Talk:Methods of computing square roots

message: I could not find any relevant research papers on the use of Lucas sequences for computing real square roots.
The closest I found is
G. Adj and F. Rodríguez-Henríquez, "Square Root Computation over Even Extension Fields," in IEEE Transactions on Computers, vol. 63, no. 11, pp. 2829-2841, Nov. 2014, doi: 10.1109/TC.2013.145.
which is concerned with square roots in finite fields and uses a different algorithm.
Should this paragraph be removed as original research?
Or it could also simply be made much shorter, by avoiding to repeat the properties of Lucas sequences. BlueRavel (talk) 23:27, 3 December 2023 (UTC)
replies: text: I have searched, and I too failed to find any relevant source for this. It was posted into the article in 2009 without any explanation, by an editor who has never made any other substantial contribution, just one other very small edit. It looks as though it may well be original research, but whether it is or not, it is unsourced, so I have removed it. JBW (talk) 21:54, 5 December 2023 (UTC)
thread_title:  Lucas sequence method - original research? 

message: I believe the section "Approximations that depend on the floating point representation" should be merged into "Initial estimate", since it is a special case of "Binary estimates". Merging would clear up the fact that the floating point trick gives an initial rough approximation, which is then typically iteratively improved.
I also believe the "Initial estimate" section should appear after the section on Heron's method, as the reader is likely more interested in the general idea of iterative refinement than in the details of how to obtain a good initial estimate in all possible ways.
Additionally, in my opinion the entirety of the article could benefit from some trimming/rewriting, as many sections contain redundant information, unnecessary details, and awkward formulations. BlueRavel (talk) 14:54, 4 December 2023 (UTC)
replies: text:  Your proposition makes sense to me, and I dont necessarily disagree.  That said though, as a pure mathematician, I am uninclined to blur the lines between programmatical issues and mathematical problems.  I think maintaining a distinction is appropriate.  An analysis of the pure mathematical problem of initial estimation in these abstract reiterative processes is a decidedly distinct discussion from considerations in this programming language, or that programming language, or this architecture, or that architecture. The former is future-proofed, the latter is not. CogitoErgoCogitoSum (talk) 21:09, 11 February 2024 (UTC)
thread_title:  Merge "Approximations that depend on the floating point representation" into "Initial estimate" 

message: Not sure if its useful, but I have found that, in general, \sqrt{x+2} \approx \frac{x+1}{\sqrt{x}}, and if  we get \sqrt{n^2+2} \approx n + \frac{1}{n}.
Similarly \sqrt{x+4} \approx \frac{x+2}{\sqrt{x}}.
I sometimes use this for quick pencil and paper calculations, if Im close enough to a convenient value.
Not sure if this is a known or established property, proven, bounded, or if its already in the article in some alternative capacity, or if its even appropriate for this article.  I do know the taylor series approximation with two terms connects these expressions.
CogitoErgoCogitoSum (talk) 21:05, 11 February 2024 (UTC)
replies: text: There is nothing special about 2 and 4: \sqrt{x+2c} \approx \frac{x+c}{\sqrt{x}} provided that c is small compared to x. This is, in fact, just the first two terms of the series given in the article under the section heading "Taylor series". JBW (talk) 01:45, 13 February 2024 (UTC)

text: I don't think they are useful.  In the first, you have replaced a square root and an addition with a square root, an addition, and a division to get an approximate answer.  Bubba73 You talkin' to me? 08:02, 13 February 2024 (UTC)
thread_title:  Useful addition??