Complex Numbers

wiki

e.g. cx = cx (+-*/) "2i", cx = cx (+-*/) 2, cx = cx (+-*/) cx2;

one can use cx^.5 to get the squareroot of a complex number; also cx^'*' to get the conjugate complex value. To get a complex number you have to call the function complex.to( num )

Download the whole package from

http://luaforge.net/projects/luamatrix

AK 3-Oct-2007: If patching the Lua core (5.1.2) itself is an option, one might consider the LNUM patch at:

http://luaforge.net/projects/lnum

It makes complex numbers the internal Lua number type (define LNUM_COMPLEX, LNUM_INT32, use a C99 compatible compiler). Fully compatible with Lua 5.1.2 otherwise, extension modules etc.

-- complex 0.3.0
-- Lua 5.1

-- 'complex' provides common tasks with complex numbers

-- function complex.to( arg ); complex( arg )
-- returns a complex number on success, nil on failure
-- arg := number or { number,number } or ( "(-)<number>" and/or "(+/-)<number>i" )
--      e.g. 5; {2,3}; "2", "2+i", "-2i", "2^2*3+1/3i"
--      note: 'i' is always in the numerator, spaces are not allowed

-- a complex number is defined as carthesic complex number
-- complex number := { real_part, imaginary_part }
-- this gives fast access to both parts of the number for calculation
-- the access is faster than in a hash table
-- the metatable is just a add on, when it comes to speed, one is faster using a direct function call

-- http://luaforge.net/projects/LuaMatrix
-- http://lua-users.org/wiki/ComplexNumbers

-- Licensed under the same terms as Lua itself.

--/////////////--
--// complex //--
--/////////////--

-- link to complex table
local complex = {}

-- link to complex metatable
local complex_meta = {}

-- complex.to( arg )
-- return a complex number on success
-- return nil on failure
local _retone = function() return 1 end
local _retminusone = function() return -1 end
function complex.to( num )
   -- check for table type
   if type( num ) == "table" then
      -- check for a complex number
      if getmetatable( num ) == complex_meta then
         return num
      end
      local real,imag = tonumber( num[1] ),tonumber( num[2] )
      if real and imag then
         return setmetatable( { real,imag }, complex_meta )
      end
      return
   end
   -- check for number
   local isnum = tonumber( num )
   if isnum then
      return setmetatable( { isnum,0 }, complex_meta )
   end
   if type( num ) == "string" then
      -- check for real and complex
      -- number chars [%-%+%*%^%d%./Ee]
      local real,sign,imag = string.match( num, "^([%-%+%*%^%d%./Ee]*%d)([%+%-])([%-%+%*%^%d%./Ee]*)i$" )
      if real then
         if string.lower(string.sub(real,1,1)) == "e"
         or string.lower(string.sub(imag,1,1)) == "e" then
            return
         end
         if imag == "" then
            if sign == "+" then
               imag = _retone
            else
               imag = _retminusone
            end
         elseif sign == "+" then
            imag = loadstring("return tonumber("..imag..")")
         else
            imag = loadstring("return tonumber("..sign..imag..")")
         end
         real = loadstring("return tonumber("..real..")")
         if real and imag then
            return setmetatable( { real(),imag() }, complex_meta )
         end
         return
      end
      -- check for complex
      local imag = string.match( num,"^([%-%+%*%^%d%./Ee]*)i$" )
      if imag then
         if imag == "" then
            return setmetatable( { 0,1 }, complex_meta )
         elseif imag == "-" then
            return setmetatable( { 0,-1 }, complex_meta )
         end
         if string.lower(string.sub(imag,1,1)) ~= "e" then
            imag = loadstring("return tonumber("..imag..")")
            if imag then
               return setmetatable( { 0,imag() }, complex_meta )
            end
         end
         return
      end
      -- should be real
      local real = string.match( num,"^(%-*[%d%.][%-%+%*%^%d%./Ee]*)$" )
      if real then
         real = loadstring( "return tonumber("..real..")" )
         if real then
            return setmetatable( { real(),0 }, complex_meta )
         end
      end
   end
end

