Package org.hipparchus.complex

Class Complex

  • All Implemented Interfaces:
    Serializable, FieldElement<Complex>
    public class Complex
extends Object
implements FieldElement<Complex>, Serializable
    Representation of a Complex number, i.e. a number which has both a real and imaginary part.

    Implementations of arithmetic operations handle NaN and infinite values according to the rules for Double, i.e. equals(java.lang.Object) is an equivalence relation for all instances that have a NaN in either real or imaginary part, e.g. the following are considered equal:

    • 1 + NaNi
    • NaN + i
    • NaN + NaNi

    Note that this contradicts the IEEE-754 standard for floating point numbers (according to which the test x == x must fail if x is NaN). The method equals for primitive double in Precision conforms with IEEE-754 while this class conforms with the standard behavior for Java object types.

    See Also:
    Serialized Form

    • Field Detail

      • I

        public static final Complex I
        The square root of -1. A number representing "0.0 + 1.0i"

      • NaN

        public static final Complex NaN
        A complex number representing "NaN + NaNi"

      • INF

        public static final Complex INF
        A complex number representing "+INF + INFi"

      • ONE

        public static final Complex ONE
        A complex number representing "1.0 + 0.0i"

      • ZERO

        public static final Complex ZERO
        A complex number representing "0.0 + 0.0i"

    • Constructor Detail

      • Complex

        public Complex​(double real)
        Create a complex number given only the real part.
        Parameters:
        real - Real part.

      • Complex

        public Complex​(double real,
               double imaginary)
        Create a complex number given the real and imaginary parts.
        Parameters:
        real - Real part.
        imaginary - Imaginary part.

    • Method Detail

      • abs

        public double abs()
        Return the absolute value of this complex number. Returns NaN if either real or imaginary part is NaN and Double.POSITIVE_INFINITY if neither part is NaN, but at least one part is infinite.
        Returns:
        the absolute value.

      • add

        public Complex add​(Complex addend)
            throws NullArgumentException
        Returns a Complex whose value is (this + addend). Uses the definitional formula

        (a + bi) + (c + di) = (a+c) + (b+d)i

        If either this or addend has a NaN value in either part, NaN is returned; otherwise Infinite and NaN values are returned in the parts of the result according to the rules for Double arithmetic.
        Specified by:
        add in interface FieldElement<Complex>
        Parameters:
        addend - Value to be added to this Complex.
        Returns:
        this + addend.
        Throws:
        NullArgumentException - if addend is null.

      • add

        public Complex add​(double addend)
        Returns a Complex whose value is (this + addend), with addend interpreted as a real number.
        Parameters:
        addend - Value to be added to this Complex.
        Returns:
        this + addend.
        See Also:
        add(Complex)

      • conjugate

        public Complex conjugate()
        Returns the conjugate of this complex number. The conjugate of a + bi is a - bi.

        NaN is returned if either the real or imaginary part of this Complex number equals Double.NaN.

        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 1 + POSITIVE_INFINITY i is 1 - NEGATIVE_INFINITY i.

        Returns:
        the conjugate of this Complex object.

      • divide

        public Complex divide​(Complex divisor)
               throws NullArgumentException
        Returns a Complex whose value is (this / divisor). Implements the definitional formula 
          
    a + bi          ac + bd + (bc - ad)i
    ----------- = -------------------------
    c + di         c2 + d2
        but uses prescaling of operands to limit the effects of overflows and underflows in the computation.

        Infinite and NaN values are handled according to the following rules, applied in the order presented:

        • If either this or divisor has a NaN value in either part, NaN is returned.
        • If divisor equals ZERO, NaN is returned.
        • If this and divisor are both infinite, NaN is returned.
        • If this is finite (i.e., has no Infinite or NaN parts) and divisor is infinite (one or both parts infinite), ZERO is returned.
        • If this is infinite and divisor is finite, NaN values are returned in the parts of the result if the Double rules applied to the definitional formula force NaN results.
        Specified by:
        divide in interface FieldElement<Complex>
        Parameters:
        divisor - Value by which this Complex is to be divided.
        Returns:
        this / divisor.
        Throws:
        NullArgumentException - if divisor is null.

