Numerical solutions of system of nonlinear algebraic equations nonlinear algebraic eq

In summary, the conversation discusses methods for solving a system of nonlinear algebraic equations. The Newton-Raphson method is recommended as a global convergence method, but it is noted that without derivatives, other methods may be less effective. The simplex method and conjugate direction methods are suggested as alternatives that do not use gradients. Another method, the continuation method, is also mentioned for finding all solutions to a system of nonlinear algebraic equations. However, it is noted that this method may be difficult to apply in cases where the equations contain hyperbolic functions.
  • #1
alexyan
16
0
Could somebody who knows well the method of numerical solutions of system of nonlinear algebraic equations nonlinear algebraic equations recommand a global convergence methods? thank you very much!
 
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  • #2
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  • #3
could you recommand the method without using Derivatives? thank you!
 
  • #4
Without derivatives the methods are typically less effective (can be 'inefficient' real quick, if it's possible to apply in your case methods utilizing gradients they are typically far more usable & efficient), but ones like the simplex method and conjugate direction methods are zeroth order methods and as such don't use gradients. The simplex method is pretty used for example in unconstrained nonlinear optimization.

http://www-fp.mcs.anl.gov/otc/Guide/OptWeb/continuous/unconstrained/nonlinsimplex.html [Broken]
 
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  • #5
For a system of nonlinear algebraic equations, if you want to find all the solutions, you can also consider the continuation method (some people call it homotopy continuation method).
Here is one link:
http://www.math.uic.edu/~jan/PHCpack/phcpack.html
 
  • #6
thank you!
 
  • #7
Chingkui, my equations is not the polynomial. it contains the hyperbolic functions and is diffificult to simpilify to hyperbolic functions. how can I do with homotopy continuation method?
 

1. What is a system of nonlinear algebraic equations?

A system of nonlinear algebraic equations is a set of equations that involve variables raised to powers other than 1 and can be written in the form of f(x1, x2, ..., xn) = 0. These equations cannot be solved using basic algebraic techniques and require numerical methods to find solutions.

2. Why do we need numerical solutions for systems of nonlinear algebraic equations?

Unlike linear equations, nonlinear equations do not have a closed-form solution and cannot be solved algebraically. Therefore, numerical methods are necessary to approximate the solutions. These solutions are important in many fields such as engineering, physics, and economics.

3. What are some common numerical methods used to solve systems of nonlinear algebraic equations?

Some commonly used methods include the Newton-Raphson method, the Secant method, and the Broyden's method. These methods involve iteratively guessing a solution and improving it until a satisfactory level of accuracy is achieved.

4. How do we determine the accuracy of a numerical solution for a system of nonlinear algebraic equations?

The accuracy of a solution can be determined by comparing it to a known exact solution, if one exists. Otherwise, we can check the residual, which is the difference between the left and right sides of the equations at the solution. A small residual indicates a more accurate solution.

5. Are there any limitations to numerical solutions for systems of nonlinear algebraic equations?

Yes, there are limitations to numerical solutions. These methods can only approximate solutions and are not guaranteed to find all solutions or the most accurate solution. They also require a good initial guess to converge to a solution. Additionally, some methods may fail to converge for certain equations or may converge to a different solution depending on the initial guess.

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