# Q: Symbolics.jl | How to convert symbolic multivariate polynomial to DynamicPolynomials type?

**URL:** <https://discourse.julialang.org/t/q-symbolics-jl-how-to-convert-symbolic-multivariate-polynomial-to-dynamicpolynomials-type/105618>\
**Category:** Modelling & Simulations\
**Tags:** question\
**Created:** [October 31, 2023, 2:35pm UTC](https://discourse.julialang.org/t/q-symbolics-jl-how-to-convert-symbolic-multivariate-polynomial-to-dynamicpolynomials-type/105618 "2023-10-31T14:35:02Z")\
**Posts on this page:** 5\
**Page:** 1

<div class="post-metadata">

**Author:** ![Audrius-St](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/audrius-st/32/24175_2.png) [@Audrius-St](https://discourse.julialang.org/u/Audrius-St)\
**Post date:** [October 31, 2023, 2:35pm UTC](https://discourse.julialang.org/t/q-symbolics-jl-how-to-convert-symbolic-multivariate-polynomial-to-dynamicpolynomials-type/105618/1 "2023-10-31T14:35:02Z")

</div>

Hello,

How would one go about converting symbolic multivariate polynomials generated in Symbolics.jl to DynamicPolynomials.jl type?

The end goal is fast numerical evaluation of multivariate polynomials using  
FixedPolynomials.jl

A highly truncated MWE of the type of symbolic multivariate polynomial that I would like to convert is in the code fragment below:

```julia
# test_symbolics_to_dynamicpolynomials.jl
using Symbolics

function main()

    @variables q[1:2], p[1:2], t

    expr = 
        p[1] - q[1]*t - (1//2)*p[1]*(t^2) - (2//1)*q[1]*q[2]*t -
        p[1]*q[2]*(t^2) - p[2]*q[1]*(t^2) + (1//6)*q[1]*(t^3) -
        (2//3)*p[1]*p[2]*(t^3)

    # How to proceed?
end

begin
    main()
end

```

---

<div class="post-metadata">

**Author:** ![sumiya11](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/sumiya11/32/207147_2.png) [@sumiya11](https://discourse.julialang.org/u/sumiya11)\
**Post date:** [October 31, 2023, 5:52pm UTC](https://discourse.julialang.org/t/q-symbolics-jl-how-to-convert-symbolic-multivariate-polynomial-to-dynamicpolynomials-type/105618/2 "2023-10-31T17:52:18Z")

</div>

Hi, as far as I know, there is no exported conversion in Symbolics.jl.

You can try using PolyForm:

```julia
using Symbolics
@variables x y t
expr = t*(x + y^2)

polyf = PolyForm(Symbolics.unwrap(expr))

polyf.p, typeof(polyf.p)
# prints
(tx + ty², DynamicPolynomials.Polynomial{DynamicPolynomials.Commutative{DynamicPolynomials.CreationOrder}, MultivariatePolynomials.Graded{MultivariatePolynomials.LexOrder}, Int64})

```

---

<div class="post-metadata">

**Author:** ![Audrius-St](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/audrius-st/32/24175_2.png) [@Audrius-St](https://discourse.julialang.org/u/Audrius-St)\
**Post date:** [October 31, 2023, 8:26pm UTC](https://discourse.julialang.org/t/q-symbolics-jl-how-to-convert-symbolic-multivariate-polynomial-to-dynamicpolynomials-type/105618/3 "2023-10-31T20:26:38Z")

</div>

Thank you for your suggestion.

Polyform takes the code partway to what is required.

```julia
# test_symbolics_to_dynamicpolynomials.jl
using Symbolics

function main()

    @variables q[1:2], p[1:2], t

    expr = 
        p[1] - q[1]*t - (1//2)*p[1]*(t^2) - (2//1)*q[1]*q[2]*t -
        p[1]*q[2]*(t^2) - p[2]*q[1]*(t^2) + (1//6)*q[1]*(t^3) -
        (2//3)*p[1]*p[2]*(t^3)

    # @show substitute(expr, Dict([p[i] => @Base.r_str(i) for i = 1:2]))

    pf_expr = PolyForm(Symbolics.unwrap(expr))

    @show pf_expr.p
    println("")
    @show typeof(pf_expr.p)

end

begin
    main()
end

```

generates

```julia
pf_expr.p = getindex_10378028488760083673 - getindex_17699249546051641148t - 2//1getindex_17699249546051641148tgetindex_1435230719039639055 - 1//2getindex_10378028488760083673t² - getindex_17699249546051641148t²getindex_13271089874976073229 + 1//6getindex_17699249546051641148t³ - getindex_10378028488760083673t²getindex_1435230719039639055 - 2//3getindex_10378028488760083673t³getindex_13271089874976073229

typeof(pf_expr.p) = DynamicPolynomials.Polynomial{DynamicPolynomials.Commutative{DynamicPolynomials.CreationOrder}, MultivariatePolynomials.Graded{MultivariatePolynomials.LexOrder}, Rational{Int64}}
DynamicPolynomials.Polynomial{DynamicPolynomials.Commutative{DynamicPolynomials.CreationOrder}, MultivariatePolynomials.Graded{MultivariatePolynomials.LexOrder}, Rational{Int64}}

```

So now I need to make the substitutions  
`q[i] => q_i, p[i] => p_i, | i = 1, 2, . . .`  
somehow before calling Polyform.

No Idea how at the moment, but I’m guessing that this may be done using metaprogramming.

[I use indexed array symbols initially as it make the generating code compact and easily extensible.]

