Numeric mathematics bessel-functions

[peacecalc]

Hello numeric fans,

does anybody knows a numerical function, which calculates every root of the bessel function first kind. I have only an algorithm for the hp 50g in USER-RPL which is able to found a root with input an estimation of the position for the root.

It would be more usefull, when one can input let's say: 1. until 5. root and/or when it is possible to find the last root.

Greetings
peacecalc

[Namir]

Hello,

Interesting that you pose this question. I am writing an article for the online HP Solve newsletter that discusses a mutli-root finder that locates roots in a range of values. The method can locate roots that are also minima or maxima (something that most root-seeking algorithms fail to do).

Namir

[peacecalc]

Hello Namir,

When your article will be published? I can't wait reading it!

It exists an integral representation for first kind functions, the integrand function can be used to find roots of the first derivate, that's a way to find the last root of the bessel function itself.

Greetings
peacecalc

[C.Ret]

The functions Bessel function J_n(z) each have an infinite number of real zeros, all of which are simple with the possible exception of z = 0.

For nonnegative n, the kth positive zeros of first kind Bessel functions are denoted j_n,k, except that z = 0 is typically counted as the first zero of J'₀(z)(Abramowitz and Stegun 1972, p. 370).

The first few roots j_n,k of the Bessel functions are given in the following table for small nonnegative integer values of n and k.

k  J0(x)    J1(x)    J2(x)    J3(x)    J4(x)    J5(x)  

1   2.4048   3.8317   5.1356   6.3802   7.5883   8.7715

2   5.5201   7.0156   8.4172   9.7610  11.0647  12.3386

3   8.6537  10.1735  11.6198  13.0152  14.3725  15.7002

4  11.7915  13.3237  14.7960  16.2235  17.6160  18.9801

5  14.9309  16.4706  17.9598  19.4094  20.8269  22.2178

Sources :

http://mathworld.wolfram.com/BesselFunctionZeros.html

http://en.wikipedia.org/wiki/Bessel_function

[peacecalc]

Hello C.Ret,

thank you for your private lesson in mathematics. I forgot the important fact you mentioned. I threw some things to disarray. The approximation for great values of z is written with a cos function, which has obviously an infinit number of roots.

But one problem is left, how to find every roots in a given range. I'm waiting for Namir's article, the suspense is killing me.

Greetings
peacecalc

[C.Ret]

Yuo are wel come.

I am also waiting on Namir's article.
This lesson is only a copy-paste from InterNet sources about Bessel functions. But it was a good (or needed) refresch for my memories on the Bessel topic too!

I am glad to see that sharing this can also help anyone else ! :-)

As you, I was a bit disapointed that no algorithm or 'approximation function or code' was available nor advertise on the Web.

[Ángel Martin]

I´d suggest you check Jean-Marc Baillard´s authoritative pages. He´s got a few programs worth studying, like the Mac Mahon expansion one, see it here:

http://hp41programs.yolasite.com/bessel.php

It is also included in the BESSEL ROM, availabe at TOS - and by extension on the CL of course; where speed is definitely not an issue.

[C.Ret]

Thanks a lot.

This Mac Mahon expansion expression si exactly the 'code' I was looking for !

[peacecalc]

Hello Martin,

it looks like a taylor expansion of the asymptotic approximation, is that right?

Greetings
peacecalc

[Namir]

Balliard's use of the MacMahon approximation seems to work for roots that are high in their sequence number.

Edited: 1 July 2012, 4:44 p.m.

[Ángel Martin]

Quote:

---

seems to work for roots that are high in their sequence number.

---

Can you give an example where it doesn´t work?

[Jean-Michel]

Hi,

MacMahon's formulae are very good for large roots.

I've used the first 5 terms of these asymptotic expansions,

but better formulas are given in the "Digital Library

of Mathematical Functions" up to the terms of order 7

cf http://dlmf.nist.gov/10.21

For the first roots, one can use a root-finding program

and ascending series or continued fractions.

Unfortunately, I don't know direct formulas in these cases.

Hope this helps,

JM.

[peacecalc]

Hello JM at al.,

thank you for interesting links and further information.

If I understand right, the formulae from Mac Mahon works even for real k parameter values. For integer values is an easy approach possible (and x --> oo).

again thank you and

Greetings
peacecalc