Class to do math on complex numbers.
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* The contents of this file are subject to the Jabber Open Source License
* Version 1.0 (the "License"). You may not copy or use this file, in either
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* Copyrights
*
* Portions created by or assigned to Cursive Systems, Inc. are
* Copyright (c) 2002 Cursive Systems, Inc. All Rights Reserved. Contact
* information for Cursive Systems, Inc. is available at http://www.cursive.net/.
*
* Portions Copyright (c) 2002 Joe Hildebrand.
*
* Acknowledgements
*
* Special thanks to the Jabber Open Source Contributors for their
* suggestions and support of Jabber.
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* --------------------------------------------------------------------------*/
namespace bedrock.util
{
using System;
/// <summary>
/// Class to do math on complex numbers. Lots of optimizations, many from
/// the numerical methods literature. Sorry, but I've lost the citations by now.
/// </summary>
public class Complex : IFormattable
{
private double m_real;
private double m_imag;
// Double.Epsilon is too small
private static double s_tolerance = 1E-15;
/// <summary>
/// Create a complex number from a real part and an imaginary part.
/// Both parts use double-precision.
/// </summary>
/// <param name="real">Real part</param>
/// <param name="imag">Imaginary part. Multiplied by "i" and added to real.</param>
public Complex(double real, double imag)
{
m_real = real;
m_imag = imag;
}
/// <summary>
/// Complex number with imaginary part of 0.
/// </summary>
/// <param name="real">Real part</param>
public Complex(double real) : this(real, 0.0)
{
}
/// <summary>
/// Create a complex number from a polar representation.
/// </summary>
/// <param name="magnitude">The magnitude of the polar representation</param>
/// <param name="radianAngle">The angle, in radians, of the polar representation</param>
public static Complex Polar(double magnitude, double radianAngle)
{
return new Complex(magnitude * Math.Cos(radianAngle),
magnitude * Math.Sin(radianAngle));
}
/// <summary>
/// The real part of the complex number
/// </summary>
public double Real
{
get { return m_real; }
set { m_real = value; }
}
/// <summary>
/// The imaginary part of the complex number
/// </summary>
public double Imaginary
{
get { return m_imag; }
set { m_imag = value; }
}
/// <summary>
/// Get a new complex number that is the conjugate (Imaginary *= -1) of the current.
/// </summary>
public Complex Conjugate()
{
return new Complex(m_real, -m_imag);
}
/// <summary>
/// Return the absoulte value of the complex number.
/// </summary>
public double Abs()
{
return Abs(m_real, m_imag);
}
/// <summary>
/// sqrt(first^2 + second^2), with optimizations
/// </summary>
/// <param name="first">first number</param>
/// <param name="second">second number</param>
private static double Abs(double first, double second)
{
// avoid double math wherever possible...
//return Math.Sqrt((first * first) + (second * second));
first = Math.Abs(first);
second = Math.Abs(second);
if (first == 0d)
{
return second;
}
if (second == 0d)
{
return first;
}
if (first > second)
{
double temp = second / first;
return first * Math.Sqrt(1d + (temp * temp));
}
else
{
double temp = first / second;
return second * Math.Sqrt(1d + (temp * temp));
}
}
/// <summary>
/// Angle, in radians, of the current value.
/// </summary>
public double Arg()
{
return Math.Atan2(m_imag, m_real);
}
/// <summary>
/// The square root of the current value.
/// </summary>
public Complex Sqrt()
{
//return Math.Sqrt(this.Abs()) *
// new Complex( Math.Cos(this.Arg()/2),
// Math.Sin(this.Arg()/2));
if ((m_real == 0d) && (m_imag == 0d))
{
return new Complex(0d, 0d);
}
else
{
double ar = Math.Abs(m_real);
double ai = Math.Abs(m_imag);
double temp;
double w;
if (ar >= ai)
{
temp = ai / ar;
w = Math.Sqrt(ar) *
Math.Sqrt(0.5d * (1d + Math.Sqrt(1d + (temp * temp))));
}
else
{
temp = ar / ai;
w = Math.Sqrt(ai) *
Math.Sqrt(0.5d * (temp + Math.Sqrt(1d + (temp * temp))));
}
if (m_real > 0d)
{
return new Complex(w, m_imag / (2d * w));
}
else
{
double r = (m_imag >= 0d) ? w : -w;
return new Complex(r, m_imag / (2d * r));
}
}
}
/// <summary>
/// Raise the current value to a power.
/// </summary>
/// <param name="exponent">The power to raise to.</param>
public Complex Pow(double exponent)
{
double real = exponent * Math.Log(this.Abs());
double imag = exponent * this.Arg();
double scalar = Math.Exp(real);
return new Complex(scalar * Math.Cos(imag), scalar * Math.Sin(imag));
}
/// <summary>
/// Raise the current value to a power.
