View source for Talk:Runge-Kutta methods

===============================
This article is submitted by an author who is well qualified to describe Runge--Kutta methods.
I have two major reservations (connected), and a number of minor reservations which I list
under Detailed Comment.

1. The article begins with a general description of Runge--Kutta methods, but fails to state
explicitly that the paragraph beginning  "Although it is not known, for arbitrary orders...."
before Table 2 now refers ONLY to Explicit Runge--Kutta methods.  I would suggest that this 
paragraph be preceded by the title  "Explicit Runge--Kutta Methods" so that this section is
distinguished from that which follows.
   
The serious reservation I have connected with this suggestion is that the article focuses
on "Implicit Runge--Kutta Methods", and much of the description and discussion of implementation
refers to the use of these methods, and the accepted fact that they are efficient methods for
treating stiff-problems.  I have given more detail below.  In particular, the examples of 
explicit Runge--Kutta methods cited are those primarily of historical interest.  Under 
implementation, there is no description or citation of development and implementation of 
Explicit Runge--Kutta pairs: such formulas are to be found as the basic formulas used in
the practical implementations of MATLAB and MAPLE, and perhaps other software environments.

2. The second serious reservation I have is connected to the first:  under the section 
describing Implementation costs, the author implies that Explicit Runge--Kutta methods are
perhaps inefficient when compared to linear multistep and predictor methods, the fact that
the former methods have an advantage over the latter when moderately changing values of  $f$
allow improvements in efficiency by changing stepsizes.  

It seems to me that a balanced article should explain that for non-stiff problems, 
implementation with step-size changes is the norm, and that some articles such as those
by Fehlberg, Verner, Dormand-Prince and others might be cited.  In doing so, the article
also locates articles giving coefficients of explicit methods up to the orders given in 
Table 2.  Hence, I suggest that the section on Implementation Costs be perhaps equally
divided between costs for implementing Explicit and Implicit methods, rather than focussed
on the latter.