Module:Complex Number/Functions
此模块的文档可以在Module:Complex Number/Functions/doc创建
local p = {}
local math_lib
local to_number
function p._init(_math_lib, _to_number)
local warp_funcs={"factorial","gamma","sec","csc","sech","csch","asec","acsc","asech","acsch","gd","cogd","arcgd",
"LambertW","norm",
"gcd","lcm","range","binomial",'minimum','maximum','average','min','max','avg','geoaverage','var','σ',
'selectlist','for','while','summation','product','if','iff','ifelse','ifelsef','diff','integral', '∫', 'limit',
'hide','exprs','lastexpr','equalexpr',--調整條目中定義不顯示的函數
'randomseed','time','nil','null','call',
'frameArg','object','string','symbols','passObject','typeof','length','array','assignMember',
'divisorsigma','findnext','findlast','divisor','primedivisor','eulerphi'
}
for i=1,#warp_funcs do
if _math_lib[ warp_funcs[i] ] == nil then
_math_lib[ warp_funcs[i] ] = p['_' .. warp_funcs[i] ]
end
end
math_lib = _math_lib
to_number = _to_number
return _math_lib
end
function p._complex_number()
return p._init(require("Module:Complex Number").cmath.init(), require("Module:Complex Number").cmath.init().toComplexNumber)
end
local noop_func = function()end
local function assertArg(val, index, func)
assert (val ~= nil, string.format(
"bad argument #%d to '%s' (number expected, got %s)", index, func, type(val)))
end
function p._binomial(cal_n,cal_k)
local r_n=tonumber(tostring(cal_n))
local r_k=tonumber(tostring(cal_k))
if r_n and r_k then
if r_n>0 and r_k>=0 then
local f_n, f_k;
_,f_n = math.modf(r_n);_,f_k = math.modf(r_k)
if math.abs(f_n) < 1e-12 and math.abs(f_k) < 1e-12 then
local result = 1
if r_n == 0 then return result end
while r_k>0 do
result = result * r_n / r_k
r_n = r_n - 1
r_k = r_k - 1
end
return result
end
end
end
local n=to_number(cal_n)
local k=to_number(cal_k)
assertArg(n, 1, 'binomial')
assertArg(k, 2, 'binomial')
return p._factorial(n) * math_lib.inverse( p._factorial(k) ) * math_lib.inverse( p._factorial(n-k) )
end
function p._factorial(cal_z)return p._gamma(to_number(cal_z) + 1)end
function p._sec(cal_z)return math_lib.inverse( math_lib.cos( to_number(cal_z) ) )end
function p._csc(cal_z)return math_lib.inverse( math_lib.sin( to_number(cal_z) ) )end
function p._sech(cal_z)return math_lib.inverse( math_lib.cosh( to_number(cal_z) ) )end
function p._csch(cal_z)return math_lib.inverse( math_lib.sinh( to_number(cal_z) ) )end
function p._asec(cal_z)return math_lib.acos( math_lib.inverse( to_number(cal_z) ) )end
function p._acsc(cal_z)return math_lib.asin( math_lib.inverse( to_number(cal_z) ) )end
function p._asech(cal_z)return math_lib.acosh( math_lib.inverse( to_number(cal_z) ) )end
function p._acsch(cal_z)return math_lib.asinh( math_lib.inverse( to_number(cal_z) ) )end
function p._gd(cal_z)return math_lib.atan( math_lib.tanh( to_number(cal_z) * 0.5 ) ) * 2 end
function p._arcgd(cal_z)return math_lib.atanh( math_lib.tan( to_number(cal_z) * 0.5 ) ) * 2 end
function p._cogd(cal_z)local x = to_number(cal_z); return - math_lib.sgn( x ) * math_lib.log( math_lib.abs( math_lib.tanh( x * 0.5 ) ) ) end
function p._range(val,vmin,vmax)
assertArg(val, 1, 'range')
local min_inf, max_inf = tonumber("-inf"), tonumber("inf")
local function get_vector(_val)
local val = to_number(_val)
if val == nil then return {} end
return math_lib.tovector(val)
end
local v_val, v_min, v_max = math_lib.tovector(to_number(val)), get_vector(vmin), get_vector(vmax)
for i=1,#v_val do
local min_v, max_v = math.min(v_min[i]or min_inf, v_max[i]or max_inf), math.max(v_min[i]or min_inf, v_max[i]or max_inf)
if v_val[i] < min_v or v_val[i] > max_v then return to_number(math_lib.nan) end
end
return to_number(val)
end
function p._calc_diff(_value, _left, _right, _expr)
local value = to_number(_value)
local left = to_number(_left)
local right = to_number(_right)
local left_val = _expr(value + left)
local right_val = _expr(value + right)
return (left_val - right_val) / (left - right)
end
function p._diff_abramowitz_stegun(_x, _h, _f)
local x = to_number(_x)
local h = to_number(_h)
local _1 = to_number(1) local _2 = to_number(2) local _8 = to_number(8) local _12 = to_number(12)
--if tonumber(math_lib.abs(math_lib.nonRealPart(x))) > 1e-8 then
-- --[[數值微分#複變的方法]] --(Martins, Sturdza et.al.)
