Rate of convergence and asymptotic error constant

This means that for a fixed number of iterations, it will produce a more accurate solution than the method with a larger C. In summary, the order p represents the speed of convergence for root finding algorithms such as secant, regula falsi, bisection, and Newton's method. A higher order p means that the number of correct significant digits is approximately doubled in each iteration. The asymptotic error constant C represents the rate of convergence for the method, with a smaller C indicating a faster convergence and a more accurate solution with a fixed number of iterations. If two methods have the same order p but different C values, the method with the smaller C will converge faster and produce a more accurate solution.
  • #1
azay
19
0
In the context of root finding algorithms such as secant, regula falsi, bisection, Newton's method:

In

[tex]
\lim_{n \to \infty} \frac{|x*-x_{n+1}|}{|x*-x_{n}|^{p}} = C

[/tex]

I understand the meaning of the order p is the speed of convergence. For example, in Newton's method the order p = 2 and thus the number of correct significant digits is approximately doubled in each iteration step. But is there an intuitive meaning to be given to the asymptotic error constant C? What does this number mean? What is the difference between two methods that have the same order p, but for a different C?
 
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  • #2
As I understand it, if they are of the same order, the method with a smaller C will converge faster.
 

1. What is the rate of convergence?

The rate of convergence refers to how quickly a sequence or series of values approaches a specific limit or target value. It is a measure of the speed at which the values are converging towards the limit.

2. How is the rate of convergence calculated?

The rate of convergence is typically calculated by finding the limit of the ratio between consecutive terms in the sequence or series. This limit is known as the asymptotic error constant.

3. What is the significance of the rate of convergence in scientific calculations?

The rate of convergence is important in scientific calculations because it gives an indication of how accurate and efficient a numerical method is in approximating a desired solution. A higher rate of convergence means the method is more accurate and requires fewer iterations to reach the desired solution.

4. Can the rate of convergence vary for different numerical methods?

Yes, the rate of convergence can vary for different numerical methods. Some methods may have a faster rate of convergence than others, making them more efficient for certain types of calculations. It is important to consider the rate of convergence when choosing a numerical method for a specific problem.

5. Is a higher rate of convergence always better?

Not necessarily. While a higher rate of convergence may indicate a more efficient method, it may also come with a higher computational cost. Sometimes, a balance between the rate of convergence and the computational cost needs to be considered when choosing a numerical method.

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