-- complex( arg )
-- same as complex.to( arg )
-- set __call behaviour of complex
setmetatable( complex, { __call = function( _,num ) return complex.to( num ) end } )

-- complex.new( real, complex )
-- fast function to get a complex number, not invoking any checks
function complex.new( ... )
   return setmetatable( { ... }, complex_meta )
end

-- complex.type( arg )
-- is argument of type complex
function complex.type( arg )
   if getmetatable( arg ) == complex_meta then
      return "complex"
   end
end

-- complex.convpolar( r, phi )
-- convert polar coordinates ( r*e^(i*phi) ) to carthesic complex number
-- r (radius) is a number
-- phi (angle) must be in radians; e.g. [0 - 2pi]
function complex.convpolar( radius, phi )
   return setmetatable( { radius * math.cos( phi ), radius * math.sin( phi ) }, complex_meta )
end

-- complex.convpolardeg( r, phi )
-- convert polar coordinates ( r*e^(i*phi) ) to carthesic complex number
-- r (radius) is a number
-- phi must be in degrees; e.g. [0° - 360°]
function complex.convpolardeg( radius, phi )
   phi = phi/180 * math.pi
   return setmetatable( { radius * math.cos( phi ), radius * math.sin( phi ) }, complex_meta )
end

--// complex number functions only

-- complex.tostring( cx [, formatstr] )
-- to string or real number
-- takes a complex number and returns its string value or real number value
function complex.tostring( cx,formatstr )
   local real,imag = cx[1],cx[2]
   if formatstr then
      if imag == 0 then
         return string.format( formatstr, real )
      elseif real == 0 then
         return string.format( formatstr, imag ).."i"
      elseif imag > 0 then
         return string.format( formatstr, real ).."+"..string.format( formatstr, imag ).."i"
      end
      return string.format( formatstr, real )..string.format( formatstr, imag ).."i"
   end
   if imag == 0 then
      return real
   elseif real == 0 then
      return ((imag==1 and "") or (imag==-1 and "-") or imag).."i"
   elseif imag > 0 then
      return real.."+"..(imag==1 and "" or imag).."i"
   end
   return real..(imag==-1 and "-" or imag).."i"
end

-- complex.print( cx [, formatstr] )
-- print a complex number
function complex.print( ... )
   print( complex.tostring( ... ) )
end

-- complex.polar( cx )
-- from complex number to polar coordinates
-- output in radians; [-pi,+pi]
-- returns r (radius), phi (angle)
function complex.polar( cx )
   return math.sqrt( cx[1]^2 + cx[2]^2 ), math.atan2( cx[2], cx[1] )
end

-- complex.polardeg( cx )
-- from complex number to polar coordinates
-- output in degrees; [-180°,180°]
-- returns r (radius), phi (angle)
function complex.polardeg( cx )
   return math.sqrt( cx[1]^2 + cx[2]^2 ), math.atan2( cx[2], cx[1] ) / math.pi * 180
end

-- complex.mulconjugate( cx )
-- multiply with conjugate, function returning a number
function complex.mulconjugate( cx )
   return cx[1]^2 + cx[2]^2
end

-- complex.abs( cx )
-- get the absolute value of a complex number
function complex.abs( cx )
   return math.sqrt( cx[1]^2 + cx[2]^2 )
end

-- complex.get( cx )
-- returns real_part, imaginary_part
function complex.get( cx )
   return cx[1],cx[2]
end

-- complex.set( cx, real, imag )
-- sets real_part = real and imaginary_part = imag
function complex.set( cx,real,imag )
   cx[1],cx[2] = real,imag
end

-- complex.is( cx, real, imag )
-- returns true if, real_part = real and imaginary_part = imag
-- else returns false
function complex.is( cx,real,imag )
   if cx[1] == real and cx[2] == imag then
      return true
   end
   return false
end

--// functions returning a new complex number

-- complex.copy( cx )
-- copy complex number
function complex.copy( cx )
   return setmetatable( { cx[1],cx[2] }, complex_meta )
end

