Re: Re: Homotopic algorithm to solve a system of equations

*To*: mathgroup at smc.vnet.net*Subject*: [mg66914] Re: [mg66875] Re: Homotopic algorithm to solve a system of equations*From*: "Chris Chiasson" <chris at chiasson.name>*Date*: Sat, 3 Jun 2006 03:26:22 -0400 (EDT)*References*: <e5mhml$kjl$1@smc.vnet.net> <200606020809.EAA18052@smc.vnet.net>*Sender*: owner-wri-mathgroup at wolfram.com

Are you talking about using differential equations to go from the solution of a system of equations to which one _does_ know the answer to the solution of a system of equations to which one _does not_ (yet) know the answer? On 6/2/06, aTn <ayottes at dms.umontreal.ca> wrote: > > Ok, some of you wrote back to me wondering if such algorithms exist and > how they work. Here is a quick explanation. > > The methods I'm referring to are know as "probability one globally > convergent homotopy algorithms" (see L. Waston's great introductory > paper) that were developped to help solve a system of equations > (polynomial with real coefficients). > > The basic result used in these algorithms is a parametrized version of > Sards theorem: > > Let U be an open set in R^m and let p: U x R^n x [0,1[ ---> R^n be a > C^2 map. If p is transversal to zero (i.e. 0 is a regular value of p), > then the set of points 'v' in the open set U such that p_v : R^n x > [0,1[ --> R^n : (x,t) ---> p(v,x,t) is transversal to 0, is of measure > one. > > The way to apply this result to equation solving is the following. > Consider a finite set of n polynomials with n indeterminates (and real > coefficients). To it corresponds a (smooth) function F:R^n---> R^n. The > idea is to find an open set U in R^m, a C^2 map p: U x R^n x [0,1[ --> > R^n which is transversal to zero, such that a solution of p_v(x,0) = 0 > is known explicitly for all v in U, p_v(x,1)= F(x) for all v in U and > all x in R^n, and finally for which p_v^(-1)(0) is bounded for all v in > U. Then (by the previous theorem) you can find a path in p_a^(-1)(0) > for some 'v' for which the differential of p_v is of full rank linking > a zero of p_v(. , 0) to a zero of p_v(. , 1) = F. We want the > differential to be of full rank so we can use algorithms such as the > Newton method to solve successive problems along the chosen path. > > I'm looking for a package implementing known probability one globally > convergent homotopy algorithms. I hope this message makes my question > clearer. > > Regards, > > Etienne (aTn) > > P.S.: I typed fast, so feel free to point out mystakes I might have > made :) > > -- http://chris.chiasson.name/

**Follow-Ups**:**Re: Re: Re: Homotopic algorithm to solve a system of equations***From:*Andrzej Kozlowski <akoz@mimuw.edu.pl>

**References**:**Re: Homotopic algorithm to solve a system of equations***From:*"aTn" <ayottes@dms.umontreal.ca>

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