[SciPy-dev] BVP code

[Anders Lyckegaard]

Hi all!

# A bit of background!
The shooting routine is for solving ordinary differential equations with 
boundary conditions given at two different points in time/space. 

In general a system of n first order equations.
y'(x) = y(x, y)

with the boundary conditions
r(y(a), y(b)) = 0

(in case of linear boundary conditions these can be written as
A y(a) + B y(b) = c )

The shooting routine is provided with an initial guess for the vector y(a).
It then uses an initial value problem solver to find y(b). (scipy.deint)
Then a it uses an interation proceedure to fit y(a) and y(b) to the boundary 
conditions. (scipy.optimize.fsolve)

# Multi-point shooting
I have also implemented multi-point shooting. It is just a simple extension 
to the simple shooting. You provide initial guesses for intermediate points 
and fit those also. 
This routine is not very well know, but it's supposed to have better 
convergence properties and better acurracy in case of moderately stiff 
problems, see:

@Book{stoer-numericalanalysis,
  Author         = {Stoer, J. and Bulirsch, R.},
  Title          = {Introduction to {N}umerical {A}nalysis},
  Publisher      = {Springer-Verlag, New York},
  isbn           = {0-387-90420-4},
  year           = 1980,
}

# Best in class?
I'm not sure if this is a best in class algorithm. Shooting is very widely 
used, and is recommended in Numerical Recipes as a good place to start. 

Stoer and Bulirsch show that multi-point shooting compares quite well to 
finite difference solutions, but it is a quite old reference and finite 
difference approaces have evolved quite a lot in recent yeas. 

Cash and Wright have implemented some algorithms base on collocation and 
deferred corrections, that might be worth considering. 
http://www.ma.ic.ac.uk/~jcash/BVP_software/readme.html

# Code
I have included a bit of code. It does not have any error checking and have 
not been optimized in any way, but is works with the example that I have 
included. 

# Future plans
I plan to implement some continuation proceedure, based on the arch-length 
method. This might be of intrest also, but more about that later!

I hope I answered some of all your questions.

-- 
Anders Lyckegaard, Ph.D student
Institute of Mechanical Engineering, Aalborg University
Pontoppidanstraede 101, DK-9220 Aalborg Oest, Denmark
Phone: +45 9635 8080 (direct: +45 9635 9325), fax: +45 9815 1675
Email: al at ime.auc.dk
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