GATE 2021 CH – Question 44
To solve an algebraic equation $f(x)=0$, an iterative scheme of the type $x_{n+1}=g(x_n)$ is proposed, where $g(x)=x-\frac{f(x)}{f\prime(x)}$. At the solution $x=s$, $g\prime(s)=0$ and $g^{\prime\prime}(s)\ne0$. The order of convergence for this iterative scheme near the solution is _____.
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Correct answer: 2
Explanation
Write the error as $e_n=x_n-s$ and expand g about its fixed point $g(s)=s$:
$$e_{n+1}=g(s+e_n)-g(s)=g\prime(s)e_n+\frac{g^{\prime\prime}(s)}{2}e_n^2+O(e_n^3).$$
The given first derivative is zero and the second derivative is nonzero. Therefore the leading error is proportional to $e_n^2$, so convergence is **quadratic: order 2**. This is the usual local behavior of Newton’s method at a simple root.