Category talk:Wren-complex

Complex numbers

This module aims to provide broadly similar functionality for complex numbers as already exists for real numbers. However, as the former - taken as a whole - are unordered, methods which require ordering have been omitted.

Complex numbers can be constructed from two separate Nums, from a pair of Nums, from a Rat object, from a complex string or (by copying) from another Complex number . These representations can also be mixed in operations as long as the first operand is itself a Complex object.

Support for complex matrices is also included though is not as extensive as for real matrices in the Wren-matrix module. It does however include some extra methods (such as conjugate transpose) which only apply to complex matrices.

Source code

<lang ecmascript>/* Module "complex.wren" */

import "/trait" for Cloneable

/* Complex represents a complex number of the form 'a + bi' where 'a' and 'b'

  are both Nums. Complex objects are immutable.

• /

class Complex is Cloneable {

   // Private helper function to check that 'o' is a suitable type and throw an error
   // otherwise. Real numbers, rationals and numeric strings are returned as complex numbers.
   static check_(o) {
       if (o is Complex) return o
       if (o is Num) return new_(o, 0)
       if ((o is List) && o.count == 2 && (o[0] is Num) && (o[1] is Num)) {
           return Complex.new_(o[0], o[1])
       }
       if (o.type.toString == "Rat") return new_(r.toFloat, 0)
       if (o is String) return fromString(o)
       Fiber.abort("Argument must be a number, pair of numbers, a rational number or a complex string.")
   }

   // Constants.
   static minusOne     { Complex.new_(-1,  0) }
   static zero         { Complex.new_( 0,  0) }
   static one          { Complex.new_( 1,  0) }
   static two          { Complex.new_( 2,  0) }
   static ten          { Complex.new_(10,  0) } 
   static imagMinusOne { Complex.new_( 0, -1) }
   static imagOne      { Complex.new_( 0,  1) }
   static imagTwo      { Complex.new_( 0,  2) }
   static imagTen      { Complex.new_( 0, 10) }
   static i            { Complex.new_( 0,  1) } // same as imagOne

   static pi           { Complex.new_(Num.pi, 0) }
   static e            { Complex.new_(2.71828182845904523536, 0) }
   static phi          { Complex.new_(1.6180339887498948482,  0) } // golden ratio
   static tau          { Complex.new_(1.6180339887498948482,  0) } // synonym for phi
   static ln2          { Complex.new_(0.69314718055994530942, 0) } // 2.log
   static ln10         { Complex.new_(2.30258509299404568402, 0) } // 10.log

   // Determines whether a Complex object is always shown as such
   // or, if purely real, as a real.
   static showAsReal     { __showAsReal }
   static showAsReal=(b) { __showAsReal = b }

   // Constructs a new Complex object by passing it real and imaginary components.
   construct new(real, imag) {
       if (real.type != Num || imag.type != Num) System.print("Argument(s) must be numbers.")
       _real = real
       _imag = imag
   }

   // Convenience method which constructs a new Complex object from a Num by passing it
   // just a real component.
   static new(real) { new(real, 0) }

   // Private constructor which avoids type checking.
   construct new_(real, imag) {
       _real = real
       _imag = imag
   }

   // Constructs a Complex object from an ordered pair of numbers [real, imag].
   static fromPair(p) {
      if (p.type != List || p.count != 2 || p[0].type != Num || p[1].type != Num) {
           Fiber.abort("Argument must be an ordered pair of numbers.")
      }
      return Complex.new_(p[0], p[1])
   }