-- complex.add( cx1, cx2 )
-- add two numbers; cx1 + cx2
function complex.add( cx1,cx2 )
   return setmetatable( { cx1[1]+cx2[1], cx1[2]+cx2[2] }, complex_meta )
end

-- complex.sub( cx1, cx2 )
-- subtract two numbers; cx1 - cx2
function complex.sub( cx1,cx2 )
   return setmetatable( { cx1[1]-cx2[1], cx1[2]-cx2[2] }, complex_meta )
end

-- complex.mul( cx1, cx2 )
-- multiply two numbers; cx1 * cx2
function complex.mul( cx1,cx2 )
   return setmetatable( { cx1[1]*cx2[1] - cx1[2]*cx2[2],cx1[1]*cx2[2] + cx1[2]*cx2[1] }, complex_meta )
end

-- complex.mulnum( cx, num )
-- multiply complex with number; cx1 * num
function complex.mulnum( cx,num )
   return setmetatable( { cx[1]*num,cx[2]*num }, complex_meta )
end

-- complex.div( cx1, cx2 )
-- divide 2 numbers; cx1 / cx2
function complex.div( cx1,cx2 )
   -- get complex value
   local val = cx2[1]^2 + cx2[2]^2
   -- multiply cx1 with conjugate complex of cx2 and divide through val
   return setmetatable( { (cx1[1]*cx2[1]+cx1[2]*cx2[2])/val,(cx1[2]*cx2[1]-cx1[1]*cx2[2])/val }, complex_meta )
end

-- complex.divnum( cx, num )
-- divide through a number
function complex.divnum( cx,num )
   return setmetatable( { cx[1]/num,cx[2]/num }, complex_meta )
end

-- complex.pow( cx, num )
-- get the power of a complex number
function complex.pow( cx,num )
   if math.floor( num ) == num then
      if num < 0 then
         local val = cx[1]^2 + cx[2]^2
         cx = { cx[1]/val,-cx[2]/val }
         num = -num
      end
      local real,imag = cx[1],cx[2]
      for i = 2,num do
         real,imag = real*cx[1] - imag*cx[2],real*cx[2] + imag*cx[1]
      end
      return setmetatable( { real,imag }, complex_meta )
   end
   -- we calculate the polar complex number now
   -- since then we have the versatility to calc any potenz of the complex number
   -- then we convert it back to a carthesic complex number, we loose precision here
   local length,phi = math.sqrt( cx[1]^2 + cx[2]^2 )^num, math.atan2( cx[2], cx[1] )*num
   return setmetatable( { length * math.cos( phi ), length * math.sin( phi ) }, complex_meta )
end

-- complex.sqrt( cx )
-- get the first squareroot of a complex number, more accurate than cx^.5
function complex.sqrt( cx )
   local len = math.sqrt( cx[1]^2+cx[2]^2 )
   local sign = (cx[2]<0 and -1) or 1
   return setmetatable( { math.sqrt((cx[1]+len)/2), sign*math.sqrt((len-cx[1])/2) }, complex_meta )
end

-- complex.ln( cx )
-- natural logarithm of cx
function complex.ln( cx )
   return setmetatable( { math.log(math.sqrt( cx[1]^2 + cx[2]^2 )),
      math.atan2( cx[2], cx[1] ) }, complex_meta )
end

-- complex.exp( cx )
-- exponent of cx (e^cx)
function complex.exp( cx )
   local expreal = math.exp(cx[1])
   return setmetatable( { expreal*math.cos(cx[2]), expreal*math.sin(cx[2]) }, complex_meta )
end

-- complex.conjugate( cx )
-- get conjugate complex of number
function complex.conjugate( cx )
   return setmetatable( { cx[1], -cx[2] }, complex_meta )
end

-- complex.round( cx [,idp] )
-- round complex numbers, by default to 0 decimal points
function complex.round( cx,idp )
   local mult = 10^( idp or 0 )
   return setmetatable( { math.floor( cx[1] * mult + 0.5 ) / mult,
      math.floor( cx[2] * mult + 0.5 ) / mult }, complex_meta )
end

