Re: fixed point _convergence _check

*To*: mathgroup at smc.vnet.net*Subject*: [mg47502] Re: fixed point _convergence _check*From*: Paul Abbott <paul at physics.uwa.edu.au>*Date*: Wed, 14 Apr 2004 07:17:11 -0400 (EDT)*Organization*: The University of Western Australia*References*: <c537go$j52$1@smc.vnet.net> <c5gflm$ao7$1@smc.vnet.net>*Sender*: owner-wri-mathgroup at wolfram.com

In article <c5gflm$ao7$1 at smc.vnet.net>, m004202002 at yahoo.com (why) wrote: > In fixed point methodes to find out > convergence we need to check > > [nonlinear system... > u(x,y)=x^2+xy-10 = 0 > v(x,y)=y+3xy^2-57 = 0 > ] > > du/dx+dV/dx < 1 d<-means partial derivative > and > du/dy+dV/dy < 1 > > means, after find out jacobian ,mathematica > will inserts x,y value into du/dx,dV/dx,du/dy,dV/dy > and show me du/dx+dV/dx =? ,du/dy+dV/dy=? > how i can do that with mathematica ? Not sure if I completely understand your question. Here goes anyway: Here is your nonlinear system. Note that space implies multiplication: u[x_, y_] = x^2 + x y - 10; v[x_, y_] = 3 x y^2 + y - 57; You can solve this system using Solve or NSolve: Solve[{u[x, y], v[x, y]}=={0,0},{x,y}] Alternatively, you can find the roots of this system using FindRoot (effectively fixed points of Newton's method), specifiying a starting point for the search: ans = FindRoot[{u[x, y], v[x, y]}, {{x, 0}, {y, 1}}] {x -> 2., y -> 3.} Substituting (using /.) this solution into the conditions shows that neither is satisfied: D[u[x, y] + v[x, y], x] < 1 /. ans False D[u[x, y] + v[x, y], y] < 1 /. ans False Alternatively, compute the Jacobian, jac = Outer[D, {u[x, y], v[x, y]}, {x, y}] {{2 x + y, x}, {3 y^2, 6 x y + 1}} and then test the conditions: Thread[Plus @@ Transpose[jac] < 1] /. ans {False, False} Cheers, Paul -- Paul Abbott Phone: +61 8 9380 2734 School of Physics, M013 Fax: +61 8 9380 1014 The University of Western Australia (CRICOS Provider No 00126G) 35 Stirling Highway Crawley WA 6009 mailto:paul at physics.uwa.edu.au AUSTRALIA http://physics.uwa.edu.au/~paul