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lib/complex.rb
187
lib/complex.rb
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@ -5,48 +5,31 @@
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# $Date: 1998/07/08 10:05:28 $
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# by Keiju ISHITSUKA(SHL Japan Inc.)
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#
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# --
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# Usage:
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# class Complex < Numeric
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# ----
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#
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# Complex(x, y) --> x + yi
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# y.im --> 0 + yi
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# complex.rb implements the Complex class for complex numbers. Additionally,
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# some methods in other Numeric classes are redefined or added to allow greater
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# interoperability with Complex numbers.
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#
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# Complex::polar
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#
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# Complex::+
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# Complex::-
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# Complex::*
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# Complex::/
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# Complex::**
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# Complex::%
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# Complex::divmod -- obsolete
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# Complex::abs
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# Complex::abs2
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# Complex::arg
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# Complex::polar
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# Complex::conjugate
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# Complex::<=>
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# Complex::==
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# Complex::to_f
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# Complex::to_r
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# Complex::to_s
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#
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# Complex::I
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#
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# Numeric::im
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#
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# Math.sqrt
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# Math.exp
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# Math.cos
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# Math.sin
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# Math.tan
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# Math.log
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# Math.log10
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# Math.atan2
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# Complex numbers can be created in the following manner:
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# - <tt>Complex(a, b)</tt>
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# - <tt>Complex.new(a, b)</tt>
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# - <tt>Complex.polar(radius, theta)</tt>
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#
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# Additionally, note the following:
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# - <tt>Complex::I</tt> (the mathematical constant <i>i</i>)
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# - <tt>Numeric#im</tt> (e.g. <tt>5.im -> 0+5i</tt>)
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#
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# The following +Math+ module methods are redefined to handle Complex arguments.
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# They will work as normal with non-Complex arguments.
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# sqrt exp cos sin tan log log10 atan2
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#
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#
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# Creates a Complex number. +a+ and +b+ should be Numeric. The result will be
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# <tt>a+bi</tt>.
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#
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def Complex(a, b = 0)
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if a.kind_of?(Complex) and b == 0
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a
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@ -63,21 +46,30 @@ def Complex(a, b = 0)
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end
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end
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#
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# The complex number class. See complex.rb for an overview.
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#
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class Complex < Numeric
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@RCS_ID='-$Id: complex.rb,v 1.3 1998/07/08 10:05:28 keiju Exp keiju $-'
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undef step
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def Complex.generic?(other)
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def Complex.generic?(other) # :nodoc:
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other.kind_of?(Integer) or
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other.kind_of?(Float) or
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(defined?(Rational) and other.kind_of?(Rational))
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end
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#
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# Creates a +Complex+ number in terms of +r+ (radius) and +theta+ (angle).
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#
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def Complex.polar(r, theta)
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Complex(r*Math.cos(theta), r*Math.sin(theta))
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end
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#
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# Creates a +Complex+ number <tt>a</tt>+<tt>b</tt><i>i</i>.
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#
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def initialize(a, b = 0)
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raise "non numeric 1st arg `#{a.inspect}'" if !a.kind_of? Numeric
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raise "non numeric 2nd arg `#{b.inspect}'" if !b.kind_of? Numeric
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@ -85,6 +77,9 @@ class Complex < Numeric
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@image = b
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end
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#
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# Addition with real or complex number.
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#
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def + (other)
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if other.kind_of?(Complex)
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re = @real + other.real
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@ -98,6 +93,9 @@ class Complex < Numeric
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end
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end
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#
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# Subtraction with real or complex number.
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#
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def - (other)
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if other.kind_of?(Complex)
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re = @real - other.real
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@ -111,6 +109,9 @@ class Complex < Numeric
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end
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end
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#
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# Multiplication with real or complex number.
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#
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def * (other)
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if other.kind_of?(Complex)
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re = @real*other.real - @image*other.image
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@ -124,6 +125,9 @@ class Complex < Numeric
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end
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end
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#
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# Division by real or complex number.
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#
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def / (other)
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if other.kind_of?(Complex)
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self*other.conjugate/other.abs2
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@ -135,6 +139,9 @@ class Complex < Numeric
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end
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end
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#
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# Raise this complex number to the given (real or complex) power.
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#
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def ** (other)
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if other == 0
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return Complex(1)
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@ -177,6 +184,9 @@ class Complex < Numeric
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end
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end
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#
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# Remainder after division by a real or complex number.
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#
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def % (other)
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if other.kind_of?(Complex)
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Complex(@real % other.real, @image % other.image)
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@ -188,6 +198,7 @@ class Complex < Numeric
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end
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end
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#--
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# def divmod(other)
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# if other.kind_of?(Complex)
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# rdiv, rmod = @real.divmod(other.real)
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@ -200,31 +211,54 @@ class Complex < Numeric
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# x.divmod(y)
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# end
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# end
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#++
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#
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# Absolute value (aka modulus): distance from the zero point on the complex
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# plane.
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#
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def abs
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Math.sqrt!((@real*@real + @image*@image).to_f)
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end
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#
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# Square of the absolute value.
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#
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def abs2
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@real*@real + @image*@image
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end
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#
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# Argument (angle from (1,0) on the complex plane).
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#
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def arg
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Math.atan2(@image.to_f, @real.to_f)
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end
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#
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# Returns the absolute value _and_ the argument.
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#
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def polar
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return abs, arg
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end
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#
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# Complex conjugate (<tt>z + z.conjugate = 2 * z.real</tt>).
