# How bad is my maths? (numerical ODE method)

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m4r35n357
TL;DR Summary
My mathematics is not rigorous, so how does this read to a real mathematician?
I have written some ODE solvers, using a method which may not be well known to many. This is my attempt to explain my implementation of the method as simply as possible, but I would appreciate review and corrections.

At various points the text mentions Taylor Series recurrences, which I only mention briefly as I am concentrating on describing the integration method. These are documented in detail here.

Bear in mind, this is a working method, it is just my description of it that requires review!

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Homework Helper
Gold Member
I have only skimmed this but it seems to me that, rather than follow the method in the paper you linked (which requires calculation of higher order derivatives of the ODE), you are approximating the higher derivatives using differences. How is this different from a Runge-Kutta method?

m4r35n357
How is this different from a Runge-Kutta method?
Entirely different. I do not use differences anywhere. Have you looked at the recurrences paper (third link), because that is a big part of the technique?
My aim is to explain the Jorba-Zou method more simply and thoroughly, because it deserves more attention.

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Homework Helper
Gold Member
Oh so you are calculating higher order derivatives of the ODE? I don't think therefore that you can 'only mention briefly' the derivation of the recurrence relation between terms of the Taylor Series because this derivation, which is specific to each system of ODEs, is fundamental to the process.

You say that you want to concentrate on describing the 'integration method' itself, but I can't see where you do that and in any case this isn't this simply the stepwise calculation of a truncated Taylor series?

You say that your aim is to explain the methods described in the Jorba and Zou paper more simply and thoroughly, but there does not seem to be any thoroughness in your explanation. For instance I would expect your description to describe and justify the calculation of step-size and provide bounds for the calculation error and roundoff error propogation.

What do you hope to achieve by using Horner's method? The calculation of the truncated Taylor series sum involves only a handful of operations each step so is not worth optimising - and won't you need to recalculate the coefficients for Horner's method at each step anyway? In any case, you need to consider the impact of roundoff error propogation in this method.

I wouldn't call myself a real mathematician, but I do know what I expect to see in a paper describing a numerical method so I hope this gives you a few pointers.

m4r35n357
Oh so you are calculating higher order derivatives of the ODE? I don't think therefore that you can 'only mention briefly' the derivation of the recurrence relation between terms of the Taylor Series because this derivation, which is specific to each system of ODEs, is fundamental to the process.
I am seeking only a review of my bit, particularly notation. The recurrences are comprehensively documented by an expert in the field (linked in the OP), what can I contribute to that? Addition is also fundamental to the process, but I am not going to explain that!
You say that you want to concentrate on describing the 'integration method' itself, but I can't see where you do that and in any case this isn't this simply the stepwise calculation of a truncated Taylor series?
Not sure what you mean by "stepwise calculation of a truncated Taylor series". My description is the only thing in the document! I am explaining how the derivative "jets" are calculated iteratively as opposed to numerical differences or symbolic differentiation.
You say that your aim is to explain the methods described in the Jorba and Zou paper more simply and thoroughly, but there does not seem to be any thoroughness in your explanation. For instance I would expect your description to describe and justify the calculation of step-size and provide bounds for the calculation error and roundoff error propogation.
No, just the basic method.
What do you hope to achieve by using Horner's method? The calculation of the truncated Taylor series sum involves only a handful of operations each step so is not worth optimising - and won't you need to recalculate the coefficients for Horner's method at each step anyway? In any case, you need to consider the impact of roundoff error propogation in this method.
Horner's method is the integration in this case; it determines the value at the next time step.
I wouldn't call myself a real mathematician, but I do know what I expect to see in a paper describing a numerical method so I hope this gives you a few pointers.
I am not writing a paper, I just want to document my code. I posted some Lorenz code in an earlier thread. I was asked by @Greg Bernhardt if I wanted to write an insights article on it, but I declined as it was above my pay grade. I saw this as the next best thing.

m4r35n357
You say that you want to concentrate on describing the 'integration method' itself, but I can't see where you do that and in any case this isn't this simply the stepwise calculation of a truncated Taylor series?
Truncated to arbitrary order, to be precise.
What do you hope to achieve by using Horner's method? The calculation of the truncated Taylor series sum involves only a handful of operations each step so is not worth optimising
Could be in the thousands as above, it is arbitrary (although 8th to 20th order is more typical).

Homework Helper
Gold Member
Could be in the thousands as above
Then I think you should definitely investigate roundoff error propogation more. I see I am not the first to hint at this problem:
You do seem to get quite different results if you use different approaches to the polynomial (such as pre-calculating the ##h^i## values and summing in order of increasing or decreasing ##i##), but I suppose that's to be expected in a system like this one.

m4r35n357
Then I think you should definitely investigate roundoff error propogation more. I see I am not the first to hint at this problem:
I know about roundoff, and I also know about arbitrary precision arithmetic.
In terms of errors, it is a taylor series, so clearly the method is "like" RK4, but with less truncation error (for order higher than 4), and no finite difference errors.
I'm trying to get someone to actually read (not skim) my stuff and point out actual mistakes in my notation, I'm not looking for homework!
Remember this technique is nearly a century old if you trace the references.

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Mentor
2022 Award
I'm trying to get someone to actually read (not skim) my stuff and point out actual mistakes in my notation
This is the official statement: I'm afraid we are not able to discuss personal work as we are no reviewers of papers.

Inofficial, I like to add a few remarks. Your ODE solver requires a login. This is a hurdle not many will be willing to take; me, for example. Then you provide a paper from Jorba and Zou from 2004 whose copyright situation is not clear, and which is possibly not peer reviewed. At least I cannot tell from the link you gave. This means, that it is hard to tell which value this paper has, i.e. whether it contains flaws or not. But let us assume that they presented a valid algorithm for the moment. In order to understand what you might have done, one has to read this paper, which is factually almost another peer review. You cannot expect anyone to do this. However, we assumed that the paper is correct. Then you are searching for someone who is familiar with its content. Something yourself said is unlikely:
My aim is to explain the Jorba-Zou method more simply and thoroughly, because it deserves more attention.
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