-- local _i = math_lib.i or _1
-- return (math_lib.im(_f(x + _i * h))) / h
--else
--[[數值微分#高階方法]] --(Abramowitz & Stegun, Table 25.2)
return (- _f(x + _2*h) + _8*_f(x + h) - _8*_f(x - h) + _f(x - _2*h)) / (_12 * h)
--end
end
function p._calc_from(_value, _left, _right, _expr)
local value = to_number(_value)
local left = to_number(_left)
local left_val = _expr(value + left)
return left_val - p._calc_diff(_value, _left, _right, _expr) * left
end
function p._get_sample_number(_value)
local num = math_lib.re(math_lib.abs(to_number(_value)))
return (tonumber(num)~=nil) and ((num < 1e-8) and 1 or num) or 1
end
function p._diff(_expr, _value)
local func_type = type(noop_func)
local value_str = tostring(_value)
if value_str == 'nan' or value_str == '-nan' or value_str == 'nil'then
error("計算失敗：無效的數值 ")
return _expr
end
if value_str == 'inf' or value_str == '-inf'then
error("計算失敗：不支援無窮微分 ")
return _expr
end
local small_scale_pow = math_lib.floor(math_lib.log(p._get_sample_number(_value)) / math_lib.log(to_number(10)))-5
local small_scale = math_lib.pow(10,small_scale_pow)
if type(_expr) == func_type then
return p._diff_abramowitz_stegun(_value, small_scale, _expr)
else
return 0
end
end
function p._integral(_a,_b,func,step)
local a = to_number(_a)
local b = to_number(_b)
local begin_str = tostring(_a)
local stop_str = tostring(_b)
if begin_str == 'nan' or begin_str == '-nan' or begin_str == 'nil' or stop_str == 'nan' or stop_str == '-nan' or stop_str == 'nil'then
error("計算失敗：無效的迴圈 ")
return func
end
if begin_str == 'inf' or begin_str == '-inf' or stop_str == 'inf' or stop_str == '-inf'then
error("計算失敗：不支援無窮求積 ")
return func
end
local n = tonumber(step) or 2000
local f = (type(func)==type(noop_func))and func or (function()return to_number(func) or 0 end)
local h = (b-a)/n
local x0, xn = a, b
local _2, _7, _12, _14, _32, _45 =
to_number(2), to_number(7), to_number(12), to_number(14), to_number(32), to_number(45)
local i0, i1, i2 = 1, 2, 4
local sumfxi0, sumfxi1, sumfxi2 = to_number(0), to_number(0), to_number(0)
for i=1,n do -- Boole's rule
if i0 > n-1 and i1 > n-2 and i2 > n-4 then break end
local xi0, xi1, xi2 = a + i0 * h, a + i1 * h, a + i2 * h
if i0 <= n-1 then sumfxi0 = sumfxi0 + f(xi0) end
if i1 <= n-2 then sumfxi1 = sumfxi1 + f(xi1) end
if i1 <= n-4 then sumfxi2 = sumfxi2 + f(xi2) end
i0 = i0 + 2
i1 = i1 + 4
i2 = i2 + 4
end
return (_2 * h / _45) * (_7 * (f(x0) + f(xn)) + _32 * sumfxi0 + _12 * sumfxi1 + _14 * sumfxi2)
end
p['_∫'] = p._integral
function p._limit(_value, _way, _expr)
local way = to_number(_way)
local func_type = type(noop_func)
local value_str = tostring(_value)
if value_str == 'nan' or value_str == '-nan' or value_str == 'nil' then
error("計算失敗：無效的數值 ")
return _expr
end
if value_str == 'inf' or value_str == '-inf'then
error("計算失敗：不支援無窮極限 ")
return _expr
end
local small_scale_pow = tonumber(math_lib.re(math_lib.floor(math_lib.log(p._get_sample_number(_value)) / math_lib.log(to_number(10))))-6)
local small_scale = math.pow(10,small_scale_pow)
local small_scale_a = small_scale / 10
local small_scale_c = small_scale * 10
if type(_expr) == func_type then
if math_lib.re(math_lib.abs(way)) < 1e-10 then
local left = p._calc_from(_value, -small_scale, -small_scale_a, _expr)
local right = p._calc_from(_value, small_scale_a, small_scale, _expr)
if math_lib.re(math_lib.abs(left - right)) < small_scale_c then
return (left + right) / 2
else
return math_lib.nan
end
else
return (math_lib.re(way) > 0) and p._calc_from(_value, small_scale_a, small_scale, _expr) or
p._calc_from(_value, -small_scale, -small_scale_a, _expr)
end
else
return _expr
end
end
p._nil = "nil"
p._null = "nil"
---------------------- 流程控制擴充 ----------------------
function p._if(_expr, _true_expr, _false_expr)
return p._ifelse_func(false, _expr, _true_expr, _false_expr)
end
function p._iff(_expr, _true_expr, _false_expr)
return p._ifelse_func(true, _expr, _true_expr, _false_expr)
end
function p._ifelse(...)
return p._ifelse_func(false, ...)
end
function p._ifelsef(...)
return p._ifelse_func(true, ...)
end
function p._ifelse_func(is_func, ...)
local func = noop_func
local exprlist = {...}
local last_else = #exprlist % 2 == 1
local max_num = (last_else and (#exprlist - 1) or #exprlist) / 2
for i=1,max_num do
local _expr = exprlist[i * 2 - 1]
local expr = (type(_expr) == type(func)) and _expr() or _expr
local expr_true = exprlist[i * 2]
local _chk_flag = math_lib.abs(to_number(expr)) > 1e-14;
if _chk_flag then
return (type(expr_true) == type(func) and is_func) and expr_true() or expr_true
end
end
if last_else then
local expr_false = exprlist[#exprlist]
return (type(expr_false) == type(func) and is_func) and expr_false() or expr_false
end
local _expr = exprlist[1]
return (type(_expr) == type(func)) and _expr() or _expr
end
local function check_while(_ifexpr)
local result = (type(_ifexpr) == type(noop_func)) and _ifexpr() or _ifexpr
if result == true then return true end
if not result then return false end
return math_lib.abs(to_number(result)) > 1e-14
end
function p._while(_ifexpr, _expr)
local result
while check_while(_ifexpr) do
result = (type(_expr) == type(noop_func)) and _expr() or _expr
if type(result) == type({}) and result['return'] then break end
end
return result
end
function p._for(_start,_end,_step,_expr)
local _begin = to_number(_start);
local _stop = to_number(_end);
local _do_step = to_number(_step);
local check_loop = (_stop - _begin) / _do_step
local begin_str = tostring(_begin)
local stop_str = tostring(_stop)
if math_lib.re(math_lib.abs(_do_step))<=1e-14 or math_lib.re(check_loop) < 0 or
begin_str == 'nan' or begin_str == '-nan' or begin_str == 'nil' or begin_str == 'inf' or begin_str == '-inf' or
stop_str == 'nan' or stop_str == '-nan' or stop_str == 'nil' or stop_str == 'inf' or stop_str == '-inf'then
error("計算失敗：無效的迴圈 ")
return _expr
end
if type(_expr) == type(noop_func) then
local it = _begin
local init = to_number(0)
while math_lib.re(it) <= math_lib.re(_stop) do
init = _expr(to_number(it),to_number(init))
if type(init) == type({}) and init['return'] then break end
it = it + _do_step
end
return init
else
return _expr
end
end
function p._summation(_start,_end,_expr)
local _begin = to_number(_start);
local _stop = to_number(_end);
local _do_step = to_number(1);
local check_loop = (_stop - _begin) / _do_step
local begin_str = tostring(_begin)
local stop_str = tostring(_stop)
if math_lib.re(math_lib.abs(_do_step))<=1e-14 or math_lib.re(check_loop) < 0 or
begin_str == 'nan' or begin_str == '-nan' or begin_str == 'nil' or stop_str == 'nan' or stop_str == '-nan' or stop_str == 'nil'then
error("計算失敗：無效的迴圈 ")
return _expr
end
if begin_str == 'inf' or begin_str == '-inf' or stop_str == 'inf' or stop_str == '-inf'then
error("計算失敗：不支援無窮求和 ")
return _expr
end
local func_type = type(noop_func)
local it = _begin
local init = to_number(0)--空和
while math_lib.re(it) <= math_lib.re(_stop) do
init = init + ((type(_expr) == func_type) and _expr(to_number(it)) or to_number(_expr))--累加
it = it + _do_step
end
return init
end
function p._product(_start,_end,_expr)
local _begin = to_number(_start);
local _stop = to_number(_end);
local _do_step = to_number(1);
local check_loop = (_stop - _begin) / _do_step
local begin_str = tostring(_begin)
local stop_str = tostring(_stop)
if math_lib.re(math_lib.abs(_do_step))<=1e-14 or math_lib.re(check_loop) < 0 or
begin_str == 'nan' or begin_str == '-nan' or begin_str == 'nil' or stop_str == 'nan' or stop_str == '-nan' or stop_str == 'nil'then
error("計算失敗：無效的迴圈 ")
return _expr
end
if begin_str == 'inf' or begin_str == '-inf' or stop_str == 'inf' or stop_str == '-inf'then
error("計算失敗：不支援無窮求積 ")
return _expr
end
local func_type = type(noop_func)
local it = _begin
local init = to_number(1)--空積
while math_lib.re(it) <= math_lib.re(_stop) do
init = init * ((type(_expr) == func_type) and _expr(to_number(it)) or to_number(_expr))--累乘
it = it + _do_step
end
return init
end
---------------------- 工具函數擴充 ----------------------
function p._randomseed(_seed)
local seed = tonumber(tostring(_seed)) or (os.time() * os.clock())
math.randomseed(math.floor(seed))
return to_number(seed)
end
function p._time() return to_number(os.time()) end
function p._call(func, ...)
