Linear multistep method: Difference between revisions

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Furthermore, if the method is convergent, the method is said to be ''strongly stable'' if <math>z=1</math> is the only root of modulus 1. If it is convergent and all roots of modulus 1 are not repeated, but there is more than one such root, it is said to be ''relatively stable''. Note that 1 must be a root for the method to be convergent; thus convergent methods are always one of these two.

To assess the performance of linear multistep methods on [[stiff equation]]s, consider the linear test equation ''y''' = λ''y''. A multistep method appledapplied to this differential equation with step size ''h'' yields a linear [[recurrence relation]] with characteristic polynomial

:<math> \pi(z; h\lambda) = (1 - h\lambda\beta_s) z^s + \sum_{k=0}^{s-1} (\alpha_k - h\lambda\beta_k) z^k = \rho(z) - h\lambda\sigma(z). </math>

This polynomial is called the ''stability polynomial'' of the multistep method. If all of its roots have modulus less than one then the numerical solution of the multistep method will converge to zero and the multistep method is said to be ''absolutely stable'' for that value of ''h''λ. The method is said to ''A-stable'' if it is absolutely stable for all ''h''λ with negative real part. The ''region of absolute stability'' is the set of all ''h''λ for which the multistep method is absolutely stable {{harv|Süli|Mayers|2003|pp=347 & 348}}. For more details, see the section on [[Stiff equation#Multistep methods|stiff equations and multistep methods]].