Prove that if $x_n \to q$ as $n \to \infty$ and if $f'(q) \neq 0$ then $q$ is a zero of $f$ for the secant method.

Question

The following is a textbook exercise that i am struggling with

In the secant method prove that if $x_n \to q$ as $n \to \infty$ and if $f'(q) \neq 0$ then $q$ is a zero of $f$

I think i have a fair idea about why intuitively it is true however I have troubles with the more rigorous proof, this is one of the first classes with proofs I am taking, so im am unsure as to what technique I should approach a problem like this with, and where to go from there?

@MaximilianJanisch no information given regarding that. – Sep 30 '19 at 21:10 — , Sep 30 '19 at 21:10
Out of curiosity, what is the name of the book? – Maximilian Janisch Sep 30 '19 at 21:11 — Maximilian Janisch, Sep 30 '19 at 21:11

score 1 · Answer 1 · answered Sep 30 '19 at 21:29

1

Under the stated assumptions, the secant root formula $$ x_{n+1}=x_n-f(x_n)\frac{x_n-x_{n-1}}{f(x_n)-f(x_{n-1})} $$ converges term wise to $$ q=q-f(q)\frac1{f'(q)} $$ so that indeed $f(q)=0$ for the limit point.

answered Sep 30 '19 at 21:29

Lutz Lehmann

131,652

$f'$ should be continuous – Maximilian Janisch Sep 30 '19 at 21:31
Yes. It would makesno sense to apply a method using linear approximation to a function that can not be linearly approximated. – Lutz Lehmann Sep 30 '19 at 21:35
True! $ \ \ \ $ – Maximilian Janisch Sep 30 '19 at 21:40

Prove that if $x_n \to q$ as $n \to \infty$ and if $f'(q) \neq 0$ then $q$ is a zero of $f$ for the secant method.

1 Answers1