Solving Numerical System of Equations for $\lambda$ and $\mu$

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In summary, the given conversation discusses a system of equations involving functions \Phi and \lambda and asks how to solve it numerically. It is stated that it is not possible to solve it numerically if \Phi is unknown. The suggestion is made to represent the solution as \lambda = Af(\Phi), where A is the numerical solution and f is some function. The question of the best method to solve non-linear systems of equations is also brought up.
  • #1
JohanL
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[tex]

e^{-2 \lambda}(2r \lambda_r -1) + 1 = 8 \pi r^2G_\Phi(r,\mu)
[/tex]
[tex]
e^{-2 \lambda}(2r \mu_r +1) - 1 = 8 \pi r^2H_\Phi(r,\mu)

[/tex]

where

[tex]

G_\Phi(r,\mu) = \frac{2\pi}{r^2}\int_{1}^{\infty}\int_{0}^{r^2(\epsilon^2-1)} \Phi(e^{\mu(r)\epsilon,L}) \frac{\epsilon}{\sqrt{\epsilon^2-1-L/r^2}}dL
d\epsilon
[/tex]
[tex]
H_\Phi(r,\mu) = \frac{2\pi}{r^2}\int_{1}^{\infty}\int_{0}^{r^2(\epsilon^2-1)} \Phi(e^{\mu(r)\epsilon,L}) \frac{\epsilon}{\sqrt{\epsilon^2-1-L/r^2}}dL
d\epsilon

[/tex]

If you have a system like this and want to solve it numerically for [tex]\lambda [/tex] and [tex] \mu [/tex] how do you deal with the function [tex] \Phi [/tex]. I mean: It can be any function...i have never solved a system like that before.

thank you.
 
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  • #2
Obviously, you can't solve a system of equations numerically if you have an unknown function (or unknown number) in it.
 
  • #3
Yepp,,,that is obvious but what's the best you can do?
Can you get a solution

[tex]

\lambda = Af(\Phi)

[/tex]

Where A is the numerical solution and f is some function.
And then you can plot lambda for the most probable [tex]\Phi[/tex]´s or something

Whats the best method to attack non-linear system of equations like this one?
 

FAQ: Solving Numerical System of Equations for $\lambda$ and $\mu$

1. What is the purpose of solving numerical system of equations for $\lambda$ and $\mu$?

The purpose of solving numerical system of equations for $\lambda$ and $\mu$ is to find the values of these variables that satisfy all of the equations in the system. These values will help determine the relationship between the unknown quantities in the equations.

2. What is the difference between solving a system of equations analytically and numerically?

When solving a system of equations analytically, the solutions are found algebraically using methods such as substitution or elimination. On the other hand, solving numerically involves using numerical methods such as the Gauss-Jordan method or the Newton-Raphson method to find approximate solutions.

3. How do you know when a numerical system of equations has a unique solution?

A numerical system of equations has a unique solution when the number of equations in the system is equal to the number of unknown variables, and the system is consistent (i.e. has at least one solution). This can be determined by checking the rank of the coefficient matrix.

4. Can a numerical system of equations have more than one solution?

Yes, a numerical system of equations can have more than one solution if it is inconsistent (i.e. has no solution) or if it is dependent (i.e. has infinitely many solutions). Inconsistent systems have contradictory equations, while dependent systems have equations that are linearly dependent on each other.

5. What are some common applications of solving numerical system of equations for $\lambda$ and $\mu$?

Solving numerical system of equations is commonly used in various fields such as engineering, physics, economics, and statistics. Some applications include finding optimal solutions in optimization problems, determining the values of unknown parameters in mathematical models, and analyzing data to make predictions or draw conclusions.

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