--Version=1.4, Date=04-Aug-2023
-- provides module for complex numbers
--Contains a modified version of the file complex.lua. It is availalbe on the link https://github.com/davidm/lua-matrix/blob/master/lua/complex.lua.  This is licensed under the same terms as Lua itself. This license allows to freely copy, modify and distribute the file for any purpose and without any restrictions. 
--This file is also licensed under the same terms as Lua itself. This license allows to freely copy, modify and distribute the file for any purpose and without any restrictions. 

frac= require("luamaths-fractions")
complex = {}
complex_meta = {}

local function parse_scalar(s, pos0)
   local x, n, pos = s:match('^([+-]?[%d%.]+)(.?)()', pos0)
   if not x then return end
   if n == 'e' or n == 'E' then
      local x2, n2, pos2 = s:match('^([+-]?%d+)(.?)()', pos)
      if not x2 then error 'number format error' end
      x = tonumber(x..n..x2)
      if not x then error 'number format error' end
      return x, n2, pos2
   else
      x = tonumber(x)
      if not x then error 'number format error' end
      return x, n, pos
   end
end
local function parse_component(s, pos0)
   local x, n, pos = parse_scalar(s, pos0)
   if not x then
      local x2, n2, pos2 = s:match('^([+-]?)(i)()$', pos0)
      if not x2 then error 'number format error' end
      return (x2=='-' and -1 or 1), n2, pos2
   end
   if n == '/' then
      local x2, n2, pos2 = parse_scalar(s, pos)
      x = x / x2
      return x, n2, pos2
   end
   return x, n, pos
end
local function parse_complex(s)
   local x, n, pos = parse_component(s, 1)
   if n == '+' or n == '-' then
      local x2, n2, pos2 = parse_component(s, pos)
      if n2 ~= 'i' or pos2 ~= #s+1 then error 'number format error' end
      if n == '-' then x2 = - x2 end
      return x, x2
   elseif n == '' then
      return x, 0
   elseif n == 'i' then
      if pos ~= #s+1 then error 'number format error' end
      return 0, x
   else
      error 'number format error'
   end
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

      if getmetatable( num) == frac_mt  then
         return setmetatable( { num, 0 }, complex_meta )
      end

      local real,imag = tonumber( num[1] ),tonumber( num[2] )
      if real and imag then
         return setmetatable( { real,imag }, complex_meta )
      end
      return
   end
   local isnum = tonumber( num )
   if isnum then
      return setmetatable( { isnum,0 }, complex_meta )
   end
   if type( num ) == "string" then
      local real, imag = parse_complex(num)
      return setmetatable( { real, imag }, complex_meta )
   end
end

setmetatable( complex, { __call = function( _,num ) return complex.to( num ) end } )

function complex.new( x,y)
   return setmetatable( {x,y}, complex_meta )
end

lcomplex = complex.new
function complex.type( arg )
   if getmetatable( arg ) == complex_meta then
      return "complex"
   end
end

function complex.convpolar( radius, phi )
   return setmetatable( { radius * math.cos( phi ), radius * math.sin( phi ) }, complex_meta )
end

function complex.convpolardeg( radius, phi )
   phi = phi/180 * math.pi
   return setmetatable( { radius * math.cos( phi ), radius * math.sin( phi ) }, complex_meta )
end

function complex.tostring( cx,formatstr )
   imunit = "\\imUnit"
   local real,imag = cx[1],cx[2]
   if type(cx[1]) ~= "table" and type(cx[2]) ~= "table" then
	  if imag == 0 and math.floor(real)==real then
         return math.floor(real)
	  end
	  if real == 0 and math.floor(imag)==imag and math.abs(math.floor(imag))~=1 then
         return math.floor(imag)..imunit
	  end
      if imag == 0 then
         return real
      elseif real == 0 then
         return ((imag==1 and "") or (imag==-1 and "-") or imag)..imunit
      elseif imag > 0 then
         return real.."+"..(imag==1 and "" or imag)..imunit
      end
      return real..(imag==-1 and "-" or imag)..imunit
   end

   if type(cx[1]) == "table" and type(cx[2]) ~= "table" then
      if cx[2] == 0 then
         return frac.tostring(cx[1])
      end
      if cx[2] > 0 then
         return frac.tostring(cx[1]).. "+"..(imag==1 and "" or imag)..imunit
      end
      if cx[2] < 0 then
         return frac.tostring(cx[1])..(imag==-1 and "-" or imag)..imunit
      end

