• 1 Vote(s) - 5 Average
• 1
• 2
• 3
• 4
• 5
 Iterative square root like square root recursion limit bo198214 Administrator Posts: 1,389 Threads: 90 Joined: Aug 2007 02/23/2009, 02:35 PM Surely you guys know the recursion limit formula (I use recursion to avoid the term iteration which is already occupied) for the square root of $b$: $x_{n+1} = \frac{1}{2}\left(x_n+\frac{b}{x_n}\right)$ $\lim_{n\to \infty} x_n = \sqrt{b}$ Now - also inspired the Newton formula analogy - I was asking myself whether we cant do a similar thing for the iterative square root $w$ of $f$, i.e. $w\circ w =f$. An analogon could be: $w_0 = f$ $w_{n+1} = \frac{1}{2}\left(w_n + {w_n}^{-1}\circ f\right)$, $w = \lim_{n\to\infty} w_n$. The numerical verification will take some more time for me and is currently too slow. Do you think that the above construction converge? For matrices the method works to compute matrix square root. But matrix multiplication is both side distributive while function composition is only distributive from the right. Because this restricted distributivity there may also be variants like $w_{n+1} = \frac{1}{2}\left(w_n + f\circ {w_n}^{-1}\right)$ or $w_{n+1} = \frac{1}{2}{w_n}^{-1}\left( w_n \circ w_n + f\right)$ will they converge to a differnt iterative square root? Tell me what you think! Gottfried Ultimate Fellow Posts: 763 Threads: 118 Joined: Aug 2007 02/23/2009, 03:01 PM (This post was last modified: 02/23/2009, 03:09 PM by Gottfried.) bo198214 Wrote:Now - also inspired the Newton formula analogy - I was asking myself whether we cant do a similar thing for the iterative square root $w$ of $f$, i.e. $w\circ w =f$. Nice idea. I've used f(x) = 2x + x^2 , then f05 = Ser(x) f05 = 1/2*(f05 + f (serreverse(f05))) \\ recurse to arrive at the same result like when using the matrix-root. :-) [update] it works also for the dxp(x) = exp(x)-1 function, matching the matrix-method. For exp(x) Pari/GP cannot give a seriesinversion, so I can't check this [/update] Gottfried Gottfried Helms, Kassel « Next Oldest | Next Newest »

 Possibly Related Threads... Thread Author Replies Views Last Post Can we get the holomorphic super-root and super-logarithm function? Ember Edison 10 3,601 06/10/2019, 04:29 AM Last Post: Ember Edison The super 0th root and a new rule of tetration? Xorter 4 4,095 11/29/2017, 11:53 AM Last Post: Xorter Super-root 3 andydude 10 11,076 01/19/2016, 03:14 AM Last Post: andydude Dangerous limits ... Tommy's limit paradox tommy1729 0 1,927 11/27/2015, 12:36 AM Last Post: tommy1729 tetration limit ?? tommy1729 40 48,738 06/15/2015, 01:00 AM Last Post: sheldonison Limit of mean of Iterations of f(x)=(ln(x);x>0,ln(-x) x<0) =-Omega constant for all x Ivars 10 14,921 03/29/2015, 08:02 PM Last Post: tommy1729 Another limit tommy1729 0 1,614 03/18/2015, 06:55 PM Last Post: tommy1729 A limit exercise with Ei and slog. tommy1729 0 1,970 09/09/2014, 08:00 PM Last Post: tommy1729 [MSE] The mick tommy limit conjecture. tommy1729 1 2,609 03/30/2014, 11:22 PM Last Post: tommy1729 tetration base conversion, and sexp/slog limit equations sheldonison 44 56,212 02/27/2013, 07:05 PM Last Post: sheldonison

Users browsing this thread: 1 Guest(s)