if type(func) == type(noop_func) then return func(...)end
return func
end
function p._hide(...)
local input_args = {...}
return to_number(input_args[#input_args])
end
p._exprs = p._hide
p._lastexpr = p._hide
p._equalexpr = p._hide
---------------------- 統計函數 ----------------------
function p._selectlist(x,...)
local input_args = {...}
local y = input_args[1]
local z = input_args[2]
local id_x = tonumber(tostring(x)) or 0
if type(y) == type("string") then return mw.ustring.sub(y,id_x,id_x) end
if type(y)==type({}) and #y >= id_x and id_x>0 then
if id_x <= 0 then id_x = id_x + #y + 1 end
return y[id_x] or tonumber('nan')
elseif type(z)==type({}) and #z >= id_x then
local id_y = tonumber(tostring(y)) or 0
if id_x <= 0 then id_x = id_x + #z + 1 end
if type(z[id_x])==type({}) and #(z[id_x]) >= id_y and id_y>0 then
if id_y <= 0 then id_y = id_y + #(z[id_x]) + 1 end
return (z[id_x][id_y]) or tonumber('nan')
end
end
id_x = tonumber(tostring(x)) or 0
if id_x <= 0 then id_x = id_x + #input_args + 1 end
return input_args[id_x] or tonumber('nan')
end
function p._minimum(...) return p.minmax('min', ...) end
function p._maximum(...) return p.minmax('max', ...) end
function p._average(...) return p.minmax('avg', ...) end
function p._geoaverage(...) return p.minmax('gavg', ...) end
function p._var(...) return p.minmax('var', ...) end
p._min = p._minimum
p._max = p._maximum
p._avg = p._average
p['_σ'] = function(...) return p.minmax('σ',...) end
local function flatten(inarray,outarray)
outarray = outarray or {}
if type(inarray) ~= type({}) then
outarray[#outarray + 1] = inarray
elseif inarray.numberType then
outarray[#outarray + 1] = inarray
elseif type(inarray.args) == type({}) then
local midarray = inarray.args
for k,v in pairs(midarray) do
local i = tonumber(k)
if i then outarray = flatten(midarray[i],outarray)end
end
if type(inarray.getParent) == type(noop_func) then
midarray = (inarray:getParent() or {}).args or {}
for k,v in pairs(midarray) do
local i = tonumber(k)
if i then outarray = flatten(midarray[i],outarray)end
end
end
elseif #inarray > 0 then
for i=1,#inarray do outarray = flatten(inarray[i],outarray)end
end
return outarray
end
function p.minmax(calc_mode,...)
local mode = calc_mode
local tonumber_lib = to_number or tonumber
local lib_math = math_lib or math
local args, tester = flatten({ ... }), {tonumber("nan")}
if type(calc_mode) == type({}) then mode = (calc_mode.args or calc_mode).mode or mode; args = flatten({args, calc_mode}) end
local sum, prod, count, sumsq, sig = tonumber_lib(0), tonumber_lib(1), 0, tonumber_lib(0), (mode =='var'or mode=='σ')
local mode_map = {}
local non_nan
for i=1,#args do
local got_number, calc_number = tonumber(tostring(args[i])) or tonumber("nan"), tonumber_lib(args[i])
if calc_number then sum, prod, count = calc_number + sum, calc_number * prod, count + 1 end
if sig == true then
local x_2 = calc_number * calc_number
if lib_math.dot then
x_2 = lib_math.dot(calc_number, lib_math.conjugate(calc_number))
end
sumsq = sumsq + x_2
end
mode_map[args[i]] = (mode_map[args[i]]or 0) + 1
if tostring(got_number):lower()~="nan" and type(non_nan) == type(nil) then
tester[1], non_nan = got_number, got_number
else tester[#tester + 1] = got_number end
end
local modes={min=math.min,max=math.max,sum=function()return sum end,prod=function()return prod end,count=function()return count end,
avg=function()return sum*tonumber_lib(1/count) end,
gavg=function()return lib_math.pow(prod,tonumber_lib(1/count)) end,
var=function()return sumsq*tonumber_lib(1/count)-sum*sum*tonumber_lib(1/(count*count)) end,
['σ']=function()return lib_math.sqrt(sumsq*tonumber_lib(1/count)-sum*sum*tonumber_lib(1/(count*count))) end,
mode=function()
local max_count, mode_data = 0, ''
for mkey, mval in pairs(mode_map) do
if mval > max_count then
max_count = mval
mode_data = mkey
end
end
return mode_data
end,
gcd=function(...)if not to_number or not math_lib then p._complex_number()end return p._gcd(...)end,
lcm=function(...)if not to_number or not math_lib then p._complex_number()end return p._lcm(...)end,
}
if tostring(tester[1]):lower()=="nan" and mode:sub(1,1)=='m' then
local error_msg = ''
for i=1,#args do if error_msg~=''then error_msg = error_msg .. '、 ' end
error_msg = error_msg .. tostring(args[i])
end
error("給定的數字 " .. error_msg .." 無法比較大小")
end
if type(modes[mode]) ~= type(tonumber) then
error("未知的統計方式 '" .. mode .."' ")
end
return modes[mode](unpack(tester))
end
local function fold(func, ...)
