Error estimate for iterative convergence

In summary, the author of the "Examples" section on the page about fixed-point iteration on Wikipedia suggests using a value of ##q=0.85##, but it is not clear how this number was determined. Despite researching, the origin of ##q## could not be found. However, it is possible that the value was chosen based on the approximation ##\sin(1) \approx 0.85##, which may have been a guess based on the use of ##\cos(1)## in the original question. However, this approximation does not match the actual value of ##\cos(1)##, leaving the source of the 0.85 value uncertain.
  • #1
member 428835
On the following page on wikipedia:

https://en.wikipedia.org/wiki/Fixed-point_iteration

the section "Examples" has a second bullet point, where the author suggests ##q=0.85##, but how did they get this number? I tried googling everything and could not find out how ##q## is determined.
 
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  • #2
Guess ##sin(1)\approx .85##.
 
  • #3
mathman said:
Guess ##sin(1)\approx .85##.
Why ##\sin 1##?
 
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  • #4
joshmccraney said:
Why ##\sin 1##?
It is guess based only on ##cos(1)## being in the original question.
 
  • #5
mathman said:
It is guess based only on ##cos(1)## being in the original question.
But ##\cos(1) = 0.54 \neq 0.85##, so where is the 0.85 coming from? Asked differently, where did you get the ##\sin## from?
 
  • #6
joshmccraney said:
But ##\cos(1) = 0.54 \neq 0.85##, so where is the 0.85 coming from? Asked differently, where did you get the ##\sin## from?
sin(1)=.85. My guess was that it more than just a coincidence.
 

1. What is iterative convergence and why is it important?

Iterative convergence is the process of finding a solution to a problem through repeated calculations, where each calculation improves upon the previous one. It is important because it allows for the efficient and accurate approximation of complex problems that cannot be solved through traditional methods.

2. How is error estimate calculated for iterative convergence?

Error estimate for iterative convergence is typically calculated by comparing the current iteration to the previous one and measuring the difference between them. This difference, also known as the residual, is then used to determine how close the current iteration is to the true solution.

3. What factors can affect the accuracy of error estimate for iterative convergence?

There are several factors that can affect the accuracy of error estimate for iterative convergence, including the initial guess, the number of iterations, and the convergence criteria. Additionally, the complexity of the problem and the precision of the calculations can also impact the accuracy of the error estimate.

4. How can one improve the accuracy of error estimate for iterative convergence?

To improve the accuracy of error estimate for iterative convergence, one can adjust the initial guess to be closer to the true solution, increase the number of iterations, and use a more stringent convergence criteria. Additionally, using more precise calculations and simplifying the problem can also improve the accuracy of the error estimate.

5. Can error estimate for iterative convergence be used for all types of problems?

No, error estimate for iterative convergence is typically used for problems that are difficult or impossible to solve through traditional methods. It is most commonly used in numerical analysis and scientific computing, but may not be applicable to all types of problems.

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