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Re: Q about Nonlinear Simul. Eq
*To*: mathgroup at smc.vnet.net
*Subject*: [mg3728] Re: [mg3716] Q about Nonlinear Simul. Eq
*From*: brucec (Bruce Carpenter)
*Date*: Thu, 11 Apr 1996 02:52:38 -0400
*Sender*: owner-wri-mathgroup at wolfram.com
>Hello,
>
>I'm a newbie in this group. I have a question about solving nonlinear
>simultaneous equation. Please forgive me if my question is trivial.
>
>Is there any built-in function solving nonlinear simultaneous equations
>in Mathematica? Or do I have to make a subroutine or a function to do
>that?
>
>Currently I'm using Mathematica v. 2.2.3 under UNIX.
>
>
>The following shows a small example for your reference:
>
>I tried to estimate x and y from this equation:
>*********************************************************************
>NSolve[{
> x - 4 (Exp[-1 - 2x] / (Exp[-1 - 2x] + Exp[-2 - y])) == 0,
> y - 4 (Exp[-2 - y] / (Exp[-1 - 2x] + Exp[-2 - y])) == 0,
> x + y == 4
> },
> {x, y}]
>*********************************************************************
>
>After hitting shift+enter, what I get is the following message and the
>output:
>
>*********************************************************************
>Solve::tdep:
> The equations appear to involve transcendental functions of
> the variables in an essentially non-algebraic way.
>
>Out[1]=
> -1 - 2 x
> -4 E
>NSolve[{------------------- + x == 0,
> -1 - 2 x -2 - y
> E + E
>
> -2 - y
> -4 E
> ------------------- + y == 0, x + y == 4}, {x, y}]
> -1 - 2 x -2 - y
> E + E
>********************************************************************
>
>Thank you for your help in advance.
>
>
>J.C.
Hi Jinchul,
NSolve is a general way to find numerical solutions to polynomial
equations. For arbitrary equations, there is FindRoot.
In[119]:=
??FindRoot
FindRoot[lhs == rhs, {x, x0}] searches for a numerical solution to the
equation lhs ==
rhs, starting with x == x0.
Attributes[FindRoot] = {HoldAll, Protected}
Options[FindRoot] =
{AccuracyGoal -> Automatic, Compiled -> True, DampingFactor -> 1,
Jacobian -> Automatic, MaxIterations -> 15, WorkingPrecision -> 16}
For your set of equations:
In[120]:=
soln = FindRoot[{x - 4 (Exp[-1 - 2 x] / (Exp[-1 - 2 x] + Exp[-2 - y])) == 0,
x + y - 4 == 0}, {x, 1.7}, {y, 2.2}]
Out[120]=
{x -> 1.75033, y -> 2.24967}
The three equations you gave overdetermine the system, so I just used two
of them. Then I checked the solution against all three equations.
In[121]:=
{x - 4 (Exp[-1 - 2x] / (Exp[-1 - 2x] + Exp[-2 - y])),
y - 4 (Exp[-2 - y] / (Exp[-1 - 2x] + Exp[-2 - y])), x + y - 4 } /. soln
Out[121]=
-16
{0., 4.44089 10 , 0.}
Best Regards,
Bruce Carpenter
-----------------------
Bruce Carpenter
Courseware Coordinator phone: (217) 398-0700
Wolfram Research, Inc. fax: (217) 398-0747
100 Trade Centre Drive email: brucec at wolfram.com
Champaign, IL 61820 web: http://www.wolfram.com
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