Error estimate for iterative convergence

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Discussion Overview

The discussion revolves around the determination of the value ##q=0.85## in the context of fixed-point iteration, specifically referencing a Wikipedia page. Participants are exploring how this value is derived and its relation to the sine and cosine functions.

Discussion Character

  • Exploratory, Debate/contested, Technical explanation

Main Points Raised

  • One participant questions how the value ##q=0.85## is determined from the Wikipedia page on fixed-point iteration.
  • Another participant suggests that ##\sin(1) \approx 0.85## as a possible estimate for ##q##.
  • A subsequent participant asks for clarification on why ##\sin(1)## is being used, prompting further discussion.
  • It is noted that the guess of ##\sin(1)## is based on ##\cos(1)## being part of the original question, but this connection is questioned.
  • One participant points out that ##\cos(1) = 0.54 \neq 0.85##, seeking to understand the origin of the 0.85 value and the rationale behind using sine.
  • Another participant reiterates the connection to ##\sin(1)## and expresses a belief that the relationship is more than coincidental.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the origin of the value ##q=0.85##, and there are competing views regarding the relevance of sine and cosine in this context.

Contextual Notes

There is uncertainty regarding the assumptions made in deriving the value of ##q## and the definitions of the functions involved. The discussion does not resolve the mathematical steps leading to the value of 0.85.

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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Guess ##sin(1)\approx .85##.
 
mathman said:
Guess ##sin(1)\approx .85##.
Why ##\sin 1##?
 
Last edited by a moderator:
joshmccraney said:
Why ##\sin 1##?
It is guess based only on ##cos(1)## being in the original question.
 
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?
 
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.
 

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