How to simplify a symbolic expression

General Usage
,
1

I want to simplify an expression by replacing a sub-expression with a new variable, called Q.

The following code doesn’t work:

# simplify an expression
using Symbolics

@variables t
@syms ω(t) Pg(t) Pge(t) Pgc(t)
@variables A R a b Γ ρ U J Q

expr = (0.5A*R*a*Γ*ρ*(U^2)) / (J*ω(t)) - J*((0.5A*Γ*ρ*(U^3)*(b + (R*a*ω(t)) / U) - Pgc(t) - Pge(t)) / ((J^2)*(ω(t)^2)))
# Q = 0.5A*R*a*Γ*ρ

# how to substitute 0.5A*R*a*Γ*ρ by Q in expr?
expr2 = substitute(expr, Dict([0.5A*R*a*Γ*ρ => Q]))

Q does not appear in expr2, the substitute command has no effect.

Why?

EDIT:
Is it possible to solve this task with Symbolics, or should I try other packages like GitHub - chakravala/Reduce.jl: Symbolic parser for Julia language term rewriting using REDUCE algebra ?

2

This needs to be done with a rule application, not substitute.

1 Like
3

Good suggestion, but not yet working.

using Symbolics

@variables t
@syms ω(t) Pg(t) Pge(t) Pgc(t)
@variables A R a b Γ ρ U J Q

expr = (0.5A*R*a*Γ*ρ*(U^2)) / (J*ω(t)) - J*((0.5A*Γ*ρ*(U^3)*(b + (R*a*ω(t)) / U) - Pgc(t) - Pge(t)) / ((J^2)*(ω(t)^2)))

r1 = @rule 0.5A*R*a*Γ*ρ => Q
# Q = 0.5A*R*a*Γ*ρ

# how to substitute 0.5A*R*a*Γ*ρ by Q in expr?
expr2 = r1(expr)

The output is empty.

Any suggestion how to do this correctly?

4

Perhaps I should ask different questions. For example:

  • is expr above actually an expression, because it is of type Num?
  • how can I iterate over the elements of such an expression?
  • what is the difference between @syms and @variables ?
5

Maybe you should use the package SymbolicUtils

6

Well, I think Symbolics re-exports SymbolicUtils, so that makes no difference…

7

On paper this is trivial, why is it so hard with Julia?

8

substitute currently only allows you to substitute variables for other expressions, not the other way around unfortunately…

To walk the expression, tree = Symbolics.unwrap(expr) which removes the Num <: Real wrapper.

Then you can use the expression walking interface: https://github.com/JuliaSymbolics/SymbolicUtils.jl/blob/master/page/interface.md

9

If I run this script:

using Symbolics

@variables t
@syms ω(t) Pg(t) Pge(t) Pgc(t) A
@variables R a b Γ ρ U J Q

expr = (0.5A*R*a*Γ*ρ*(U^2)) / (J*ω(t)) - J*((0.5A*Γ*ρ*(U^3)*(b + (R*a*ω(t)) / U) - Pgc(t) - Pge(t)) / ((J^2)*(ω(t)^2)))

tree = Symbolics.unwrap(expr)

function walk(tree, level=0)
    for arg in arguments(tree)
        indent = " "^level
        println(indent, arg)
        if istree(arg)
            level += 1
            walk(arg, level)
            level -= 1
        end
    end
end

walk(tree)

the output is:

(Pgc(t) + Pge(t) - 0.5A*Γ*ρ*(U^3)*(b + (R*a*ω(t)) / U)) / (J*(ω(t)^2))
 Pgc(t) + Pge(t) - 0.5A*Γ*ρ*(U^3)*(b + (R*a*ω(t)) / U)
  -0.5A*Γ*ρ*(U^3)*(b + (R*a*ω(t)) / U)
   -0.5
   A
   Γ
   ρ
   U^3
    U
    3
   b + (R*a*ω(t)) / U
    b
    (R*a*ω(t)) / U
     R*a*ω(t)
      R
      a
      ω(t)
       t
     U
  Pgc(t)
   t
  Pge(t)
   t
 J*(ω(t)^2)
  J
  ω(t)^2
   ω(t)
    t
   2
(0.5A*R*a*Γ*ρ*(U^2)) / (J*ω(t))
 0.5A*R*a*Γ*ρ*(U^2)
  0.5
  A
  R
  a
  Γ
  ρ
  U^2
   U
   2
 J*ω(t)
  J
  ω(t)
   t

So I can walk the tree. :slight_smile:

Next questions are how to identify the expression I want to replace, how to replace it, and then how to re-assemble it.

10

Next question: How can I check if two expressions are equal?

Q=0.5A*R*a*Γ*ρ*(U^2)
# now check if another expression is equal to Q
Q==0.5A*R*a*Γ*ρ*(U^2)
# the output is another expression, and not true or false
(0.5A*R*a*Γ*ρ*(U^2)) == (0.5A*R*a*Γ*ρ*(U^2))

EDIT:
I am using now:

repr(subexpr) == "0.5A*R*a*Γ*ρ*(U^2)"

Definitly not a robust solution, but works for now…

11

My current solution:

using Symbolics

@variables t
@syms ω(t) Pg(t) Pge(t) Pgc(t)
@variables R a b Γ ρ U J Q A

expr = (0.5A*R*a*Γ*ρ*(U^2)) / (J*ω(t)) - J*((0.5A*Γ*ρ*(U^3)*(b + (R*a*ω(t)) / U) - Pgc(t) - Pge(t)) / ((J^2)*(ω(t)^2)))

Qs = "0.5A*R*a*Γ*ρ"

# how to substitute 0.5A*R*a*Γ*ρ by Q in expr?

function subst(expr, from="", to="")
    sexpr = repr(expr)
    sres = replace(sexpr, from => to)
    eval(Meta.parse(sres))
end

simple = subst(expr, Qs, "Q")

Output:

(Pgc(t) + Pge(t) - 0.5A*Γ*ρ*(U^3)*(b + (R*a*ω(t)) / U)) / (J*(ω(t)^2)) + (Q*(U^2)) / (J*ω(t))

Not a good solution because it uses eval and requires to the symbolic variables to be defined globally.

I will create two issues on this topic: