smooth "splice" fxn between two lines
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From: Ray Koopman on 17 May 2010 20:27 On May 17, 3:42 pm, Ray Koopman <koop...(a)sfu.ca> wrote: > On May 17, 1:10 pm, kj <no.em...(a)please.post> wrote: >> In <rbisrael.20100516174609$2...(a)news.acm.uiuc.edu> Robert Israel >> <isr...(a)math.MyUniversitysInitials.ca> writes: >>> Ray Koopman <koop...(a)sfu.ca> writes: >>>> On May 14, 4:57 pm, Ray Koopman <koop...(a)sfu.ca> wrote: >>>>> >>>>> f(x) = (exp(-1/x) - x exp(-1/(2-x)))/(exp(-1/x) - exp(-1/(2-x))) >>>>> also works. >>>> >>>> f(x) = 1 - exp(2x/(x-2)) is simpler, and the plot is nicer. >>> >>> But it doesn't join smoothly to x at x=0: f'''(0) = -1/2. >>> Well, maybe kj would be satisfied with a C^2 join, >>> but I interpreted "smooth" as C^infty. >> >> Actually, I did have C^infty in mind. >> >> I also had in mind, but neglected to specify, that the second >> derivative of the resulting curve would be <= 0. > > If we call Robert's function f1 and my first function f2 then f(x) = > (r*f1(x) + f2(x))/(r + 1) will have C^infty splices at both 0 & 2. > Taking r = .8812 will approximately maximize the minimum f", > subject to f" <= 0. One the other hand, if we take r = 1 then f simplifies to (x - exp(2/(2-x)-2/x))/(1 - exp(2/(2-x)-2/x)), f" is still <= 0 (which condition requires 2/3 <= r <= 1), and the plot is barely distinguishable from that with r = .8812 .
From: kj on 18 May 2010 10:52
In <444e5c87-254d-46f5-ac64-9e40b81dc085(a)r21g2000prr.googlegroups.com> Ray Koopman <koopman(a)sfu.ca> writes: >On May 17, 3:42 pm, Ray Koopman <koop...(a)sfu.ca> wrote: >> On May 17, 1:10 pm, kj <no.em...(a)please.post> wrote: >>> In <rbisrael.20100516174609$2...(a)news.acm.uiuc.edu> Robert Israel >>> <isr...(a)math.MyUniversitysInitials.ca> writes: >>>> Ray Koopman <koop...(a)sfu.ca> writes: >>>>> On May 14, 4:57 pm, Ray Koopman <koop...(a)sfu.ca> wrote: >>>>>> >>>>>> f(x) = (exp(-1/x) - x exp(-1/(2-x)))/(exp(-1/x) - exp(-1/(2-x))) >>>>>> also works. >>>>> >>>>> f(x) = 1 - exp(2x/(x-2)) is simpler, and the plot is nicer. >>>> >>>> But it doesn't join smoothly to x at x=0: f'''(0) = -1/2. >>>> Well, maybe kj would be satisfied with a C^2 join, >>>> but I interpreted "smooth" as C^infty. >>> >>> Actually, I did have C^infty in mind. >>> >>> I also had in mind, but neglected to specify, that the second >>> derivative of the resulting curve would be <= 0. >> >> If we call Robert's function f1 and my first function f2 then f(x) = >> (r*f1(x) + f2(x))/(r + 1) will have C^infty splices at both 0 & 2. >> Taking r = .8812 will approximately maximize the minimum f", >> subject to f" <= 0. >One the other hand, if we take r = 1 then f simplifies to >(x - exp(2/(2-x)-2/x))/(1 - exp(2/(2-x)-2/x)), f" is still <= 0 >(which condition requires 2/3 <= r <= 1), and the plot is barely >distinguishable from that with r = .8812 . Thanks! ~K |