      • divide

        public Complex divide​(double divisor)
        Returns a Complex whose value is (this / divisor), with divisor interpreted as a real number.
        Parameters:
        divisor - Value by which this Complex is to be divided.
        Returns:
        this / divisor.
        See Also:
        divide(Complex)

      • reciprocal

        public Complex reciprocal()
        Returns the multiplicative inverse of this element.
        Specified by:
        reciprocal in interface FieldElement<Complex>
        Returns:
        the inverse of this.

      • equals

        public boolean equals​(Object other)
        Test for equality with another object. If both the real and imaginary parts of two complex numbers are exactly the same, and neither is Double.NaN, the two Complex objects are considered to be equal. The behavior is the same as for JDK's Double:
        • All NaN values are considered to be equal, i.e, if either (or both) real and imaginary parts of the complex number are equal to Double.NaN, the complex number is equal to NaN.
        • Instances constructed with different representations of zero (i.e. either "0" or "-0") are not considered to be equal.
        Overrides:
        equals in class Object
        Parameters:
        other - Object to test for equality with this instance.
        Returns:
        true if the objects are equal, false if object is null, not an instance of Complex, or not equal to this instance.

      • equals

        public static boolean equals​(Complex x,
                             Complex y,
                             int maxUlps)
        Test for the floating-point equality between Complex objects. It returns true if both arguments are equal or within the range of allowed error (inclusive).
        Parameters:
        x - First value (cannot be null).
        y - Second value (cannot be null).
        maxUlps - (maxUlps - 1) is the number of floating point values between the real (resp. imaginary) parts of x and y.
        Returns:
        true if there are fewer than maxUlps floating point values between the real (resp. imaginary) parts of x and y.
        See Also:
        Precision.equals(double,double,int)

      • equals

        public static boolean equals​(Complex x,
                             Complex y)
        Returns true iff the values are equal as defined by equals(x, y, 1).
        Parameters:
        x - First value (cannot be null).
        y - Second value (cannot be null).
        Returns:
        true if the values are equal.

      • equals

        public static boolean equals​(Complex x,
                             Complex y,
                             double eps)
        Returns 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 false if either of the arguments is NaN.
        Parameters:
        x - First value (cannot be null).
        y - Second value (cannot be null).
        eps - Amount of allowed absolute error.
        Returns:
        true if the values are two adjacent floating point numbers or they are within range of each other.
        See Also:
        Precision.equals(double,double,double)

      • equalsWithRelativeTolerance

        public static boolean equalsWithRelativeTolerance​(Complex x,
                                                  Complex y,
                                                  double eps)
        Returns 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 false if either of the arguments is NaN.
        Parameters:
        x - First value (cannot be null).
        y - Second value (cannot be null).
        eps - Amount of allowed relative error.
        Returns:
        true if the values are two adjacent floating point numbers or they are within range of each other.
        See Also:
        Precision.equalsWithRelativeTolerance(double,double,double)

      • hashCode

        public int hashCode()
        Get a hashCode for the complex number. Any Double.NaN value in real or imaginary part produces the same hash code 7.
        Overrides:
        hashCode in class Object
        Returns:
        a hash code value for this object.

      • getImaginary

        public double getImaginary()
        Access the imaginary part.
        Returns:
        the imaginary part.

      • getReal

        public double getReal()
        Access the real part.
        Returns:
        the real part.

      • isNaN

        public boolean isNaN()
        Checks whether either or both parts of this complex number is NaN.
        Returns:
        true if either or both parts of this complex number is NaN; false otherwise.

      • isInfinite

        public boolean isInfinite()
        Checks whether either the real or imaginary part of this complex number takes an infinite value (either Double.POSITIVE_INFINITY or Double.NEGATIVE_INFINITY) and neither part is NaN.
        Returns:
        true if one or both parts of this complex number are infinite and neither part is NaN.

      • multiply

        public Complex multiply​(Complex factor)
                 throws NullArgumentException
        Returns a Complex whose value is this * factor. Implements preliminary checks for NaN and infinity followed by the definitional formula:

        (a + bi)(c + di) = (ac - bd) + (ad + bc)i

        Returns NaN if either this or factor has one or more NaN parts.

