# Symbolics and substitution?

**URL:** <https://discourse.julialang.org/t/symbolics-and-substitution/57460>\
**Category:** General Usage\
**Tags:** symbolic\
**Created:** [March 18, 2021, 11:01am UTC](https://discourse.julialang.org/t/symbolics-and-substitution/57460 "2021-03-18T11:01:56Z")\
**Posts on this page:** 20\
**Page:** 1

<div class="post-metadata">

**Author:** ![BLI](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/bli/32/37206_2.png) [@BLI](https://discourse.julialang.org/u/BLI)\
**Post date:** [March 18, 2021, 11:01am UTC](https://discourse.julialang.org/t/symbolics-and-substitution/57460/1 "2021-03-18T11:01:56Z")

</div>

I’m trying to use `Symbolics.jl` to fit a polynomial to some data points (\ldots,(x\_j,y\_j),\ldots) by Lagrange interpolation, then take the derivative of the polynomial, and substitute some relationships between the data points.

I get an error message when I do the substitution. To make some sense of the syntax, I use `D` to denote \Delta x, `xim3d2` to denote x\_{i-\frac{3}{2}}, etc. [Suggestions for better notation would be appreciated…]

So here is what happens:

```julia
using Symbolics
#
@variables x D
@variables xim5d2 xim3d2 xim1d2 xip1d2
@variables uim2 uim1 ui
#
xd = [xim5d2,xim3d2,xim1d2,xip1d2]
yd = [0,uim2*D, (uim2+uim1)*D, (uim2+uim1+ui)*D]
#
function lagrange_polynomial(xd,yd)
    n = length(xd)
    LL = 0
    for j in 1:n
        L = 1
        for k in setdiff(1:n,j)
            L = L*(x-xd[k])/(xd[j]-xd[k])
        end
        LL = LL + yd[j]*L
    end
    return LL            
end
#
L = lagrange_polynomial(xd,yd)
u = derivative(L,x)
#
xsubs = Dict([xim3d2=>xim5d2+D, xim1d2=>xim3d2+D, xip1d2=>xim1d2+D])
#
substitute(u,xsubs)

```

The last line gives an error message:

```julia
TypeError: non-boolean (Num) used in boolean context.....

```

_Question_: What is it that I do wrong in the `substitute` command?

---

<div class="post-metadata">

**Author:** ![rafael.guerra](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/rafael.guerra/32/216610_2.png) [@rafael.guerra](https://discourse.julialang.org/u/rafael.guerra)\
**Post date:** [March 18, 2021, 11:42am UTC](https://discourse.julialang.org/t/symbolics-and-substitution/57460/2 "2021-03-18T11:42:52Z")

</div>

It seems to work in Julia Version 1.6.0-beta1.0 on Win 10.  
_PS: had to specify:_`u = Symbolics.derivative(L,x)`

```julia
julia> substitute(u,xsubs)
uim2*(x - xim5d2)*(x - (D + xim1d2))*((D + xim5d2 - (D + xim1d2))^-1)*((D + xim5d2 - (D + xim3d2))^-1) + uim2*(x - xim5d2)*(x - (D + xim3d2))*((D + xim5d2 - (D + xim1d2))^-1)*((D + xim5d2 - (D + xim3d2))^-1) + uim2*(x - (D + xim1d2))*(x - (D + xim3d2))*((D + xim5d2 - (D + xim1d2))^-1)*((D + xim5d2 - (D + xim3d2))^-1) + D*(uim1 + uim2)*(x - xim5d2)*(x - (D + xim1d2))*((D + xim3d2 - xim5d2)^-1)*((D + xim3d2 - (D + xim1d2))^-1)*((D + xim3d2 - (D + xim5d2))^-1) + D*(uim1 + uim2)*(x - xim5d2)*(x - (D + xim5d2))*((D + xim3d2 - xim5d2)^-1)*((D + xim3d2 - (D + xim1d2))^-1)*((D + xim3d2 - (D + xim5d2))^-1) + D*(uim1 + uim2)*(x - (D + xim1d2))*(x - (D + xim5d2))*((D + xim3d2 - xim5d2)^-1)*((D + xim3d2 - (D + xim1d2))^-1)*((D + xim3d2 - (D + xim5d2))^-1) + D*(x - xim5d2)*(x - (D + xim3d2))*(ui + uim1 + uim2)*((D + xim1d2 - xim5d2)^-1)*((D + xim1d2 - (D + xim3d2))^-1)*((D + xim1d2 - (D + xim5d2))^-1) + D*(x - xim5d2)*(x - (D + xim5d2))*(ui + uim1 + uim2)*((D + xim1d2 - xim5d2)^-1)*((D + xim1d2 - (D + xim3d2))^-1)*((D + 
xim1d2 - (D + xim5d2))^-1) + D*(x - (D + xim3d2))*(x - (D + xim5d2))*(ui + uim1 + uim2)*((D + xim1d2 - xim5d2)^-1)*((D + xim1d2 - (D + xim3d2))^-1)*((D + xim1d2 - (D + xim5d2))^-1)      

