Timeline for Formula for n-th iteration of dx/dt=B(x)
Current License: CC BY-SA 4.0
11 events
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| Jul 5, 2021 at 18:01 | comment | added | Tom Copeland | Can any of the upvoters please specify where the OP's polynomials are found in the refs mentioned? | |
| Jul 5, 2021 at 16:31 | history | edited | Daniele Tampieri | CC BY-SA 4.0 |
Changed link to Wikipedia entry on the "tree" concept in graph theory, as the former link seems not working
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| Oct 5, 2010 at 23:47 | comment | added | Gerry Myerson | Oops, I completely missed the di Bruno reference in the original question. No wonder no one has been impressed by my comments here! | |
| Oct 5, 2010 at 12:44 | comment | added | resolvent | As Hans Lundmark correctly points out, it is more than just Faa di Bruno or just Lagrange Inversion Formula (I use the LIF as presented by G.P. Egorychev). In my original post, I mention that I use Faa di Bruno to express derivatives of A(x) in terms of those of B(x) where A(x)=1/B(x) and substitute into the LIF for the inversion of t=\integral_A(x)dx. I have also considered the generalizations of LIF which express f(x,t) for an arbitrary function, f, in terms of x when inverting a series G(x,t)=0, but all paths lead to massively complex calculations. | |
| Oct 5, 2010 at 12:06 | comment | added | Gerry Myerson | @Hans, I knew it had to have something to do with di Bruno - I suggested as much in an unnoticed comment on the question - thanks for clarifying the relation. | |
| Oct 5, 2010 at 8:05 | comment | added | Hans Lundmark | @Denis Serre: To be precise, it's not only Faà di Bruno, but rather what you get if you compute $x''(t)$, $x'''(t)$, etc., using Faà di Bruno, and recursively substitute the previously computed derivatives into the right-hand side at each stage, in order to get the answer in terms of $B$ and its derivatives alone (no $x$'s). | |
| Oct 5, 2010 at 7:57 | comment | added | Hans Lundmark | This theory is described in Chapter 3 of the book "Geometric Numerical Integration" by Hairer, Lubich & Wanner. books.google.com/books?id=T1TaNRLmZv8C&printsec=frontcover | |
| Oct 5, 2010 at 5:26 | comment | added | Denis Serre | The formula you look for bears the name of Faà di Bruno. | |
| Oct 5, 2010 at 4:31 | comment | added | resolvent | Thank you! Your links led me to en.wikipedia.org/wiki/Butcher_group While it is not a final answer (they give a recursive formula by Cayley of the 1860s using Butcher's notation of the 1960s) perhaps it can lead me to it or will set me on the right path to figure it out for myself. | |
| Oct 5, 2010 at 4:27 | vote | accept | resolvent | ||
| Oct 4, 2010 at 23:31 | history | answered | jvkersch | CC BY-SA 2.5 |