Date: Tue, 4 Sep 2018 03:56:28 +0000 From: "Montgomery-Smith, Stephen" <stephen@missouri.edu> To: "freebsd-numerics@freebsd.org" <freebsd-numerics@freebsd.org>, "sgk@troutmask.apl.washington.edu" <sgk@troutmask.apl.washington.edu> Subject: Re: j0 (and y0) in the range 2 <= x < (p/2)*log(2) Message-ID: <ae55265d8d544cdc8a3ebc4314e1d01c@missouri.edu> In-Reply-To: <20180903235724.GA95333@troutmask.apl.washington.edu> References: <20180903235724.GA95333@troutmask.apl.washington.edu>
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A quick google search turned up this https://www.cl.cam.ac.uk/~jrh13/papers/bessel.pdf which has the functions p0 and q0. Maybe this was the basis of this code. ________________________________ From: owner-freebsd-numerics@freebsd.org <owner-freebsd-numerics@freebsd.or= g> on behalf of Steve Kargl <sgk@troutmask.apl.washington.edu> Sent: Monday, September 3, 2018 6:57:24 PM To: freebsd-numerics@freebsd.org Subject: j0 (and y0) in the range 2 <=3D x < (p/2)*log(2) Anyone know where the approximations for j0 (and y0) come from? msun/src/e_j0.c states * for x in (2,inf) * j0(x) =3D sqrt(2/(pi*x))*(p0(x)*cos(x0)-q0(x)*sin(x0)) * where x0 =3D x-pi/4. It is better to compute sin(x0),cos(x0) * as follow: * cos(x0) =3D cos(x)cos(pi/4)+sin(x)sin(pi/4) * =3D 1/sqrt(2) * (cos(x) + sin(x)) * sin(x0) =3D sin(x)cos(pi/4)-cos(x)sin(pi/4) * =3D 1/sqrt(2) * (sin(x) - cos(x)) * (To avoid cancellation, use * sin(x) +- cos(x) =3D -cos(2x)/(sin(x) -+ cos(x)) * to compute the worse one.) p0(x) and q0(x) are divergent asymptotic series. If I extract pzero() and qzero() from e_j0.c and compare the results against summing truncated versions of p0(x) and q0(x), there are no obvious connections. Reading the documentation for the algorithms used in MPFR suggests that x >=3D p/2*log(2), where p is precision of x, is required for use of the large argument approximation for j0(x). In double precision, p =3D 53, so we have x >=3D 18.368... Consider x=3D18.4 and sum up to N =3D 31 in the asymptotic series: % ./pq 30 18.4 p =3D 9.997932830701132e-01, q =3D -6.781826311540553e-03 <-- series pp =3D 9.997932830701132e-01, qq =3D -6.781826311540509e-03 <-- pzero,qze= ro ulp(p, pp) =3D 0.000000e+00 ulp(q, qq) =3D 2.550000e+01 This is almost reasonable if 25.5 ULP is acceptable in q0(x). Note the series are computed in long double with 64 bits of precision. Now, comparing x =3D 2 and summing N =3D 4 (best results). % ./pq 4 2 p =3D 9.894313812255859e-01, q =3D -5.334472656250000e-02 pp =3D 9.862158212188928e-01, qq =3D -5.647769967932505e-02 ulp(p, pp) =3D 1.448159e+13 ulp(q, qq) =3D 2.257545e+14 For values of N > 4, the series start to diverge! So, how does msun use the large argument approximation for j0(x)? -- Steve _______________________________________________ freebsd-numerics@freebsd.org mailing list https://lists.freebsd.org/mailman/listinfo/freebsd-numerics To unsubscribe, send any mail to "freebsd-numerics-unsubscribe@freebsd.org"
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