Is an Inverse-Free Newton's Method Possible for Matrix Inversions?

In summary, the conversation discusses the search for a Newton's method derivative that does not require an inverted Jacobian, as well as the preference to avoid relying on outside libraries and the search for a problem-specific method. The link provided is considered as a potential solution, but there is difficulty in applying it to the specific problem at hand. The existing equations are already in a suitable form and only the input parameters change. The method of using linear algebra and MATLAB's built-in routines or coding a pivot method is also mentioned as a possible approach.
  • #1
tangodirt
54
1
Does anyone know of a Newton's method derivative that does not require an inverted Jacobian? I am attempting to port my code from one language to another, and rather than rely on outside libraries for matrix inversions (like I am now), I would prefer to simply do away with the inverted matrix altogether. Also, porting outside "general-purpose" libraries from one language to another is not something I want to do, especially when I just need a "problem specific" method. I.E. reusability is not high on my requirement list.

The easier to implement in code, the better.

http://benisrael.net/InverseFreeMethod.pdf" is the closest thing I can find, but I am having a hard time making the leap from the "theory" to "application."

All of my equations are already in the form f(x1, x2, x3, etc.) = 0 and do not change. Only the input parameters (value assigned to x1, x2, x3, etc.) change.
 
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  • #2
I have been using this method for a while now, the basic equation for Newtons method with lots of dimensions is:
[tex]
(x_{n+1}-x_{n})J(x_{n})=-f(x_{n})
[/tex]
Now this is basically a linear algebra problem. MATLAB has inbuilt routines for this, or you can code up your own pivot method.
 

FAQ: Is an Inverse-Free Newton's Method Possible for Matrix Inversions?

1. What is Inverse-Free Newton's Method?

Inverse-free Newton's method is an iterative algorithm used to find the roots of a nonlinear equation. It is a variation of the traditional Newton's method, but instead of computing the inverse of the Jacobian matrix, it approximates it using other techniques.

2. How is Inverse-Free Newton's Method different from traditional Newton's method?

In inverse-free Newton's method, the inverse of the Jacobian matrix is not computed, which can be computationally expensive. Instead, the matrix is approximated using other methods, such as the secant method or the Broyden method.

3. What are the advantages of using Inverse-Free Newton's Method?

One advantage of inverse-free Newton's method is that it can be more computationally efficient than traditional Newton's method. It also allows for the use of different matrix approximations, which can be more suitable for certain types of equations.

4. What are the limitations of Inverse-Free Newton's Method?

One limitation of inverse-free Newton's method is that the convergence rate may be slower compared to traditional Newton's method. It also requires more initial guesses and has a higher risk of encountering convergence issues.

5. In what situations is Inverse-Free Newton's Method most useful?

Inverse-free Newton's method is most useful when the Jacobian matrix is too expensive or difficult to compute. It can also be beneficial when the matrix is ill-conditioned, as it allows for the use of different approximations that may improve convergence.

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