迭代方法收敛加速的含义是什么？有无正式定义与判定标准？

Great question—this is such a common sticking point when you’re first wading into numerical analysis papers! Let’s break down the formal definition and how to spot a valid convergence acceleration method:

At its core, convergence acceleration is a formal concept in numerical analysis: given a convergent sequence ${x_n}$ that approaches a limit $L$, we construct a new sequence ${y_n}$ such that ${y_n}$ converges to the same limit $L$, but does so with a faster rate of convergence than the original sequence.

The key criteria to confirm a method counts as convergence acceleration boil down to two main points:

• Limit consistency: The accelerated sequence must converge to the exact same limit as the original sequence—you can’t "accelerate" by changing what the sequence is converging to.

• Faster convergence rate: This is where formal metrics come in, usually measured via the order of convergence:

  • For the original sequence ${x_n}$, if there exists $C>0$ and $p\geq1$ such that $\lim_{n\to\infty} \frac{|x_{n+1}-L|}{|x_n-L|^p} = C$, then $p$ is the order of convergence.
  • A valid acceleration method will produce a sequence ${y_n}$ with an order of convergence $p' > p$, or for linearly convergent sequences ($p=1$, $C<1$), it will boost convergence to superlinear ($p>1$) or even quadratic ($p=2$) levels.

  For example, if your original sequence has linear convergence (error decays like $C^n$ where $0<C<1$), an accelerated method might make the error decay like $C^{n2}$—a massive speedup in how quickly you get close to the limit.

经典例子：Aitken加速 #

A perfect example is Aitken's $\Delta^2$ method, designed for linearly convergent sequences. The accelerated sequence is defined as:
$\hat{x}_n = x_n - \frac{(x_{n+1}-x_n)^2}{x_{n+2}-2x_{n+1}+x_n}$
This transforms a linearly convergent sequence into one that converges superlinearly—clear evidence of convergence acceleration.

Sometimes, you’ll also see acceleration measured via the asymptotic error constant: if the original sequence has error constant $C$, the accelerated sequence has a smaller $C' < C$, meaning error shrinks faster even if the order stays the same. But the biggest win is always increasing the convergence order.

内容的提问来源于stack exchange，提问作者Zduff