        Returns INF if neither this nor factor has one or more NaN parts and if either this or factor has one or more infinite parts (same result is returned regardless of the sign of the components).

        Returns finite values in components of the result per the definitional formula in all remaining cases.

        Specified by:
        multiply in interface FieldElement<Complex>
        Parameters:
        factor - value to be multiplied by this Complex.
        Returns:
        this * factor.
        Throws:
        NullArgumentException - if factor is null.

      • multiply

        public Complex multiply​(int factor)
        Returns a Complex whose value is this * factor, with factor interpreted as a integer number.
        Specified by:
        multiply in interface FieldElement<Complex>
        Parameters:
        factor - value to be multiplied by this Complex.
        Returns:
        this * factor.
        See Also:
        multiply(Complex)

      • multiply

        public Complex multiply​(double factor)
        Returns a Complex whose value is this * factor, with factor interpreted as a real number.
        Parameters:
        factor - value to be multiplied by this Complex.
        Returns:
        this * factor.
        See Also:
        multiply(Complex)

      • negate

        public Complex negate()
        Returns a Complex whose value is (-this). Returns NaN if either real or imaginary part of this Complex number is Double.NaN.
        Specified by:
        negate in interface FieldElement<Complex>
        Returns:
        -this.

      • subtract

        public Complex subtract​(Complex subtrahend)
                 throws NullArgumentException
        Returns a Complex whose value is (this - subtrahend). Uses the definitional formula

        (a + bi) - (c + di) = (a-c) + (b-d)i

        If either this or subtrahend has a NaN] value in either part, NaN is returned; otherwise infinite and NaN values are returned in the parts of the result according to the rules for Double arithmetic.
        Specified by:
        subtract in interface FieldElement<Complex>
        Parameters:
        subtrahend - value to be subtracted from this Complex.
        Returns:
        this - subtrahend.
        Throws:
        NullArgumentException - if subtrahend is null.

      • subtract

        public Complex subtract​(double subtrahend)
        Returns a Complex whose value is (this - subtrahend).
        Parameters:
        subtrahend - value to be subtracted from this Complex.
        Returns:
        this - subtrahend.
        See Also:
        subtract(Complex)

      • acos

        public Complex acos()
        Compute the inverse cosine of this complex number. Implements the formula:

        acos(z) = -i (log(z + i (sqrt(1 - z<sup>2</sup>))))

        Returns NaN if either real or imaginary part of the input argument is NaN or infinite.
        Returns:
        the inverse cosine of this complex number.

      • asin

        public Complex asin()
        Compute the inverse sine of this complex number. Implements the formula:

        asin(z) = -i (log(sqrt(1 - z<sup>2</sup>) + iz))

        Returns NaN if either real or imaginary part of the input argument is NaN or infinite.

        Returns:
        the inverse sine of this complex number.

      • atan

        public Complex atan()
        Compute the inverse tangent of this complex number. Implements the formula:

        atan(z) = (i/2) log((i + z)/(i - z))

        Returns NaN if either real or imaginary part of the input argument is NaN or infinite.

        Returns:
        the inverse tangent of this complex number

      • cos

        public Complex cos()
        Compute the cosine of this complex number. Implements the formula:

        cos(a + bi) = cos(a)cosh(b) - sin(a)sinh(b)i

        where the (real) functions on the right-hand side are FastMath.sin(double), FastMath.cos(double), FastMath.cosh(double) and FastMath.sinh(double).

        Returns NaN if either real or imaginary part of the input argument is NaN.

        Infinite values in real or imaginary parts of the input may result in infinite or NaN values returned in parts of the result.

          Examples:
  
   cos(1 ± INFINITY i) = 1 ∓ INFINITY i
   cos(±INFINITY + i) = NaN + NaN i
   cos(±INFINITY ± INFINITY i) = NaN + NaN i
        Returns:
        the cosine of this complex number.