-- Use a function on all supplied arguments, and return the result. The function must accept two numbers as parameters,
-- and must return a number as an output. This number is then supplied as input to the next function call.
local vals = {...}
local count = #vals -- The number of valid arguments
if count == 0 then return
-- Exit if we have no valid args, otherwise removing the first arg would cause an error.
nil, 0
end
local ret = table.remove(vals, 1)
for _, val in ipairs(vals) do
ret = func(ret, val)
end
return ret, count
end
--[[
Fold arguments by selectively choosing values (func should return when to choose the current "dominant" value).
]]
local function binary_fold(func, ...)
local value = fold((function(a, b) if func(a, b) then return a else return b end end), ...)
return value
end
---------------------- 伽瑪函數 ----------------------
local Reciprocal_gamma_coeff = {1,0.577215664901532860607,-0.655878071520253881077,-0.0420026350340952355290,0.166538611382291489502,-0.0421977345555443367482,-0.00962197152787697356211,0.00721894324666309954240,-0.00116516759185906511211,-0.000215241674114950972816,0.000128050282388116186153,-0.0000201348547807882386557,-1.25049348214267065735e-6,1.13302723198169588237e-6,-2.05633841697760710345e-7,6.11609510448141581786e-9,5.00200764446922293006e-9,-1.18127457048702014459e-9,1.04342671169110051049e-10,7.78226343990507125405e-12,-3.69680561864220570819e-12,5.10037028745447597902e-13,-2.05832605356650678322e-14,-5.34812253942301798237e-15,1.22677862823826079016e-15,-1.18125930169745876951e-16,1.18669225475160033258e-18,1.41238065531803178156e-18,-2.29874568443537020659e-19,1.71440632192733743338e-20}
--https://oeis.org/A001163 、 https://oeis.org/A001164
local stirling_series_coeff = {1,0.0833333333333333333333333,0.00347222222222222222222222,-0.00268132716049382716049383,-0.000229472093621399176954733,0.000784039221720066627474035,0.0000697281375836585777429399,-0.000592166437353693882864836,-0.0000517179090826059219337058,0.000839498720672087279993358,0.0000720489541602001055908572,-0.00191443849856547752650090,-0.000162516262783915816898635,0.00640336283380806979482364,0.000540164767892604515180468,-0.0295278809456991205054407,-0.00248174360026499773091566,0.179540117061234856107699,0.0150561130400264244123842,-1.39180109326533748139915,-0.116546276599463200850734}
function p._gamma_high_imag(cal_z)
local z = to_number(cal_z)
if z ~= nil and math_lib.abs(math_lib.nonRealPart(z)) > 2 then
local inv_z = math_lib.inverse(z)
return math_lib.sqrt((math_lib.pi * 2) * inv_z) * math_lib.pow(z * math_lib.exp(-1) *
math_lib.sqrt( (z * math_lib.sinh(inv_z) ) + math_lib.inverse(to_number(810) * z * z * z * z * z * z) ),z)
end
return nil
end
function p._gamma_morethen_lua_int(cal_z)
local z = to_number(cal_z) - to_number(1)
local lua_int_term = 18.1169 --FindRoot[ Factorial[ x ] == 2 ^ 53, {x, 20} ]
if math_lib.abs(z) > (lua_int_term - 1) or (math_lib.re(z) < 0 and math_lib.abs(math_lib.nonRealPart(z)) > 1 ) then
local sum = 1
for i = 1, #stirling_series_coeff - 1 do
local a, n = to_number(z), tonumber(i) local y, k, f = to_number(1), n, to_number(a)
while k ~= 0 do
if k % 2 == 1 then y = y * f end
k = math.floor(k / 2); f = f * f
end
sum = sum + stirling_series_coeff[i + 1] * math_lib.inverse(y)
end
return math_lib.sqrt( (2 * math.pi) * z ) * math_lib.pow( z * math.exp(-1), z ) * sum
end
return nil
end
function p._gamma_abs_less1(cal_z)
local z = to_number(cal_z)
if (math.abs(math_lib.re(z)) <=1.001) then
if math_lib.abs(math_lib.nonRealPart(z)) < 1e-14 and ( (math.abs(math_lib.re(z) - 1) < 1e-14) or (math.abs(math_lib.re(z) - 2) < 1e-14) ) then return to_number(1)end
return math_lib.inverse(p._recigamma_abs_less1(z))
end
return nil
end
function p._recigamma_abs_less1(z)
local result = to_number(0)
for i=1,#Reciprocal_gamma_coeff do
result = result + Reciprocal_gamma_coeff[i] * math_lib.pow(z,i)
end
return result
end
function p._gamma(cal_z)
local z = to_number(cal_z)
if math_lib.abs(math_lib.nonRealPart(z)) < 1e-14 and ((math_lib.re(z) < 0 or math.abs(math_lib.re(z)) < 1e-14)
