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Jitse Niesen (talk | contribs) →Multistep Method Families: add very short section on BDF |
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The exact solution at <math> t = t_4 = 2 </math> is <math> \mathrm{e}^2 = 7.3891\ldots </math>, so the two-step Adams–Bashforth method is more accurate than Euler's method. This is always the case if the step size is small enough.
==Families of multistep methods==
Three families of linear multistep methods are commonly used: Adams–Bashforth methods, Adams–Moulton methods, and the [[backward differentiation formula]]s (BDFs).
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The Adams–Moulton methods are solely due to [[John Couch Adams]], like the Adams–Bashforth methods. The name of [[Forest Ray Moulton]] became associated with these methods because he realized that they could be used in tandem with the Adams–Bashforth methods as a [[Predictor-corrector method|predictor-corrector]] pair {{harv|Moulton|1926}}; {{harvtxt|Milne|1926}} had the same idea. Adams used [[Newton's method]] to solve the implicit equation {{harv|Hairer|Nørsett|Wanner|1993|loc=§III.1}}.
=== Backward differentiation formulas (BDF) ===
:{{main|Backward differentiation formula}}
The BDF methods are implicit methods with <math> b_{s-1} = \cdots = b_0 = 0 </math> and the other coefficients chosen such that the method attains order ''s'' (the maximum possible). These methods are especially used for the solution of [[stiff equation|stiff differential equation]]s.
== Analysis ==
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