Function note

BESSELI and BESSELK are low-order approximations, rarely correctly rounded

FUNC.BESSELI · FUNC.BESSELK

FN-BESSELI-002: BESSELI and BESSELK are low-order approximations, rarely correctly rounded

Excel differs from the mathematical answer · measured · recorded 2026-11-04

BESSELI and BESSELK are low-order polynomial approximations. Over the recorded numeric rows with non-negative order, BESSELI gives the correctly rounded value on 2 of 1163 rows and BESSELK on 0 of 1155, with largest relative errors 5.0e-04 and 1.7e-07. BESSELI(-1,0) is 1.2660658480342601 where I0(1) is 1.2660658777520084, about 2.3E-8 relative.

Call and input routeInputsExcelCorrect valueRecorded in
BESSELI(-1, 0)
cell-value2		x: -1
0xbff0000000000000
n: 0
0x0000000000000000		1.2660658480342601
0x3ff441ce433e3c05		1.2660658777520084
0x3ff441ce4b386c2d		Excel 16.0 build 20430, Windows 64-bit, CPU not recorded in this capture
BESSELI(12.514410892637624, 3.987394155038235)
cell-value2		x: 12.514410892637624
0x40290760dd512b66
n: 3.987394155038235
0x400fe62ee82134c8		21360.000787344507
0x40d4dc000ce65cba		21360.00150112302
0x40d4dc0018982a92		Excel 16.0 build 20430, Windows 64-bit, CPU not recorded in this capture
BESSELK(1.8447401994787955, 0.15119840195885237)
cell-value2		x: 1.8447401994787955
0x3ffd840e4ca60c0e
n: 0.15119840195885237
0x3fc35a781fcf7430		0.1380029049573409
0x3fc1aa1445c5bb34		0.13800289523526038
0x3fc1aa1430e4f602		Excel 16.0 build 20430, Windows 64-bit, CPU not recorded in this capture

Correct value: mpmath besseli at 800 bits, order the integer part of n, correctly rounded.

Correct value: mpmath besseli at 800 bits, order the integer part of n, correctly rounded.

Correct value: mpmath besselk at 800 bits, order the integer part of n, correctly rounded.

Basis: Every numeric row of the extended capture of each function, compared with mpmath at 200 bits.

Evidence and capture sources

Machine-readable note records