```

---

<div class="post-metadata">

**Author:** ![BLI](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/bli/32/37206_2.png) [@BLI](https://discourse.julialang.org/u/BLI)\
**Post date:** [March 18, 2021, 11:56am UTC](https://discourse.julialang.org/t/symbolics-and-substitution/57460/3 "2021-03-18T11:56:40Z")

</div>

Thanks – good to know. I’m on Julia v1.5.0 on my job desktop… I’m kind of waiting for v1.6.0 official release.

Still, I’m curious as to whether there is a known reason why it doesn’t work on v1.5.x.

---

<div class="post-metadata">

**Author:** ![oheil](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/oheil/32/220745_2.png) [@oheil](https://discourse.julialang.org/u/oheil)\
**Post date:** [March 18, 2021, 12:40pm UTC](https://discourse.julialang.org/t/symbolics-and-substitution/57460/4 "2021-03-18T12:40:16Z")

</div>

I don’t get an error on Julia 1.5.3 on my Windows 10 PC.  
Perhaps it is some kind of name conflict?

```julia
julia> versioninfo()
Julia Version 1.5.3
Commit 788b2c77c1 (2020-11-09 13:37 UTC)
Platform Info:
  OS: Windows (x86_64-w64-mingw32)
  CPU: AMD Ryzen 9 3900X 12-Core Processor
  WORD_SIZE: 64
  LIBM: libopenlibm
  LLVM: libLLVM-9.0.1 (ORCJIT, znver2)

julia> substitute(u,xsubs)
uim2*(x - xim5d2)*(x - (D + xim1d2))*((D + xim5d2 - (D + xim1d2))^-1)*((D + xim5d2 - (D + xim3d2))^-1) + uim2*(x - xim5d2)*(x - (D + xim3d2))*((D + xim5d2 - (D + xim1d2))^-1)*((D + xim5d2 - (D + xim3d2))^-1) + uim2*(x - (D + xim1d2))*(x - (D + xim3d2))*((D + xim5d2 - (D + xim1d2))^-1)*((D + xim5d2 - (D + xim3d2))^-1) + D*(uim1 + uim2)*(x - xim5d2)*(x - (D + xim1d2))*((D + xim3d2 - xim5d2)^-1)*((D + xim3d2 - (D + xim1d2))^-1)*((D + xim3d2 - (D + xim5d2))^-1) + D*(uim1 + uim2)*(x - xim5d2)*(x - (D + xim5d2))*((D + xim3d2 - xim5d2)^-1)*((D + xim3d2 - (D + xim1d2))^-1)*((D + xim3d2 - (D + xim5d2))^-1) + D*(uim1 + uim2)*(x - (D + xim1d2))*(x - (D + xim5d2))*((D + xim3d2 - xim5d2)^-1)*((D + xim3d2 - (D + xim1d2))^-1)*((D + xim3d2 - (D + xim5d2))^-1) + D*(x - xim5d2)*(x - (D + xim3d2))*(ui + uim1 + uim2)*((D + xim1d2 - xim5d2)^-1)*((D + xim1d2 - (D + xim3d2))^-1)*((D + xim1d2 - (D + xim5d2))^-1) + D*(x - xim5d2)*(x - (D + xim5d2))*(ui + uim1 + uim2)*((D + xim1d2 - xim5d2)^-1)*((D + xim1d2 - (D + xim3d2))^-1)*((D + xim1d2 - (D + xim5d2))^-1) + D*(x - (D + xim3d2))*(x - (D + xim5d2))*(ui + uim1 + uim2)*((D + xim1d2 - xim5d2)^-1)*((D + xim1d2 - (D + xim3d2))^-1)*((D + xim1d2 - (D + xim5d2))^-1)

(@v1.5) pkg> st
Status `C:\Users\oheil\.julia\environments\v1.5\Project.toml`
...
  [0c5d862f] Symbolics v0.1.8
...

