The main symbolic solver for Symbolics.jl is symbolic_solve. Symbolic solving
means that it only uses symbolic (algebraic) methods and outputs exact solutions.
Symbolics.symbolic_solve
One other symbolic solver is symbolic_linear_solve which is limited compared to
symbolic_solve as it only solves linear equations.
Symbolics.symbolic_linear_solve
symbolic_solve only supports symbolic, i.e. non-floating point computations, and thus prefers equations
where the coefficients are integer, rational, or symbolic. Floating point coefficients are transformed into
rational values and BigInt values are used internally with a potential performance loss, and thus it is recommended
that this functionality is only used with floating point values if necessary. In contrast, symbolic_linear_solve
directly handles floating point values using standard factorizations.
The symbolic_solve function uses 4 hidden solvers in order to solve the user's input. Its base,
solve_univar, uses analytic solutions up to polynomials of degree 4 and factoring as its method
for solving univariate polynomials. The function's solve_multipoly uses GCD on the input polynomials then throws passes the result
to solve_univar. The function's solve_multivar uses Groebner basis and a separating form in order to create linear equations in the
input variables and a single high degree equation in the separating variable 1. Each equation resulting from the basis is then passed
to solve_univar. We can see that essentially, solve_univar is the building block of symbolic_solve. If the input is not a valid polynomial and can not be solved by the algorithm above, symbolic_solve passes
it to ia_solve, which attempts solving by attraction and isolation 2. This only works when the input is a single expression
and the user wants the answer in terms of a single variable. Say log(x) - a == 0 gives us [e^a].
Symbolics.solve_univar
Symbolics.solve_multivar
Symbolics.ia_solve
Symbolics.ia_conditions!
Symbolics.is_periodic
Symbolics.fundamental_period
using Symbolics, Nemo;
@variables x;
Symbolics.symbolic_solve(9^x + 3^x ~ 8, x)
@variables x y z;
Symbolics.symbolic_linear_solve(2//1*x + y - 2//1*z ~ 9//1*x, 1//1*x)
using Groebner;
@variables x y z;
eqs = [x^2 + y + z - 1, x + y^2 + z - 1, x + y + z^2 - 1]
Symbolics.symbolic_solve(eqs, [x,y,z])
- Linear and polynomial equations
- Systems of linear and polynomial equations
- Some transcendental functions
- Systems of linear equations with parameters (via
symbolic_linear_solve) - Equations with radicals
- Systems of polynomial equations with parameters and positive dimensional systems
- Inequalities
# Mathematica
In[1]:= Reduce[x^2 - x - 6 > 0, x]
Out[1]= x < -2 || x > 3
In[2]:= Reduce[x+a > 0, x]
Out[2]= a \[Element] Reals && x > -a
In[3]:= Solve[x^(x) + 3 == 0, x]
Out[3]= {{x -> (I \[Pi] + Log[3])/ProductLog[I \[Pi] + Log[3]]}}