   // Constructs a Complex object from a string of the form '±a±bi', '±a' or '±bi'
   // where 'a' and 'b' are string representations of Nums.
   static fromString(s) {
       if (s.type != String) Fiber.abort("Argument must be a complex string.")
       s = s.trim()
       if (s == "") Fiber.abort("Invalid complex string.")
       s = s.replace("--", "+")
       if (s[0] == "+") s = s[1..-1]
       if (s == "") Fiber.abort("Invalid complex string.")
       s = s.replace("e+", "e")
       var neg = s[0] == "-"
       if (neg) {
           s = s[1..-1]
           if (s == "") Fiber.abort("Invalid complex string.")
       }
       var negExp = s.indexOf("e-") >= 0
       if (negExp) s = s.replace("e-", "\v")
       if (s.indexOf("+") >= 0) {
           var split = s.split("+")
           if (split.count != 2) Fiber.abort("Invalid complex string.")
           if (negExp) for (i in 0..1) split[i] = split[i].replace("\v", "e-")
           var real = Num.fromString(split[0])
           if (!real) Fiber.abort("Invalid real part.")
           if (neg) real = -real
           if (!split[1].endsWith("i")) Fiber.abort("Invalid complex string.")
           var imag = Num.fromString(split[1][0..-2])
           if (!imag) Fiber.abort("Invalid imaginary part.")
           return Complex.new_(real, imag)
       } else if (s.indexOf("-") >= 0) {
           var split = s.split("-")
           if (split.count != 2) Fiber.abort("Invalid complex string.")
           if (negExp) for (i in 0..1) split[i] = split[i].replace("\v", "e-")
           var real = Num.fromString(split[0])
           if (!real) Fiber.abort("Invalid real part.")
           if (neg) real = -real
           if (!split[1].endsWith("i")) Fiber.abort("Invalid complex string.")
           var imag = Num.fromString(split[1][0..-2])
           if (!imag) Fiber.abort("Invalid imaginary part.")
           return Complex.new_(real, -imag)
       } else if (s.endsWith("i")) {
           if (negExp) s = s.replace("\v", "e-")
           var imag = Num.fromString(s[0..-2])
           if (!imag) Fiber.abort("Invalid imaginary part.")
           if (neg) imag = -imag
           return Complex.new_(0, imag)
       } else {
           if (negExp) s = s.replace("\v", "e-")
           var real = Num.fromString(s)
           if (!real) Fiber.abort("Invalid real part.")
           if (neg) real = -real
           return Complex.new_(real, 0)
       }
   }

   // Constructs a Complex object from a Rat object.
   static fromRat(r) {
       if (r.type.toString != "Rat") Fiber.abort("Argument must be a rational number.")
       return Complex.new_(r.toFloat, 0)
   }

   // Constructs a Complex object from polar coordinates (r, theta).
   static fromPolar(r, theta)  {
       if (r.type != Num || theta.type != Num) {
           Fiber.abort("Arguments must be numbers.")      
       }
       return Complex.new_(r * theta.cos, r * theta.sin)
   }

   // Basic properties.
   real { _real }  // real component
   imag { _imag }  // imaginary component

   isInfinity { _real.isInfinity || _imag.isInfinty } // true if either part is infinite
   isNan      { _real.isNan || imag.isNan }           // true if either part is nan

   // Returns whether real component is an integer and imaginary component is zero.
   isRealInteger { _real.isInteger && _imag == 0 }

   // Returns whether imaginary component is an integer and real component is zero.
   isImagInteger { _imag.isInteger && _real == 0 }

   // Returns the ordered pair [_real, _imag].
   toPair { [_real, _imag] }

   // Returns the polar coordinates of this instance [modulus, phase].
   toPolar { [abs, phase] }

   // Returns a new instance which negates the current one.
   - { Complex.new_(-_real, -_imag) }

   // Returns the inverse or reciprocal of this instance.
   inverse {
       var denom = _real * _real + _imag * _imag
       return Complex.new_(_real/denom, -_imag/denom)
   }

   // Arithmetic operators (work with real numbers, rational numbers, complex strings
   // as well as other complex numbers). Always return a new instance.
   +(o) {
       o = Complex.check_(o)
       return Complex.new_(_real + o.real, _imag + o.imag)
   }

   -(o) { this + (-o) }

   *(o) {
       o = Complex.check_(o)
       return Complex.new_(
           _real * o.real - _imag * o.imag,
           _real * o.imag + _imag * o.real
       )
   }

   /(o) { 
       o = Complex.check_(o)
       var i = o.inverse
       return Complex.new_(
           _real * i.real - _imag * i.imag,
           _real * i.imag + _imag * i.real
       )
   }

   // Returns the absolute value or modulus of this instance.
   abs { (_real*_real + _imag*_imag).sqrt }

   // Returns the phase or argument of the current instance in the range [-π, π].
   phase { _imag.atan(_real) }

   // Returns the complex conjugate of this instance
   conj { Complex.new_(_real, -_imag) }