```

---

<div class="post-metadata">

**Author:** ![BLI](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/bli/32/37206_2.png) [@BLI](https://discourse.julialang.org/u/BLI)\
**Post date:** [March 18, 2021, 12:55pm UTC](https://discourse.julialang.org/t/symbolics-and-substitution/57460/5 "2021-03-18T12:55:50Z")

</div>

Hm. So it could be that Julia 1.5.0 downgrades the package, perhaps. The code I included above is the entire code I have in my Jupyter Notebook.

[I can try to run it in Visual Code to see if there is a problem with Jupyter Lab.]

I ran the code in the REPL, and I get the same error message, so the problem must be with my Julia set-up…perhaps.

---

<div class="post-metadata">

**Author:** ![John\_Gibson](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/john_gibson/32/5321_2.png) [@John\_Gibson](https://discourse.julialang.org/u/John_Gibson)\
**Post date:** [March 18, 2021, 1:00pm UTC](https://discourse.julialang.org/t/symbolics-and-substitution/57460/6 "2021-03-18T13:00:46Z")

</div>

The Polynomials package does Lagrange interpolation. If you’re just trying to fit and manipulate polynomials that might be a better alternative. Of course you might be exploring Symbolics, learning Lagrange interpolation, or something like that.

```julia
julia> using Polynomials

julia> x = [-1.0; 0.0; 1.0];

julia> y = [2.0; 3.0; 6.0]; # numerial values of x^2 + 2x + 3

julia> f = fit(x,y)
Polynomial(3.0 + 2.0*x + 1.0*x^2)

julia> f(2)
11.0

julia> dfdx = derivative(f)
Polynomial(2.0 + 2.0*x)

julia> dfdx(0)
2.0

```

---

<div class="post-metadata">

**Author:** ![BLI](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/bli/32/37206_2.png) [@BLI](https://discourse.julialang.org/u/BLI)\
**Post date:** [March 18, 2021, 1:10pm UTC](https://discourse.julialang.org/t/symbolics-and-substitution/57460/7 "2021-03-18T13:10:03Z")

</div>

I’m looking into symbolic expressions for reconstruction of interpolation values in PDE discretization.

---

<div class="post-metadata">

**Author:** ![John\_Gibson](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/john_gibson/32/5321_2.png) [@John\_Gibson](https://discourse.julialang.org/u/John_Gibson)\
**Post date:** [March 18, 2021, 1:45pm UTC](https://discourse.julialang.org/t/symbolics-and-substitution/57460/8 "2021-03-18T13:45:28Z")

</div>

Ah. I’ve used Polynomials for a similar purpose, interpolating numerical integrations of ODEs to compute intersections of trajectories with Poincare sections. For PDEs you’d probably need MultivariatePolynomials.jl. I haven’t used that but it might work for your purpose.

---

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**Author:** ![BLI](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/bli/32/37206_2.png) [@BLI](https://discourse.julialang.org/u/BLI)\
**Post date:** [March 18, 2021, 1:59pm UTC](https://discourse.julialang.org/t/symbolics-and-substitution/57460/9 "2021-03-18T13:59:02Z")

</div>

I’m looking into symbolic schemes, so I need symbolic capabilities. I already did it by hand for a simple case, and I got the “text book” correct solution. I just want to automate it using Symbolics, so that I can find more schemes and for higher order without doing the “boring” manual computation.

In essence, I’m looking for interpolation schemes for the interface between nodes, and based on solutions at spatial nodes.

I’ve now tested the code on my laptop, too (Julia 1.5.0), with the same problem. So it seems that I need to update my version to Julia 1.5.3, or wait for the release of Julia 1.6.0.

---

<div class="post-metadata">

**Author:** ![BLI](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/bli/32/37206_2.png) [@BLI](https://discourse.julialang.org/u/BLI)\
**Post date:** [March 18, 2021, 2:00pm UTC](https://discourse.julialang.org/t/symbolics-and-substitution/57460/10 "2021-03-18T14:00:42Z")

</div>

My set-up is on Symbolics v.0.1.2.

