Convergence criterion for Newton-Raphson method

[shyjuu]

|f(x0)f''(x0)|<|f'(x0)|^2 where I is the interval containing the approximate root x0, is the convergence criterion of ...

(a) Newton - Raphson method
(b) Iteration method
(c) Secant method
(d) False position method

According to me its (a), but I confused because this formula is not directly given anywhere I had searched a lot on the internet, can anybody confirm what is the correct answer

Thanks a lot in advance

 

[chisigma]

|f(x0)f''(x0)|<|f'(x0)|^2 where I is the interval containing the approximate root x0, is the convergence criterion of ...

(a) Newton - Raphson method
(b) Iteration method
(c) Secant method
(d) False position method

According to me its (a), but I confused because this formula is not directly given anywhere I had searched a lot on the internet, can anybody confirm what is the correct answer

Thanks a lot in advance

Wellcome on MHB $SS^{2}$!...

... remembering the Newton iterations...

$$x_{n+1}= x_{n} - \frac{f(x_{n})}{f^{\ '} (x_{n})}\ (1) $$

... if is... $$\frac{|f(x_{0})\ f^{\ '}(x_{0})|}{|f^{\ '\ 2}(x_{0})|}= \frac{|f(x_{0})|}{|f^{\ '}(x_{0}|}= a < 1\ (2)$$

... and that holds for any $x_{0}$. then the sequence of $x_{n}$ obeys to the difference equation... $$ x_{n+1} - x_{n}= (\pm a)^{n}\ (3)$$... and it converges... Kind regards $\chi$ $\sigma$

 

[shyjuu]

Wellcome on MHB $SS^{2}$!...

... remembering the Newton iterations...

$$x_{n+1}= x_{n} - \frac{f(x_{n})}{f^{\ '} (x_{n})}\ (1) $$

... if is... $$\frac{|f(x_{0})\ f^{\ '}(x_{0})|}{|f^{\ '\ 2}(x_{0})|}= \frac{|f(x_{0})|}{|f^{\ '}(x_{0}|}= a < 1\ (2)$$

... and that holds for any $x_{0}$. then the sequence of $x_{n}$ obeys to the difference equation... $$ x_{n+1}= \pm a\ x_{n}\ (3)$$... and it converges... Kind regards $\chi$ $\sigma$

so I was right answer is Newton Rapshon option a, Thanks a lot for explanation