For slowly convergent sequences, the Aitken extrapolation...

For slowly convergent sequences, the Aitken extrapolation formula (3.52) can greatly accelerate the convergence. Use the following algorithm: $$ \begin{aligned} &x_{1}=g\left(x_{0}\right) \\ &x_{2}=g\left(x_{1}\right) \end{aligned} $$ $x_{3}=$ Aitken extrapolate of $x_{0}, x_{1}$, and $x_{2}$ $$ \begin{aligned} &x_{4}=g\left(x_{3}\right) \\ &x_{5}=g\left(x_{4}\right) \end{aligned} $$ $x_{6}=$ Aitken extrapolate of $x_{3}, x_{4}$, and $x_{5}$ Continue this process in the same manner. Apply it to the following iterations: (a) $x_{n+1}=2 e^{-x_{n}}, \quad x_{0}=0.8$ (b) $\quad x_{n+1}=\frac{0.9}{1+x_{n}^{4}}, \quad x_{0}=0.75$ (c) $x_{n+1}=6.28+\sin \left(x_{n}\right), \quad x_{0}=6$

For slowly convergent sequences, the Aitken extrapolation formula (3.52) can greatly accelerate the convergence. Use the following algorithm:
$$
\begin{aligned}
&x_{1}=g\left(x_{0}\right) \\
&x_{2}=g\left(x_{1}\right)
\end{aligned}
$$
$x_{3}=$ Aitken extrapolate of $x_{0}, x_{1}$, and $x_{2}$
$$
\begin{aligned}
&x_{4}=g\left(x_{3}\right) \\
&x_{5}=g\left(x_{4}\right)
\end{aligned}
$$
$x_{6}=$ Aitken extrapolate of $x_{3}, x_{4}$, and $x_{5}$
Continue this process in the same manner. Apply it to the following iterations:
(a) $x_{n+1}=2 e^{-x_{n}}, \quad x_{0}=0.8$
(b) $\quad x_{n+1}=\frac{0.9}{1+x_{n}^{4}}, \quad x_{0}=0.75$
(c) $x_{n+1}=6.28+\sin \left(x_{n}\right), \quad x_{0}=6$

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Key Concepts

Aitken Extrapolation

Aitken extrapolation, also known as the Delta-Squared process, is a numerical technique used to expedite the convergence of a sequence, particularly for sequences exhibiting linear convergence. The method computes a new estimate by using three successive iterates, effectively canceling out the dominant error term to reach a closer approximation of the limit more quickly.

Fixed-Point Iteration

Fixed-point iteration is an iterative process used to find a solution to an equation expressed in the form x = g(x). Starting from an initial guess, the method generates a sequence by repeatedly applying the function g, with the aim of converging to a fixed point that satisfies the equation.

Convergence Acceleration

Convergence acceleration refers to a class of techniques designed to improve the rate of convergence of an iterative process. When sequences converge slowly, methods like Aitken extrapolation are applied to reduce the number of iterations required to achieve a desired level of accuracy, making computational methods more efficient.

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