Should AbstractSeries be <:Real instead of <:Number?

[jonniedie]

I would like to do some calculations involving complex numbers with TaylorModel1 (from TaylorModels.jl) real and imaginary components. As far as I can tell, the only thing preventing this is that Complex requires the component types to be <:Real, but TaylorModel1 <: AbstractSeries <: Number. Is there any reason AbstractSeries is <:Number instead of <:Real? Or rather, is there any time an AbstractSeries wouldn't be <:Real?

[lbenet]

Thanks for opening this issue.

The reason we have <:Number is because Taylor series are naturally defined over the complexes, where the radius of convergence or analytic functions are properly defined; if you like, this is mathematical convenience. Once said this, there are cases where to have it <:Real seems convenient.

Regarding Taylor models, your point is well taken a probably they should be <:Real. Taylor models are defined for one (real) variable as a (real) Taylor expansion around x0 in a domain D, plus a remainder, which is an interval that rigorously bounds the true (mathematical) function in D. I haven't thought about the extension to complexes, but I guess it would need to have a "complex remainder", i.e. some sort of (two dimensional) interval box. Have you tried to use TaylorModelN with two variables for your application?

[jonniedie]

Ah, that makes sense. I probably should have opened this over at TaylorModels.

As for my use case, I have a function that takes one real input (frequency) and returns two real outputs (phase and gain of the frequency response of a given linear system). The need for a Complex{<:TaylorModel1} is just to perform intermediate calculations. I could probably make my own version of Complex with only the arithmetic I need defined (should just be +, -, *, /, ^, abs, real, and imag), I was just hoping to avoid that.

[lbenet]

Thanks for the explanation!

Do the independent variable in your (complex) expansion depends on the same real parameter? Perhaps combining two TaylorModel1s avoids redefining a bunch of functions...

[jonniedie]

To be honest, I'm still trying to learn how Taylor models fit in with interval methods, so I'm not 100% sure what I need yet. To give you a little more detail on what I'm trying to do:

I have a rational polynomial function G(s) = n(s) / d(s) where n(s) and d(s) are polynomial functions of s. The coefficients of these polynomials have uncertain parameters, which are represented by Intervals. For a given (real) value ω, I would like to evaluate the response G(ω*im) and get the argument and magnitude of the resulting complex number.

When I tried evaluating this directly using Intervals, the resulting bounds were too wide. @mforets recommended I give Taylor models a try, which is when I noticed Complex{<:TaylorModel}s wouldn't work because Complex requires <:Real parameters.

[jonniedie]

So I suppose we would have multiple parameters; one for each uncertain coefficient of n(s) and d(s).

[lbenet]

Thanks for the detailed explanation @jonniedie. Let me propose to close this issue (feel free to do it), and since this discussion belongs to TaylorModels, continue it there.