Proposal for ODE package: multistep methods

[aestriplex]

Proposal: multistep methods

Dear maintainers,
I am a Matlab user, and I am familiar with the ode solvers in Matlab.
In order to make gonum's ode package more complete (and more useful for people who need to solve a wide variety of ODEs), I would like to implement some multistep methods. To start, I aim to implement these three families of methods:

• Adams-Bashfort,
• Adams-Moulton,
• Backward Difference Formula methods.

Of course, I'm going to keep the same structure of DormandPrince5 (i.e., implementing the Integrator interface).

Methods to implement

These are the coefficients for the methods that I wish to implement:

1. Adams-Bashfort (explicit) methods
   

2. Adams-Moulton (implicit) methods
   

3. Backward Difference Formula methods
   

Thanks for your time,
Matteo Nicoli

[sbinet]

I am quite ignorant wrt ODEs.
but as I see nobody responded yet, let me say your proposal sounds okay - to my ignorant ears.

to get the ball rolling, I'd say: try to implement the easiest of the 3 methods, together with an example (and tests).
and see what gives.

unless @kortschak @btracey @romanwerpachowski @vladimir-ch or @soypat have more insightful comments?

[soypat]

Sounds good!
I'll throw a couple related discussions your way in case you want some history on the ode package.

• mailing list discussion about API design, explains why it looks this way today.
• issue regarding jacobian calculation. diff/fd uses slices while mat types seem more appropiate for ode.
• proposal: Initial Value Problem integration facilties for ordinary differential equations. gonum#1761 first issue on the matter

[aestriplex]

@sbinet @soypat Ok,
I'll start working on AB4 then