and math.abs(math.floor(math_lib.re(z)) - math_lib.re(z)) < 1e-14 ) then return tonumber("nan") end
local pre_result = p._gamma_morethen_lua_int(z) or p._gamma_high_imag(z) or p._gamma_abs_less1(z)
if pre_result then return pre_result end
local real_check = math_lib.re(z)
local loop_count = math.floor(real_check)
local start_number, zero_flag = z - loop_count, false
if math_lib.abs(start_number) <= 1e-14 then start_number = to_number(1);zero_flag = true end
local result = math_lib.inverse(p._recigamma_abs_less1(start_number))
if math_lib.abs(math_lib.nonRealPart(z)) < 1e-14 and ((math_lib.re(z) > 1e-14 )and math.abs(math.floor(math_lib.re(z)) - math_lib.re(z)) < 1e-14 ) then result = to_number(1) end
local j = to_number(start_number)
for i=1,math.abs(loop_count) do
if loop_count > 0 then result = result * j else result = result * math_lib.inverse(j-1) end
if zero_flag==true and loop_count > 0 then zero_flag=false else if loop_count > 0 then j = j + 1 else j = j - 1 end end
end
if math_lib.abs(math_lib.nonRealPart(z)) < 1e-14 and ((math_lib.re(z) > 1e-14 )and math.abs(math.floor(math_lib.re(z)) - math_lib.re(z)) < 1e-14 ) then return math_lib.floor(result) end
return result
end
---------------------- 最大公因數與最小公倍數 ----------------------
local function findGcd(a, b)
local r, oldr = to_number(b), to_number(a)
while math_lib.abs(r) > 1e-6 do local mod_val = oldr % r oldr, r = to_number(r), mod_val end
if math_lib.abs(math_lib.nonRealPart(oldr)) < 1e-14 and (math_lib.re(oldr) < 0 ) then oldr = -oldr end
return oldr
end
function p._gcd(...)
local result, count = fold(findGcd, ...)
return result
end
function p._lcm(...)
local function findLcm(_a, _b)
local a, b = to_number(_b), to_number(_a)
return math_lib.abs(a * b) / findGcd(a, b)
end
local result, count = fold(findLcm, ...)
return result
end
---------------------- 字串與物件擴充 (提供Module:Complex Number/Calculate使用) ----------------------
function p._symbols(name)
return({
comma = ',', space = ' ', colon = ':', dot = '.', squot = "'", dquot = '"', semicolon = ';', underline = '_',
lcbracket = '{', rcbracket = '}', lsbracket = '[', rsbracket = ']', lpbracket = '(', rpbracket = ')',
plus = '+', minus = '-', mul = '*', div = '/', ['pow'] = '^', equal = '=',
lt = '<', gt = '>', money = '$', percent = '%', ['and'] = '&', exclamation = '!', at = '@', hashtag = '#', to = '~', slash ='\\'
})[name]
end
function p._frameArg(str)
local frame = mw.getCurrentFrame()
local working_frame = frame:getParent() or frame
local argname = tonumber(tostring(str)) or tostring(str)
return working_frame.args[argname] or frame.args[argname]
end
function p._string(str,...)
local result = tostring(str)
local str_list = {...}
for i = 1,#str_list do
result = result .. '' .. tostring(str_list[i])
end
return result
end
function p._passObject(obj)
return obj
end
function p._assignMember(obj, member, value)
local input_obj = obj
if type(obj) == type("string") then input_obj = _G[obj] end
if type(obj) == type(0) or type(input_obj) == type(0) then error("無法傳值給數字", 2) end
if type(obj) == type(noop_func) or type(input_obj) == type(noop_func) then error("無法傳值給函數", 2) end
if input_obj == nil then error("無法傳值給空值", 2) end
input_obj[member] = value
return value
end
function p._object(obj,...)
local input_obj = obj
if type(obj) == type("string") then input_obj = _G[obj] end
if type(obj) == type(0) then return obj end
if type(obj) == type(noop_func) then return obj end
if input_obj == nil then return nil end
local members = {...}
if #members > 0 then
local it_obj = input_obj
for i = 1,#members do
if type(it_obj) ~= type({}) then return nil end
it_obj = (it_obj or {})[members[i]]
if it_obj == nil then return nil end
end
return it_obj
end
return input_obj
end
function p._typeof(obj)
if type(obj) == type({}) then
if obj.numberType then return type(0) end
local is_array = true
for index, data in pairs(obj) do
if not tonumber(index) and index ~= 'metatable' then
is_array = false
break
end
end
if is_array then return 'array' end
end
return type(obj)
end
function p._array(...)