   // Returns the square of this instance.
   square { Complex.new_(_real * _real - _imag * _imag, _real * _imag * 2) }

   // Returns the square root of this instance.
   sqrt {
       var m = abs
       var r = ((m + _real)/2).sqrt
       var i = ((m - _real)/2).sqrt
       if (_imag < 0) i = -i
       return Complex.new_(r, i)
   }

   // Returns the base 'e' exponential of this instance.
   exp {
       var e = Complex.e.real.pow(_real) /* change to _real.exp from version 0.4.0 */
       return Complex.new_(e * _imag.cos, e * _imag.sin)
   }

   // Returns the natural logarithm of the current instance.
   log {
       var p = phase
       if (p > Num.pi) p = p - Num.pi*2
       return Complex.new_(abs.log, p)
   }

   // Returns the logarithm to the base 2 of the current instance.
   log2  { log / Complex.two.log }

   // Returns the logarithm to the base 10 of the current instance.
   log10 { log / Complex.ten.log }

   // Returns this instance to the power of the complex number 'e'.
   pow(e) {
       e = Complex.check_(e)
       return (log * e).exp
   }

   // Returns the cosine of the current instance.
   cos {
       var i = Complex.i
       return ((i * this).exp + (i * (-this)).exp) / Complex.two
   }

   // Returns the sine of the current instance.
   sin {
       var i = Complex.i
       return ((i * this).exp - (i * (-this)).exp) / Complex.imagTwo
   }

   // Returns the tangent of the current instance.
   tan { sin / cos }

   // Returns the arc cosine of the current instance.
   acos {
       var c = (Complex.one - square).sqrt
       c = this + c * Complex.imagMinusOne
       return c.log * Complex.i
   }

   // Returns the arc sine of the current instance.
   asin {
       var c = (Complex.one - square).sqrt
       c = c + this * Complex.imagMinusOne
       return c.log * Complex.i
   }

   // Returns the arc tangent of the current instance.
   atan {
       var a = Complex.new_(_real, _imag - 1)
       var b = Complex.new_(-_real, -_imag - 1)
       return (Complex.imagMinusOne * (a/b).log) / Complex.two
   }

   // Returns the hyperbolic cosine of the current instance.
   cosh { (this.exp + (-this).exp)/Complex.two }

   // Returns the hyperbolic sine of the current instance.
   sinh { (this.exp - (-this).exp)/Complex.two }

   // Returns the hyperbolic tangent of the current instance.
   tanh { sinh/cosh }

   // Returns the inverse hyperbolic cosine of the current instance.
   acosh { (this + (square - Complex.one).sqrt).log }

   // Returns the inverse hyperbolic sine of the current instance.
   asinh { (this + (square + Complex.one).sqrt).log }

   // Returns the inverse hyperbolic tangent of the current instance.
   atanh {
       var c = (this + Complex.one).log
       c = c - (-(this - Complex.one)).log
       return c / Complex.two
   }

   // The inherited 'clone' method just returns 'this' as Complex objects are immutable.
   // If you need an actual copy use this method instead.
   copy() { Complex.new_(_real, _imag ) }

   // Equality operators.
   ==(o) { 
       o = Complex.check_(o)
       return _real == o.real && _imag == o.imag 
   }
   !=(o) { !(this == o) }

   // Returns the string representation of this Complex object depending on 'showAsReal'.
   toString {
       if (_real == -0) _real = 0
       if (_imag == -0) _imag = 0
       var s = (_imag >= 0) ? "%(_real) + %(_imag)" : "%(_real) - %(-_imag)"
       s = (_imag.abs != 1) ? s + "i" : s[0..-2] + "i"
       if (s.endsWith("- 0i")) s = s[0..-5] + "+ 0i"
       return (Complex.showAsReal && _imag == 0) ? s = s[0..-6] : s
   }

}

/* Complexes contains routines applicable to lists of complex numbers. */ class Complexes {

   static sum(a)  { a.reduce(Complex.zero) { |acc, x| acc + x } }
   static mean(a) { sum(a)/a.count }
   static prod(a) { a.reduce(Complex.one)  { |acc, x| acc * x } }

}

// Type aliases for classes in case of any name clashes with other modules. var Complex_Complex = Complex var Complex_Complexes = Complexes var Complex_Cloneable = Cloneable // in case imported indirectly</lang>