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Re: Can I solve this system of nonlinear equations?
*To*: mathgroup at smc.vnet.net
*Subject*: [mg125287] Re: Can I solve this system of nonlinear equations?
*From*: Dana DeLouis <dana01 at me.com>
*Date*: Sun, 4 Mar 2012 04:36:15 -0500 (EST)
*Delivered-to*: l-mathgroup@mail-archive0.wolfram.com
> I'm dealing with systems of nonlinear equations that have 8 equations
> and 8 unknowns. Here's an example:
> ... How can I tell if this is unsolvable?
Hi. Variable h is common to each equation, and appears with each other variable just once.
I believe this is almost similar to having 7 variables with 8 equations.
Rationalize your equation, and multiply by 3 those lines that have a denominator of 3.
Move the constant to the Right Hand side. (out of the way for now)
What you have is:
-500 a+500 c-b h
-500 b+500 d-c h
-500 c+500 e-d h
-500 d+500 f-e h
-500 e+500 g-f h
250 a-2000 b+2000 d-250 e-3 c h
250 b-2000 c+2000 e-250 f-3 d h
250 c-2000 d+2000 f-250 g-3 e h
Look at the coefficient list
var={a,b,c,d,e,f,g,h};
m = Table[Coefficient[lhs,var[[j]]],{j,8}] //Transpose ;
{-500,-h,500,0,0,0,0,-b}
{0,-500,-h,500,0,0,0,-c}
{0,0,-500,-h,500,0,0,-d}
{0,0,0,-500,-h,500,0,-e}
{0,0,0,0,-500,-h,500,-f}
{250,-2000,-3 h,2000,-250,0,0,-3 c}
{0,250,-2000,-3 h,2000,-250,0,-3 d}
{0,0,250,-2000,-3 h,2000,-250,-3 e}
On the first line, h is in b column, and b is in h column.
This won't work, as you would have doubles.
{-500,-h,500,0,0,0,0,-b}.{a,b,c,d,e,f,g,h}
-500 a+500 c-2 b h
Dropping the h column to 0 works.
{-500,-h,500,0,0,0,0,0}.var
-500 a+500 c-b h
The second line would also require dropping the -c in the h column to 0.
{0,-500,-h,500,0,0,0,0}.var
-500 b+500 d-c h
You would have to make the last column of the matrix all zero's for this matrix.
Hence, a zero column has no solution.
Just another technique to get a close guess...
Solve the first equation for h
h -> (-85218681 - 42822650000 a + 42822650000 c)/(85645300 b)
Substitute this h for the remaining 7 equations.
Now, you have 7 equations, with 7 unknowns.
Using NMinimize gets you close to what others have mentioned.
= = = = = = = = = =
HTH :>)
Dana DeLouis
Mac & Math 8
= = = = = = = = = =
On Feb 29, 7:28 am, Andy <andy7... at gmail.com> wrote:
> I'm dealing with systems of nonlinear equations that have 8 equations
> and 8 unknowns. Here's an example:
>
> Solve[{(((c - a)/0.002) - (0.995018769272803 + h*b)) == 0,
> (((d - b)/0.002) - (0.990074756047929 + h*c)) == 0,
> (((e - c)/0.002) - (0.985167483257382 + h*d)) == 0,
> (((f - d)/0.002) - (0.980296479563062 + h*e)) == 0,
> (((g - e)/0.002) - (0.975461279165159 + h*f)) == 0,
> (((-1*e + 8*d - 8*b + a)/(12*0.001)) - (0.990074756047929 + h*c)) ==
> 0,
> (((-1*f + 8*e - 8*c + b)/(12*0.001)) - (0.985167483257382 + h*d)) ==
> 0,
> (((-1*g + 8*f - 8*d + c)/(12*0.001)) - (0.980296479563062 + h*e)) ==
> 0}, {a, b, c, d, e, f, g, h}]
>
> Whenever I try this, Mathematica 7 just returns the empty set {}. How
> can I tell if this is unsolvable? Shouldn't I at least be able to get
> a numerical approximation with NSolve? I've tried using stochastic
> optimization to get approximate answers but every method gives poor
> results, and that's why I would like to at least approximately solve
> this if possible. Thanks very much for any help~
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