existence of real root(s) of a transcendental equation
in [Math]
|
From: Narasimham on 9 Feb 2010 14:04 On Feb 9, 1:44 am, simonsong <pengcheng.si...(a)gmail.com> wrote: > Hi, > Is there an algorithm for checking the existence of real root(s) of > an arbitrary (uni-variate) transcendental equation (not polynomial > equation)? I'd appreciate it if you can provide references to the > algorithm or the proof of the non-existence of such an algorithm (if > either of them exist). Thanks. > > Simon For F,G transcendents, y(x) = F(x) - G(x) = 0, then for y broadly : for real root to exist, function changes sign, but not its derivative. y1'*y2' > 0, y1*y2 < 0. for complex root to exist, derivative changes sign, but not the function. y1*y2 > 0, y1' *y2' < 0. Narasimham
From: Robert Israel on 9 Feb 2010 22:03 simonsong <pengcheng.simon(a)gmail.com> writes: > Hi, > Is there an algorithm for checking the existence of real root(s) of > an arbitrary (uni-variate) transcendental equation (not polynomial > equation)? I'd appreciate it if you can provide references to the > algorithm or the proof of the non-existence of such an algorithm (if > either of them exist). Thanks. > > Simon See e.g. M. Laczkovich, "The Removal of Pi from Some Undecidable Problems Involving Elementary Functions", Proc. AMS 131 (2002) 2235-2240, <http://www.ams.org/proc/2003-131-07/S0002-9939-02-06753-9/S0002-9939-02-06753-9.pdf>. He shows that in the ring generated by the integers and the functions x, sin (x^n) and sin(x sin(x^n)) (n = 1, 2,...) de fined on R it is undecidable whether or not a function has a positive value or has a root. -- Robert Israel israel(a)math.MyUniversitysInitials.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada
From: spudnik on 9 Feb 2010 22:20 could you supply a "sixth grade math / sixth grade english" explanation of the "removal of pi?" thank you! > See e.g. M. Laczkovich, "The Removal of Pi from Some Undecidable Problems > Involving Elementary Functions", Proc. AMS 131 (2002) 2235-2240, > <http://www.ams.org/proc/2003-131-07/S0002-9939-02-06753-9/S0002-9939-...>. thus: aether; may not mean what y'think it be!... quantum foam, either. redshift, assuredly not neccesarily Dopplerian; anyone in space physics knows, there ain't no absolute/ Pascalian plenum -- not his fault, either, although Hubble succumbed to the einsteinmaniacs! > > And b4 he UNDERSTOOD what they physically OR mathematically require > > or impose or mean. > I say, I can prove redshift but not black hole. Black holes violate Special > Relativity's speed limit while falling in. There is a two time aether > rate field. Two times are immatterial and flow. thus: yeah, there is at least one pair of such "fixed" points, but I would apply Boyles law, minimally, to that situation. > It does not imply that the temperature at every point is the same as > at its antipode. Nor does it imply (contrary to Tom's claim) that > "identical weather conditions" occur at some point and its antipode -- > unless "identical weather conditions" simply means the same in some > one continuous function (temperature or wind speed or ...). --les OEuvres! http://wlym.com
From: Waldek Hebisch on 9 Feb 2010 23:52
simonsong <pengcheng.simon(a)gmail.com> wrote: > On Feb 8, 4:27?pm, "Dave L. Renfro" <renfr...(a)cmich.edu> wrote: > > What functions are you allowing? > > Actually the function I encountered is of form: > \sum_i a_i exp( -exp(x+i*h) ) = b > > where a_i, h, and b are constants, and x is the unknown. > > However my interest is not limited to this function. I am curious to > know whether an algorithm for checking the existence of real roots > exists for arbitrary continuous functions. If not, for what subset of > the continuous functions does such an algorithm exist. > Do not expect algorithms unless you have rather restrictive class of functions. The sums above are in such class, but Richardson showed that for quite simply looking class of functions there is no algorithm. -- Waldek Hebisch hebisch(a)math.uni.wroc.pl |