Math 6610 • Analysis of Numerical Methods I

\textbf{a)} Find the order/rate of convergence of Newton's method to a multiple
root \(r\) of multiplicity \(m>1\) of the scalar equation \(f(x)=0\).

\textbf{b)} What method is obtained by applying Aitken's acceleration in this
situation? Show that it is at least quadratically convergent.

\textbf{c)} How would you extrapolate an integration formula whose error behaved
asymptotically as
\[
  I-I_n=c_3h^3+c_6h^6+\cdots
\]
where \(I\equiv\int_a^b f(x)\,dx\) is the exact value, \(I_n\) is the approximation
with \(n\) subintervals, and \(h\equiv\frac{b-a}{n}\).