1

I have an equation:

$$\operatorname{sinc}(x) = \frac{\sin(x)}{x} = 0.5$$

How do I find $x$ from this?

I realise there's probably not a simple equation to describe the inverse, but is there a numerical method I can use to solve this?

Michael Hardy
  • 1
Edd
  • 147
  • 3
    Fixed point iteration $x = 2 \sin x$ is one approach and it yields http://www.wolframalpha.com/input/?i=x+%3D+2+sin+x – Amzoti Jan 10 '14 at 17:48
  • https://en.wikipedia.org/wiki/Root-finding_algorithm Try Newton's method. –  Jan 10 '14 at 17:53
  • @clairharrison: Newton's Method and others also work and see the same result as the Fixed Point above: See: WA. It converges in 4-steps for a start value of 2. – Amzoti Jan 10 '14 at 18:04
  • you can draw functions sin(x) and 0.5x and you can estimate its root – Bobby Jan 10 '14 at 18:13

1 Answers1

4

We cannot find a closed form for this, so you are right, numerical methods are needed.

We can use any root finding approach.

For Fixed Point Iteration, we have:

$$x = 2 \sin x$$

This leads to this root found using WA

We can also use Newton's Method with (converges in 4-steps for a start value of 2. ):

$$f(x) = 2 \sin x - x$$

Here are the same results using WA.

The root is:

$$x = \pm ~1.895494267033981\ldots$$

Amzoti
  • 56,093