Re: Upper bounding a transcendental function with Polynomials.
blah12_at_mail.com
Date: 08/16/04
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Date: 16 Aug 2004 04:48:33 -0700
"Jan C. Hoffmann" <jch@arcore.de> wrote in message news:<411f924f$0$7318$9b4e6d93@newsread2.arcor-online.net>...
> "Jan C. Hoffmann" <jch@arcore.de> schrieb im Newsbeitrag
> news:411de411$0$7311$9b4e6d93@newsread2.arcor-online.net...
> >
> > <blah12@mail.com> schrieb im Newsbeitrag
> > news:75ec2d78.0408131702.6e4dd207@posting.google.com...
> > > "Jan C. Hoffmann" <jch@arcore.de> wrote in message
> news:<411c62ae$0$7312$9b4e6d93@newsread2.arcor-online.net>...
> > > > <blah12@mail.com> schrieb im Newsbeitrag
> > > > news:75ec2d78.0408111757.2dbcbe7d@posting.google.com...
> > > > > Hello everyone,
> > > > >
> > > > > I am trying to solve the following problem :
> > > > >
> > > > > I have a function f[x] given as an unsolvable integral
> > > > > which is :
> > > > >
> > > > > f[x] = - Integrate[Exp[-y^2/2]/Sqrt[2*Pi] *
> > > > > Ln[ (1-B) + B/D * Exp[(y+A*x)^2/2/(1+C)] ],
> > > > > {y,-Infinity,Infinity}]
> > > > >
> > > > > where A>0 , 0 < B < 1 , C = 1/A^2 , D = Sqrt[1+A^2]
> > > > >
> > > > > now define g[x] = f[x] - f[0].
> > > > >
> > > > > It is easy to see that f[x] and g[x] are even functions of x.
> > > > > Also g[x=0] = 0 , and it can be seen that g[x] < 0 for all
> > > > > other x ( g'[x] < 0 for all x>0) and finally
> > > > > lim_{x->Infinity} (g[x]) -> -Infinity
> > > > >
> > > > > g[x] is monotonically decreasing for x>=0 , from 0 to
> > > > > -Infinity.
> > > > >
> > > > >
> > > > > Now I'm looking for the LARGEST number E>0 such that
> > > > > (I need to do this for SPECIFIC values of A,B)
> > > > >
> > > > > g[x] + E*x^2 <= 0 for all x.
> > > > >
> > > > >
> > > > > Does anyone have any idea how to attack this problem please ?
> > > > >
> > > > > If f[x] was a "simple" function then by common calculus tools
> > > > > this could be solved, BUT the unsolvable integral complicates
> > > > > things a great deal.
> > > > >
> > > > > EVEN if I can find a number that is "close" enough to this optimal
> > > > > E from BELOW it will be great (just so the above equation is NEVER
> > > > > larger than 0).
> > > > >
> > > > >
> > > > > If anyone knows how to find a polynomial that will be an upper
> > > > > bound on g[x] at least in the interval [0, z) , for z close to 1 or
> > > > > smaller, this can be very helpful as well (I couldn't do this using
> > > > > taylor series because knowing the sign of the the error is very
> > > > > problematic).
