multistep iterative methods
- To: mathgroup at smc.vnet.net
- Subject: [mg40324] multistep iterative methods
- From: Selwyn Hollis <hollisse at mail.armstrong.edu>
- Date: Tue, 1 Apr 2003 04:52:09 -0500 (EST)
- Sender: owner-wri-mathgroup at wolfram.com
I'd like to throw this out as a challenge to the group: What's the most
efficient way to implement in Mathematica an explicit multistep
iterative method such as, say, the 4-step Adams-Bashforth method for
solving y' = f(t,y):
y[k+1]:= y[k] + (h/24)*(55*f[k] - 59*f[k-1] + 37*f[k-2] - 9*f[k-3])
where y[0], y[1], y[2], y[3] are "given," and f[i] denotes f[t0 +i*h,
y[i]]. The desired output would be the list
{y[0], y[1], y[2], ... , y[n]}.
A suitable toy problem is
y' = -2t*y^2, y(0) = 1,
with h = 0.01, n = 1000 (?), and the starting values taken from the
exact solution y = 1/(1+t^2):
y[0]=1, y[1] = 0.9999, y[2] = 0.9996, y[3] = .999101.
Thanks in advance.
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Selwyn Hollis