Note that if the equation really is (in traditional notation) 4^x - 2^x - 8 = 0
then it can be rewritten as y^2 - y - 8 = 0, y = 2^x and solved in closed form as well, yielding a countably infinite set of solutions aligned along one (or two) vertical lines in the complex plane. (If I am not mistaken!) Louis > On 29 Jan 2018, at 07:19, Rob Hodgkinson <[email protected]> wrote: > > @Skip et al … > > also apologies for my sill definitions, I should have used y inside the > definitions not x (!!!), sorry if I confused the issue… > > as in here for the first interpretation … > > x,"0 (3 : '8+(2^y)-((2^2)^y)') x=:1.6+0.05*i.8 > > …/Rob > >> On 29 Jan 2018, at 3:49 pm, Jose Mario Quintana >> <[email protected]> wrote: >> >> In that case, >> >> (X=. (8 + (2 ^ ]) - (2 ^ 2) ^ ])New (^:22) 1) >> 1.75372489 >> >> is a root, >> >> (8 + (2 ^ ]) - (2 ^ 2) ^ ])X >> 0 >> >> but, it is not the only one, >> >> (X=. (8 + (2 ^ ]) - (2 ^ 2) ^ ])New (^:22) 0.5j_0.5) >> 1.24627511j4.53236014 >> >> (8 + (2 ^ ]) - (2 ^ 2) ^ ])X >> 8.8817842e_16j7.72083702e_15 >> >> >> >>> On Sun, Jan 28, 2018 at 9:27 PM, Raul Miller <[email protected]> wrote: >>> >>> Hmm... >>> >>> I had originally thought about calling out the (2^2)^x interpretation >>> as a possibility, because rejected that, because that would be better >>> expressed as 4^x >>> >>> But it's possible that Skip got the 1.75379 number from someone who >>> thought different about this. >>> >>> And, to be honest, it is an ambiguity in the original expression - >>> just one that I thought should be rejected outright, rather than >>> suggested. >>> >>> Which gets us into another issue, which is that what one person would >>> think is obviously right is almost always what some other person would >>> think is obviously wrong... (and this issue crops up all over, not >>> just in mathematic and/or programming contexts). >>> >>> -- >>> Raul >>> >>> >>>> On Sun, Jan 28, 2018 at 9:08 PM, Rob Hodgkinson <[email protected]> wrote: >>>> @Skip >>>> >>>> Skip, I am a confused in your original post… your actual post read; >>>> >>>> ================================================ >>>> What is the best iterative way to solve this equation: >>>> (-2^2^x) + (2^x) +8 =0 >>>> then later to Raul, >>>> 0 = 8 + (2^x) - 2^2^x NB. Is correct, and the answer is real >>>> The answer is close to 1.75379 >>>> ================================================ >>>> >>>> However I suspect your original syntax was not J syntax (could it have >>> been math type syntax ?) as it differs on the J style right-to-left syntax >>> on the 2^2^x expression. >>>> >>>> The 2 possible interpretations are shown below and only the Excel type >>> syntax seems to get close to your expected answer. >>>> >>>> NB. Excel interpretation >>>> x,"0 (3 : '8+(2^x)-((2^2)^x)') x=:1.6+0.05*i.8 >>>> 1.6 1.84185 >>>> 1.65 1.28918 >>>> 1.7 0.692946 >>>> 1.75 0.0498772 NB. intercept seems close to your >>> expected value of 1.75379 >>>> 1.8 _0.64353 >>>> 1.85 _1.39104 >>>> 1.9 _2.19668 >>>> 1.95 _3.06478 >>>> >>>> NB. J style interpretation >>>> x,"0 (3 : '8+(2^x)-(2^2^x)') x=:1.6+0.05*i.8 >>>> 1.6 2.85522 >>>> 1.65 2.33325 >>>> 1.7 1.74188 >>>> 1.75 1.07063 >>>> 1.8 0.307207 NB. But in this model the intercept is >>> above 1.8, but this is the model that has been coded in responses to your >>> post ?? >>>> 1.85 _0.562844 >>>> 1.9 _1.5566 >>>> 1.95 _2.69431 >>>> >>>> Please clarify, thanks, Rob >>>> >>>> >>>>> On 29 Jan 2018, at 12:48 pm, Jose Mario Quintana < >>> [email protected]> wrote: >>>>> >>>>> Moreover, apparently there is at least another solution, >>>>> >>>>> ((-2^2^X) + (2^X) +8 ) [ X=. 