---

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**Author:** ![Audrius-St](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/audrius-st/32/24175_2.png) [@Audrius-St](https://discourse.julialang.org/u/Audrius-St)\
**Post date:** [November 2, 2023, 6:59pm UTC](https://discourse.julialang.org/t/q-symbolics-jl-how-to-convert-symbolic-multivariate-polynomial-to-dynamicpolynomials-type/105618/4 "2023-11-02T18:59:06Z")

</div>

[Edit] The following test code solves the above variable substitution issue using a bit of trivial metaprogramming.

```julia
# test_symbolics_to_dynamicpolynomials.jl
using FixedPolynomials
using Symbolics

function main()

    @variables q[1:2], p[1:2], t

    # Generate the numbered variables {qi, pi | i = 1, 2, . . .}
    numbered_cv_str = "@variables"

    for i = 1:2
        numbered_cv_str = string(numbered_cv_str, " ", "q", string(i), " ", "p", string(i))
    end

    @show eval(Meta.parse(numbered_cv_str))

    # Test expression
    expr = 
        p[1] - q[1]*t - (1//2)*p[1]*(t^2) - (2//1)*q[1]*q[2]*t -
        p[1]*q[2]*(t^2) - p[2]*q[1]*(t^2) + (1//6)*q[1]*(t^3) -
        (2//3)*p[1]*p[2]*(t^3)

    # Substitute {q[i] => qi | i = 1, 2, . . .}
    expr = 
        substitute(
            expr, 
            Dict(
                [q[i] => Num(eval(Symbol("q$(i)"))) for i = 1:2]
            )
        )

    # Substitute {p[i] => pi | i = 1, 2, . . .}
    expr = 
        substitute(
            expr, 
            Dict(
                [p[i] => Num(eval(Symbol("p$(i)"))) for i = 1:2]
            )
        )

    @show expr

    pf_expr = PolyForm(Symbolics.unwrap(expr))

    @show pf_expr.p

    @show Polynomial{Float64}(pf_expr.p)
end

begin
    main()
end

```

yields

`pf_expr.p = p1 - q1t - 2//1q1tq2 - 1//2p1t² - q1t²p2 + 1//6q1t³ - p1t²q2 - 2//3p1t³p2`

and

```julia
Polynomial{Float64}(pf_expr.p) = -0.6666666666666666p₁t³p₂-p₁t²q₂+0.16666666666666666q₁t³-q₁t²p₂-0.5p₁t²-2.0q₁tq₂-q₁t+p₁
-0.6666666666666666p₁t³p₂-p₁t²q₂+0.16666666666666666q₁t³-q₁t²p₂-0.5p₁t²-2.0q₁tq₂-q₁t+p₁

```

as desired.

---

<div class="post-metadata">

**Author:** ![Audrius-St](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/audrius-st/32/24175_2.png) [@Audrius-St](https://discourse.julialang.org/u/Audrius-St)\
**Post date:** [November 5, 2023, 3:29am UTC](https://discourse.julialang.org/t/q-symbolics-jl-how-to-convert-symbolic-multivariate-polynomial-to-dynamicpolynomials-type/105618/5 "2023-11-05T03:29:16Z")

</div>

Turns out FixedPolynomials.jl has been supersedes by StaticPolynomials.jl

```julia
# test_symbolics_to_staticpolynomials.jl
using StaticArrays
using StaticPolynomials
using Symbolics

function main()
    # Indexed array variables
    @variables q[1:2], p[1:2]

    # Time variable
    @variables t

    # Generate the numbered variables {qi, pi | i = 1, 2, . . .}
    variables_cv_str = "variables_array = @variables"

    for i = 1:2
        variables_cv_str = string(variables_cv_str, " ", "q", string(i), " ", "p", string(i))
    end
    variables_cv_str = string(variables_cv_str, " t")

    eval(Meta.parse(variables_cv_str))
    @show typeof(variables_array)

    # Test expression
    expr = 
        p[1] - q[1]*t - (1//2)*p[1]*(t^2) - (2//1)*q[1]*q[2]*t -
        p[1]*q[2]*(t^2) - p[2]*q[1]*(t^2) + (1//6)*q[1]*(t^3) -
        (2//3)*p[1]*p[2]*(t^3)

    # Substitute {q[i] => qi | i = 1, 2, . . .}
    expr = 
        substitute(
            expr, 
            Dict(
                [q[i] => Num(eval(Symbol("q$(i)"))) for i = 1:2]
            )
        )

    # Substitute {p[i] => pi | i = 1, 2, . . .}
    expr = 
        substitute(
            expr, 
            Dict(
                [p[i] => Num(eval(Symbol("p$(i)"))) for i = 1:2]
            )
        )

    @show expr
    println("")

    # Convert Symbolics polynomial to DynamicPolynomials form
    @show pf_expr = PolyForm(Symbolics.unwrap(expr))
    @show typeof(pf_expr)
    @show typeof(pf_expr.p)
    println("")

    # Convert the DynamicPolynomials form to StaticPolynomials
    @show p_static = StaticPolynomials.Polynomial(pf_expr.p)
    println("")

    # Test array [q1, p1, q2, p2, t]
    @time x = @SVector rand(5)

    # Evaluate
    @show @time StaticPolynomials.evaluate(p_static, x)
end

begin
    main()
end

```

which yields

```julia
p_static = StaticPolynomials.Polynomial(pf_expr.p) = p₁ - q₁t-1//2p₁t² + 1//6q₁t³-2//1q₁tq₂ - p₁t²q₂ - q₁t²p₂-2//3p₁t³p₂
typeof(p_static) = Polynomial{Rational{Int64}, SExponents{40}(a1dc5e8c62fc751a), Nothing}

```