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I need to solve $f(x) = e^{-\sin (4x)} - 3/4$ with 3 digits after the decimal point correction, but cannot find out how.

I'd really appreciate it if anyone could point me to the solution. I think I may need to use iterative methods , but it would be very helpful to get some guiding about choosing and using the iterative method.

Thanks in advance.

Lee David Chung Lin
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itamar
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  • @Amzoti, thanks for the answer, but could you be please more detailed about the process itself ? – itamar Jun 28 '14 at 14:22

2 Answers2

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Using logarithms and inverse trigonometric functions, the very first positive solution of the equation is $$x=\frac{1}{4} \sin ^{-1}\left(\log \left(\frac{4}{3}\right)\right)$$ (remember that you have an infinite number of solutions to this equation). You know that $\log\left(\frac{4}{3}\right)$ is a rather small number (since close to $1$). Let us call it $a$, then $$x=\frac{1}{4} \sin ^{-1}(a)$$ Now, use Taylor expansion of $\sin ^{-1}(a)$ built at $a=0$; it is $$\sin ^{-1}(a)=a+\frac{a^3}{6}+O\left(a^4\right)$$ which I suppose will be sufficient.

Now, let us approximate $a$ and, for that, I suggest you use $$\log\frac{1+y}{1-y}=2 \Big(y + \frac{y^3}{3}+O\left(y^4\right)\Big)$$ Since we want to compute $\log\left(\frac{4}{3}\right)$, we need to use $y=\frac{1}{7}$ so $a \simeq \frac{296}{1029}$. So now, $\sin^{-1}(a) \simeq \frac{953217976}{3268642167}$ and so $x=\frac{238304494}{3268642167} \simeq 0.07290626561$ while the exact solution is $x=0.07295142852$. This is not too bad.

If instead of using two terms in each expansion, you only retain the first, you then have $a \simeq \frac{2}{7}$, $\sin^{-1}(a) \simeq \frac{2}{7}$ and then $x\simeq \frac{1}{14}=0.07142857143$ which is not very accurate.

There is a more direct way which require composition of series. If you look at the expansion of $$\frac{1}{4} \sin ^{-1}(\log (1+b))=\frac{b}{4}-\frac{b^2}{8}+\frac{b^3}{8}-\frac{b^4}{8}+O\left(b^5\right)$$ Now, replace $b$ by $\frac{1}{3}$. But, surprizingly, this will require more calculations.

I used as long as possible the rational numbers but you do not need to (and please, notice that all of that can be done with a pocket calculator or with your phone).

Claude Leibovici
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  • Well donne! Of course, there are other solutions$$x=\frac{1}{4} \sin ^{-1}\left(\log \left(\frac{4}{3}\right)+2k\pi\right)$$ and $$x=\frac{1}{4} \sin ^{-1}\left(\pi-\log \left(\frac{4}{3}\right)+2k\pi\right)$$ – JJacquelin Jun 28 '14 at 15:03
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I make another answer to address the problem of the use of iterative methods as you initially asked in your post.

For solving nonlinear equations, a simple and efficient root-finding method is Newton which, starting from a "reasonable" (this word would require rather extended discussions) estimate $x_0$ will update it according to $$x_{n+1}=x_n-\frac{f(x_n)}{f'(x_n)}$$ Suppose that we start iterations at $x_0=0$ (we are lazy); then the following iterates generated by the above process will be :$0.062500$,$0.0726861$,$0.0729513$,$0.0729514$ which the solution for six significant figures.

Claude Leibovici
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