> > > > >
> > > >
> > > >
> > > > Your function
> > > >
> > > > z = -Exp(-y ^ 2 / 2) / Sqrt(2 * pi) * Log((1 - B_) + B_ / D_ * Exp((y
> +
> A_ *
> > > > x) ^ 2 / 2 / (1 + C_)))
> > > >
> > > > Example values
> > > >
> > > > A_ = 0.5
> > > > B_ = 0.5
> > > > C_ = 1 / A_ ^ 2
> > > > D_ = Sqrt(1 + A_ ^ 2)
> > > >
> > > > Have a least square polynomial function approximation
> > > >
> > > > z = f (x, y) = 4.06440168454817*10^-03*x^0*y^0
> > > > + -3.09292015753578*10^-07*x^1*y^0 + -3.33405592787114*10^-03*x^2*y^0
> > > > + -1.29496691491341*10^-06*x^3*y^0 + -1.85929697392827*10^-05*x^4*y^0
> +
> > > > 6.21558610087785*10^-09*x^0*y^1 + -1.89758446536286*10^-03*x^1*y^1 +
> > > > 4.48123783330665*10^-07*x^2*y^1 + -2.15564446786301*10^-05*x^3*y^1 +
> > > > 3.25800980134267*10^-07*x^4*y^1 + -7.77554611994807*10^-04*x^0*y^2 +
> > > > 2.77828932482955*10^-08*x^1*y^2 + 3.55854425430838*10^-04*x^2*y^2 +
> > > > 1.15351834872955*10^-07*x^3*y^2 + 2.10917380269816*10^-06*x^4*y^2
> > > > + -2.19336285936576*10^-10*x^0*y^3 + 6.73395796611381*10^-05*x^1*y^3
> > > > + -1.5798490603598*10^-08*x^2*y^3 + 7.68155639890942*10^-07*x^3*y^3
> > > > + -1.14608524858889*10^-08*x^4*y^3 + 2.08631471802857*10^-05*x^0*y^4
> > > > + -5.56456152799723*10^-10*x^1*y^4 + -7.87273816815265*10^-06*x^2*y^4
> > > > + -2.29770862224741*10^-09*x^3*y^4 + -4.79893346366327*10^-08*x^4*y^4
> > > >
> > > > Integral ~ -3.0267E-02
> > > >
> > > > Graph and boundaries you find in
> > > > http://homepages.compuserve.de/Jan390906/news/z-ng-04-08-13-07.htm
> > >
> > >
> > > Hi Jan,
> > > Thanks a lot for answering.
> > >
> > > I'm sorry but I fail to see how this approximation helps me in
> > > solving my initial problem ?
> > > I don't need just an approximation of g[x], I also MUST know if this
> > > approximation constitutes an upper bound on my real function.
> > >
> > > The problem I'm trying to solve is to find the largest E, such that
> > > g[x] + E*x^2 <= 0 for all x.
> >
> >
> > So far I integrated f[x] for a specified domain.
> >
> > For finding integration f[x=0] I had to set x=10^-6 for avoiding trouble
> > with integration.
> >
> > f[10^-6]= 3.296E-09 ~ 0.
> >
> > That is g[x] ~ f[x]
> >
> > Now you could compute table values like
> >
> > x g[x]
> >
> > 0 ~ 0
> > 1 -1.26685267106262E-03
> > 2 -3.02674846613907E-02
> > 3 -0.116469436214612
> > 4 -0.292684349167664
> > 5 -0.596404538356658
> > 10 -5.724148314286670
> >
> > Least square approximated data for [0, 10]
> >
> > g[x] = 3.718173667589*10^-03 + -1.919570149511*10^-02 * x^1 +
> > 1.457880028900*10^-02 * x^2 + -6.993542610978*10^-03 * x^3
> >
> > Find E
> >
> > g[x] + E*x^2 = 0
> >
> > E = -g[x]/x^2
> >
> > For this example you get a local max E(0.417) = 0.0129
> >
> > See also graph and max
> > http://homepages.compuserve.de/Jan390906/news/z-ng-04-08-14-11.htm
> >
> > Remark
> > You stated {y,-Infinity,Infinity}. You can reduce infinity to a plausible
> > domain that can be evaluated by an increase of the result e.g. +/-10^-6.
> >
>
>
> CORRECTION:
>
> My program didn't find the solution via polynomial (degree 4). The program
> was accidenly swiched to direct numerical integration to your function.
> Sorry, I wasn't aware of that. But the result is ok.
>
> A polynomial solution is possible. But you need a much higher degree.
Sorry but I'm not looking for an approximation polynomial,
I'm looking for a polynomial that is a good UPPER BOUND on
the function g[x] which I defined above.
I'm still waiting for some assistance on this, if anyone
has any idea what I can do please advise.
Thanks.
- Next message: Henrik Bergstr?m: "Re: LinAlg: Vector Spaces - need help understanding an example"
- Previous message: Joseph Shtok: "Re: Open Letter to Serre Regarding Grothendieck"
- In reply to: Jan C. Hoffmann: "Re: Upper bounding a transcendental function with Polynomials."
- Messages sorted by: [ date ] [ thread ]
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