Number of Solutions of a Nonlinear System

In summary: However, it may not be as efficient as methods for finding real roots only.Summary: There are methods available for finding both real and complex roots of polynomials with real coefficients, but they may not be as efficient as methods for finding only real roots.
  • #1
Wledig
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TL;DR Summary
How can one determine the number of solutions of a nonlinear system?
I developed a program to solve the nonlinear system below through the method of successive approximations and was only able to find one solution, namely ##x_1 = 0.93377## and ##x_2 =0.88417##, even though I tried many different starting points. I was wondering if there's a way to determine if there are other solutions to this system, and if so, how many more there would be.

\begin{cases} 3x_1^2 + x_2 = \frac{7}{2} \\ x_1 + x_2^3 = \frac{13}{8} \end{cases}
 
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  • #2
Wledig said:
Summary: How can one determine the number of solutions of a nonlinear system?

I developed a program to solve the nonlinear system below through the method of successive approximations and was only able to find one solution, namely ##x_1 = 0.93377## and ##x_2 =0.88417##, even though I tried many different starting points. I was wondering if there's a way to determine if there are other solutions to this system, and if so, how many more there would be.

\begin{cases} 3x_1 + x_2 = \frac{7}{2} \\ x_1 + x_2^3 = \frac{13}{8} \end{cases}

if you use elimination, you can subtract 1/3 of the first equation from the second to get

##x_2^3 - \frac{1}{3}x_2 = \frac{13}{8} - \frac{7}{6}##
or
##x_2^3 - \frac{1}{3}x_2 - \frac{11}{24} = 0##

it's a cubic, so you can solve exactly (the other two roots are complex). the solution for ##x_2## is a bit different than what you've stated, though
 
  • #3
Oops, my bad. The first term in the first equation is actually ##x_1^2##, I corrected it now.
 
  • #4
According to Wikipedia there is a thing called Bezout’s theorem, which states that the number of intersection points of two plane curves is at most the product of the degrees, with equality when counting complex points and points at infinity, weighted by multiplicity.

So basically, you can expect at most 6 solutions, since you have a quadratic and a cubic.
 
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  • #5
I see. There's probably no way to determine the complex solutions numerically though, I believe. Thanks for the insight.
 
  • #6
Wledig said:
Summary: How can one determine the number of solutions of a nonlinear system?...

\begin{cases} 3x_1^2 + x_2 = \frac{7}{2} \\ x_1 + x_2^3 = \frac{13}{8} \end{cases}

using this updated equation, you can do substitution:
\begin{cases} 3x_1^2 + x_2 = \frac{7}{2} \\ x_1 = \frac{13}{8}- x_2^3 \end{cases}
and substitute in the top line. Degree 6 polynomial in ##x_2##. Easy numeric solutions.
 
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  • #7
StoneTemplePython said:
Degree 6 polynomial in x2x2x_2. Easy numeric solutions.
Are they? I could use Newton-Raphson's method for instance to determine the real roots, but what about the complex ones? I'm not aware of any method for finding those numerically.
 
  • #8
Wledig said:
Are they? I could use Newton-Raphson's method for instance to determine the real roots, but what about the complex ones? I'm not aware of any method for finding those numerically.
There's different schools of thought on this but one of the big ones is:

some the most heavily optimized numerical algorithms out there are for eigenvalues (QR Algorithm and various other flavors).

Numpy and Matlab and many other languages will give you eigenvalues (roots of a monic single variable polynomial) if you input them in a vector. In fact, my understanding is that if you ask Matlab to give you polynomial roots, it will input the polynomial into a companion matrix and run an eigenvalue routine on it. Numpy does the same thing with this routine:

https://docs.scipy.org/doc/numpy/reference/generated/numpy.roots.html
 
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  • #9
I would find it hard to believe that no algorithm exists for finding roots of real polynomials numerically. In fact, I have just learned from a Google search about Bairstow’s method, which computes both real and complex roots for polynomials with real coefficients.
 
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Related to Number of Solutions of a Nonlinear System

1. What is a nonlinear system?

A nonlinear system is a set of equations that cannot be solved by using standard algebraic methods. This means that the equations contain variables that are raised to powers other than 1, or that are multiplied together. Nonlinear systems are often more complex and difficult to solve than linear systems.

2. How do you find the number of solutions of a nonlinear system?

The number of solutions of a nonlinear system can be determined by graphing the equations and finding the points where they intersect. If there is only one point of intersection, then there is one solution. If there are multiple points of intersection, then there are either infinite solutions or no solutions.

3. Can a nonlinear system have more than one solution?

Yes, a nonlinear system can have more than one solution. This occurs when the equations have multiple points of intersection when graphed. In this case, there are infinite solutions to the system.

4. What is the difference between a unique solution and infinite solutions?

A unique solution means that there is only one point of intersection between the equations, and therefore only one solution to the system. Infinite solutions means that there are multiple points of intersection, and therefore an infinite number of solutions to the system.

5. Can a nonlinear system have no solutions?

Yes, a nonlinear system can have no solutions. This occurs when the equations do not intersect when graphed, meaning there is no point that satisfies both equations simultaneously. In this case, the system is said to be inconsistent, and there are no solutions.

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