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I'm doing some computation that requires the use of Spherical Bessel Functions of the 1st kind, at high orders and values.

So, I managed to find this, while running it over a wide range of values. I hit an unexpected 0:

In[209]:= SphericalBesselJ[210, (1/1.5)*2*Pi*1.5*1000/40] // InputForm

Out[209]//InputForm=
0.

For explaining this, (1/1.5)*2*Pi*1.5*1000/40 is about 157.08.

That was strange to me, so I plotted it from 157 to 157.1. It looks smooth, no crazy behavior or zeros.

Trying it slightly on either side of this supposed 0:

In[212]:= SphericalBesselJ[210, (1/1.5)*2*Pi*1.5*1000/40 - .0001] // InputForm

Out[212]//InputForm=
9.476205413946214*^-16


In[213]:= SphericalBesselJ[210, (1/1.5)*2*Pi*1.5*1000/40 + .0001] // InputForm

Out[213]//InputForm=
9.477897706291978*^-16

In fact, I just realized that it does the same thing for all n (the first argument) greater than 200. I can't find anything on the Mathematica page for this function... am I missing something obvious, like I usually am?

thanks!

edit: Some further poking has revealed that I can also get it to work for n = 71 (for some different values of the second argument). I'm very confused now.

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SphericalBesselJ[210, (1/1.5)*2*Pi*1.5*1000/40] // InputForm

0.

Use higher precision input

SphericalBesselJ[210, (1/1.5`20)*2*Pi*1.5`20*1000/40] // InputForm

9.4770515229477837927439`0.13632911832271324*^-16

SphericalBesselJ[210, (2/3)*2*Pi*(3/2)*1000/40] // N[#, 20] & // InputForm

9.47705152294778379274395028349340334928589`20.*^-16

SphericalBesselJ[210, Rationalize[(1/1.5)*2*Pi*1.5*1000/40, 0]] // 
  N[#, 20] & // InputForm

9.477051522947880778530041024948616289348122`20.*^-16

Plot[SphericalBesselJ[210, k], {k, 156, 158}, WorkingPrecision -> 20, 
 PlotPoints -> 51]

enter image description here

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  • 2
    $\begingroup$ Maybe it's worth pointing to the documentation of SetPrecision. (+1). $\endgroup$ – Jens Apr 15 '15 at 23:15
  • $\begingroup$ Ahhh, thank you, I should have known. I've been tricked by this before, but I thought I was being careful, checking with InputForm. But I'm still a little confused, why is it going to 0? Why do we find some anomalous point like that? $\endgroup$ – YungHummmma Apr 15 '15 at 23:16
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    $\begingroup$ See reference.wolfram.com/language/tutorial/… "In most cases, built-in Wolfram Language functions will give you results that have as much precision as can be justified on the basis of your input. ... If you give higher-precision input, the Wolfram Language will use higher precision in its internal calculations, and you will usually be able to get a higher-precision result." You gave very low precision inputs and the resulting precision only justified a 0. result. $\endgroup$ – Bob Hanlon Apr 15 '15 at 23:58

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