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How would you prove that Newton's Method applied to $f(x) = ax + b$ converges in one step? Would it be because the derivative of $f(x)$ is simply $a$?

Mark McClure
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Jesus
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1 Answers1

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Write down the formula $$N(x) = x-\frac{f(x)}{f'(x)}$$ for this specific function and simplify.

In the course of that procedure, you will certainly use the fact that $f'(x)=a$.

Mark McClure
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  • So, it would just converge to b/a after one step, correct? – Jesus Jan 23 '14 at 19:28
  • If the expression simplifies to just a constant then, yes, the convergence takes just one step. However, the constant should be the root, which is $-b/a$. – Mark McClure Jan 23 '14 at 19:37