/// </summary>
/// <param name="exponent">The power to raise to.</param>
public Complex Pow(Complex exponent)
{
double real = Math.Log(this.Abs());
double imag = this.Arg();
double r2 = (real * exponent.m_real) - (imag * exponent.m_imag);
double i2 = (real * exponent.m_imag) + (imag * exponent.m_real);
double scalar = Math.Exp(r2);
return new Complex(scalar * Math.Cos(i2), scalar * Math.Sin(i2));
}
/// <summary>
/// Returns e raised to the specified power.
/// </summary>
public Complex Exp()
{
return Math.Exp(m_real) *
new Complex( Math.Cos(m_imag), Math.Sin(m_imag));
}
/// <summary>
/// 1 / (the current value)
/// </summary>
public Complex Inverse()
{
double scalar;
double ratio;
if (Math.Abs(m_real) >= Math.Abs(m_imag))
{
ratio = m_imag / m_real;
scalar = 1d / (m_real + m_imag * ratio);
return new Complex(scalar, -scalar * ratio);
}
else
{
ratio = m_real / m_imag;
scalar = 1d / (m_real * ratio + m_imag);
return new Complex(scalar * ratio, -scalar);
}
}
/// <summary>
/// Returns the natural (base e) logarithm of the current value.
/// </summary>
public Complex Log()
{
return new Complex(Math.Log(this.Abs()), this.Arg());
}
/// <summary>
/// Returns the sine of the current value.
/// </summary>
public Complex Sin()
{
Complex iz = this * Complex.i;
Complex izn = -iz;
return (iz.Exp() - izn.Exp()) / new Complex(0,2);
}
/// <summary>
/// Returns the cosine of the current value.
/// </summary>
public Complex Cos()
{
Complex iz = this * Complex.i;
Complex izn = -iz;
return (iz.Exp() + izn.Exp()) / 2.0;
}
/// <summary>
/// Returns the tangent of the current value.
/// </summary>
public Complex Tan()
{
return this.Sin() / this.Cos();
}
/// <summary>
/// Returns the hyperbolic sin of the current value.
/// </summary>
public Complex Sinh()
{
return (this.Exp() - (-this).Exp()) / 2d;
}
/// <summary>
/// Returns the hyperbolic cosine of the current value.
/// </summary>
public Complex Cosh()
{
return (this.Exp() + (-this).Exp()) / 2d;
}
/// <summary>
/// Returns the hyperbolic tangent of the current value.
/// </summary>
public Complex Tanh()
{
return this.Sinh() / this.Cosh();
}
/// <summary>
/// Returns the arc sine of the current value.
/// </summary>
public Complex Asin()
{
// TODO: if anyone cares about this function, some of it
// should probably be inlined and streamlined.
Complex I = i;
return -I * ((this*I) + (1 - (this * this)).Sqrt()).Log();
}
/// <summary>
/// Returns the arc cosine of the current value.
/// </summary>
public Complex Acos()
{
// TODO: if anyone cares about this function, some of it
// should probably be inlined and streamlined.
Complex I = i;
return -I * (this + I * (1 - (this*this)).Sqrt()).Log();
}
/// <summary>
/// Returns the arc tangent of the current value.
/// </summary>
public Complex Atan()
{
// TODO: if anyone cares about this function, some of it
// should probably be inlined and streamlined.
Complex I = i;
return -I/2 * ((I - this)/(I + this)).Log();
}
/// <summary>
/// Returns the arc hyperbolic sine of the current value.
/// </summary>
public Complex Asinh()
{
return (this + ((this*this) + 1).Sqrt()).Log();
}
/// <summary>
/// Returns the arc hyperbolic cosine of the current value.
/// </summary>
public Complex Acosh()
{
return 2d * (((this+1d) / 2d).Sqrt() +
((this-1) / 2d).Sqrt()).Log();
// Gar. This one didn't work. Perhaps it isn't returning the
// "pricipal" value.
//return (this + ((this*this) - 1).Sqrt()).Log();
}
/// <summary>
/// Returns the arc hyperbolic tangent of the current value.
/// </summary>
public Complex Atanh()
{
return ((1+this) / (1-this)).Log() / 2d;
}
/// <summary>
/// Is the current value Not a Number?
/// </summary>
public bool IsNaN()
{
return Double.IsNaN(m_real) || Double.IsNaN(m_imag);
}
/// <summary>
/// Is the current value infinite?
/// </summary>
public bool IsInfinity()
{
return Double.IsInfinity(m_real) || Double.IsInfinity(m_imag);
}
/// <summary>
///
/// </summary>
/// <returns></returns>
public override int GetHashCode()
{
return ((int)m_imag << 16) ^ (int) m_real;
}
/// <summary>
/// Format as string like "x + yi".
