Systems of equations

[evykassirer]

This is blocked on #128 (new parser) because we need to support multiple equations.

It'd be great to have another module, like simplifyExpression and solveEquation that solved a system of equations

e.g. y = 2x + 2, x + y = 5, solve for x

[hmaurer]

With the work being done by @kevinbarabash on https://github.com/kevinbarabash/math-ast maybe we could model systems of equations with the "AND" logical operator at the top level? e.g.

y = 2x + 2 AND x = y = 5

We would then have tree-rewriting rules like for the rest which handle this type of structure.
I am unsure on how CAS systems do it these days, @tkosan might have insights on the topic!

[tkosan]

This is the algorithm I prototyped in MathPiper for solving systems of equations:

equations := MapSingle("MathParse",
                ["x - 2y + 3z = 7",
                  "2x + y + z = 4",
               "-3x + 2y - 2z = -10"]);
solutions := [];
While(equations !=? []) {
  equation := Pop(equations, 1);
  solution := Solve(equation, UnderscoreConstants(equation)[1]);
  If(solution !=? []) {
    variable := solution[1][1];
    value := solution[1][2];
    equations := Substitute(variable, value) equations;
    solutions := Substitute(variable, value) solutions;
    Append!(solutions, variable == value);
  };
};
MapSingle("MetaToObject", solutions /:: [a_ == b_ <- a == Eval(b)]);

Result: [x == 2, y == (-1), z == 1]

Currently, the algorithm holds the equations in a list. However, flattening an expression with respect to a binary operator is straightforward, so modeling systems of equations with the "AND" logical operator should not cause problems:

In> Flatten(a &? b &? c, "&?")
Result: [a, b, c]

[kevinbarabash]

@hmaurer I added a List node in math-ast with the intent of using it for system of equations or multiple solutions (in the case of quadratic formula). Unfortunately, it feels to general. What differentiates a List of multiple solutions from a List that's a system of equations?

I like that "AND" indicates that all equations need to hold and "OR" indicates that each of the multiple solutions is valid. I am worried though about overloading logical operators. Logic operators should be used in expression for logic proofs.

Maybe we can introduce any and all nodes or add a property to the List node that indicates any or all. Another option is to have an explicit System node that can contain multiple equations and inequalities. And use List for multiple solutions.

[kevinbarabash]

I've added tentative support for system of equation and sequences to math-parser. Here are a couple of test cases:

• 2x + 3 = y, x - y = 10
• a < x < b
• x/2, x/3, x/4

semantic-math/math-ast#12 has some additional thoughts about converting things like a < x < b to Bounds nodes.

(edit: links fixed)