Re: Limit of nested function
- To: mathgroup at smc.vnet.net
- Subject: [mg122171] Re: Limit of nested function
- From: dimitris <dimmechan at yahoo.com>
- Date: Tue, 18 Oct 2011 07:41:14 -0400 (EDT)
- Delivered-to: l-mathgroup@mail-archive0.wolfram.com
- References: <j7ffsb$j8b$1@smc.vnet.net>
Hi. What is beautiful with this forum is that apart of improving his Mathematica skills, one can improve deeply his mathematical understanding! There are persons here like Daniel Lichtblau and Andrzej Kozlowski (to name just a few!) that their answers are like complete lectures! Also most times when you ask something, somebody has asked something similar in the past!! See this link from an old post that arose from a similar question of mine, back in 2006. http://groups.google.com/group/comp.soft-sys.math.mathematica/browse_thread/thread/b33bb33fc86f296c/9fa4d93bc30ef9f7?lnk=gst&q=limit+of+nested+function#9fa4d93bc30ef9f7 You will find it quite useful. Let's see therefore your nested function. "The limit of a nested expression of the form: Nest[f,...], if it exists, is a fixed point of the function f. A fixed point of f is a root of f[x]-x==0. " In[12]:= Solve[(x == Sqrt[#1*Sqrt[#1]] & )[x]] Out[12]= {{x -> 0}, {x -> 1}} In[30]:= Take[Table[N[Nest[Sqrt[#1*Sqrt[#1]] & , 3, n], 100], {n, 2, 500}], -4] Out[30]= {1.0000000000000000000000000000000000000000000000000000000000000088367457204909355234061713384536523777, 1.0000000000000000000000000000000000000000000000000000000000000066275592903682016425546285038402392833, 1.0000000000000000000000000000000000000000000000000000000000000049706694677761512319159713778801794625, 1.0000000000000000000000000000000000000000000000000000000000000037280021008321134239369785334101345968} In[35]:= FixedPoint[N[Sqrt[#1*Sqrt[#1]] & , 100], N[3, 100], SameTest -> (Abs[#1 - #2] < 1/10000000000 & )] Out[35]= 1.0000000002633672286467555600009138543119629410021971435246219138655285754491\ 500282111126864472584129519268862 In Daniel Lichtblau's words (from above mentioned link): "Limit is designed to work on functions defined on a continuum, and known in a closed form. The function in question satisfies neither requirement... Bottom line is that the Limit function was not designed or intended to handle functions defined on only a sequence of points, or to handle functions not given in explicit closed form...For examples such as this one gets results exactly... using Solve to find an exact representation of the fixed point for the iteration." Dimitris Anagnostou