---

<div class="post-metadata">

**Author:** ![oheil](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/oheil/32/220745_2.png) [@oheil](https://discourse.julialang.org/u/oheil)\
**Post date:** [March 18, 2021, 2:02pm UTC](https://discourse.julialang.org/t/symbolics-and-substitution/57460/11 "2021-03-18T14:02:27Z")

</div>

What happens if you do  
`] up`

---

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**Author:** ![BLI](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/bli/32/37206_2.png) [@BLI](https://discourse.julialang.org/u/BLI)\
**Post date:** [March 18, 2021, 2:03pm UTC](https://discourse.julialang.org/t/symbolics-and-substitution/57460/12 "2021-03-18T14:03:09Z")

</div>

I’m trying to do it right now… it takes some time on my 3 year old laptop…

---

<div class="post-metadata">

**Author:** ![BLI](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/bli/32/37206_2.png) [@BLI](https://discourse.julialang.org/u/BLI)\
**Post date:** [March 18, 2021, 2:48pm UTC](https://discourse.julialang.org/t/symbolics-and-substitution/57460/13 "2021-03-18T14:48:47Z")

</div>

OK – Symbolics was updated to v 0.1.9 … I didn’t even think of this: I installed Symbolics 1-2 weeks ago. I guess I should have known that the developers work quickly… Anyway, now `substitute` works.

Next: my construct `xsubs` …

```julia
xsubs = Dict([xim3d2=>xim5d2+D, xim1d2=>xim3d2+D, xip1d2=>xim1d2+D])

```

seems to be ill chosen. The `substitute` command doesn’t understand that it should do a recursive substitute. So I need to change this to:

```julia
xsubs = Dict([xim3d2=>xim5d2+D, xim1d2=>xim3d2+2D, xip1d2=>xim1d2+3D])

```

When I do this, I get the desired result:

```julia
julia> xsubs = Dict(xim3d2=>xim5d2+D, xim1d2=>xim5d2+2D, xip1d2=>xim5d2+3D);
julia> uu = simplify(substitute(u,xsubs);polynorm=true);
julia> simplify(substitute(uu,Dict(x=>xim5d2+3D));polynorm=true)
(11//6)∗𝑢𝑖+(1//3)∗𝑢𝑖𝑚2−((7//6)∗𝑢𝑖𝑚1)

```

which is consistent with

u\_{i+\frac{1}{2}}^{(1)} = \frac {1}{3}\bar{u}\_{i-2} - \frac{7}{6}\bar{u}\_{i-1} + \frac{11}{6}\bar{u}\_i

in Eq. 5, p. 106 of Y.-T. Zhang and C.-W. Shu: “ENO and WENO Schemes”, Chapter 5 in Handbook of Numerical Analysis, Vol. 5, Elsevier, 2016. [Redirecting](http://dx.doi.org/10.1016/bs.hna.2016.09.009)

It should be relatively simple to extend my code above to generate such reconstruction expressions for general stencils.

---

<div class="post-metadata">

**Author:** ![oheil](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/oheil/32/220745_2.png) [@oheil](https://discourse.julialang.org/u/oheil)\
**Post date:** [March 18, 2021, 3:05pm UTC](https://discourse.julialang.org/t/symbolics-and-substitution/57460/14 "2021-03-18T15:05:47Z")

</div>

So I did a `] up` now (after installing it a today) and got the 0.1.9 too, crazy speed 😉

---

<div class="post-metadata">

**Author:** ![j\_verzani](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/j_verzani/32/8551_2.png) [@j\_verzani](https://discourse.julialang.org/u/j_verzani)\
**Post date:** [March 18, 2021, 6:06pm UTC](https://discourse.julialang.org/t/symbolics-and-substitution/57460/15 "2021-03-18T18:06:02Z")

</div>

Well, `Polynomials` struggled a bit out of the box with this, but if that is the hammer you have, then well, you can get there through the `Lagrange` polynomial type, as follows:

```julia

using Polynomials, SpecialPolynomials

fit(Polynomial, xd, yd); # error

u = fit(Lagrange, xd, yd); # show error with `Num` coefficients...
uu = convert(Polynomial, u);
uu′ = Polynomials.derivative(uu);

xsubs = Dict(xim3d2=>xim5d2+D, xim1d2=>xim5d2+2D, xip1d2=>xim5d2+3D);
v = map(pⱼ -> simplify(substitute(pⱼ, xsubs), polynorm=true), uu′);
vv = v(xim5d2+3D)

simplify(vv, polynorm=true)

```

This results in the same answer as already shown:

```julia
(11//6)*ui + (1//3)*uim2 - ((7//6)*uim1)

