Math 6610 • Analysis of Numerical Methods I

{Absolute Stability} Consider solving the initial value problem
\[
  y'=f(x,y),\qquad y(a)=y_0 \tag{4}
\]
by the linear multistep method (LMM)
\[
  \sum_{j=0}^k \alpha_j y_{n+j}
  =
  h\sum_{j=0}^k \beta_j f_{n+j}. \tag{5}
\]

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where we use the standard notation
\[
  x_n=a+nh,\qquad y_n\approx y(x_n),\qquad f_n=f(x_n,y_n). \tag{6}
\]
Define what it means that the LMM is \textit{explicit}. Define its
\textit{region of absolute stability}. Show that an explicit linear multistep
method cannot have an infinite region of absolute stability.