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MathGroup Archive 2004

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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


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