Am working on code to derive mathematical relations, its initial purpose was in the field of non-commutative geometry however it appears to have become generally useful, feels like a good idea if we make the base self contained and release it as a separate package upon which more specialized ones can be built.
The code is here: https://github.com/jhlq/fys/tree/master/ncg quin@Nicol:~/fys/ncg$ julia -L equations -q Let's start from the top, the most recent file is matchers.jl which currently contains one matcher function that checks if an equation matches the pattern ax^2+bx+c=0. By writing such matchers (and introducing types for new operators) the base can readily be extended to solve custom equations. The quadratic matcher in action: julia> @time solve(:a*(:x*:y*:z)^2+:b*:y*:z*:x+:c,Function)[end] #the type specifies which kind of operator to apply elapsed time: 1.779767594 seconds (426648608 bytes allocated, 25.89% gc time) Equation(Expression({:x,:z,:y}),Expression({-1,:b,÷(Expression({2,:a})),:+,-1,Sqrt(Expression({:b,:b,:+,-4,:a,:c})),÷(Expression({2,:a}))})) The Equation type is specified in equations.jl, it has two fields, lhs and rhs. The solve function when given an expression assumes it is the lhs of an equation with rhs=0 and finds solutions by calling the matches function which so far also accepts the Sqrt and Div types as ops, these types implement their own simplify methods. The original matches passes terms and factors from left to right: julia> matches(:x+:y) 6-element Array{Equation,1}: Equation(:y,Expression({-1,:x})) Equation(1,Expression({-1,:x,÷(:y)})) Equation(0,Expression({-1,:x,:+,-1,:y})) Equation(:x,Expression({-1,:y})) Equation(1,Expression({-1,:y,÷(:x)})) Equation(0,Expression({-1,:y,:+,-1,:x})) Now we descend into the base at common.jl. The function simplify is essential and called everywhere (probably too manywhere), it resolves nested expressions and sums it all up: julia> ex=(:x+3+9*:x)^3 Expression({Expression({:x,:+,3,:+,Expression({9,:x})}),Expression({Expression({:x,:+,3,:+,Expression({9,:x})}),Expression({:x,:+,3,:+,Expression({9,:x})})})}) julia> simex=simplify(ex) Expression({1000,:x,:x,:x,:+,900,:x,:x,:+,270,:x,:+,27}) julia> evaluate(ex,[:x=>1])==evaluate(simex,[:x=>1]) true A crucial component of simplify is componify which is the trustable cookie unwrapper (technical term), it extracts components without eating anyting: julia> ex=:x-(:x-3)+(:y+3)^2 Expression({:x,:+,-1,:x,:+,-1,-1,3,:+,Expression({Expression({:y,:+,3}),Expression({:y,:+,3})})}) julia> ex=componify(ex) Expression({:x,:+,-1,:x,:+,-1,-1,3,:+,:y,:y,:+,:y,3,:+,3,:y,:+,3,3}) One final frequently used internal function is addparse, it returns an array of arrays with factors. For many cases the iterative shorthand will be sufficient: julia> for term in ex;print(term);end {:x}{-1,:x}{-1,3}{:y,:y}{:y,3}{3,:y}{3,3} There are a whole host of additional details that optimization pros presumably can have many field days, camping trips and holidays with. Will continue nurturing these wines to climb toward quantum gravity and desire the community to be involved so everyone is very welcome to hack away at the jungle, add branches and tread the paths created. We have a choice to make, representing a structural birth for the code: What shall this package be named?
