Numerical Solution to System of Non-linear Equations

NSolve::infsolns" indicates that there are an infinite number of solutions. Therefore, there is no unique solution to this set of equations. The solutions are determined by a complicated set of equations that involve the variable C.
  • #1
Maximilien
5
0

Homework Statement



I am having problems solving this system of non-linear equations in Matlab and Mathematica. a,b,c,d,e,f are all independent variables.

In Matlab I use fsolve() and get one solution, in Mathematica I use NSolve[] and get another totally different solution. Mathematica also reports this error: "NSolve::infsolns: Infinite solution set has dimension at least 1."

What is the problem here? Is it that there is no unique solution to this set of equations?

Homework Equations



[tex]
-\frac{21 a b c}{200000}+\frac{21 a^2 d}{400000}-\frac{21 b^2 d}{400000}+\frac{e}{100}=0
[/tex]
[tex]
\frac{21 a^2 c}{400000}-\frac{21 b^2 c}{400000}+\frac{21 a b d}{200000}+\frac{f}{100}=0
[/tex]
[tex]
\frac{c}{100}-\frac{19 a b e}{200000}+\frac{19 a^2 f}{400000}-\frac{19 b^2 f}{400000}=0
[/tex]
[tex]
\frac{d}{100}+\frac{19 a^2 e}{400000}-\frac{19 b^2 e}{400000}+\frac{19 a b f}{200000}=0
[/tex]
[tex]
\frac{a}{100}+\frac{b c e}{10000}-\frac{a d e}{10000}-\frac{a c f}{10000}-\frac{b d f}{10000}-\frac{1}{5 \sqrt{2}}=0
[/tex]
[tex]
\frac{b}{100}+\frac{a c e}{10000}+\frac{b d e}{10000}+\frac{b c f}{10000}-\frac{a d f}{10000}=0
[/tex]
 
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  • #2
Maximilien said:

Homework Statement



I am having problems solving this system of non-linear equations in Matlab and Mathematica. a,b,c,d,e,f are all independent variables.

In Matlab I use fsolve() and get one solution, in Mathematica I use NSolve[] and get another totally different solution. Mathematica also reports this error: "NSolve::infsolns: Infinite solution set has dimension at least 1."

What is the problem here? Is it that there is no unique solution to this set of equations?

Homework Equations



[tex]
-\frac{21 a b c}{200000}+\frac{21 a^2 d}{400000}-\frac{21 b^2 d}{400000}+\frac{e}{100}=0
[/tex]
[tex]
\frac{21 a^2 c}{400000}-\frac{21 b^2 c}{400000}+\frac{21 a b d}{200000}+\frac{f}{100}=0
[/tex]
[tex]
\frac{c}{100}-\frac{19 a b e}{200000}+\frac{19 a^2 f}{400000}-\frac{19 b^2 f}{400000}=0
[/tex]
[tex]
\frac{d}{100}+\frac{19 a^2 e}{400000}-\frac{19 b^2 e}{400000}+\frac{19 a b f}{200000}=0
[/tex]
[tex]
\frac{a}{100}+\frac{b c e}{10000}-\frac{a d e}{10000}-\frac{a c f}{10000}-\frac{b d f}{10000}-\frac{1}{5 \sqrt{2}}=0
[/tex]
[tex]
\frac{b}{100}+\frac{a c e}{10000}+\frac{b d e}{10000}+\frac{b c f}{10000}-\frac{a d f}{10000}=0
[/tex]

I used a re-scaled system (which you should always do, as a matter of course): I set x = 100*X for x = a,b,c,d,e,f and X = A,B,C,D,E,F, then solved the system exactly in Maple. Here is the solution:


> sol:=%;

sol := {A = %1, B = 0, C = C,

2 2 2 1/2
D = 1/20 RootOf(_Z + 400 C + 190 %1 - 19 %1 2 ),

2 2 2 2 1/2
E = -21/8 %1 RootOf(_Z + 400 C + 190 %1 - 19 %1 2 ),

2
F = -105/2 %1 C},

1/2
2
{A = ----, B = 0, C = 0, D = 0, E = 0, F = 0}, {A = %1, B = 0,
10

2 2 1/2
C = 1/20 RootOf(_Z + 190 %1 - 19 %1 2 ), D = 0, E = 0,

2 2 2 1/2
F = -21/8 %1 RootOf(_Z + 190 %1 - 19 %1 2 )}

4
%1 := RootOf(9975 _Z - 4)

It is a bit hard to read here, but there are two distinct solution types, one being
{A = .14150988295122837392, B = 0., C = .24393682254305351184e-2*I, D = 0., E = 0., F = -.25645443231646279921e-2*I} (where I = sqrt(-1)) and the other contained in the first curly brackets. In this "other" solution, C is arbitrary and all the other variables are determined as functions of C, but of a complicated kind: C appears in the coefficients of 4th degree polynomials, and A, B, etc., involve roots of these polynomials. So, no, the solution is certainly not unique.

RGV
 

1. What is a system of non-linear equations?

A system of non-linear equations is a set of equations where the unknown variables are raised to powers greater than one or are multiplied together. These equations cannot be solved by simple algebraic methods.

2. What is a numerical solution to a system of non-linear equations?

Numerical solution to a system of non-linear equations is a method of finding approximate solutions to the equations using numerical algorithms. This involves breaking down the equations into smaller, simpler equations and using iterative methods to approach the solution.

3. When is a numerical solution necessary for a system of non-linear equations?

A numerical solution is necessary when the equations cannot be solved analytically using algebraic methods. This is often the case for complex systems of equations or when there are multiple unknown variables involved.

4. What are the advantages of using a numerical solution for a system of non-linear equations?

One major advantage of using a numerical solution is that it can provide approximate solutions for complex systems of equations that cannot be solved analytically. It is also a useful tool for solving systems of equations with many unknown variables.

5. What are some common numerical methods used for solving systems of non-linear equations?

Some common numerical methods used for solving systems of non-linear equations include Newton's method, the Broyden method, and the secant method. These methods use iterative techniques to approach the solution and may require initial guesses for the unknown variables.

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