--// metatable functions

complex_meta.__add = function( cx1,cx2 )
   local cx1,cx2 = complex.to( cx1 ),complex.to( cx2 )
   return complex.add( cx1,cx2 )
end
complex_meta.__sub = function( cx1,cx2 )
   local cx1,cx2 = complex.to( cx1 ),complex.to( cx2 )
   return complex.sub( cx1,cx2 )
end
complex_meta.__mul = function( cx1,cx2 )
   local cx1,cx2 = complex.to( cx1 ),complex.to( cx2 )
   return complex.mul( cx1,cx2 )
end
complex_meta.__div = function( cx1,cx2 )
   local cx1,cx2 = complex.to( cx1 ),complex.to( cx2 )
   return complex.div( cx1,cx2 )
end
complex_meta.__pow = function( cx,num )
   if num == "*" then
      return complex.conjugate( cx )
   end
   return complex.pow( cx,num )
end
complex_meta.__unm = function( cx )
   return setmetatable( { -cx[1], -cx[2] }, complex_meta )
end
complex_meta.__eq = function( cx1,cx2 )
   if cx1[1] == cx2[1] and cx1[2] == cx2[2] then
      return true
   end
   return false
end
complex_meta.__tostring = function( cx )
   return tostring( complex.tostring( cx ) )
end
complex_meta.__concat = function( cx,cx2 )
   return tostring(cx)..tostring(cx2)
end
-- cx( cx, formatstr )
complex_meta.__call = function( ... )
   print( complex.tostring( ... ) )
end
complex_meta.__index = {}
for k,v in pairs( complex ) do
   complex_meta.__index[k] = v
end

return complex

--///////////////--
--// chillcode //--
--///////////////--

Testcode:

local complex = require "complex"

local cx,cx1,cx2,re,im

-- complex.to / complex call
cx = complex { 2,3 }
assert( tostring( cx ) == "2+3i" )
cx = complex ( 2 )
assert( tostring( cx ) == "2" )
assert( cx:tostring() == 2 )
cx = complex "2^2+3/2i"
assert( tostring( cx ) == "4+1.5i" )
cx = complex ".5-2E-3i"
assert( tostring( cx ) == "0.5-0.002i" )
cx = complex "3i"
assert( tostring( cx ) == "3i" )
cx = complex "2"
assert( tostring( cx ) == "2" )
assert( cx:tostring() == 2 )
assert( complex "2 + 4i" == nil )

-- complex.new
cx = complex.new( 2,3 )
assert( tostring( cx ) == "2+3i" )

-- complex.type
assert( complex.type( cx ) == "complex" )
assert( complex.type( {} ) == nil )

-- complex.convpolar( radius, phi )
assert( complex.convpolar( 3, 0 ):round(10) == complex "3" )
assert( complex.convpolar( 3, math.pi/2 ):round(10) == complex "3i" )
assert( complex.convpolar( 3, math.pi ):round(10) == complex "-3" )
assert( complex.convpolar( 3, math.pi*3/2 ):round(10) == complex "-3i" )
assert( complex.convpolar( 3, math.pi*2 ):round(10) == complex "3" )

-- complex.convpolardeg( radius, phi )
assert( complex.convpolardeg( 3, 0 ):round(10) == complex "3" )
assert( complex.convpolardeg( 3, 90 ):round(10) == complex "3i" )
assert( complex.convpolardeg( 3, 180 ):round(10) == complex "-3" )
assert( complex.convpolardeg( 3, 270 ):round(10) == complex "-3i" )
assert( complex.convpolardeg( 3, 360 ):round(10) == complex "3" )

-- complex.tostring( cx,formatstr )
cx = complex "2+3i"
assert( complex.tostring( cx ) == "2+3i" )
assert( complex.tostring( cx, "%.2f" ) == "2.00+3.00i" )
-- does not support a second argument
assert( tostring( cx, "%.2f" ) == "2+3i" )

