110 lines
2.5 KiB
Lua
110 lines
2.5 KiB
Lua
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--Bespoke symbolic differentiation.
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--Little prototypes for building a syntax tree containing only
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--multiplication, addition, and composition,
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--evaluating its value, and expanding it according to the chain rule.
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local op;
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local function d( x ) return x:diff() end
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local function D( x ) return setmetatable( { ["n"] = x }, op.atom ) end
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local computedDerivatives = {}
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local __call = D
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local __add = function( a, b ) return setmetatable( { a, b }, op.add ) end
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local __mul = function( a, b ) return setmetatable( { a, b }, op.mul ) end
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local __pow = function( a, b ) return setmetatable( { a, b }, op.compose ) end
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op = {
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atom = {
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__add = __add,
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__mul = __mul,
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__pow = __pow,
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__call = __call,
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__tostring = function( self ) return tostring( self.n ) end,
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__index = {
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eval = function( self )
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return computedDerivatives[ self.n ]
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end,
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diff = function( self )
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return D( self.n + 1 )
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end,
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}},
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compose = {
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__add = __add,
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__mul = __mul,
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__pow = __pow,
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__call = __call,
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__tostring = function( self ) return "("..tostring( self[1] ).." o "..tostring( self[2] )..")" end,
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__index = {
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eval = function( self )
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--All compositions are of the form f^(n) o f,
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--whose value at a fixed point p is f^(n)(p)
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return self[1]:eval()
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end,
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diff = function( self )
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--All compositions are of the form f^(n) o f,
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--whose derivatives are always ( f^(n+1) o f )
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return d( self[1] ) ^ D(0) * D(1)
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end,
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}},
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add = {
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__add = __add,
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__mul = __mul,
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__pow = __pow,
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__call = __call,
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__tostring = function( self ) return tostring( self[1] ).." + "..tostring( self[2] ) end,
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__index = {
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name = "add",
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eval = function( self )
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return self[1]:eval() + self[2]:eval()
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end,
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diff = function( self )
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return d(self[1]) + d(self[2])
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end,
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}},
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mul = {
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__add = __add,
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__mul = __mul,
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__pow = __pow,
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__call = __call,
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__tostring = function( self ) return tostring( self[1] ).." x "..tostring( self[2] ) end,
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__index = {
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eval = function( self )
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return self[1]:eval() * self[2]:eval()
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end,
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diff = function( self )
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return d(self[1]) * self[2] + self[1] * d(self[2])
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end,
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}},
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}
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op.New = function( nodeType, a, b )
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return setmetatable( { a, b }, op[nodeType] or error( "Node Type Not Recognized" ) )
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end
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local a = D(1) ^ D(0)
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for i = 1, 5 do
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print( a )
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a = a:diff()
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end
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return op
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