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--- @class Rational
--- Represents an element of the field of rational numbers or rational functions.
--- @field numerator Ring
--- @field denominator Ring
--- @field ring RingIdentifier
Rational = {}
local __Rational = {}
--------------------------
-- Static functionality --
--------------------------
-- Metatable for ring objects.
local __obj = {__index = Rational, __eq = function(a, b)
return a["ring"] == b["ring"] and
(a["child"] == b["child"] or a["child"] == nil or b["child"] == nil) and
(a["symbol"] == b["symbol"] or a["child"] == nil or b["child"] == nil)
end, __tostring = function(a)
if a.symbol then
return tostring(a.child.child) .. "(" .. a.symbol .. ")"
end
if a.child then
return "QQ"
end
return "(Generic Fraction Field)"
end}
--- @param symbol SymbolExpression
--- @param child RingIdentifier
--- @return RingIdentifier
function Rational.makering(symbol, child)
local t = {ring = Rational}
t.symbol = symbol
t.child = child
t = setmetatable(t, __obj)
return t
end
--- Converts a string of the form -?[0-9]+ or -?[0-9]+\/[0-9]+ to a rational number.
--- @param str string
--- @return Rational|Integer
function Rational.fromstring(str)
local divloc = string.find(str, "/");
if not divloc then
return Integer(str)
end
return Rational(Integer(string.sub(str, 1, divloc - 1)), Integer(string.sub(str, divloc + 1, #str)))
end
----------------------------
-- Instance functionality --
----------------------------
-- So we don't have to copy the field operations each time.
__RationalOperations = Copy(__FieldOperations)
__RationalOperations.__index = Rational
__RationalOperations.__tostring = function(a)
if a.ring.symbol then
return "(" .. tostring(a.numerator)..")/("..tostring(a.denominator) .. ")"
end
return tostring(a.numerator).."/"..tostring(a.denominator)
end
--- Creates a new rational given a numerator and denominator that are part of the same ring.
--- Rational numbers are represented uniquely.
--- @param n Ring
--- @param d Ring
--- @param keep boolean
function Rational:new(n, d, keep)
local o = {}
o = setmetatable(o, __RationalOperations)
if n:getring() == PolynomialRing.getring() then
o.symbol = n.symbol
end
if d:getring() == PolynomialRing.getring() then
o.symbol = d.symbol
end
if d == Integer(0) then
error("Arithmetic error: division by zero")
end
n = n or Integer.zero()
d = d or Integer.one()
o.numerator = n
o.denominator = d
if not keep then
o:reduce()
end
if o.numerator:getring() == Integer.getring() then
o.ring = Integer.getring()
elseif o.numerator:getring() == PolynomialRing.getring() then
o.ring = Ring.resultantring(o.numerator:getring(), o.denominator:getring())
end
if (not keep) and o.denominator == Integer.one() or (not keep) and o.numerator == Integer.zero() then
return o.numerator
end
return o
end
--- Reduces a rational expression to standard form. This method mutates its object.
function Rational:reduce()
if self.numerator:getring() == Integer.getring() then
if self.denominator < Integer.zero() then
self.denominator = -self.denominator
self.numerator = -self.numerator
end
local gcd = Integer.gcd(self.numerator, self.denominator)
self.numerator = self.numerator//gcd
self.denominator = self.denominator//gcd
elseif self.numerator:getring() == PolynomialRing.getring() then
local lc = self.denominator:lc()
self.denominator = self.denominator/lc
self.numerator = self.numerator/lc
local gcd = PolynomialRing.gcd(self.numerator, self.denominator)
self.numerator = self.numerator//gcd
self.denominator = self.denominator//gcd
end
end
--- @return RingIdentifier
function Rational:getring()
local t = {ring=Rational}
if self then
t.child = self.ring
t.symbol = self.symbol
end
t = setmetatable(t, __obj)
return t
end
--- @param ring RingIdentifier
--- @return Ring
function Rational:inring(ring)
if ring == self:getring() then
return self
end
if ring == Rational:getring() and ring.symbol then
if not self:getring().symbol then
return Rational(self:inring(ring.child), self:inring(ring.child):one(), true)
end
return Rational(self.numerator:inring(ring.child), self.denominator:inring(ring.child), true)
end
if ring == PolynomialRing:getring() then
return PolynomialRing({self:inring(ring.child)}, ring.symbol)
end
error("Unable to convert element to proper ring.")
end
--- @return boolean
function Rational:isconstant()
if self.symbol then
return false
end
return true
end
--- @return Expression
function Rational:tocompoundexpression()
return BinaryOperation(BinaryOperation.DIV, {self.numerator:tocompoundexpression(), self.denominator:tocompoundexpression()})
end
--- Returns this rational as a floating point number. Can only approximate the value of most rationals.
--- @return number
function Rational:asnumber()
return self.numerator:asnumber() / self.denominator:asnumber()
end
function Rational:add(b)
return Rational(self.numerator * b.denominator + self.denominator * b.numerator, self.denominator * b.denominator)
end
function Rational:neg()
return Rational(-self.numerator, self.denominator, true)
end
function Rational:mul(b)
return Rational(self.numerator * b.numerator, self.denominator * b.denominator)
end
-- function Rational:inv(b)
-- return Rational(self.numerator * b.numerator, self.denominator * b.denominator)
-- end
function Rational:pow(b)
return (self.numerator ^ b) / (self.denominator ^ b)
end
function Rational:div(b)
return Rational(self.numerator * b.denominator, self.denominator * b.numerator)
end
function Rational:eq(b)
return self.numerator == b.numerator and self.denominator == b.denominator
end
function Rational:lt(b)
if self.numerator < Integer.zero() and b.numerator > Integer.zero() then
return true
end
if self.numerator > Integer.zero() and b.numerator < Integer.zero() then
return false
end
if (self.numerator >= Integer.zero() and b.numerator >= Integer.zero()) or (self.numerator <= Integer.zero() and b.numerator <= Integer.zero()) then
return self.numerator * b.denominator < self.denominator * b.numerator
end
return self.numerator * b.denominator > self.denominator * b.numerator
end
function Rational:le(b)
return self:eq(b) or self:lt(b)
end
function Rational:zero()
return Integer.zero()
end
function Rational:one()
return Integer.one()
end
function Rational:tolatex()
if string.sub(self.numerator:tolatex(),1,1) == '-' then
return "- \\frac{" .. string.sub(self.numerator:tolatex(),2,-1) .. "}{" .. self.denominator:tolatex() .. "}"
end
return "\\frac{" .. self.numerator:tolatex() .."}{".. self.denominator:tolatex().. "}"
end
-----------------
-- Inheritance --
-----------------
__Rational.__index = Field
__Rational.__call = Rational.new
Rational = setmetatable(Rational, __Rational)
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