Newton-Raphson Method

Question

just want somebody to help me verify if I'm doing using Newton-Raphson method correctly by checking the result of an equation.

$f(x) = e^{x-1} + x^2 - 7$

I'm trying to find the zero of f(x) with initial guess $x_0 = 1.5$, and by applying the method TWICE, I got result 2.0466.

Could somebody please help me verify my answer is correct?

Thanks in advance.

I’d run the process once more to see if the new value of $f(x)$ is really tiny, as it should be. — Lubin, Sep 16 '13 at 18:45
Sorry, I just substituted into $f$ to see that the value was moderately small. I did not run the process (too lazy!). — Lubin, Sep 16 '13 at 18:51

score 3 · Accepted Answer · answered Sep 16 '13 at 18:47

Your answer is almost correct. You have a rounding error. Try not to round the previous approximations. Use the "answer-key" on your calculator to store the exact answer. Only round at the very end. The correct answer is $2.0467$ to three decimal places.

The recurrence relation is given by $$x_{n+1} = x_n - \frac{\operatorname{f}(x_n)}{\operatorname{f}'(x_n)}$$ In your case $\operatorname{f}'(x) = \operatorname{e}^{x-1}+2x$ and so $$x_{n+1} = x_n - \frac{\operatorname{e}^{x_n-1}+x_n^2-7}{\operatorname{e}^{x_n-1}+2x_n}$$

If $x_0 = 1.5$ then

$$x_1 = 1.5 - \frac{\operatorname{e}^{0.5}+1.5^2-7}{\operatorname{e}^{0.5}+3} \approx 2.167$$

If $x_1 = 2.167...$ then

$$x_2 = 2.167... - \frac{\operatorname{e}^{1.167...}+(2.167...)^2-7}{\operatorname{e}^{1.167...}+3} \approx 2.0466567$$

great, that's exactly what I have on my board,thanks a lot! – Vol_Smile Sep 16 '13 at 18:51 — Vol_Smile, Sep 16 '13 at 18:51
@Vol_Smile No worries. I'm glad I could help. – Fly by Night Sep 16 '13 at 18:54 — Fly by Night, Sep 16 '13 at 18:54

Newton-Raphson Method

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