Generate Taylor series

[hersle]

Is something like this suited for Symbolics.jl?

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[hersle]

One problem is that the coefficients come out as floats (I want them to be rationals) because of substitute(..., x => 0). A minimal example:

julia> @variables x
1-element Vector{Num}:
 x

julia> y = expand_derivatives(Differential(x)(sin(x)) / 2)
(1//2)*cos(x)

julia> substitute(y, x => 0)
0.5

Using fold does not help:

julia> substitute(y, x => 0, fold=false)
(1//2)*cos(0)

julia> substitute(y, x => 0, fold=true)
0.5

The issue is that e.g. cos(0) evaluates to the float 1.0, so (1//2) * 1.0 is converted to 0.5 (see #1299). Is there an easy way to fix this?

[ChrisRackauckas]

This does seem like a good thing for Symbolics.

@shashi do you know how to make substitute not convert to float?

[hersle]

One (far from perfect) solution is to rationalize() float coefficients into rational numbers. This seems to work well for the tests, except for e.g. the first irrational term in the series for acos(x) = π/2 + ..., which becomes 122925461//78256779.

[hersle]

There is also TaylorSeries.jl, which could perhaps compose with Symbolics.jl.

[hersle]

I have made quite a hefty rewrite of the perturbation example. My goal has been to significantly shorten it and better display the features of Symbolics.jl, and particularly the power of the new Taylor series functions.

I hope I have not lost the spirit of the old tutorial. Please let me know if you want me to restore some of it.

[hersle]

I would appreciate it if someone looks at this :) Suggestions are welcome! Particularly regarding how you think the public Taylor series interface should work, i.e. the current series, taylor and taylor_coeff functions.

cc @karlwessel and #1292 and SciML/ModelingToolkit.jl#3098

[karlwessel]

I love to take a more closer look at this, but not before mid of next week.

[karlwessel]

There is also TaylorSeries.jl, which could perhaps compose with Symbolics.jl.

This is what makes evaluating the usefulness of this PR complicated for me.
The package TaylorSeries and Symbolics + this PR seem to overlap pretty heavily:

• both implement some notion of a variable,
• both implement derivatives for basic julia functions,
• both allow to define Taylor series using this functionality.

What does TaylorSeries provide that wouldn't be easily doable using this PR?

• probably a lot of ease of use functions like evaluating Taylor series
• multivariate Taylor series
• integration of Taylor series

What does Symbolics + this PR provide that wouldn't be easily doable with TaylorSeries?

• expansion of user defined functions around different variables
• the whole Symbolics ecosystem

In a perfect World:

1. should TaylorSeries use Symbolics to implement all of its stuff
2. should TaylorSeries just work with Symbolics?
3. should this PR use Symbolics to implement its functionality?

In our world: Is it ok for them to just live happily side by side? If yes: is this really worth adding to Symbolics, or would it be enough to add a tutorial that shows how to expand a function as a Taylor series?

[karlwessel]

This won't work for making a taylor series for functions, like for the perturbation theory for differential equations example. Maybe adding a method like this would work:

series(coeff, x) = sum([x^(i-1)*y for (i, y) in enumerate(coeff)])

That could be used like

@variables t, (y(t))[1:3] epsilon
series(y, epsilon)

I think that function could also be used to implement all of the other use cases.

• series with $x_0 \ne 0$: series(y .+ x0, epsilon)
• series with first coeff zero: series([0; y], epsilon)

[karlwessel]

I am mixed about the changes. A lot of the explanations in the text have become much clearer, especially on describing perturbation theory and its advantages (for CAS and in general).
It also does a good job to show of the new implemented features for working with Taylor series.

However, since the previous version had to implement all of the taylor specific features by hand it was easier for me to understand the procedure and adapt it for my own problem. Now if one of the new library methods fails for my own specific problem I wouldn't really know what to do. That of course wouldn't be a problem if the new methods just work for everything.

[karlwessel]

I think the comments are not necessary here. This is self explanatory enough:

using Symbolics
@variables x
quintic = x^5 + x ~ 1

[karlwessel]

I think introducing a fully blown, automatic solver function is to much at this point. I think it would be better to keep solving it step by step and then maybe introduce and use this method to solve the second example.

Or maybe even do all of the steps again and let the reader realize the generality of the procedure and let them implement a generic method themself.

[karlwessel]

I would suggest to split this PR into multiple steps/stages.

In the first step introduce and implement easy to understand simple versions of the Taylor series methods in the perturbation example, specific to the perturbation example.

In the next step see how those methods can be used or adapted to work for the example on perturbation theory for ODEs in ModelingToolkit.

And last, if those methods have proven to work in the examples and maybe have been used to solve other perturbation problems (multivariate equations, PDEs, field equations) we can use that experience to decide how and where to add those methods to the Symbolics library or its ecosystem.