BesselZeros and Working Precision?

• To: mathgroup at smc.vnet.net
• Subject: [mg33841] BesselZeros and Working Precision?
• From: aes <siegman at stanford.edu>
• Date: Fri, 19 Apr 2002 02:29:00 -0400 (EDT)
• Organization: Stanford University
• Sender: owner-wri-mathgroup at wolfram.com

```I'm puzzled by why the four alternative forms

BesselJZeros[0,  1][[1]]

N[ BesselJZeros[0,  1][[1]], 12]

BesselJZeros[0,  1,  WorkingPrecision ->12][[1]]

N[ BesselJZeros[0, 1, WorkingPrecision ->12][[1]], 12]

all give the same 6-digit answer, namely 2.40483, whereas

BesselJZeros[0, 1, WorkingPrecision -> 30][[1]]

2.4048255576957727686216318793264541313535027

I'm also *really* puzzled by why

BesselJZeros[0, 1, WorkingPrecision -> 6][[1]]

This last one really seems like a bug; it's certainly a "gotcha".  I'm not
arguing that there's any good reason for coding something with a reduced
MachinePrecision like this -- but nonetheless I don't think it should give that

Actually, here's a table of  BesselJZeros[0, 1, WorkingPrecision -> k][[1]] for
k from 1 to 30.  Note erroneous answers at k = 7 and 8 also.

{"1", "0"},
{"2", "0"},
{"3", "0"},
{"4", "0"},
{"5", "0"},
{"6", "0"},
{"7", "2.398095666212098`"},
{"8", "2.404944945798332`"},
{"9", "2.404825723910026`"},
{"10", "2.404825723910026`"},
{"11", "2.404825723910026`"},
{"12", "2.404825723910026`"},
{"13", "2.404825723910026`"},
{"14", "2.4048255576957467`"},
{"15", "2.4048255576957467`"},
{"16", "2.4048255576957467`"},
{"17", "2.4048255576957467`"},
{"18", "2.4048255576957467`"},
{"19", "2.4048255576957467`"},
{"20", "2.404825557695772768622813251300758`20"},
{"21", "2.404825557695772768622529251832104`21"},
{"22", "2.404825557695772768622529251842381`22"},
{"23", "2.404825557695772768622530855769674`23"},
{"24", "2.404825557695772768622530855769332`24"},
{"25", "2.40482555769577276862253084774527500835`25"},
{"26", "2.40482555769577276862253084774527504299`26"},
{"27", "2.40482555769577276862253084778108108087`27"},
{"28", "2.40482555769577276862163187932659770515`28"},
{"29", "2.4048255576957727686216318793265975336787736`29"},
{"30", "2.4048255576957727686216318793264541313535027`30"}}

```

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