   end

   if type(cx[1]) ~= "table" and type(cx[2]) == "table" then
      if frac.toFnumber(cx[2])==0 then return cx[1] end
      return cx[1].."+"..frac.tostring(cx[2])..imunit
   end

   if type(cx[1]) == "table" and type(cx[2]) == "table" then
      if frac.toFnumber(cx[1]) == 0 and frac.toFnumber(cx[2]) ~= 0 then
         return frac.tostring(cx[2])..imunit
      end

      if frac.toFnumber(cx[2]) == 0 then
         return frac.tostring(cx[1] + 0)
      end

      if cx[2].d == 1 then
         if cx[2].n > 0 then
            return frac.tostring(cx[1]).. "+"..(cx[2].n==1 and "" or math.floor(cx[2].n))..imunit
         end
         if cx[2].n < 0 then
            return frac.tostring(cx[1])..(cx[2].n==-1 and "-" or math.floor(cx[2].n))..imunit
         end
      end
   end
   return frac.tostring(cx[1]).. "+"..frac.tostring(cx[2])..imunit
end

function complex.print( ... )
   print( complex.tostring( ... ) )
end

function complex.polar( cx )
   return math.sqrt( cx[1]^2 + cx[2]^2 ), math.atan2( cx[2], cx[1] )
end

function complex.polardeg( cx )
   return math.sqrt( cx[1]^2 + cx[2]^2 ), math.atan2( cx[2], cx[1] ) / math.pi * 180
end

function complex.norm2( cx )
   return cx[1]^2 + cx[2]^2
end

function complex.abs( cx )
   return math.sqrt( cx[1]^2 + cx[2]^2 )
end

function complex.get( cx )
   return cx[1],cx[2]
end

function complex.set( cx,real,imag )
   cx[1],cx[2] = real,imag
end

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

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

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

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

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

function complex.mulnum( cx,num )
   return setmetatable( { cx[1]*num,cx[2]*num }, complex_meta )
end

function complex.div( cx1,cx2 )
   local val = cx2[1]*cx2[1] + cx2[2]*cx2[2]
   return setmetatable( { (cx1[1]*cx2[1]+cx1[2]*cx2[2])/val,(cx1[2]*cx2[1]-cx1[1]*cx2[2])/val }, complex_meta )
end

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

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
   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

function complex.sqrt( cx )
   local h
   local k

   if type(cx[1]) ~= "table" then h = cx[1] end
   if type(cx[2]) ~= "table" then k = cx[2] end

   if type(cx[1]) == "table" then h = frac.toFnumber(cx[1]) end
   if type(cx[2]) == "table" then k = frac.toFnumber(cx[2]) end
   local len = math.sqrt( h^2 + k^2 )
   local sign = ( h<0 and -1) or 1
   return setmetatable( { math.sqrt((h +len)/2), sign*math.sqrt((len-h)/2) }, complex_meta )
end

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

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

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

function Xround(num, numDecimalPlaces)
   if type(num)=="number" then
      if num==math.floor(num) then
         return math.floor(num)
      end
   end
   if type(num)=="number" then
      local mult = 10^(numDecimalPlaces or 0)
      return math.floor(num * mult + 0.5) / mult
   end
   return num
end

function complex.round( cx,idp )
   local mult =10^( idp or 0 )
   if type(cx[1]) ~= "table" and type(cx[2]) ~= "table" then
      return setmetatable( {Xround(cx[1],idp), Xround(cx[2],idp)}, complex_meta )
   end
   if type(cx[1]) ~= "table" and type(cx[2]) == "table" then
      return setmetatable( {Xround(cx[1],idp), cx[2]}, complex_meta )
   end
   if type(cx[1]) == "table" and type(cx[2]) ~= "table" then
      return setmetatable({cx[1],Xround(cx[2],idp)}, complex_meta )
   end
   if type(cx[1]) == "table" and type(cx[2]) == "table" then
      return setmetatable({cx[1],cx[2]}, complex_meta )
   end
end

complex.zero = complex.new(0, 0)
complex.one  = complex.new(1, 0)

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