Newton's method for a system of equations

In summary, the conversation discussed the convergence of Newton's method for a system of nonlinear equations. It was suggested to refer to a lecture note from the Technical University of Trondheim for more information. It was also mentioned that there are criteria, such as Kantorovich's Theorem, that guarantee convergence for Newton's method. It was advised to apply the same principles as fixed-point iteration for single variable case to the multivariable case. It was noted that this topic may not be covered in introductory numerical analysis classes.
  • #1
defunc
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Under what circumstances will Newtons method for a system of nonlinear equations converge? Are there any criteria at all which guarantees convergence?

Regards
 
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  • #2
Interesting question!

I googled a bit, and found a lecture note that might be of interest:
http://www.math.ntnu.no/emner/TMA4122/2006h/notat-src/nr-systems-a4.pdf

(This is from the Technical University of Trondheim (NTNU), which has overall good standards, also internationally)
NOTE OF CAUTION:

I haven't as yet read it through; so I cannot say whether it is mathematically acceptable.
 
  • #3
http://planetmath.org/encyclopedia/KantorovitchsTheorem.html will tell you that Newton's method applied to a vector-valued multivariable function converges if started from an input which satisfies some inequalities. There are several versions for stronger types of convergence.
 
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  • #4
Remember, Newtons method is basically the same as fixed-point iteration.

So, I'm assuming you know what needs to happen for convergence of a single variable in the fixed point case. Simply extend that to the multivariable case.

i.e, There exists a 'K' between 0 < K < 1, where you will start adding absolute values of partial derivatives, which will end up being less than K. If you have a textbook, hopefully it will either mention it in the main text, or as an exercise.

It's kind of important. If this is for an introductory numerical analysis class, it might not be covered in an introductory text.
 
  • #5
Thank you all for your replies!
 

1. What is Newton's method for a system of equations?

Newton's method for a system of equations is an iterative algorithm used to find the roots of a system of nonlinear equations. It involves using the Jacobian matrix and the gradient vector to approximate the solution to the system of equations.

2. How does Newton's method for a system of equations work?

Newton's method works by making an initial guess for the solution to the system of equations and then using the Jacobian matrix and gradient vector to find an improved guess. This process is repeated until the solution converges to a desired level of accuracy.

3. What are the advantages of using Newton's method for a system of equations?

One advantage of using Newton's method is that it can converge to the solution of a system of equations faster than other methods, such as the bisection method or the secant method. It also allows for the simultaneous solution of multiple equations in a system.

4. What are the limitations of Newton's method for a system of equations?

One limitation of Newton's method is that it requires the Jacobian matrix to be invertible, which may not always be the case for certain systems of equations. It also relies heavily on the initial guess for the solution, which may lead to convergence to a wrong solution if the initial guess is not close enough to the actual solution.

5. In what fields is Newton's method for a system of equations commonly used?

Newton's method is commonly used in fields such as engineering, economics, and physics to solve complex systems of nonlinear equations. It is also used in computer graphics and optimization problems to find the roots of a system of equations.

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