-- 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.

---MODIFIED TO SUIT THE MTA SCRIPTING SYSTEM

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

-- link to complex table
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

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