/// </summary>
public override string ToString()
{
return this.ToString(null, null);
}
/// <summary>
///
/// </summary>
/// <param name="format"></param>
/// <param name="sop"></param>
/// <returns></returns>
public string ToString(string format, IFormatProvider sop)
{
if (this.IsNaN())
return "NaN";
if (this.IsInfinity())
return "Infinity";
if (m_imag == 0d)
return m_real.ToString(format, sop);
if (m_real == 0d)
return m_imag.ToString(format, sop) + "i";
if (m_imag < 0.0)
{
return m_real.ToString(format, sop) + " - " +
(-m_imag).ToString(format, sop) + "i";
}
return m_real.ToString(format, sop) + " + " +
m_imag.ToString(format, sop) + "i";
}
/// <summary>
/// Do a half-assed job of assessing equality, using the current Tolerance value.
/// Will work with other Complex numbers or doubles.
/// </summary>
/// <param name="other">The other object to compare against. Must be double or Complex.</param>
public override bool Equals(object other)
{
if (other is Complex)
{
Complex o = (Complex) other;
// performance optimization for "identical" numbers"
if ((o.m_real == m_real) && (o.m_imag == m_imag))
return true;
return Equals(o, s_tolerance);
}
double d = (double) other; // can fire exception
if (m_imag != 0.0)
return false;
return Math.Abs(m_real - d) < s_tolerance;
}
/// <summary>
/// Is this number within a tolerance of being equal to another Complex number?
/// </summary>
/// <param name="other">The other Complex to comapare against.</param>
/// <param name="tolerance">The tolerance to be within.</param>
public bool Equals(Complex other, double tolerance)
{
return (this - other).Abs() < tolerance;
}
/// <summary>
/// Calls Equals().
/// </summary>
/// <param name="first">Complex</param>
/// <param name="second">Complex</param>
public static bool operator==(Complex first, Complex second)
{
return first.Equals(second);
}
/// <summary>
/// Calls !Equals().
/// </summary>
/// <param name="first">Complex</param>
/// <param name="second">Complex</param>
public static bool operator!=(Complex first, Complex second)
{
return !first.Equals(second);
}
/// <summary>
/// Adds two complex numbers.
/// </summary>
/// <param name="first">Complex</param>
/// <param name="second">Complex</param>
public static Complex operator+(Complex first, Complex second)
{
return new Complex(first.m_real + second.m_real,
first.m_imag + second.m_imag);
}
/// <summary>
/// Subtracts two complex numbers.
/// </summary>
/// <param name="first">Complex</param>
/// <param name="second">Complex</param>
public static Complex operator-(Complex first, Complex second)
{
return new Complex(first.m_real - second.m_real,
first.m_imag - second.m_imag);
}
/// <summary>
/// Negates a complex number.
/// </summary>
/// <param name="first">Complex</param>
public static Complex operator-(Complex first)
{
return new Complex(-first.m_real, -first.m_imag);
}
/// <summary>
/// Multiplies two complex numbers.
/// </summary>
/// <param name="first">Complex</param>
/// <param name="second">Complex</param>
public static Complex operator*(Complex first, Complex second)
{
return new Complex((first.m_real * second.m_real) -
(first.m_imag * second.m_imag),
(first.m_real * second.m_imag) +
(first.m_imag * second.m_real));
}
/// <summary>
/// Multiplies a complex number and a Complex.
/// </summary>
/// <param name="first">Complex</param>
/// <param name="second">double</param>
public static Complex operator*(Complex first, double second)
{
return new Complex(first.m_real * second, first.m_imag * second);
}
/// <summary>
/// Divides two Complex numbers.
/// </summary>
/// <param name="first">Complex</param>
/// <param name="second">Complex</param>
public static Complex operator/(Complex first, Complex second)
{
//return (first * second.Conjugate()) /
// ((second.m_real * second.m_real) +
// (second.m_imag * second.m_imag));
double scalar;
double ratio;
if (Math.Abs(second.m_real) >= Math.Abs(second.m_imag))
{
ratio = second.m_imag / second.m_real;
scalar = 1d / (second.m_real + (second.m_imag * ratio));
return new Complex(scalar * (first.m_real + (first.m_imag*ratio)),
scalar * (first.m_imag - (first.m_real*ratio)));
}
else
{
ratio = second.m_real / second.m_imag;
scalar = 1d / ((second.m_real * ratio) + second.m_imag);
return new Complex(scalar * (first.m_real*ratio + first.m_imag),
scalar * (first.m_imag*ratio - first.m_real));
}
}
/// <summary>
/// Divides a Complex number by a double.
/// </summary>
/// <param name="first">Complex</param>
/// <param name="second">double</param>
public static Complex operator/(Complex first, double second)
{
return new Complex(first.m_real / second, first.m_imag / second);
}
/// <summary>
/// Converts a double to a real Complex number.
/// </summary>
/// <param name="real">Real part</param>
public static implicit operator Complex(double real)
{
return new Complex(real);
}
/// <summary>
/// Constant for sqrt(-1).
/// </summary>
public static Complex i
{
get { return new Complex(0, 1); }
}
/// <summary>
/// Tolerance value for Equals().
/// </summary>
public static double Tolerance
{
get { return s_tolerance; }
set
{
if (value <= 0)
throw new ArgumentOutOfRangeException
("Tolerance must be greater than 0");
s_tolerance = value;
}
}
}
}
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