```

---

<div class="post-metadata">

**Author:** ![BLI](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/bli/32/37206_2.png) [@BLI](https://discourse.julialang.org/u/BLI)\
**Post date:** [March 25, 2021, 9:49am UTC](https://discourse.julialang.org/t/symbolics-and-substitution/57460/16 "2021-03-25T09:49:49Z")

</div>

> [@j\_verzani](#):
>
> `u = fit(Lagrange, xd, yd); # show error with `Num` coefficients..`

A couple of questions:

1. `fit(Lagrange, xd, yd)` doesn’t work for my set-up. I use Julia v. 1.5.0, with Symbolics v. 0.1.11, Polynomials v. 1.2.0, and SpecialPolynomials v. 0.1.10. What could be wrong?
2. My own `lagrange_polynomial()` function works. Would it be possible to instead convert the output from that function to a polynomial? (`convert(Polynomial,u)`? If so, how do I specify the _variable_ in the polynomial (`x`) in contrast to my symbolic parameters?
3. Would it be possible to _integrate_ the resulting polynomial using Polynomials?

---

<div class="post-metadata">

**Author:** ![j\_verzani](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/j_verzani/32/8551_2.png) [@j\_verzani](https://discourse.julialang.org/u/j_verzani)\
**Post date:** [March 25, 2021, 11:10am UTC](https://discourse.julialang.org/t/symbolics-and-substitution/57460/17 "2021-03-25T11:10:06Z")

</div>

I have a slightly updated set of packages

```julia
(fit) pkg> st
Status `/private/tmp/fit/Project.toml`
  [f27b6e38] Polynomials v1.2.0
  [a25cea48] SpecialPolynomials v0.1.2
  [0c5d862f] Symbolics v0.1.11

```

And the fitting worked.

As for how to convert the expression to a polynomial, I had thought it would be easy. I tried something along the lines of this:

```julia
@variables a b c x
p = a*x^2 + b*x + c
z = variable(Polynomial)
q = substitute(p, Dict(x => z))

```

But that produced a `Num` expression, not a polynomial, as I had hoped. I would guess you would need someway to `collect` the expression (in SymPy’s use of that word) and then pick off each coefficient. This then could be passed along to `Polynomials`.

As to integrating, before this call `vv = v(xim5d2+3D)`, `v` is a polynomial which can be integrated. But `vv` becomes a symbolic expression after the polynomial evaluation.

---

<div class="post-metadata">

**Author:** ![BLI](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/bli/32/37206_2.png) [@BLI](https://discourse.julialang.org/u/BLI)\
**Post date:** [March 25, 2021, 12:10pm UTC](https://discourse.julialang.org/t/symbolics-and-substitution/57460/18 "2021-03-25T12:10:03Z")

</div>

- OK: I upgraded SpecialPolynomials to version v. 0.1.2, and now have the same version numbering as you report (except perhaps for the Julia v.). I still get an error message when I do `fit(Lagrange,xd,yd)`.

- I haven’t found a `collect` method in Symbolics. Since we are talking polynomials, I can always differentiate a number of times to find the coefficients of the polynomial, although this is somewhat clumsy.

---

<div class="post-metadata">

**Author:** ![j\_verzani](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/j_verzani/32/8551_2.png) [@j\_verzani](https://discourse.julialang.org/u/j_verzani)\
**Post date:** [March 25, 2021, 12:45pm UTC](https://discourse.julialang.org/t/symbolics-and-substitution/57460/19 "2021-03-25T12:45:57Z")

</div>

There is only 0.1.0, 0.1.1, and 0.1.2 tagged. To test the above, I activated a new environment and just installed the three packages with `add`. (I too am waiting for 1.6 to sort out my mess of a default environment 🙂 As for collect, SymPy’s `collect` finds the common powers of a term in an expression. I’m not sure how that is implemented, perhaps through pattern matching, but differentiating might be one way to go about it if you know you have a polynomial with non-negative powers only.

---

<div class="post-metadata">

**Author:** ![j\_verzani](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/j_verzani/32/8551_2.png) [@j\_verzani](https://discourse.julialang.org/u/j_verzani)\
**Post date:** [March 25, 2021, 12:48pm UTC](https://discourse.julialang.org/t/symbolics-and-substitution/57460/20 "2021-03-25T12:48:18Z")

</div>

Is the error in the fitting or the display? I have a semicolon to prevent calling `show` which needs updating to accommodate the `Num` type.

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