return {...}
end
function p._length(obj)
if type(obj) == type({}) then
if obj.numberType then return 1 end
local max_index = 0
for key, data in pairs(obj) do
local index = tonumber(key)
if (index or 0) > max_index then
max_index = index
end
end
return max_index
elseif type(obj) == type("string") then
return mw.ustring.len(obj)
else
return 1
end
end
---------------------- 數論相關 ----------------------
function p._findnext(func, x)
local it = to_number(x) + 1
if type(func) ~= type(noop_func) then
if math_lib.abs(to_number(func)) < 1e-14 then
return to_number("inf")
else
return it
end
end
local checker = func(it)
while math_lib.abs(to_number(checker)) < 1e-14 do
it = it + 1
checker = func(it)
end
return it
end
function p._findlast(func, x)
local it = to_number(x) - 1
if type(func) ~= type(noop_func) then
if math_lib.abs(to_number(func)) < 1e-14 then
return to_number("-inf")
else
return it
end
end
local checker = func(it)
while math_lib.abs(to_number(checker)) < 1e-14 do
it = it - 1
checker = func(it)
end
return it
end
local function key_sort(t)
if type(t) ~= type({"table"}) then return {t} end
local key_list = {}
for k,v in pairs(t) do key_list[#key_list + 1] = k end
table.sort(key_list)
return key_list
end
local function get_divisor(n, combination)
local is_complex = math_lib.abs(math_lib.nonRealPart(n)) > 1e-14
local factors = {}
if math_lib.abs(math_lib.floor(n)-n) < 1e-14 then
local lib_factor = require('Module:Factorization')
factors = (lib_factor[is_complex and "_gaussianFactorization" or "_factorization"])(
is_complex and n or tonumber(tostring(n))
)
else return combination and {{n}} or {} end
if not combination then return factors end
local gened=require('Module:Combination').getCombinationGenerator()
gened:init(factors,0)
return gened:findSubset()
end
function p._primedivisor(_n, _x)
local n = to_number(_n)
if math_lib.abs(n) < 1e-14 then return 0 end
local is_complex = math_lib.abs(math_lib.nonRealPart(n)) > 1e-14
if not is_complex then n = math_lib.abs(n) end
local primedivisors = key_sort(get_divisor(n, false))
return primedivisors[math_lib.abs(to_number(_x or #primedivisors))] or 0
end
function p._eulerphi(_n, _x)
local n = to_number(_n)
if math_lib.abs(n) < 1e-14 then return 0 end
local is_complex = math_lib.abs(math_lib.nonRealPart(n)) > 1e-14
if not is_complex and math_lib.re(n) < 1e-14 then return 0 end
local primedivisors = get_divisor(n, false)
local result = 1
for p,k in pairs(primedivisors) do
local p_r = to_number(p)
local k_r = to_number(k)
result = result * math_lib.pow(p_r, k_r-1) * (p_r-1)
end
return result
end
function p._divisor(_n, _x)
local n = to_number(_n)
local function _index(total) return (total <= 2) and total or (total - 1) end
if math_lib.abs(n) < 1e-14 then return 0 end
local is_complex = math_lib.abs(math_lib.nonRealPart(n)) > 1e-14
if not is_complex then n = math_lib.abs(n) end
local combination = get_divisor(n, true)
local divisors = {}
for i=1,#combination do
local divisor = to_number(1)
for j=1,#(combination[i]) do
divisor = divisor * to_number(combination[i][j])
end
divisors[#divisors+1] = divisor
end
table.sort(divisors, function(a,b) return math_lib.abs(a) < math_lib.abs(b) end)
return divisors[math_lib.abs(to_number(_x or _index(#divisors)))] or 0
end
function p._divisorsigma(_x, _n)
local x = to_number(1), n
if _n == nil then
n = to_number(_x)
else
x = to_number(_x)
n = to_number(_n)
end
if math_lib.abs(n) < 1e-14 then return 0 end
local is_complex = math_lib.abs(math_lib.nonRealPart(n)) > 1e-14
if not is_complex then n = math_lib.abs(n) end
local combination = get_divisor(n, true)
local sum = to_number(0)
for i=1,#combination do
local divisor = to_number(1)
for j=1,#(combination[i]) do
divisor = divisor * to_number(combination[i][j])
end
sum = sum + math_lib.pow(divisor, x)
end
return sum
end
---------------------- 朗伯W函數 ----------------------
local function zexpz(z) return math_lib.exp(z) * z end
--The derivative of z * exp(z) = exp(z) + z * exp(z)
local function zexpz_d(z) return math_lib.exp(z) + math_lib.exp(z) * z end
--The second derivative of z * exp(z) = 2. * exp(z) + z * exp(z)