-- complex.polar( cx )
local r,phi = complex.polar( {3,0} )
assert( r == 3 )
assert( phi == 0 )

local r,phi = complex.polar( {0,3} )
assert( r == 3 )
assert( phi == math.pi/2 )

local r,phi = complex.polar( {-3,0} )
assert( r == 3 )
assert( phi == math.pi )

local r,phi = complex.polar( {0,-3} )
assert( r == 3 )
assert( phi == -math.pi/2 )

-- complex.polardeg( cx )
local r,phi = complex.polardeg( {3,0} )
assert( r == 3 )
assert( phi == 0 )

local r,phi = complex.polardeg( {0,3} )
assert( r == 3 )
assert( phi == 90 )

local r,phi = complex.polardeg( {-3,0} )
assert( r == 3 )
assert( phi == 180 )

local r,phi = complex.polardeg( {0,-3} )
assert( r == 3 )
assert( phi == -90 )

-- complex.mulconjugate( cx )
cx = complex "2+3i"
assert( complex.mulconjugate( cx ) == 13 )

-- complex.abs( cx )
cx = complex "3+4i"
assert( complex.abs( cx ) == 5 )

-- complex.get( cx )
cx = complex "2+3i"
re,im = complex.get( cx )
assert( re == 2 )
assert( im == 3 )

-- complex.set( cx, re, im )
cx = complex "2+3i"
complex.set( cx, 3, 2 )
assert( cx == complex "3+2i" )

-- complex.is( cx, re, im )
cx = complex "2+3i"
assert( complex.is( cx, 2, 3 ) == true )
assert( complex.is( cx, 3, 2 ) == false )

-- complex.copy( cx )
cx = complex "2+3i"
cx1 = complex.copy( cx )
complex.set( cx, 1, 1 )
assert( cx1 == complex "2+3i" )

-- complex.add( cx1, cx2 )
cx1 = complex "2+3i"
cx2 = complex "3+2i"
assert( complex.add(cx1,cx2) == complex "5+5i" )

-- complex.sub( cx1, cx2 )
cx1 = complex "2+3i"
cx2 = complex "3+2i"
assert( complex.sub(cx1,cx2) == complex "-1+1i" )

-- complex.mul( cx1, cx2 )
cx1 = complex "2+3i"
cx2 = complex "3+2i"
assert( complex.mul(cx1,cx2) == complex "13i" )

-- complex.mulnum( cx, num )
cx = complex "2+3i"
assert( complex.mulnum( cx, 2 ) == complex "4+6i" )

-- complex.div( cx1, cx2 )
cx1 = complex "2+3i"
cx2 = complex "3-2i"
assert( complex.div(cx1,cx2) == complex "i" )

-- complex.divnum( cx, num )
cx = complex "2+3i"
assert( complex.divnum( cx, 2 ) == complex "1+1.5i" )

-- complex.pow( cx, num )
cx = complex "2+3i"
assert( complex.pow( cx, 3 ) == complex "-46+9i" )

cx = complex( -121 )
cx = cx^.5
-- we have to round here due to the polar calculation of the squareroot
cx = cx:round( 10 )
assert( cx == complex "11i" )

cx = complex"2+3i"
assert( cx^-2 ~= 1/cx^2 )
assert( cx^-2 == (cx^-1)^2 )
assert( tostring( cx^-2 ) == tostring( 1/cx^2 ) )

-- complex.sqrt( cx )
cx = complex( -121 )
assert( complex.sqrt( cx ) == complex "11i" )
cx = complex "2-3i"
cx = cx^2
assert( cx:sqrt() == complex "2-3i" )

-- complex.ln( cx )
cx = complex "3+4i"
assert( cx:ln():round( 4 ) == complex "1.6094+0.9273i" )

-- complex.exp( cx )
cx = complex "2+3i"
assert( cx:ln():exp() == complex "2+3i" )

-- complex.conjugate( cx )
cx = complex "2+3i"
assert( cx:conjugate() == complex "2-3i" )

-- metatable

-- __add
cx = complex "2+3i"
assert( cx+2 == complex "4+3i" )

-- __unm
cx = complex "2+3i"
assert( -cx == complex "-2-3i" )