Re: Wrong formulae for complex elementary functions
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Hi everybody, On 11/27/2010 08:37 AM, Michael Kerrisk wrote:
Hi Andries, Since you are the mathematician, can you comment? Thanks, Michael On Fri, Nov 26, 2010 at 9:57 AM, Richard B. Kreckel<kreckel@xxxxxxxx> wrote:Hi! The man pages for cacos, cacosf, cacosl, catan, catanf, catanl, cacosh, cacoshf, cacoshl, catanh, catanhf, and catanhl contain wrong maths. cacos, cacosf, cacosl: The formula given in the man page cacos(z) = -i clog(z + csqrt(z * z - 1)) gives wrong results in second and fourth quadrant of complex plain. The formula cacos(z) = -i clog(z + I*csqrt(1 - z * z)) gives correct results. catan, catanf, catanl: The formula given in the man page catan(z) = 1 / 2i clog((1 + iz) / (1 - iz)) gives wrong results on the negative imaginary axis beginning at -I (along one of the two branch cuts). Besides, the formula is written in an ambiguous way. The formula catan(z) = (clog(1 + iz) - clog(1 - iz)) / 2i gives correct results. cacosh, cacoshf, cacoshl: The formula given in the man page cacosh(z) = (0.5) * clog((1 + z) / (1 - z)) gives wrong results everywhere in the complex plain. (The formula seems to be copied from the one for catanh, where it is sometimes correct.) The formula cacosh(z) = 2 * clog(csqrt((z + 1)/2) + csqrt((z - 1)/2)) gives correct results. catanh, catanhf, catanhl: The formula given in the man page catanh(z) = 0.5 * clog((1 + z) / (1 - z)) gives wrong results on the positive real axis beginning at 1 (along one of the two branch cuts). The formula catanh(z) = 0.5 * (clog(1 + z) - clog(1 - z)) gives correct results. I've also checked casin, casinf, casinl, casinh, casinhf, and casinhl and the formulae given there casin(z) = -i clog(iz + csqrt(1 - z * z)) casinh(z) = clog(z + csqrt(z * z + 1)) are actually correct. I suspect that some of these errors are due to somebody trying to "simplify" these formulae. Since this can ruin the behavior, I recommend to add a comment to the upstream sources that warns against attempt of simplification. For the sake of reference, I am looking at manpages-dev 3.25-1 on Debian/squeeze. Bye! -richy. -- Richard B. Kreckel <http://www.ginac.de/~kreckel/>
This doesn't have seem to have been fixed in the Linux man pages, does it? May I suggest a non-mathematical approach to moving forward? Just write a little C program and compare the complex results of a) the library implementation of catan, cacos, etc., b) the formula written in the man page, and c) the formula I proposed above.If you include a couple of values from all four quadrants, you'll see that my formula always agrees with the implementation while the one from the man page doesn't.
Is there anything else I can do to convince you that this should be fixed? If yes, pretty please, let me know! I would really appreciate getting this fixed.
Best wishes -richy. -- Richard B. Kreckel <http://www.ginac.de/~kreckel/> -- To unsubscribe from this list: send the line "unsubscribe linux-man" in the body of a message to majordomo@xxxxxxxxxxxxxxx More majordomo info at http://vger.kernel.org/majordomo-info.html