local function zexpz_dd(z)return math_lib.exp(z) * 2 + math_lib.exp(z) * z end
--Determine the initial point for the root finding
local function LWInitPoint(_z, k)
local z = to_number(_z)
local two_pi_k_I = math_lib.i * 2 * math_lib.pi * k
local ip = math_lib.log(z) + two_pi_k_I - math_lib.log(math_lib.log(z) + two_pi_k_I) --initial point coming from the general asymptotic approximation
local p = math_lib.sqrt((math_lib.e * z + 1) * 2) --used when we are close to the branch cut around zero and when k=0,-1
if math_lib.abs(-(-math_lib.exp(-1)) + z) <= 1 then --we are close to the branch cut, the initial point must be chosen carefully
if k == 0 then ip = -math_lib[1] + p - 1/3 * math_lib.pow(p, 2) + 11/72 * math_lib.pow(p, 3) end
if k == 1 and math_lib.im(z) < 0 then ip = -math_lib[1] - p - 1/3 * math_lib.pow(p, 2) - 11/72 * math_lib.pow(p, 3) end
if k == -1 and math_lib.im(z) > 0 then ip = -math_lib[1] - p - 1/3 * math_lib.pow(p, 2) - 11/72 * math_lib.pow(p, 3) end
end
if k == 0 and math_lib.abs(z - 0.5) <= 0.5 then ip = ((z * 7.061302897 + 0.1237166) * 0.35173371) / ((z * 2 + 1) * 0.827184 + 2) end-- (1,1) Pade approximant for W(0,a)
if k == -1 and math_lib.abs(z - 0.5) <= 0.5 then ip = -(((math_lib.i * 4.22096 +
2.2591588985) * ((-math_lib.i * 33.767687754 - 14.073271) * z - (-math_lib.i * 19.071643 +
12.7127) * (z*2 + 1))) / (-(-math_lib.i*10.629721 + 17.23103) * (z*2 + 1) + 2)) end -- (1,1) Pade approximant for W(-1,a)
return ip;
end
function p._LambertW(_z, _k)
local z = to_number(_z)
local k = to_number(_k) or to_number(0)
local _2 = math_lib[1] * 2
if math_lib.abs(math_lib.nonRealPart(k)) > 1e-14 then error("朗伯W函数的k只能是實數") end
k = math_lib.re(k)
--For some particular z and k W(z,k) has simple value:
if math_lib.abs(z) == 0 then return (k == 0) and 0 or to_number(-math.huge) end
if z == -math_lib.exp(-1) and (k == 0 or k == -1) then return -math_lib[1] end
if z == math_lib.exp(1) and k == 0 then return math_lib[1]+0 end
--Halley method begins
local w, wprev = LWInitPoint(z, k), LWInitPoint(z, k) --intermediate values in the Halley method
local maxiter = 30 --max number of iterations. This eliminates improbable infinite loops
local iter = 0 --iteration counter
local prec = 1e-30; --difference threshold between the last two iteration results (or the iter number of iterations is taken)
wprev = w
w = w - _2 *((zexpz(w) - z) * zexpz_d(w)) /
(_2*math_lib.pow(zexpz_d(w),2) - (zexpz(w) - z)*zexpz_dd(w))
iter = iter + 1
while ((math_lib.abs(w - wprev) > prec) and iter < maxiter) do
wprev = w
w = w - _2 *((zexpz(w) - z) * zexpz_d(w)) /
(_2*math_lib.pow(zexpz_d(w),2) - (zexpz(w) - z)*zexpz_dd(w))
iter = iter + 1
end
return w
end
---------------------- 範數 ----------------------
function p._norm(_z, _p)
local p_value = to_number(_p or 2)
local check_inf = tostring(_p):match("[Ii][Nn][Ff]")
local abs_p, re_p = math_lib.abs(p_value), math_lib.re(p_value)
local value_list = {}
if type(_z) == type(0) or type(_z) == type("string") or (type(_z) == type({}) and _z.numberType) then
local z = to_number(_z)
if type(math_lib.dot) == type(noop_func) then
if type(math_lib.elements) == type({}) and #(math_lib.elements) > 0 then
for i=1,#(math_lib.elements) do value_list[#value_list + 1] = math_lib.dot(z, math_lib.elements[i]) end
else return math_lib.abs(z) end
else return math_lib.abs(z) end
elseif type(_z) == type({}) and #_z > 0 then
for i=1,#_z do value_list[#value_list + 1] = to_number(_z[i]) or to_number(0) end
end
if #value_list > 0 then
local norm_sum, norm_max, norm_min, non_zero_count = 0, -1, tonumber("inf"), 0
for i=1,#value_list do
local abs_value = math_lib.abs(value_list[i])
if abs_value > norm_max then norm_max = abs_value end
if abs_value < norm_min then norm_min = abs_value end
if abs_value ~= 0 then norm_sum = math_lib.pow(abs_value, p_value) + norm_sum end
if abs_value > 1e-14 then non_zero_count = non_zero_count + 1 end
end
return check_inf and (re_p > 0 and norm_max or norm_min) or
(abs_p >= 1 and math_lib.pow(norm_sum, math_lib.inverse(p_value)) or
(abs_p ~= 0 and norm_sum or non_zero_count))
end
error("無效的范數")
end
return p