All zeros for Bessel function

I dont want to find all zeros of this function by using 'bessel function' besselj(nu, Z) in matlab toolbox. I would like to create a matlab function to calculate all roots. I create a function on newton iteration method, but it calculated just one root. thanks for your helps in advance.

[David Goodmanson]

Hi Zeynep,

J_1/2(x) equals zero if and only if the right hand side of that equation is zero. So you should consider when the r.h.s. is zero.

[Zeynep Toprak]

hello David, thank you for your reply at first . Yes I know. Firstly, I look at its graph. And I see that there are 4 roots in the interval [0, 10]. First root is near zero. Second root is near 3. third one is near 6 and finally last root is near 9. By using newton rapson method, I assign initial value x0= 2, 7 and 10 respectively. I have found three roots except for zero. But my question is that how can I find these exact roots without looking graph? And for that, I dont want to use besselj function

[Walter Roberson]

You examine the formula and see that the first term is 0 only if x is infinite, which cannot occur for that range. You then examine the second term and see that it is zero when sin(x) is 0, which happens exactly every π/2 . No need to consult a graph or use Newton's method.

[David Goodmanson]

Hi Walter, you meant to say every pi

[Walter Roberson]

You are right, should be every π

[Zeynep Toprak]

that's, as a result, is my solution way right? I have found each root one by one. is it right?

[Walter Roberson]

That is one way; there are other approaches.

[Zeynep Toprak]

well, can I learn other approaches? please explain me a bit clearly? many thanks dear Walter.

[Walter Roberson]

https://en.m.wikipedia.org/wiki/Root-finding_algorithm

There are also techniques that involve splitting up the interval into a number of subintervals and running a Newton type algorithm on the entire vector of starting points, and then at the end taking the unique values (taking into account round-off error). This approach only really works if you have information about the minium separation of the zeros.

[Zeynep Toprak]

dear Walter this is a great help! many thanks!