HP Forum Archive 21

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Numeric mathematics bessel-functions Message #1 Posted by peacecalc on 30 June 2012, 2:43 p.m. 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

Re: Numeric mathematics bessel-functions Message #2 Posted by Namir on 30 June 2012, 3:02 p.m., in response to message #1 by peacecalc 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

Re: Numeric mathematics bessel-functions Message #3 Posted by peacecalc on 30 June 2012, 5:32 p.m., in response to message #2 by Namir 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

Re: Numeric mathematics bessel-functions Message #4 Posted by C.Ret on 1 July 2012, 1:21 a.m., in response to message #1 by peacecalc The functions Bessel function Jn(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 jn,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 jn,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

Re: Numeric mathematics bessel-functions Message #5 Posted by peacecalc on 1 July 2012, 2:44 a.m., in response to message #4 by C.Ret 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

Re: Numeric mathematics bessel-functions Message #6 Posted by C.Ret on 1 July 2012, 2:57 a.m., in response to message #5 by peacecalc 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.

Re: Numeric mathematics bessel-functions Message #7 Posted by 聲gel Martin on 1 July 2012, 3:30 a.m., in response to message #6 by C.Ret I悲 suggest you check Jean-Marc Baillard愀 authoritative pages. He愀 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.

Re: Numeric mathematics bessel-functions Message #8 Posted by C.Ret on 1 July 2012, 5:42 a.m., in response to message #7 by 聲gel Martin Thanks a lot. This Mac Mahon expansion expression si exactly the 'code' I was looking for !

Re: Numeric mathematics bessel-functions Message #9 Posted by peacecalc on 1 July 2012, 10:06 a.m., in response to message #7 by 聲gel Martin Hello Martin, it looks like a taylor expansion of the asymptotic approximation, is that right? Greetings peacecalc

Re: Numeric mathematics bessel-functions Message #10 Posted by Namir on 1 July 2012, 4:43 p.m., in response to message #7 by 聲gel Martin 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.

Re: Numeric mathematics bessel-functions Message #11 Posted by 聲gel Martin on 2 July 2012, 2:28 a.m., in response to message #10 by Namir Quote: seems to work for roots that are high in their sequence number. Can you give an example where it doesn愒 work?

Re: Numeric mathematics bessel-functions Message #12 Posted by JMBaillard on 2 July 2012, 7:43 a.m., in response to message #11 by 聲gel Martin 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.

Re: Numeric mathematics bessel-functions Message #13 Posted by peacecalc on 5 July 2012, 1:37 p.m., in response to message #12 by JMBaillard 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

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