Enter the current approximation, previous approximation, and the approximation before the previous approximation into the calculator to determine the rate of convergence.

Rate Of Convergence Formula

The following formula is used to calculate the rate of convergence. 

ROC = |(xn+1 - xn) / (xn - xn-1)|

Variables:

  • ROC is the rate of convergence xn+1 is the current approximation
  • xn is the previous approximation
  • xn-1 is the approximation before the previous approximation

To calculate the rate of convergence, subtract the previous approximation from the current approximation. Then, subtract the approximation before the previous approximation from the previous approximation. Divide the first result by the second result and take the absolute value of the quotient.

What is the Rate Of Convergence?

A rate of convergence refers to the speed at which a sequence of numbers approaches a certain value or limit. It is a concept used in numerical analysis and is particularly important in the field of computational mathematics. The rate of convergence helps to determine how many iterations are needed to achieve a certain level of accuracy, and thus can be used to assess the efficiency of numerical methods. Different methods may have different rates of convergence, with faster rates being more desirable as they require fewer iterations to reach the desired precision.

How to Calculate Rate Of Convergence?

The following steps outline how to calculate the Rate of Convergence (ROC).

  1. First, determine the current approximation (xn+1).
  2. Next, determine the previous approximation (xn).
  3. Next, determine the approximation before the previous approximation (xn-1).
  4. Next, subtract the previous approximation (xn) from the current approximation (xn+1).
  5. Next, subtract the approximation before the previous approximation (xn-1) from the previous approximation (xn).
  6. Next, divide the result from step 4 by the result from step 5.
  7. Finally, take the absolute value of the quotient obtained in step 6 to calculate the Rate of Convergence (ROC).

Example Problem:

Use the following variables as an example problem to test your knowledge.

current approximation (xn+1) = 5

previous approximation (xn) = 3

approximation before the previous approximation (xn-1) = 2