Talk:Linear multistep method
Overall this is an excellent article so I only have very minor quibbles.
The animated demos are very good, but at first I found the first one (Explicit Adams) puzzling until I read to the end and saw that it said “… continue from item 3.” Since I assume that the articles are intended for an audience that is not particularly knowledgeable, I am concerned that at first this gives the impression that one has to do the slope evaluation on every step. I think this would be clearer if there was a clear indication that the “repeat” started at the third step.
Just before the discussion of A-stability it refers to “Gear’s site on BDF methods.” This should be changed to something like “the entry on BDF” since in the future someone else could become the curator.
There is a need for a reference to “symplectic” – preferably one in Scholarpedia, but if not, to some other source for the reader who doesn’t know what it is.
Inclusion of a short introduction of *order arrows* with a reference to the book "Numerical methods for ordinary differential equations" by John C. Butcher and the article "General linear methods", Acta Numer. 15 (2006), 157--256, by John C. Butcher, would be a valuable complement to the article. A picture with order arrows for BDF3 would introduce another way for observing that BDF3 is not A-stable.
Explicit Adams: i) The x-axis in Fig 1 should be labeled with t-values (not x-values) to match the analysis.
ii) An overall weakness is the assumption that interpolation is well--understood. So many of the examples and so much of the theory depend upon interpolation that it may be helpful to explain this briefly at the start. Failing this, a link is required to a good introduction to this topic. Here, in the 3rd bullet point for Fig 1, perhaps `interpolate (t_n,f_n) would be better. (The definition of f_n seems to be below this in my viewer. If this is generally the case, perhaps the definition should precede Fig 1.)
iii) An overall weakness is a lack of links or references to the literature, though I admit that this is somewhat balanced because this is an almost self-contained piece of self-evident mathematics. Here, perhaps some of the historical references from the authors' 1987 book would be appropriate, to indicate who Adams was and where the methods originated.
iv) `Newton's formula for polynomial interpolation...' In fact, these formulae can be derived as in bullet point 3 of Fig 1 by integrating any representation of the interpolant. Perhaps a link needs to be made to Newton's formula if this is important --- probably a book reference is less ideal in an electronic article.
v) In my viewer at least, the equation numbers are not centred correctly for multiple equations. I am not sure if this can be corrected.
i) I think the last bullet point of Fig 2 should be punctuated with a full stop and not a semi-colon.
General Formulation and Consistency
i) Equation (6) does not use the f_n notation introduced earlier. Is this deliberate?
Stability and Convergence
i) Line 1: `developped' has only one p. `exposed' is not commonly used in English --- perhaps `presented' would be more idiomatic.
ii) Line 3. The authors choose to use `stable' instead of the more conventional `zero--stable'.
iii) Last line of first para: `However these methods are of no practical use...' There are low order exceptions to this general statement, when 2k=k+2 or 2k=k+1.
Convergence Theorem: i) Line 1: There is a spurious space between r and the comma.
ii) Line 3: Should read: Compact intervals [0, nh], where nh=<T.
Stiff stability analysis
i) Fig 3: Caption for upper graph should read: Explicit Adams _for_ stiff problem. Again the x-axis values should be times, I think.
ii) Line preceding (10): Should read: _the_ unit circle...
iii) Perhaps it would be useful to give more details of the derivation of (8) for a diagonalisable matrix system. Cetainly, as things stand, it is not exactly clear what f(y), immediately after (8), refers to. (Have we reduced to exclusively autonomous systems here?)
iv) The root-locus treatment is nice, but perhaps reference should be made to Jeltsch & Nevanlinna to justify the method rigorously.
i) Line 1: Should read: This class of methods _was_ discovered... Line 2: ... for stiff problems _has_been_ recognised...
ii) For eqn (12), reference to the NIP is more essential than earlier. Although the nabla symbol is called the backward difference operator, it is not clear that a non--specialist would know what this means.
iii) Before (13): `developed' should perhaps be changed to `expanded'.
iv) Close to Fig 6, it is claimed that BDF k work _perfectly_ for stiff problems. This absolute statement is an over-simplification.
v) Later in the paragraph, the phrase `Newton-like procedures' is used. I think this would be opaque to a non-specialist.
Second Dahlquist Barrier
i) Second paragraph: This reflects the original motivation for designing these methods, rather than the fuller understanding of such methods that has now been arrived at using the theory of general linear methods. I believe that GLMs should be mentioned as a class here.
ii) The word `order' is spelled `oder' in the penultimate line.
i) This is a fairly brief account, considering its importance. Perhaps an insightful example would be appropriate.
ii) The reference to a Lipschitz constant here would be clearer, if in the earlier subsection on Adams methods applied to stiff systems, the eigenvalue lambda had been associated with such a constant.
iii) First bullet point: Change `constant' for `error'
iv) Second bullet point: Lipschitz _condition_
Methods for Hamiltonian Systems
i) The terms Hamiltonian and symplectic should be referenced, ideally with electronic links.
ii) In (16) we are told that the alpha's and beta's are different. Is it also true that k is different. Perhaps further emphasis that rho and sigma are different is also required.
iii) I wonder whether Eirola & Sanz-Serna 1991 is also a key reference for symmetric LMMs.
iv) Last sentence before Long-time behavior: The term `numerical resonances' is unclear.
i) Second bullet point: Is dot(q)'Aq intended?
Implementation and software
i) Line 3: adjust ---> choose
ii) Bullet point 1: Shampine & Watson needs a date and to be in the list of references at the end.
iii) Bullet points 2 and 3: Perhaps the use of first names is informal and inconsistent with the other parts of the article.