Variable Step Multistep Methods

1. What?

   A Predictor-Corrector Method that uses variable step sizes for error control.

2. Why?

   Predictor-corrector techniques always generate two approximations at each step, so they are natural candidates for error-control adaptation.

3. Derivation:
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4. Choosing ‘\(q\)’:
   
   • \(q\) is, generally, chosen conservatively:
     
5. Properties:
   
   1. A change in step size for a multistep method is more costly in terms of function evaluations than for a one-step method, because new equally-spaced starting values must be computed.
   2. Consequently, we ignore the step-size change whenever the local truncation error is between \(\dfrac{\epsilon}{10}\) and \(\epsilon\), that is, when
   3. \(q\) is given an upper bound to ensure that a single unusually accurate approximation does not result in too large a step size.
6. Algorithm:
   Show Derivation