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Solving a non-linear system to find coefficients in Runge-Kutta method

• To: mathgroup at smc.vnet.net
• Subject: [mg112994] Solving a non-linear system to find coefficients in Runge-Kutta method
• From: Allamarein <matteo.diplomacy at gmail.com>
• Date: Sun, 10 Oct 2010 06:41:31 -0400 (EDT)

I have just launched:

RungeKuttaOrderConditions[4, 4, ButcherRowSum -> True,
RungeKuttaMethod -> Explicit]

Well done: I get a set of  11 non-linear equations with 13 unknows,
like theory predicts.
I can choice c(2) = 1/2 and a(3,1) = 0, so I have 11 eqs. and 11unkns.
Anyway, how can I solve this system in Mathematica? I need to find the
elements of matrix a and of vector b and c.

This is only an example.
If I could solve the previous problem, I would extend that procedure
to RK6 or RK8