/*
* 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 $Id: Complex.java 1131229 2011-06-03 20:49:25Z luc $
*/
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();
}
}