Fixed Point Iteration: Why Change g(x) to Find Other Roots?

In summary, the conversation discusses the concept of fixed points in the Theory of Fixed Point Iteration, where a fixed point is a number that satisfies the equation x = g(x). It is not the same as a root of the equation 0 = g(x). To find different fixed points, the form of g(x) may need to be changed, as seen in the example with the function x3+3x2+x+4. This can result in different convergence values and is not a mistake, as each form represents a different point of intersection between the curve y = g(x) and the line y = x.
  • #1
281
0
Hi,
in the method -Theory of Fixed Point Iteration of x = g(x)

If the function g(x) has several roots, why sometimes we need to change the form of g(x) to find the other roots?

For example we can have x3+3x2+x+4,
one form can be x= -x3-3x2-4
or
another form can be
x=sqrt(-x3-3x2-4) / sqrt (3)

those two forms converge into diffrent values.WHY?

Thank you

Brad
 
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  • #2
You've misunderstood the whole thing!

Definition: A fixed point of a function g(x) is a number p such that p = g(p).

Caution. A fixed point is not a root of the equation 0 = g(x), it is a solution of the equation x = g(x).

Geometrically, the fixed points of a function g(x) are the point(s) of intersection of the curve y = g(x) and the line y = x.
 
  • #3
iNCREDiBLE said:
You've misunderstood the whole thing!

Definition: A fixed point of a function g(x) is a number p such that p = g(p).

Caution. A fixed point is not a root of the equation 0 = g(x), it is a solution of the equation x = g(x).

Geometrically, the fixed points of a function g(x) are the point(s) of intersection of the curve y = g(x) and the line y = x.

Ok that makes more sense now.

Thank you very much
 

1. What is fixed point iteration and how does it relate to finding roots?

Fixed point iteration is an iterative method used to approximate the roots of a function. It involves repeatedly applying a function to a starting value until the value converges to a fixed point, which is a value where the function takes on the same value as the input. This method is applicable to finding roots because the fixed points of a function correspond to its roots.

2. Why is it necessary to change the function g(x) in fixed point iteration?

The function g(x) is used to generate the sequence of values that will converge to the fixed point. In order for this method to work, g(x) must satisfy certain conditions, such as being continuous and having a derivative with absolute value less than 1. If these conditions are not met, the iteration will not converge to the correct root. Therefore, changing g(x) can help ensure that the method will produce an accurate result.

3. How does changing g(x) affect the accuracy of the root approximation?

Changing g(x) can have a significant impact on the accuracy of the root approximation. If the new g(x) is closer to the root, the iteration will converge faster and produce a more accurate result. On the other hand, if the new g(x) is further from the root, the iteration may not converge at all or converge to a different root.

4. Are there any limitations to using fixed point iteration to find roots?

Yes, there are some limitations to using this method. It may not work for functions that do not have a fixed point or for functions with multiple roots. Additionally, the method may converge slowly or not at all if the initial guess is too far from the root. In these cases, other numerical methods may be more effective for finding roots.

5. Can fixed point iteration be used to find complex roots?

Yes, fixed point iteration can be used to find complex roots. However, the function g(x) must be defined for complex numbers and the iteration must be performed in the complex plane. This can make the method more complex and may require additional considerations, such as choosing an appropriate initial guess and checking for convergence in the complex plane.

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