wolfram.com/SquareRoot.html" TARGET="_top"> + * square root</a> of this complex number. + * <p> + * Implements the following algorithm to compute <code>sqrt(a + bi)</code>: + * <ol><li>Let <code>t = sqrt((|a| + |a + bi|) / 2)</code></li> + * <li><pre>if <code> a ≥ 0</code> return <code>t + (b/2t)i</code> + * else return <code>|b|/2t + sign(b)t i </code></pre></li> + * </ol> + * where <ul> + * <li><code>|a| = {@link Math#abs}(a)</code></li> + * <li><code>|a + bi| = {@link Complex#abs}(a + bi) </code></li> + * <li><code>sign(b) = {@link MathUtils#indicator}(b) </code> + * </ul></p> + * <p> + * Returns {@link Complex#NaN} if either real or imaginary part of the + * input argument is <code>NaN</code>.</p> + * <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></p> + * + * @return the square root of this complex number + * @since 1.2 + */ + public Complex sqrt() { + if (isNaN()) { + return Complex.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), + MathUtils.indicator(imaginary) * t); + } + } + + /** + * Compute the + * <a href="http://mathworld.wolfram.com/SquareRoot.html" TARGET="_top"> + * square root</a> of 1 - <code>this</code><sup>2</sup> for this complex + * number. + * <p> + * Computes the result directly as + * <code>sqrt(Complex.ONE.subtract(z.multiply(z)))</code>.</p> + * <p> + * Returns {@link Complex#NaN} if either real or imaginary part of the + * input argument is <code>NaN</code>.</p> + * <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> + * + * @return the square root of 1 - <code>this</code><sup>2</sup> + * @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. + * <p> + * 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 java.lang.Math#sin}, {@link java.lang.Math#cos}, + * {@link MathUtils#cosh} and {@link MathUtils#sinh}.</p> + * <p> + * Returns {@link Complex#NaN} if either real or imaginary part of the + * input argument is <code>NaN</code>.</p> + * <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(1 ± INFINITY i) = 0 + NaN i + * tan(±INFINITY + i) = NaN + NaN i + * tan(±INFINITY ± INFINITY i) = NaN + NaN i + * tan(±π/2 + 0 i) = ±INFINITY + NaN i</code></pre></p> + * + * @return the tangent of this complex number + * @since 1.2 + */ + public Complex tan() { + if (isNaN()) { + return Complex.NaN; + } + + double real2 = 2.0 * real; + double imaginary2 = 2.0 * imaginary; + double d = FastMath.cos(real2) + MathUtils.cosh(imaginary2); + + return createComplex(FastMath.sin(real2) / d, MathUtils.sinh(imaginary2) / d); + } + + /** + * Compute the + * <a href="http://mathworld.wolfram.com/HyperbolicTangent.html" TARGET="_top"> + * hyperbolic tangent</a> of this complex number. + * <p> + * 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 java.lang.Math#sin}, {@link java.lang.Math#cos}, + * {@link MathUtils#cosh} and {@link MathUtils#sinh}.</p> + * <p> + * Returns {@link Complex#NaN} if either real or imaginary part of the + * input argument is <code>NaN</code>.</p> + * <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(1 ± INFINITY i) = NaN + NaN i + * tanh(±INFINITY + i) = NaN + 0 i + * tanh(±INFINITY ± INFINITY i) = NaN + NaN i + * tanh(0 + (π/2)i) = NaN + INFINITY i</code></pre></p> + * + * @return the hyperbolic tangent of this complex number + * @since 1.2 + */ + public Complex tanh() { + if (isNaN()) { + return Complex.NaN; + } + + double real2 = 2.0 * real; + double imaginary2 = 2.0 * imaginary; + double d = MathUtils.cosh(real2) + FastMath.cos(imaginary2); + + return createComplex(MathUtils.sinh(real2) / d, FastMath.sin(imaginary2) / d); + } + + + + /** + * <p>Compute the argument of this complex number. + * </p> + * <p>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> + * <p>If either real or imaginary part (or both) is NaN, NaN is returned. Infinite parts are handled + * as java.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 java.Math.atan2 for full details.</p> + * + * @return the argument of this complex number + */ + public double getArgument() { + return FastMath.atan2(getImaginary(), getReal()); + } + + /** + * <p>Computes the n-th roots of this complex number. + * </p> + * <p>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</code></i>, where <code>abs</code> and <code>phi</code> are + * respectively the {@link #abs() modulus} and {@link #getArgument() argument} of this complex number. + * </p> + * <p>If one or both parts of this complex number is NaN, a list with just one element, + * {@link #NaN} is returned.</p> + * <p>if neither part is NaN, but at least one part is infinite, the result is a one-element + * list containing {@link #INF}.</p> + * + * @param n degree of root + * @return List<Complex> all nth roots of this complex number + * @throws IllegalArgumentException if parameter n is less than or equal to 0 + * @since 2.0 + */ + public List<Complex> nthRoot(int n) throws IllegalArgumentException { + + if (n <= 0) { + throw MathRuntimeException.createIllegalArgumentException( + LocalizedFormats.CANNOT_COMPUTE_NTH_ROOT_FOR_NEGATIVE_N, + n); + } + + List<Complex> result = new ArrayList<Complex>(); + + if (isNaN()) { + result.add(Complex.NaN); + return result; + } + + if (isInfinite()) { + result.add(Complex.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 the real part + * @param imaginaryPart the imaginary part + * @return a new complex number instance + * @since 1.2 + */ + protected Complex createComplex(double realPart, double imaginaryPart) { + return new Complex(realPart, imaginaryPart); + } + + /** + * <p>Resolve the transient fields in a deserialized Complex Object.</p> + * <p>Subclasses will need to override {@link #createComplex} to deserialize properly</p> + * @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(); + } + +} |