2.9992934709539156j_13.597080425481581 >>>>> 5.19549681e_16j_2.92973749e_15 >>>>> >>>>> >>>>> On Sun, Jan 28, 2018 at 7:28 PM, Jose Mario Quintana < >>>>> [email protected]> wrote: >>>>> >>>>>> Are you sure? >>>>>> >>>>>> u New >>>>>> - (u %. u D.1) >>>>>> >>>>>> ,. (8 + (2 ^ ]) - 2 ^ 2 ^ ])New (^:(<22)) 1 >>>>>> 1 >>>>>> 3.44167448 >>>>>> 3.25190632 >>>>>> 3.03819348 >>>>>> 2.7974808 >>>>>> 2.53114635 >>>>>> 2.25407823 >>>>>> 2.00897742 >>>>>> 1.86069674 >>>>>> 1.82070294 >>>>>> 1.81842281 >>>>>> 1.81841595 >>>>>> 1.81841595 >>>>>> 1.81841595 >>>>>> 1.81841595 >>>>>> 1.81841595 >>>>>> 1.81841595 >>>>>> 1.81841595 >>>>>> 1.81841595 >>>>>> 1.81841595 >>>>>> 1.81841595 >>>>>> 1.81841595 >>>>>> >>>>>> (8 + (2 ^ ]) - 2 ^ 2 ^ ]) 1.81841595 >>>>>> _8.21739086e_8 >>>>>> >>>>>> (8 + (2 ^ ]) - 2 ^ 2 ^ ]) 1.75379 >>>>>> 1.01615682 >>>>>> >>>>>> PS. Notice that New is a tacit version (not shown) of Louis' VN >>> adverb. >>>>>> >>>>>> >>>>>> >>>>>> >>>>>> On Sun, Jan 28, 2018 at 5:52 PM, Skip Cave <[email protected]> >>>>>> wrote: >>>>>> >>>>>>> Raul, >>>>>>> >>>>>>> You had it right in the first place. >>>>>>> >>>>>>> 0 = 8 + (2^x) - 2^2^x NB. Is correct, and the answer is real >>>>>>> >>>>>>> The answer is close to 1.75379 >>>>>>> >>>>>>> I wanted to know how to construct the Newton Raphson method using the >>>>>>> iteration verb N described in the link: http://code.jsoftware. >>>>>>> com/wiki/NYCJUG/2010-11-09 >>>>>>> under "A Sampling of Solvers - Newton's Method" >>>>>>> >>>>>>> N=: 1 : '- u % u d. 1' >>>>>>> >>>>>>> Skip >>>>>>> >>>>>>> >>>>>>> >>>>>>> >>>>>>> >>>>>>> Skip Cave >>>>>>> Cave Consulting LLC >>>>>>> >>>>>>> On Sun, Jan 28, 2018 at 4:38 PM, Raul Miller <[email protected]> >>>>>>> wrote: >>>>>>> >>>>>>>> Eh... I *think* you meant what would be expressed in J as: >>>>>>>> >>>>>>>> 0 = 8 + (2^x) - 2^2^x >>>>>>>> >>>>>>>> I'd probably try maybe a few hundred rounds of newton's method first, >>>>>>>> and see where that leads. >>>>>>>> >>>>>>>> But there's an ambiguity where the original expression (depending on >>>>>>>> the frame of reference of the poster) could have been intended to be: >>>>>>>> >>>>>>>> 0 = 8 + (2^x) + _2^2^x >>>>>>>> >>>>>>>> [if that is solvable, x might have to be complex] >>>>>>>> >>>>>>>> Thanks, >>>>>>>> >>>>>>>> -- >>>>>>>> Raul >>>>>>>> >>>>>>>> On Sun, Jan 28, 2018 at 5:25 PM, Skip Cave <[email protected]> >>>>>>>> wrote: >>>>>>>>> What is the best iterative way to solve this equation: >>>>>>>>> >>>>>>>>> (-2^2^x) + (2^x) +8 =0 >>>>>>>>> >>>>>>>>> >>>>>>>>> Skip Cave >>>>>>>>> Cave Consulting LLC >>>>>>>>> ------------------------------------------------------------ >>>>>>> ---------- >>>>>>>>> For information about J forums see http://www.jsoftware.com/forum >>>>>>> s.htm >>>>>>>> ------------------------------------------------------------ >>> ---------- >>>>>>>> For information about J forums see http://www.jsoftware.com/ >>> forums.htm >>>>>>> ------------------------------------------------------------ >>> ---------- >>>>>>> For information about J forums see http://www.jsoftware.com/ >>> forums.htm >>>>>>> >>>>>> >>>>>> >>>>> ---------------------------------------------------------------------- >>>>> For information about J forums see http://www.jsoftware.com/forums.htm >>>> >>>> ---------------------------------------------------------------------- >>>> For information about J forums see http://www.jsoftware.com/forums.htm >>> ---------------------------------------------------------------------- >>> For information about J forums see http://www.jsoftware.com/forums.htm >>> >> ---------------------------------------------------------------------- >> For information about J forums see http://www.jsoftware.com/forums.htm > > ---------------------------------------------------------------------- > For information about J forums see http://www.jsoftware.com/forums.htm ---------------------------------------------------------------------- For information about J forums see http://www.jsoftware.com/forums.htm
