       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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