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People seem to say that the Adams-Bashforth method requires some "boot strapping" because it needs two initial conditions:

$y_{n+1}=y_n+\frac{\Delta t}{2}[3f(t_n,y_n) - f(t_{n-1}, y_{n-1})]$

I understand that, if only one initial condition is provided, then another initial condition must be derived. So, are we saying that it is "boot strapping" because we have to just use one initial condition to get another "initial condition"? For example, using just regular Euler's or something to get another "initial condition" using the first one?

If not, what is meant by "boot strapping" here? Thanks.

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    $\begingroup$ Boot strapping in this setting just means doing a separate procedure that sets up the conditions for your method to work when such conditions are not initially given. So for instance getting a second initial condition when you are only given one. One thing to be careful of is to not have your order of convergence drop as a result of the choice of the bootstrapping method. $\endgroup$ – Ian Oct 2 '16 at 21:11
  • $\begingroup$ Yeah, so this doesn't seem particularly special to me, it just means using a method that requires only one initial condition, to get another "initial value". But, we do that all the time with Euler's method? The next value we predict becomes the initial value for next iteration? So....unless I'm missing something this doesn't seem that special $\endgroup$ – Candic3 Oct 2 '16 at 21:32
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    $\begingroup$ The point is that you can't actually run the very first step as an Adams-Bashforth step. The very first step has to be done some other way. $\endgroup$ – Ian Oct 2 '16 at 21:56
  • $\begingroup$ yes.....thank you!!!!! $\endgroup$ – Candic3 Oct 2 '16 at 21:56
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Your differential equation initial value problem is given as $y'=f(t,y)$, $y(t_0)=y_0$.

For the multistep method the first computable value is $y_2$. However, to compute that you also need $y_1$ which can not be computed via the multistep method and thus needs some other method to provide that value.

Note that if the multi-step method has error order $p$, then the initialization method should have at least error order $p-1$ so that the propagation of these initial local errors does not contribute a worse order term to the global error.

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