      • cosh

        public Complex cosh()
        Compute the hyperbolic cosine of this complex number. Implements the formula: 
          
   cosh(a + bi) = cosh(a)cos(b) + sinh(a)sin(b)i
        where the (real) functions on the right-hand side are FastMath.sin(double), FastMath.cos(double), FastMath.cosh(double) and FastMath.sinh(double).

        Returns NaN if either real or imaginary part of the input argument is NaN.

        Infinite values in real or imaginary parts of the input may result in infinite or NaN values returned in parts of the result. 
          Examples:
  
   cosh(1 ± INFINITY i) = NaN + NaN i
   cosh(±INFINITY + i) = INFINITY ± INFINITY i
   cosh(±INFINITY ± INFINITY i) = NaN + NaN i
        Returns:
        the hyperbolic cosine of this complex number.

      • exp

        public Complex exp()
        Compute the exponential function of this complex number. Implements the formula: 
          
   exp(a + bi) = exp(a)cos(b) + exp(a)sin(b)i
        where the (real) functions on the right-hand side are FastMath.exp(double), FastMath.cos(double), and FastMath.sin(double).

        Returns NaN if either real or imaginary part of the input argument is NaN.

        Infinite values in real or imaginary parts of the input may result in infinite or NaN values returned in parts of the result. 
          Examples:
  
   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
        Returns:
        ethis.

      • log

        public Complex log()
        Compute the natural logarithm of this complex number. Implements the formula: 
          
   log(a + bi) = ln(|a + bi|) + arg(a + bi)i
        where ln on the right hand side is FastMath.log(double), |a + bi| is the modulus, abs(), and arg(a + bi) = FastMath.atan2(double, double)(b, a).

        Returns NaN if either real or imaginary part of the input argument is NaN.

        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. 
          Examples:
  
   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
        Returns:
        the value ln   this, the natural logarithm of this.

      • pow

        public Complex pow​(Complex x)
            throws NullArgumentException
        Returns of value of this complex number raised to the power of x. Implements the formula: 
          
   yx = exp(x·log(y))
        where exp and log are exp() and log(), respectively.

        Returns NaN if either real or imaginary part of the input argument is NaN or infinite, or if y equals ZERO.

        Parameters:
        x - exponent to which this Complex is to be raised.
        Returns:
        thisx.
        Throws:
        NullArgumentException - if x is null.

      • pow

        public Complex pow​(double x)
        Returns of value of this complex number raised to the power of x.
        Parameters:
        x - exponent to which this Complex is to be raised.
        Returns:
        thisx.
        See Also:
        pow(Complex)

      • sin

        public Complex sin()
        Compute the sine of this complex number. Implements the formula: 
          
   sin(a + bi) = sin(a)cosh(b) - cos(a)sinh(b)i
        where the (real) functions on the right-hand side are FastMath.sin(double), FastMath.cos(double), FastMath.cosh(double) and FastMath.sinh(double).

        Returns NaN if either real or imaginary part of the input argument is NaN.

        Infinite values in real or imaginary parts of the input may result in infinite or NaN values returned in parts of the result. 

          Examples:
  
   sin(1 ± INFINITY i) = 1 ± INFINITY i
   sin(±INFINITY + i) = NaN + NaN i
   sin(±INFINITY ± INFINITY i) = NaN + NaN i
        Returns:
        the sine of this complex number.

      • sinh

        public Complex sinh()
        Compute the hyperbolic sine of this complex number. Implements the formula: 
          
   sinh(a + bi) = sinh(a)cos(b)) + cosh(a)sin(b)i
        where the (real) functions on the right-hand side are FastMath.sin(double), FastMath.cos(double), FastMath.cosh(double) and FastMath.sinh(double).

        Returns NaN if either real or imaginary part of the input argument is NaN.

        Infinite values in real or imaginary parts of the input may result in infinite or NaN values returned in parts of the result. 

          Examples:
  
   sinh(1 ± INFINITY i) = NaN + NaN i
   sinh(±INFINITY + i) = ± INFINITY + INFINITY i
   sinh(±INFINITY ± INFINITY i) = NaN + NaN i
        Returns:
        the hyperbolic sine of this.

