Do you have a good estimate for how many terms are needed in the exponential series? That is, given epsilon>0 and real y>0, what is a good estimate for the smallest n such that epsilon > (y^n)%!n ?
Such an n must exist because the limit of the sequence (y^n)%!n is 0. My current approach is to use Stirling's approximation for !n . ----- Original Message ----- From: John Randall <[EMAIL PROTECTED]> Date: Saturday, December 2, 2006 4:55 pm Subject: Re: [Jgeneral] Bug? Sin of pi. > Roger Hui wrote: > > For example, what is sin 0.8 to 40 digits? > > > 0j40 ": -/ (y1^i) % ! i=: 2 * i. 17x > > 0.7173560908995227616271746105813853661928 > > > > In comparison, Abramowitz & Stegun says (Table 4.6, > > page 158) that sin 0.8 is > > > > 0.71735 60908 99522 76162 718 > > > > I believe the last digit in the A&S figure is > > incorrect. > > I agree. > > (a) Direct calculation with series for sin, and a crude error > bound using > (|0.8<1) and (|cos 0.8)<:1 . > > x=:4r5 > sin=:4 : '-/ (y ^ i) % ! i=. 1 + 2 * i. x: x' > NB. x sin y evaluates x (nonzero) terms of the Taylor series > e=:3 : '%! +:>: y' > NB. error estimate using 1>:|cos y and 1>:|x. > > e 16 NB. not enough > 3.38716e_39 > e 17 NB. enough > 2.68822e_42 > 0j40": 17 sin 4r5 > 0.7173560908995227616271746105813853661928 > > (b) Using Maple. > > > Digits:=45; > Digits := 45 > > > sin(0.8); > 0.717356090899522761627174610581385366192785238 > > The value of Digits sets the internal precision. I have set it to > 45 to > allow for roundoff error, and the last few are not to be relied upon, > although I believe them to be correct. > > Check: > > e 19 > 1.22562e_48 > 0j45": 19 sin 4r5 > 0.717356090899522761627174610581385366192785238 ---------------------------------------------------------------------- For information about J forums see http://www.jsoftware.com/forums.htm