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#
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def conjugate
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Complex(@real, -@image)
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end
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#
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# Compares the absolute values of the two numbers.
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#
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def <=> (other)
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self.abs <=> other.abs
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end
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#
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# Test for numerical equality (<tt>a == a + 0<i>i</i></tt>).
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#
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def == (other)
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if other.kind_of?(Complex)
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@real == other.real and @image == other.image
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@ -236,6 +270,9 @@ class Complex < Numeric
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end
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end
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#
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# Attempts to coerce +other+ to a Complex number.
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#
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def coerce(other)
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if Complex.generic?(other)
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return Complex.new(other), self
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@ -244,16 +281,25 @@ class Complex < Numeric
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end
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end
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#
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# FIXME
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#
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def denominator
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@real.denominator.lcm(@image.denominator)
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end
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#
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# FIXME
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#
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def numerator
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cd = denominator
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Complex(@real.numerator*(cd/@real.denominator),
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@image.numerator*(cd/@image.denominator))
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end
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#
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# Standard string representation of the complex number.
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#
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def to_s
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if @real != 0
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if defined?(Rational) and @image.kind_of?(Rational) and @image.denominator != 1
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@ -278,35 +324,65 @@ class Complex < Numeric
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end
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end
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#
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# Returns a hash code for the complex number.
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#
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def hash
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@real.hash ^ @image.hash
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end
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#
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# Returns "<tt>Complex(<i>real</i>, <i>image</i>)</tt>".
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#
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def inspect
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sprintf("Complex(%s, %s)", @real.inspect, @image.inspect)
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end
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#
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# +I+ is the imaginary number. It exists at point (0,1) on the complex plane.
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#
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I = Complex(0,1)
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# The real part of a complex number.
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attr :real
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# The imaginary part of a complex number.
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attr :image
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end
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#
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# Numeric is a built-in class on which Fixnum, Bignum, etc., are based. Here
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# some methods are added so that all number types can be treated to some extent
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# as Complex numbers.
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#
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class Numeric
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#
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# Returns a Complex number <tt>(0,<i>self</i>)</tt>.
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#
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def im
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Complex(0, self)
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end
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#
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# The real part of a complex number, i.e. <i>self</i>.
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#
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def real
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self
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end
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#
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# The imaginary part of a complex number, i.e. 0.
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#
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def image
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0
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end
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#
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# See Complex#arg.
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#
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def arg
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if self >= 0
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return 0
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@ -315,20 +391,28 @@ class Numeric
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end
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end
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#
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# See Complex#polar.
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#
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def polar
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return abs, arg
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end
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#
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# See Complex#conjugate (short answer: returns <i>self</i>).
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#
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def conjugate
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self
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end
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end
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class Fixnum
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if not defined? Rational
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alias power! **
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end
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# Redefined to handle a Complex argument.
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def ** (other)
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if self < 0
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Complex.new(self) ** other
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@ -367,6 +451,7 @@ module Math
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alias log10! log10
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alias atan2! atan2
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# Redefined to handle a Complex argument.
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def sqrt(z)
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if Complex.generic?(z)
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if z >= 0
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@ -379,6 +464,7 @@ module Math
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end
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end
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# Redefined to handle a Complex argument.
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def exp(z)
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if Complex.generic?(z)
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exp!(z)
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@ -387,14 +473,21 @@ module Math
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end
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end
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#
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# Hyperbolic cosine.
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#
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def cosh!(x)
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(exp!(x) + exp!(-x))/2.0
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end
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#
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# Hyperbolic sine.
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#
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def sinh!(x)
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(exp!(x) - exp!(-x))/2.0
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end
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# Redefined to handle a Complex argument.
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def cos(z)
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if Complex.generic?(z)
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cos!(z)
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@ -404,6 +497,7 @@ module Math
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end
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end
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# Redefined to handle a Complex argument.
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def sin(z)
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if Complex.generic?(z)
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sin!(z)
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@ -413,6 +507,7 @@ module Math
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end
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end
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# Redefined to handle a Complex argument.
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def tan(z)
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if Complex.generic?(z)
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tan!(z)
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@ -421,6 +516,7 @@ module Math
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end
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end
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# Redefined to handle a Complex argument.
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def log(z)
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if Complex.generic?(z) and z >= 0
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log!(z)
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@ -430,6 +526,7 @@ module Math
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end
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end
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# Redefined to handle a Complex argument.
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def log10(z)
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if Complex.generic?(z)
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log10!(z)
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@ -438,6 +535,8 @@ module Math
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end
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end
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# FIXME: I don't know what the point of this is. If you give it Complex
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# arguments, it will fail.
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def atan2(x, y)
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if Complex.generic?(x) and Complex.generic?(y)
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atan2!(x, y)
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@ -446,10 +545,14 @@ module Math
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end
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end
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#
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# Hyperbolic arctangent.
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#
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def atanh!(x)
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log((1.0 + x.to_f) / ( 1.0 - x.to_f)) / 2.0
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end
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# Redefined to handle a Complex argument.
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def atan(z)
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if Complex.generic?(z)
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atan2!(z, 1)
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@ -490,3 +593,11 @@ module Math
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module_function :atanh!
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end
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# Documentation comments:
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# - source: original (researched from pickaxe)
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# - a couple of fixme's
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# - Math module methods sinh! etc. a bit fuzzy. What exactly is the intention?
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# - RDoc output for Bignum etc. is a bit short, with nothing but an
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# (undocumented) alias. No big deal.
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