      • sqrt

        public Complex sqrt()
        Compute the square root of this complex number. Implements the following algorithm to compute sqrt(a + bi):
        1. Let t = sqrt((|a| + |a + bi|) / 2)
        2. if  a &#8805; 0 return t + (b/2t)i
  else return |b|/2t + sign(b)t i
        where

        Returns NaN if either real or imaginary part of the input argument is NaN.

        Infinite values in real or imaginary parts of the input may result in infinite or NaN values returned in parts of the result. 
          Examples:
  
   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
        Returns:
        the square root of this.

      • sqrt1z

        public Complex sqrt1z()
        Compute the square root of 1 - this2 for this complex number. Computes the result directly as sqrt(ONE.subtract(z.multiply(z))).

        Returns NaN if either real or imaginary part of the input argument is NaN.

        Infinite values in real or imaginary parts of the input may result in infinite or NaN values returned in parts of the result.
        Returns:
        the square root of 1 - this2.

      • tan

        public Complex tan()
        Compute the tangent of this complex number. Implements the formula: 
          
   tan(a + bi) = sin(2a)/(cos(2a)+cosh(2b)) + [sinh(2b)/(cos(2a)+cosh(2b))]i
        where the (real) functions on the right-hand side are FastMath.sin(double), FastMath.cos(double), FastMath.cosh(double) and FastMath.sinh(double).

        Returns NaN if either real or imaginary part of the input argument is NaN.

        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. 
          Examples:
  
   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
        Returns:
        the tangent of this.

      • tanh

        public Complex tanh()
        Compute the hyperbolic tangent of this complex number. Implements the formula: 
          
   tan(a + bi) = sinh(2a)/(cosh(2a)+cos(2b)) + [sin(2b)/(cosh(2a)+cos(2b))]i
        where the (real) functions on the right-hand side are FastMath.sin(double), FastMath.cos(double), FastMath.cosh(double) and FastMath.sinh(double).

        Returns NaN if either real or imaginary part of the input argument is NaN.

        Infinite values in real or imaginary parts of the input may result in infinite or NaN values returned in parts of the result. 
          Examples:
  
   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
        Returns:
        the hyperbolic tangent of this.

      • getArgument

        public double getArgument()
        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.

        If either real or imaginary part (or both) is NaN, NaN is returned. Infinite parts are handled as 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 Math.atan2 for full details.

        Returns:
        the argument of this.

      • nthRoot

        public List<Complex> nthRoot​(int n)
                      throws MathIllegalArgumentException
        Computes the n-th roots of this complex number. The nth roots are defined by the formula: 
          
   zk = abs1/n (cos(phi + 2πk/n) + i (sin(phi + 2πk/n))
        for k=0, 1, ..., n-1, where abs and phi are respectively the modulus and argument of this complex number.

        If one or both parts of this complex number is NaN, a list with just one element, NaN is returned. if neither part is NaN, but at least one part is infinite, the result is a one-element list containing INF.

        Parameters:
        n - Degree of root.
        Returns:
        a List of all n-th roots of this.
        Throws:
        MathIllegalArgumentException - if n <= 0.

      • createComplex

        protected Complex createComplex​(double realPart,
                                double imaginaryPart)
        Create a complex number given the real and imaginary parts.
        Parameters:
        realPart - Real part.
        imaginaryPart - Imaginary part.
        Returns:
        a new complex number instance.
        See Also:
        valueOf(double, double)

      • valueOf

        public static Complex valueOf​(double realPart,
                              double imaginaryPart)
        Create a complex number given the real and imaginary parts.
        Parameters:
        realPart - Real part.
        imaginaryPart - Imaginary part.
        Returns:
        a Complex instance.

      • valueOf

        public static Complex valueOf​(double realPart)
        Create a complex number given only the real part.
        Parameters:
        realPart - Real part.
        Returns:
        a Complex instance.

      • readResolve

        protected final Object readResolve()
        Resolve the transient fields in a deserialized Complex Object. Subclasses will need to override createComplex(double, double) to deserialize properly.
        Returns:
        A Complex instance with all fields resolved.