Solving for (or approximating) $x$ in $\frac{x-\sin x}{x^{0.4}} = c$

Question

I'm an engineer and I'm deducting a formula, and I found a BIG problem: I can't isolate $x$.

The formula is: $$\frac{x-\sin x}{x^{0.4}} = c$$ where $c$ is a constant.

How to find the value for $x$, or at least a good approximation?

What is the range of $c$ in your application? It makes a difference if you look for solutions for small $c$, or for rather large $c$. — Lutz Lehmann, Nov 09 '20 at 19:52
I'm sorry, C can vary a lot but in most of my tests i'd say that C > 2 — Rafael Araujo, Nov 09 '20 at 20:19
for $c>5.278$ there are more than one value for $x$, I mean that the function in the RHS is not invertible. — Raffaele, Nov 09 '20 at 20:27

score 2 · Answer 1 · answered Nov 09 '20 at 19:56

2

The leading term of $x-\sin x$ is $\frac{x^3}6$. This would result in a first approximation of $x=(6c)^{1/2.6}$ for smallish values of $c$.

For larger values of $c$ one would use that $|\sin x|\le 1$ to get approximately $x=c^{1/0.6}$.

answered Nov 09 '20 at 19:56

Lutz Lehmann

126,666

score 1 · Answer 2 · answered Nov 09 '20 at 20:01

1

I would use Newton-Raphson, with $c^{\frac{5}{3}}$ as initial guess. If $c$ is close to $0$ (e.g. in $(-1, 1)$), you could use $\sqrt[2.6]{6c}$ instead.

The function oscillates around $x^{\frac{3}{5}}$ and is nice and smooth. This should work fine.

I tested it for $c= 10$ and within 3 iterations the error was tiny. Depending on the required accuracy, you can easily add more iterations.

answered Nov 09 '20 at 20:01

Robby the Belgian

1,322

Hello robby, thanks for the answer i'm really happy you could find an good approximation. Could you show me the precise method and final formula to find the value of x? – Rafael Araujo Nov 09 '20 at 20:17
Sure, have a look here: https://www.desmos.com/calculator/ftfaw0w5aq . The function $a$ takes an initial guess ($x$) and a target value ($c$) and returns a better approximation. – Robby the Belgian Nov 09 '20 at 20:26
Thanks very much!! One more question: Why you had c^5/3 as an inital guess? – Rafael Araujo Nov 09 '20 at 20:32
See Lutz Lehmann's answer. Since, for values "far" from 0, $\frac{x - \sin x}{x^{0.4}} \approxeq \frac{x}{x^{0.4}}$, that approximation works well enough for an initial guess. – Robby the Belgian Nov 09 '20 at 20:47
Thanks robby, i've tested many iterations and it's a sucess!! But the error suddenly stops decreasing after some iterations, is that normal? – Rafael Araujo Nov 09 '20 at 21:57
That is mostly due to numerical issues. A computer or calculator has a limit on its precision. – Robby the Belgian Nov 09 '20 at 22:03

score 1 · Answer 3 · answered Nov 09 '20 at 20:07

There is no analytical function that will give you x = f(c). I can recommend Wolfram Alpha to get a quick insight into the problem: (https://www.wolframalpha.com/input/?i=%28x+-+sinx%29+%2Fx%5E0.4+%3D+10+solve+for+x).

You get immediately a plot of your function. Hope that helps a bit

score 1 · Answer 4 · answered Nov 09 '20 at 20:41

Here is the graph of the inverse of $$f(x)=\frac{x-\sin x}{x^{0.4}}$$ as you can see it is not a function. For some value of $c$ there are more than one $x$, even three in some intervals.

For $c=3$ we have $x_1\approx 4.5;\;x_2\approx 6.35;\;x_3\approx 7.83$.

For $c=3.3$ we have $x\approx 8.55$.

I used GeoGebra, plotted $f(x)$ and to get the inverse I took its symmetric with respect to the bisector of the first quadrant. This gives fine approximations for any $c$.

$$ $$

Solving for (or approximating) $x$ in $\frac{x-\sin x}{x^{0.4}} = c$

4 Answers4